Discovering Theorems in Game Theory Two-Person Games with Unique Nash Equilibria

合集下载

40_game_theory_ch_2

Chapter2Nim and combinatorial games2.1Aims of the chapterThis chapter•introduces the basics of combinatorial games,and explains the central role of the game nim.A detailed summary of the chapter is given in section2.5.Furthermore,this chapter•demonstrates the use of abstract mathematics in game theory.This chapter is written more formally than the other chapters,in parts in thetraditional mathematical style of deﬁnitions,theorems and proofs.One reason fordoing this,and why we start with combinatorial games,is that this topic and styleserves as a warning shot to those who think that game theory,and this unit inparticular,is‘easy’.If we started with the well-known‘prisoner’s dilemma’(whichmakes its due appearance in Chapter3),the less formally inclined student might belulled into a false sense of familiarity and‘understanding’.We therefore startdeliberately with an unfamiliar topic.This is a mathematics unit,with great emphasis on rigour and clarity,and on usingmathematical notions precisely.As mathematical prerequisites,game theory requiresonly the very basics of linear algebra,calculus and probability theory.However,gametheory provides its own conceptual tools that are used to model and analyseinteractive situations.This unit emphasises the mathematical structure of theseconcepts,which belong to‘discrete mathematics’.Learning a number of newmathematical concepts is exempliﬁed by combinatorial game theory,and it willcontinue in the study of classical game theory in the later chapters.2.2Learning objectivesAfter studying this chapter,you should be able to:•play nim optimally;•explain the concepts of game-sums,equivalent games,nim values and the mex rule;•apply these concepts to play other impartial games like those described in the exercises.40Game theory2.3Essential readingThis chapter of the guide.2.4Further readingVery few textbooks on game theory deal with combinatorial games.An exception ischapter1of the following book:•Mendelson,Elliot Introducing Game Theory and Its Applications.(Chapman& Hall/CRC,2004)[ISBN1584883006].The winning strategy for the game nim based on the binary system wasﬁrst describedin the following article,which is available electronically from the JSTOR archive:•Bouton,Charles‘Nim,a game with a complete mathematical theory.’The Annals of Mathematics,2nd Ser.,Vol.3,No.1/4(1902),pp.35–39.The deﬁnitive text on combinatorial game theory is the set of volumes‘WinningWays’by Berlekamp,Conway and Guy.The material of this chapter appears in theﬁrst volume:•Berlekamp,Elwyn R.,John H.Conway and Richard K.Guy Winning Ways for Your Mathematical Plays,Volume1,second edition.(A.K.Peters,2001)[ISBN1568811306].Some small pieces of that text have been copied here nearly verbatim,for example inSections2.7,2.9,and2.12below.The four volumes of‘Winning Ways’are beautiful books.However,they are notsuitable reading for a beginner,because the mathematics is hard,and the reader isconfronted with a wealth of material.The introduction to combinatorial game theorygiven here represents a very small fraction of that body of work,but may invite youto study it further.A very informative and entertaining mathematical tour of parlour games is•Bewersdorﬀ,J¨o rg Logic,Luck and White Lies.(A.K.Peters,2005)[ISBN 1568812108].Combinatorial games are treated in part II of that book.2.5What is combinatorial game theory?This chapter is on the topic of combinatorial games.These are games with twoplayers,perfect information,and no chance moves,speciﬁed by certain rules.Familiargames of this sort are chess,go,checkers,tic-tac-toe,dots-and-boxes,and nim.Suchgames can be played perfectly in the sense that either one player can force a win orboth can force a draw.In reality,games like chess and go are too complex toﬁnd anoptimal strategy,and they derive their attraction from the fact that(so far)it is notknown how to play them perfectly.We will,however,learn how to play nim perfectly.There is a‘classical’game theory with applications in economics which is verydiﬀerent from combinatorial game theory.The games in classical game theory aretypically formal models of conﬂict and co-operation which cannot only be lost orWhat is combinatorial game theory? won,and in which there is often no perfect information about past and future moves.To the economist,combinatorial games are not very interesting.Chapters3–6of theunit are concerned with classical game theory.Why,then,study combinatorial games at all in a unit that is mostly about classicalgame theory,and which aims to provide an insight into the theory of games as usedin economics?The reason is that combinatorial games have a rich and interesting mathematical theory.We will explain the basics of that theory,in particular thecentral role of the game nim for impartial games.It is non-trivial mathematics,it isfun,and you,the student,will have learned something that you would most likelynot have learned otherwise.Theﬁrst‘trick’from combinatorial game theory is how to win in the game nim,using the binary system.Historically,that winning strategy was discoveredﬁrst(published by Charles Bouton in1902).Only later did the central importance of nim,in what is known as the Sprague–Grundy theory of impartial games,becomeapparent.It also revealed why the binary system is important(and not,say,theternary system,where numbers are written in base three),and learning that is more satisfying than just learning how to use it.In this chapter,weﬁrst deﬁne the game nim and more general classes of games withperfect information.These are games where every player knows exactly the state ofthe game.We then deﬁne and study the concepts listed in the learning outcomesabove,which are the concepts of game-sums,equivalent games,nim values and themex rule.It is best to learn these concepts by following the chapter in detail.Wegive a brief summary here,which will make more sense,and should be re-consulted,after aﬁrst study of the chapter(so do not despair if you do not understand this summary).Mathematically,any game is deﬁned by other‘games’that a player can reach in hisﬁrst move.These games are called the options of the game.This seemingly circulardeﬁnition of a‘game’is sound because the options are simpler games,which needfewer moves in total until they end.The deﬁnition is therefore not circular,butrecursive,and the mathematical tool to argue about such games is that ofmathematical induction,which will be used extensively(it will also recur inchapter3as‘backward induction’for game trees).Here,it is very helpful to befamiliar with mathematical induction for proving statements about natural numbers.We focus here on impartial games,where the available moves are the same nomatter whether player I or player II is the player to make a move.Games are‘combined’by the simple rule that a player can make a move in exactly one of thegames,which deﬁnes a sum of these games.In a‘losing game’,theﬁrst player tomove loses(assuming,as always,that both players play as well as they can).Animpartial game added to itself is always losing,because any move can be copied inthe other game,so that the second player always has a move left.This is known asthe‘copycat’principle(lemma2.6).An important observation is that a losing gamecan be‘added’(via the game-sum operation)to any game without changing thewinning or losing properties of the original game.In section2.11,the central theorem2.10explains the winning strategy in nim.The importance of nim for impartial games is then developed in section2.12via thebeautiful mex rule.After the comparatively hard work of the earlier sections,wealmost instantly obtain that any impartial game is equivalent to a nim heap(corollary2.13).At the end of the chapter,the sizes of these equivalent nim heaps(called nim values)are computed for some examples of impartial games.Many other examples arestudied in the exercises.40Game theoryOur exposition is distinct from the classic text‘Winning Ways’in the followingrespects:First,we only consider impartial games,even though many aspects carryover to more general combinatorial games.Secondly,we use a precise deﬁnition ofequivalent games(see section2.10),because a game where you are bound to loseagainst a smart opponent is not the same as a game where you have already lost.Two such games are merely equivalent,and the notion of equivalent games is helpfulin understanding the theory.So this text is much more restricted,but to some extentmore precise than‘Winning Ways’,which should help make this topic accessible andenjoyable.2.6Nim–rulesThe game nim is played with heaps(or piles)of chips(or counters,beans,pebbles,matches).Players alternate in making a move,by removing some chips from one ofthe heaps(at least one chip,possibly the entire heap).Theﬁrst player who cannotmove any more loses the game.The players will be called,rather unimaginatively,player I and player II,with player Ito start the game.For example,consider three heaps of size1,1,2.What is a good move?Removingone of the chips from the heap with two chips will create the position1,1,1,thenplayer II must move to1,1,then player I to1,and then player II takes the last chipand wins.So this is not a good opening move.The winning move is to remove allchips from the heap of size2,to reach position1,1,and then player I will win.Hence we call1,1,2a winning position,and1,1a losing position.When moving in a winning position,the player to move can win by playing well,bymoving to a losing position of the other player.In a losing position,the player tomove will lose no matter what move she chooses,if her opponent plays well.Thismeans that all moves from a losing position lead to a winning position of theopponent.In contrast,one needs only one good move from a winning position thatgoes to a losing position of the next player.Another winning position consists of three nim heaps of sizes1,1,1.Here all movesresult in the same position and player I always wins.In general,a player in a winningposition must play well by picking the right move.We assume that players play well,forcing a win if they can.Suppose nim is played with only two heaps.If the two heaps have equal size,forexample in position4,4,then theﬁrst player to move loses(so this is a losingposition),because player II can always copy player I’s move by equalising the twoheaps.If the two heaps have diﬀerent sizes,then player I can equalise them byremoving an appropriate number of chips from the larger heap,putting player II in alosing position.The rule for2-heap nim is therefore:Lemma2.1The nim position m,n is winning if and only if m=n,otherwise losing,for all m,n≥0.This lemma applies also when m=0or n=0,and thus includes the cases that oneor both heap sizes are zero(meaning only one heap or no heap at all).With three or more heaps,nim becomes more diﬃcult.For example,it is notimmediately clear if,say,positions1,4,5or2,3,6are winning or losing positions.⇒At this point,you should try exercise2.1(a)on page28.Combinatorial games,in particular impartial games 2.7Combinatorial games,in particular impartial gamesThe games we study in this chapter have,like nim,the following properties:1.There are just two players.2.There are several,usuallyﬁnitely many,positions,and sometimes a particularstarting position.3.There are clearly deﬁned rules that specify the moves that either player canmake from a given position to the possible new positions,which are called theoptions of that position.4.The two players move alternately,in the game as a whole.5.In the normal play convention a player unable to move loses.6.The rules are such that play will always come to an end because some player willbe unable to move.This is called the ending condition.So there can be nogames which are drawn by repetition of moves.7.Both players know what is going on,so there is perfect information.8.There are no chance moves such as rolling dice or shuﬄing cards.9.The game is impartial,that is,the possible moves of a player only depend onthe position but not on the player.As a negation of condition5,there is also the mis`e re play convention where a playerunable to move wins.In the surrealist(and unsettling)movie‘Last year atMarienbad’by Alain Resnais from1962,mis`e re nim is played,several times,withrows of matches of sizes1,3,5,7.If you have a chance,try to watch that movie andspot when the other player(not the guy who brought the matches)makes a mistake!Note that this is mis`e re nim,not nim,but you will be able toﬁnd out how to play itonce you know how to play nim.(For games other than nim,normal play and mis`e reversions are typically not so similar.)In contrast to condition9,games where the available moves depend on the player(as in chess where one player can only move white pieces and the other only blackpieces)are called partisan games.Much of combinatorial game theory is aboutpartisan games,which we do not consider to keep matters simple.Chess,and the somewhat simpler tic-tac-toe,also fail condition6because they mayend in a tie or draw.The card game poker does not have perfect information(asrequired in7)and would lose all its interest if it had.The analysis of poker,althoughit is also a win-or-lose game,leads to the‘classical’theory of zero-sum games(withimperfect information)that we will consider later.The board game backgammon is agame with perfect information but with chance moves(violating condition8)because dice are rolled.We will be relatively informal in style,but our notions are precise.In condition3above,for example,the term option refers to a position that is reachable in onemove from the current position;do not use‘option’when you mean‘move’.Similarly,we will later use the term strategy to deﬁne a plan of moves,one for everyposition that can occur in the game.Do not use‘strategy’when you mean‘move’.However,we will take some liberty in identifying a game with its starting positionwhen the rules of the game are clear.40Game theory⇒Try now exercises2.2and2.3starting on page28.2.8Simpler games and notation for nim heapsA game,like nim,is deﬁned by its rules,and a particular starting position.Let G besuch a particular instance of nim,say with the starting position1,1,2.Knowing therules,we can identify G with its starting position.Then the options of G are1,2,and1,1,1,and1,1.Here,position1,2is obtained by removing either theﬁrst or thesecond heap with one chip only,which gives the same result.Positions1,1,1and1,1are obtained by making a move in the heap of size two.It is useful to list the optionssystematically,considering one heap to move in at a time,so as not to overlook anyoption.Each of the options of G is the starting position of another instance of nim,deﬁningone of the new games H,J,K,say.We can also say that G is deﬁned by the movesto these games H,J,K,and we call these games also the options of G(byidentifying them with their starting positions;recall that the term‘option’has beendeﬁned in point3of section2.7).That is,we can deﬁne a game as follows:Either the game has no move,and theplayer to move loses,or a game is given by one or several possible moves to newgames,in which the other player makes the initial move.In our example,G isdeﬁned by the possible moves to H,J,or K.With this deﬁnition,the entire game iscompletely speciﬁed by listing the initial moves and what games they lead to,because all subsequent use of the rules is encoded in those games.This is a recursive deﬁnition because a‘game’is deﬁned in terms of‘game’itself.We have to add the ending condition that states that every sequence of moves in agame must eventually end,to make sure that a game cannot go on indeﬁnitely.This recursive condition is similar to deﬁning the set of natural numbers as follows:(a)0is a natural number;(b)if n is a natural number,then so is n+1;and(c)allnatural numbers are obtained in this way,starting from0.Condition(c)can beformalised by the principle of induction that says:if a property P(n)is true for n=0,and if the property P(n)implies P(n+1),then it is true for all natural numbers.We use the following notation for nim heaps.If G is a single nim heap with nchips,n≥0,then we denote this game by∗n.This game is completely speciﬁed byits options,and they are:options of∗n:∗0,∗1,∗2,...,∗(n−1).(2.1) Note that∗0is the empty heap with no chips,which allows no moves.It is invisiblewhen playing nim,but it is useful to have a notation for it because it deﬁnes themost basic losing position.(In combinatorial game theory,the game with no moves,which is the empty nim heap∗0,is often simply denoted as0.)We could use(2.1)as the deﬁnition of∗n;for example,the game∗4is deﬁned by itsoptions∗0,∗1,∗2,∗3.It is very important to include∗0in that list of options,because it means that∗4has a winning move.Condition(2.1)is a recursivedeﬁnition of the game∗n,because its options are also deﬁned by reference to suchgames∗k,for numbers k smaller than n.This game fulﬁls the ending conditionbecause the heap gets successively smaller in any sequence of moves.If G is a game and H is a game reachable by one or more successive moves from thestarting position of G,then the game H is called simpler than G.We will oftenprove a property of games inductively,using the assumption that the property appliesto all simpler games.An example is the–already stated and rather obvious–Sums of games property that one of the two players can force a win.(Note that this applies togames where winning or losing are the only two outcomes for a player,as implied bythe‘normal play’convention in5above.)Lemma2.2In any game G,either the starting player I can force a win,or player IIcan force a win.Proof.When the game has no moves,player I loses and player II wins.Now assumethat G does have options,which are simpler games.By inductive assumption,ineach of these games one of the two players can force a win.If,in all of them,thestarting player(which is player II in G)can force a win,then she will win in G byplaying accordingly.Otherwise,at least one of the starting moves in G leads to agame G where the second-moving player in G (which is player I in G)can force awin,and by making that move,player I will force a win in G.If in G,player I can force a win,its starting position is a winning position,and wecall G a winning game.If player II can force a win,G starts with a losing position,and we call G a losing game.2.9Sums of gamesWe continue our discussion of nim.Suppose the starting position has heap sizes1,5,5.Then the obvious good move is to option5,5,which is losing.What about nim with four heaps of sizes2,2,6,6?This is losing,because2,2and6,6independently are losing positions,and any move in a heap of size2can becopied in the other heap of size2,and similarly for the heaps of size6.There is asecond way of looking at this example,where it is not just two losing games puttogether:consider the game with heap sizes2,6.This is a winning game.However,two such winning games,put together to give the game2,6,2,6,result in a losinggame,because any move in one of the games2,6,for example to2,4,can be copiedin the other game,also to2,4,giving the new position2,4,2,4.So the secondplayer,who plays‘copycat’,always has a move left(the copying move)and hencecannot lose.Deﬁnition2.3The sum of two games G and H,written G+H,is deﬁned asfollows:The player may move in either G or H as allowed in that game,leaving theposition in the other game unchanged.Note that G+H is a notation that applies here to games and not to numbers,evenif the games are in some way deﬁned using numbers(for example as nim heaps).The result is a new game.More formally,assume that G and H are deﬁned in terms of their options(via movesfrom the starting position)G1,G2,...,G k and H1,H2,...,H m,respectively.Then theoptions of G+H are given asoptions of G+H:G1+H,...,G k+H,G+H1,...,G+H m.(2.2) Theﬁrst list of options G1+H,G2+H,...,G k+H in(2.2)simply means that theplayer makes his move in G,the second list G+H1,G+H2,...,G+H m that hemakes his move in H.We can deﬁne the game nim as a sum of nim heaps,where any single nim heap isrecursively deﬁned in terms of its options by(2.1).So the game nim with heaps ofsize1,4,6is written as∗1+∗4+∗6.40Game theoryThe‘addition’of games with the abstract+operation leads to an interestingconnection of combinatorial games with abstract algebra.If you are somewhatfamiliar with the concept of an abstract group,you will enjoy this connection;if not,you do not need to worry,because this connection it is not essential for ourdevelopment of the theory.A group is a set with a binary operation+that fulﬁls three properties:1.The operation+is associative,that is,G+(J+K)=(G+J)+K holds for allG,J,K.2.The operation+has a neutral element0,so that G+0=G and0+G=G forall G.3.Every element G has an inverse−G so that G+(−G)=0.Furthermore,4.The group is called commutative(or‘abelian’)if G+H=H+G holds for allG,H.Familiar groups in mathematics are,for example,the set of integers with addition,orthe set of positive real numbers with multiplication(where the multiplicationoperation is written as·,the neutral element is1,and the inverse of G is written asG−1).The games that we consider form a group as well.In the way the sum of two gamesG and H is deﬁned,G+H and H+G deﬁne the same game,so+is commutative.Moreover,when one of these games is itself a sum of games,for example H=J+K,then G+H is G+(J+K)which means the player can make a move in exactly one ofthe games G,J,or K.This means obviously the same as the sum of games(G+J)+K,that is,+is associative.The sum G+(J+K),which is the same as(G+J)+K,can therefore be written unambiguously as G+J+K.An obvious neutral element is the empty nim heap∗0,because it is‘invisible’(itallows no moves),and adding it to any game G does not change the game.However,there is no direct way to get an inverse operation because for any game Gwhich has some options,if one adds any other game H to it(the intention beingthat H is the inverse−G),then G+H will have some options(namely at least theoptions of moving in G and leaving H unchanged),so that G+H is not equal to theempty nim heap.The way out of this is to identify games that are‘equivalent’in a certain sense.Wewill see shortly that if G+H is a losing game(where theﬁrst player to move cannotforce a win),then that losing game is‘equivalent’to∗0,so that H fulﬁls the role ofan inverse of G.2.10Equivalent gamesThere is a neutral element that can be added to any game G without changing it.By deﬁnition,because it allows no moves,it is the empty nim heap∗0:G+∗0=G.(2.3)However,other games can also serve as neutral elements for the addition of games.We will see that any losing game can serve that purpose,provided we considercertain games as equivalent according to the following deﬁnition.Equivalent games Deﬁnition2.4Two games G,H are called equivalent,written G≡H,if and only iffor any other game J,the sum G+J is losing if and only if H+J is losing.In deﬁnition2.4,we can also say that G≡H if for any other game J,the sum G+Jis winning if and only if H+J is winning.In other words,G is equivalent to H if,whenever G appears in a sum G+J of games,then G can be replaced by H without changing whether G+J is winning or losing.One can verify easily that≡is indeed an equivalence relation,meaning it is reﬂexive(G≡G),symmetric(G≡H implies H≡G),and transitive(G≡H and H≡K implyG≡K;all these conditions hold for all games G,H,K).Using J=∗0in deﬁnition2.4and(2.3),G≡H implies that G is losing if and only ifH is losing.The converse is not quite true:just because two games are winning doesnot mean they are equivalent,as we will see shortly.However,any two losing gamesare equivalent,because they are all equivalent to∗0:Lemma2.5If G is a losing game(the second player to move can force a win),thenG≡∗0.Proof.Let G be a losing game.We want to show G≡∗0By deﬁnition2.4,this istrue if and only if for any other game J,the game G+J is losing if and only if∗0+Jis losing.According to(2.3),this holds if and only if J is losing.So let J be any other game;we want to show that G+J is losing if and only if J islosing.Intuitively,adding the losing game G to J does not change which player in Jcan force a win,because any intermediate move in G by his opponent is simplycountered by the winning player,until the moves in G are exhausted.Formally,weﬁrst prove by induction the simpler claim that for all games J,if J islosing,then G+J is losing.(So weﬁrst ignore the‘only if’part.)Our inductive assumptions for this simpler claim are:for all losing games G that are simplerthan G,if J is losing,then G +J is losing;and for all games J that are simplerthan J,if J is losing,then G+J is losing.So suppose that J is losing.We want to show that G+J is losing.Any initial movein J leads to an option J which is winning,which means that there is acorresponding option J of J (by player II’s reply)where J is losing.Hence,whenplayer I makes the corresponding initial move from G+J to G+J ,player II cancounter by moving to G+J .By inductive assumption,this is losing because J islosing.Alternatively,player I may move from G+J to G +J.Because G is a losinggame,there is a move by player II from G to G where G is again a losing game,and hence G +J is also losing,by inductive assumption,because J is losing.Thiscompletes the induction and proves the claim.What is missing is to show that if G+J is losing,so is J.If J was winning,then therewould be a winning move to some option J of J where J is losing,but then,by ourclaim(the‘if’part that we just proved),G+J is losing,which would be a winningoption in G+J for player I.But this is a contradiction.This completes the proof.The preceding lemma says that any losing game Z,say,can be added to a game Gwithout changing whether G is winning or losing(in lemma2.5,Z is called G).Thatis,extending(2.3),Z losing=⇒G+Z≡G.(2.4)As an example,consider Z=∗1+∗2+∗3,which is nim with three heaps of sizes1,2,3.To see that Z is losing,we examine the options of Z and show that all ofthem are winning games.Removing an entire heap leaves two unequal heaps,whichis a winning position by lemma2.1.Any other move produces three heaps,two of40Game theorywhich have equal size.Because two equal heaps deﬁne a losing nim game Z,they can be ignored by(2.4),meaning that all these options are like single nim heaps and therefore winning positions,too.So Z=∗1+∗2+∗3is losing.The game G=∗4+∗5is clearly winning.By(2.4), the game G+Z is equivalent to G and is also winning.However,verifying directly that∗1+∗2+∗3+∗4+∗5is winning would not be easy to see without using(2.4). It is an easy exercise to show that in sums of games,games can be replaced by equivalent games,resulting in an equivalent sum.That is,for all games G,H,J,G≡H=⇒G+J≡H+J.(2.5)Note that(2.5)is not merely a re-statement of deﬁnition2.4,because equivalence of the games G+J and H+J means more than just that the games are either both winning or both losing(see the comments before lemma2.9below).Lemma2.6(The copycat principle)G+G≡∗0for any impartial game G. Proof.Given G,we assume by induction that the claim holds for all simpler games G .Any option of G+G is of the form G +G for an option G of G.This is winning by moving to the game G +G which is losing,by inductive assumption.So G+G is indeed a losing game,and therefore equivalent to∗0by lemma2.5.We now come back to the issue of inverse elements in abstract groups,mentioned at the end of section2.9.If we identify equivalent games,then the addition+of games deﬁnes indeed a group operation.The neutral element is∗0,or any equivalent game (that is,a losing game).The inverse of a game G,written as the negative−G,fulﬁlsG+(−G)≡∗0.(2.6) Lemma2.6shows that for an impartial game,−G is simply G itself.Side remark:For games that are not impartial,that is,partisan games,−G exists also.It is G but with the roles of the two players exchanged,so that whatever move was available to player I is now available to player II and vice versa.As an example, consider the game checkers(with the rule that whoever can no longer make a move loses),and let G be a certain conﬁguration of pieces on the checkerboard.Then−G is the same conﬁguration with the white and black pieces interchanged.Then in the game G+(−G),player II(who can move the black pieces,say),can also play‘copycat’.Namely,if player I makes a move in either G or−G with a white piece, then player II copies that move with a black piece on the other board(−G or G, respectively).Consequently,player II always has a move available and will win the game,so that G+(−G)is indeed a losing game for the starting player I,that is,G+(−G)≡∗0.However,we only consider impartial games,where−G=G.The following condition is very useful to prove that two games are equivalent. Lemma2.7Two impartial games G,H are equivalent if and only if G+H≡∗0.Proof.If G≡H,then by(2.5)and lemma2.6,G+H≡H+H≡∗0.Conversely,G+H≡∗0implies G≡G+H+H≡∗0+H≡H.Sometimes,we want to prove equivalence inductively,where the following observation is useful.Lemma2.8Two games G and H are equivalent if all their options are equivalent, that is,for every option of G there is an equivalent option of H and vice versa.。

Game Theory Steven J Brams

Game TheorySteven J. BramsNew York UniversityDecember 2005(International Encyclopedia of the Social Sciences, 2nd ed., forthcoming)Game theory is a branch of mathematics used to analyze competitive situations whose outcomes depend not only on one’s own choices, and perhaps chance, but also on the choices made by other parties, or “players.” Because the outcome of a game is dependent on what all players do, each player tries to anticipate the choices of other players in order to determine its own best choice. How these interdependent strategic calculations are made is the subject of the theory.Game theory was created in practically one stroke with the publication of Theory of Games and Economic Behavior in 1944 by mathematician John von Neumann and economist Oskar Morgenstern. This was a monumental intellectual achievement and has given rise to hundreds of books and thousands of articles in a variety of disciplines.The theory has several major divisions, the following being the most important:• 2-person versus n-person: the 2-person theory deals with theoptimal strategic choices of two players, whereas the n-persontheory (n > 2) mostly concerns what coalitions, or subsets ofplayers, will form and be stable, and what constitute reasonablepayments to their members.• zero-sum versus nonzero-sum: the payoffs to all players sum tozero (or some other constant) at each outcome in zero-sum gamesbut not in nonzero-sum games, wherein the sums are variable; zero-sum games are games of total conflict, in which what one playergains the others lose, whereas nonzero-sum games permit theplayers to gain or lose together.• cooperative versus noncooperative: cooperative games are thosein which players can make binding and enforceable agreements,whereas noncooperative games may or may not allow forcommunication among the players but do assume that anyagreement reached must be in equilibrium—that is, it is rationalfor a player not to violate it if other players do not, because theplayer would be worse off if it did.Games can be described by several different forms, the three most important being:(l) extensive (game tree)—indicates sequences of choices that players (and possibly chance, according to nature or some random device) can make, with payoffs defined at the end of each sequence of choices;(2) normal/strategic (payoff matrix)—indicates strategies, or complete plans contingent on other players’ choices, for each player, with payoffs defined at the intersection of each set of strategies in a matrix;(3) characteristic function—indicates values that all possible coalitions (subsets) of players can ensure for their members, whatever the other players do.These different game forms, or representations, give less and less detailed information about a game—with the sequences in form l dropped from form 2, and the strategies to implement particular outcomes in form 2 dropped from form 3—to highlight different aspects of a strategic situation.Common to all areas of game theory is the assumption that players are rational: They have goals, can rank outcomes (or, more stringently, attach utilities, or values, tothem), and choose better over worse outcomes. Complications arise from the fact that there is generally no dominant, or unconditionally best, strategy for a player because of the interdependency of player choices. (Games in which there is only one player are sometimes called “games against nature” and are the subject of decision theory.)A game is sometimes defined as the sum-total of its rules. Common parlor games, like chess or poker, have well-specified rules and are generally zero-sum games, making cooperation with the other player(s) unprofitable. Poker differs from chess in being not only an n-person game (though two players can also play it) but also a game of incomplete information, because the players do not have full knowledge of each other’s hands, which depend in part on chance.The rules of most real-life games are equivocal; indeed, the “game” may be about the rules to be used (or abrogated). In economics, rules are generally better known and followed than in politics, which is why game theory has become the theoretical foundation of economics, especially microeconomics. But game-theoretic models also play a major role in other subfields of economics, including industrial organization, public economics, and international economics. Even in macroeconomics, in which fiscal and monetary policies are studied, questions about setting interest rates and determining the money supply have a strong strategic component, especially with respect to the timing of such actions. It is little wonder that economics, more than any of the other social sciences, uses game theory at all levels.Game-theoretic modeling has made major headway in political science, including international relations, in the last generation. While international politics is considered to be quite anarchistic, there is certainly some constancy in the way conflicts develop and may, or may not, be resolved. Arms races, for instance, are almost always nonzero-sum games in which two nations can benefit if they reach some agreement on limitingweapons, but such agreements are often hard to verify or enforce and, consequently, may be unstable.Since the demise of the superpower conflict around 1990, interest has shifted to whether a new “balance of power”—reminiscent of the political juggling acts of European countries in the nineteenth and early twentieth century—may emerge in different regions or even worldwide. For example, will China, as it become more and more a superpower in Asia, align itself with other major Asian countries, like India and Japan, or will it side more with Western powers to compete against its Asian rivals? Game theory offers tools for studying the stability of new alignments, including those that might develop on political-economy issues.Consider, for example, the World Trade Organization (WTO), whose durability is now being tested by regional trading agreements that have sprung up among countries in the Americas, Europe, and Asia. The rationality of supporting the WTO, or joining a regional trading bloc, is very much a strategic question that can be illuminated by game theory. Game theory also provides insight into how the domestic politics of a country impinges on its foreign policy, and vice versa, which has led to a renewed interest in the interconnections between these two levels of politics.Other applications of game theory in political science have been made to strategic voting in committees and elections, the formation and disintegration of parliamentary coalitions, and the distribution of power in weighted voting bodies. On the normative side, electoral reforms have been proposed to lessen the power of certain parties (e.g., the religious parties in Israel), based on game-theoretic analysis. Similarly, the voting weights of members of the European Union Council of Ministers, and its decision rules for taking action (e.g., simple majority or qualified majority), have been studied with an eye to making the body both representative of individual members’ interests and capable of taking collective action.As game-theoretic models have become more prominent in political science, they have, at the same time, created a good deal of controversy. Some critics charge that they abstract too much from strategic situations, reducing actors to hyper-rational players or bloodless automatons that do not reflect the emotions or the social circumstances of people caught up in conflicts. Moreover, critics contend, game-theoretic models are difficult to test empirically, in part because they depend on counterfactuals that are never observed. That is, they assume that players take into account contingencies that are hard to reconstruct, much less model precisely.But proponents of game theory counter that the theory brings rigor to the study of strategic choices that no other theory can match. Furthermore, they argue that actors are, by and large, rational—they choose better over worse means, even if the goals that they seek to advance are not always apparent.When information is incomplete, so-called Bayesian calculations can be made that take account of this incompleteness. The different possible goals that players may have can also be analyzed and their consequences assessed.Because such reconstruction is often difficult to do in real-life settings, laboratory experiments—in which conditions can be better controlled—are more and more conducted. In fact, experiments that test theories of bargaining, voting, and other political-economic processes have become commonplace in economics and political science. Although they are less common in the other social sciences, social psychology has long used experiments to investigate the choices of subjects in games like Prisoners’ Dilemma.This infamous game captures a situation in which two players have dominant strategies of not cooperating, as exemplified by an arms race or a price war. But doing so results in an outcome worse for both than had they cooperated. Because mutualcooperation is not a “Nash equilibrium,” however, each player has an incentive to defect from cooperation.Equally vexing problems confront the players in another well-known game, Chicken. Not only is cooperation unstable, but non-cooperation leads to a disastrous outcome. It turns out that each player should defect if and only if the other player cooperates, but anticipating when an opponent will do so is no mean feat.Since the invention of game theory more than 60 years ago, its development has been remarkable. Two Nobel prizes in economics were awarded to a total of five game theorists in 1994 and 2005 (including John Nash of “beautiful mind” fame), but many other recipients of this prize have used game theory extensively. In addition, game-theoretic modeling has progressed rapidly in political science—and, to a less extent, in the other social sciences—as well as in a variety of other disciplines, including biology, business, and law.BibliographyAumann, Robert J., and Sergiu Hart (eds.). Handbook of Game Theory with Economic Applications. Amsterdam: Elsevier, 1992 (vol. 1), 1994 (vol. 2), 2002 (vol. 3). A comprehensive treatment of game theory and its applications, developed in long review chapters written by leading experts.Brams, Steven J. Theory of Moves. New York: Cambridge University Press, 1994. A dynamic approach to game theory that is applied to a wide variety of conflicts. Dixit, Avinash, and Susan Skeath. Games of Strategy, 2nd ed. New York: W.W. Norton, 2005. An elementary game-theory text that requires only a minimal mathematical background.Nasar, Sylvia. A Beautiful Mind. New York: Simon & Schuster, 1998. A biography of John Nash that also offers an account of the early history of game theory. Afictionalized version of this biography was made into a movie in 2001. Osborne, Martin J. An Introduction to Game Theory. New York: Oxford University Press, 2004. An intermediate text with many applications.Von Neumann, John, and Oskar Morgenstern. Theory of Games and Economic Behavior, 3rd ed. Princeton, NJ: Princeton University Press, 1953. The founding work ongame theory by its inventors.。

Mathematical Models Game Theory 博弈论数学模型

Mathematical Models: Game TheoryMark FeyUniversity of RochesterThis course is designed to teach graduate students in political science the tools of game theory. The course will cover the standard group of essential concepts and some additional topics that are particularly important in formal theory. In addition, we will cover some specific applications of game theory in political science.Students should have, at a minimum, a mathematical background of algebra (solving equations, graphing functions, etc.) and a basic knowledge of probability. Development of the theory does not require calculus, although some applications will be presented that utilize basic calculus (i.e., derivatives). A very brief summary of the necessary mathematics is in Appendix One of Morrow.Game theory, as with most mathematical topics, is best learned by doing, rather than reading. Thus, an important part of the course will be the problem sets covering the lecture material and readings. These problem sets will count for 60% of the final grade, and a take-home exam at the end of the course will count for 40% of the final grade. Solutions to the problem sets will be covered in class. Auditors are welcome, and those who complete the problem sets and keep up with the lectures and reading will be entitled to seek help with problems and with the material. There are two required texts for the course:James Morrow, 1995. Game Theory for Political Scientists. Princeton University Press.Robert Gibbons, 1992. Game Theory for Applied Economists. Princeton University Press. Other readings will be made available to you for photocopying.ScheduleJune 27Introduction, Basic Assumptions of Rational ChoiceMorrow: Chs. 1-2June 28 Decision Theory, OptimizationJune 29 Representing Games, Strategic Form GamesMorrow: Chs. 3-4June 30 Strategic Form Games129July 3 & 4 No class, but lots of fireworks!July 5 Strategic Form Games, DominanceGibbons: Sec. 1.1July 6 Nash Equilibrium, Mixed StrategiesGibbons: Sec. 1.3July 7 Zero-sum Games, ApplicationsJuly 10 Extensive Form Games, Backwards Induction Morrow: Ch. 5July 11 Subgame Perfection, Forward InductionGibbons: Ch. 2July 12 Bayesian Games, Bayesian EquilibriumMorrow: Ch. 6July 13 Bayesian EquilibriumGibbons: Ch. 3July 14 Perfect Bayesian Equilibrium and Sequential Equilibrium Morrow: Ch. 7Gibbons: Sec. 4.1July 17 Sequential EquilibriumJuly 18 Signaling GamesMorrow: Ch. 8Gibbons: Sec. 4.2July 19 Cheap Talk GamesGibbons: Sec. 4.3July 20 Repeated GamesMorrow: Ch. 9July 21 Applications, Wrap-up130。

游戏理论概括英文作文

游戏理论概括英文作文英文：Game theory is a branch of applied mathematics and economics that studies strategic interactions between rational decision-makers. It provides a framework for analyzing and understanding how individuals, firms, and governments make decisions in competitive situations. The central concept of game theory is the "game," which is a formal model of a strategic interaction between two or more players. Each player in the game has a set of possible strategies, and the outcome of the game depends on the strategies chosen by all players.One of the most famous games in game theory is the prisoner's dilemma, which illustrates the concept of Nash equilibrium, named after the mathematician John Nash. In the prisoner's dilemma, two suspects are arrested and placed in separate cells. They are given the option to cooperate with each other by remaining silent, or to betrayeach other by confessing. The outcome of their decision depends on the choices made by both suspects. The Nash equilibrium in this game occurs when both suspects choose to betray each other, even though they would be better off if they both remained silent. This example demonstrates how rational individuals may not always make the best decisions when they are in a competitive situation.Another important concept in game theory is the notion of a dominant strategy, which is a strategy that always provides the best outcome for a player, regardless of the strategies chosen by other players. For example, in the game of rock-paper-scissors, if one player always chooses "rock" regardless of the other player's choice, they have a dominant strategy. Understanding dominant strategies can help players make optimal decisions in competitive games.Game theory has applications in various fields, including economics, political science, biology, and computer science. For example, it is used to analyze voting behavior in elections, to model the behavior of firms in oligopoly markets, and to study the evolution ofcooperation in biological systems. In computer science, game theory is used to design algorithms for multi-agent systems and to analyze the behavior of autonomous agents.In conclusion, game theory provides a powerful framework for analyzing strategic interactions and understanding the behavior of rational decision-makers in competitive situations. By studying game theory, we cangain insights into a wide range of phenomena in the social and natural sciences.中文：博弈论是应用数学和经济学的一个分支，研究理性决策者之间的战略互动。

夸美纽斯的游戏理论

夸美纽斯的游戏理论Kuhn's Game Theory is a mathematical approach to studying the interactions between two or more players. It is based on the idea that each player has a set of strategies that they can use to maximize their chances of winning in a given situation. The theory is used to analyze the decisions that players make in a game, and to determine the optimal strategy for each player to maximize their chances of winning. This theory has been applied to a wide variety of games, including poker, chess, and Go. It has also been used to study economic and political interactions.Krashen's Game Theory is a theory of second language acquisition developed by Stephen Krashen, a linguist and educational researcher. The theory posits that language acquisition is a subconscious process that occurs when learners are exposed to comprehensible input. It suggests that learners can acquire language when they are exposed to meaningful input that they can understand, even if they are not actively attempting to learn the language. Krashen's Game Theory proposes that the best way to learn a second language is to play games in the target language. Games provide an enjoyable and engaging way for learners to interact with the language and learn it in a natural way. Games also provide an opportunity for learners to practice using the language in a low-stakes environment. Krashen's Game Theory suggests that playing games in the target language can be an effective way to learn a second language.Kuhn's game theory is a theory that explains how players in a game interact with each other. It is based on the idea that players are motivated by a desire to maximize their own benefit, and that they will make decisions based on their own interests. The theory states that players will make decisions that are in their own best interest, regardless of the consequences for other players. The theory also suggests that players will try to anticipate the moves of their opponents and adjust their strategies accordingly. This theory is used to explain a wide range of phenomena in economics, politics, and other fields.KMUNS's game theory is a theory of strategic interaction between two or more players. It is based on the idea that players choose strategies that maximize their benefits given the strategies chosen by other players. The theory has been applied to a wide range of fields, including economics, politics, and military strategy. KMUNS's game theory is a powerful tool for understanding the behavior of individuals and organizations in situations where the outcomes depend on the decisions of multiple players. It allows us to analyze how different strategies can lead to different outcomes, and how players can use different strategies to achieve their goals.Kleiner-Minsky's game theory is a mathematical approach to understanding how different players interact and make decisions in a game. It is based on the idea that players have different objectives and will make decisions based on their own preferences. The theory is used to analyze the behavior of players in a game, and to predict how they will act in different situations. It canalso be used to develop strategies for players to maximize their chances of winning. Kleiner-Minsky's game theory has been used in a variety of fields, including economics, political science, and artificial intelligence.Kauffman's Game Theory is a theory of evolutionary dynamics that was developed by American biologist and complexity theorist Stuart Kauffman. The theory is based on the idea that complex systems can self-organize and evolve over time in response to their environment. Kauffman's Game Theory is used to explain how complex systems such as ecosystems, social networks, and markets can emerge and evolve without the need for outside intervention. The theory suggests that the emergence of complexity is a result of the interactions between the elements of the system, and that the system is capable of self-organization and adaptation. Kauffman's Game Theory has been used to explain a wide range of phenomena, from the evolution of species to the emergence of new technologies.。

game theory

What is Game theory
is a branch of applied mathematics that is used in the social science, most notably in economics, as well as in biology , engineering, political science, international relations, computer science and philosophy.
Game
theory attempts to mathematically capture behavior in strategic situations, in which an individual's success in making choices depends on the choices of others.

. Robert Gplied fields within economics who want a serious and thorough discussion of game theory but who may have found other works overly abstract. Gibbons emphasizes the economic applications of the theory at least as much as the pure theory itself; formal arguments about abstract games play a minor role.

这个理论中有个经典的例子——囚徒困境，说的是：A 和B合伙抢了银行，被抓获。他们被关进分隔的牢房。检察官分别告诉他们：如果两人都不坦白，他们会因非法携带枪支的罪名各判刑1年；如果其中一人招供而另一人不招，坦白者作为证人将不会被起诉，另一人将会被重判10年；如果两人都招供，则两人都会以抢劫罪名各判5年。结果，两个人都招供了。故事背后的原理是这样的：A和B都是理性的，首先关心的是减少自己刑期，而不在乎对方被判多少年。由于信息是不对称的，双方没有串谋的机会，所以，A会这样推理： “假如B不招，我只要一招供，立马可以获得自由，而不招却要坐牢一年，显然招比不招好；假如B招了，我若不招，则要坐牢10年，招了只坐5年，显然还是以招认为好。可见无论B招与不招，我的最佳选择都是招认。”同理，B也是理性的人，也会如此推理。于是两人都作出招供的选择，这对他们个人来说都最佳的，即最符合他们个体理性的选择。但现在的问题是，两个人都招供，并不是集体最优的选择。无论对哪个人来说，两个人都不招供，要比两个人都招供好得多。最符合个体理性的选择，却是集体非理性的。

game theory became a formal topic of study in the

game theory became a formal topic ofstudy in theGame theory, also known as the science of strategizing, has a rich and intriguing history that spans across multiple disciplines. However, it wasn't until the mid-20th century that game theory truly emerged as a formal topic of academic study.The foundations of game theory can be traced back to the early works of mathematicians such as Emile Borel and John von Neumann. Borel's concept of strategy in chess, published in 1921, introduced the idea of optimal play in games of perfect information. Von Neumann, on the other hand, is credited with developing the minimax theorem, a fundamental principle in game theory that deals with zero-sum games.It was the seminal work of John von Neumann and Oskar Morgenstern in 1944, titled "Theory of Games and Economic Behavior," that marked the official birth of game theory as a formal field of study. This book provided a rigorous mathematical framework for analyzing strategic interactions between rational players and established game theory as a branch of applied mathematics with widespread applications in economics, political science, and beyond.The subsequent decades saw significant contributions from other notable scholars like John Nash, who introduced the concept of the Nash equilibrium, a fundamental solution concept in non-cooperative games. This equilibrium, which describes a stable state where no player can improve their outcome by unilaterally changing their strategy, has become a cornerstone of modern game theory.Today, game theory is a thriving field of research with applications in various domains such as economics, finance, politics, sociology, and even biology. It provides a powerful toolbox for analyzing complex strategic interactions and understanding how rational individuals make decisions in competitive environments. The formalization of game theory has not only deepened our understanding of strategic behavior but has also opened up new avenues for exploring the intricacies of humandecision-making.。

gametheory6博弈论英文精品PPT课件

3 / 27
Review Dynamic Games Centipede Game Ultimatum Game Summary
• action is a decision in one particular node (confess, remain silent, head, tail,…) • strategy is a plan of actions for every possible situation that might occur, for every possible node (AF-Accept if Albert goes In, Fight if Albert plays Out) • strategy – it is deciding about the action in each decision node prior to the game • it is like as if you want your friend to play the game instead of you, you have to tell him in advance what to do in each situation
OUT
0 2
IN
FA
-3
2
-1
1
5 / 27
Review Dynamic Games Centipede Game Ultimatum Game Summary
Dynamic Game (tree):
OUT
IN
0 2
Static game (table):
IN OUT
FA
-3
2
-1
1
F
NEA:
-3,-1

Game Theory

Game Theory
Game theory is a study of how to mathematically determine the best strategy for given conditions in order to optimize the outcome.
Basic Elements
Pirates A B C D E
1
100
2
0
100
3
1
0
99
4
0
1
0
99
5
1
0
1
0
98
This is just The Game Theory
While you are making decisions, other people’s consideration should also be involved in your plan. Whether you can be successful or not lies not only on yourself but also on your competitors’ choice. Game theory is a method of decision-making and you choose the optimal one which maximizes your own interest on the thorough analysis of mutual strategies.
The problem
• Imagine you are the oldest in a band of five pirates, where no two pirates in the group are of the same age. It is your responsibility as the oldest pirate to decide how to divide the group's one hundred pieces of gold.

无处不在的博弈

中文摘要最近四五十年，经济学经历了一场博弈论革命。

1994年度的诺贝尔经济学奖授予三位博弈论专家，2005年度的诺贝尔经济学奖又授予两位博弈论专家，可以看做博弈论成熟的标志。

这也更大激发了人们了解博弈论的热情。

现在的博弈论研究，特别是国内的应用研究，只知道在数学上折腾，不知道博弈的思想更加重要，不知道还可以通过博弈思维去破译诸多社会现象、文化现象。

“当代最后一个经济学全才”保罗.萨缪尔森教授，在他生命的最后年月，告诫我们说：“要想在现代社会做一个有文化的人，你必须对博弈论有一个大致的了解。

”在许多人纠结于关于博弈论一系列的表格、图形、模型时，他们忽视了很重要的一点，其实博弈就存在于我们身边，存在于生活中的各个角落。

商业竞争、政治选举、职场生存、婚姻经营、朋友相处，就像两人对弈，常常是相当人格化的竞争。

博弈贯穿于我们的生活。

本文将通过几个经典的例子及日常生活中的现象对博弈进行分析。

关键词：博弈论；模型；无处不在ABSTRACTThe last forty or fifty years, has experienced a revolution in game theory of economics. 1994 Nobel economics prize was awarded to three game-theory experts, 2005 Nobel Prize in economics awarded two experts on game theory, game theory can be thought of as a sign of maturity. Larger fires the enthusiasm of people about the game.Now, on game theory, especially applied research, just tossing in mathematics, does not know the games are more important, don't know what else can be done by game thought to decipher the many social and cultural phenomena. "Contemporary last generalist in economics" Professor Paul.Samuelson, in the last years of his life, told us: "If you want to become a literate person in modern society, you have to have a general understanding of game theory. ”Many people struggling with on a series of tables, graphics, game theory models, they ignore a very important point, in fact games around, exists in every corner of life. Business competition, political elections, survival, marriage and business career, friends, like two people plays chess, personification is often quite competitive. Game runs through our lives. This article through several classic examples of analysis and phenomena of everyday life on the game.Key words:Game theory, model and prevasiveness无处不在的博弈一、引言“博弈论”原本是数学的一个分支但由于它较好地解决了对竞争等问题的可操作性分析，成为经济学中激荡人心的一个研究领域。

Game Theory 2

GAME THEORYThomas S.Ferguson Part II.Two-Person Zero-Sum Games1.The Strategic Form of a Game.1.1Strategic Form.1.2Example:Odd or Even.1.3Pure Strategies and Mixed Strategies.1.4The Minimax Theorem.1.5Exercises.2.Matrix Games.Domination.2.1Saddle Points.2.2Solution of All2by2Matrix Games.2.3Removing Dominated Strategies.2.4Solving2×n and m×2Games.2.5Latin Square Games.2.6Exercises.3.The Principle of Indiﬀerence.3.1The Equilibrium Theorem.3.2Nonsingular Game Matrices.3.3Diagonal Games.3.4Triangular Games.3.5Symmetric Games.3.6Invariance.3.7Exercises.4.Solving Finite Games.4.1Best Responses.4.2Upper and Lower Values of a Game.4.3Invariance Under Change of Location and Scale.4.4Reduction to a Linear Programming Problem.4.5Description of the Pivot Method for Solving Games.4.6A Numerical Example.4.7Exercises.5.The Extensive Form of a Game.5.1The Game Tree.5.2Basic Endgame in Poker.5.3The Kuhn Tree.5.4The Representation of a Strategic Form Game in Extensive Form.5.5Reduction of a Game in Extensive Form to Strategic Form.5.6Example.5.7Games of Perfect Information.5.8Behavioral Strategies.5.9Exercises.6.Recursive and Stochastic Games.6.1Matrix Games with Games as Components.6.2Multistage Games.6.3Recursive Games. -Optimal Strategies.6.4Stochastic Movement Among Games.6.5Stochastic Games.6.6Approximating the Solution.6.7Exercises.7.Continuous Poker Models.7.1La Relance.7.2The von Neumann Model.7.3Other Models.7.4Exercises.References.Part II.Two-Person Zero-Sum Games1.The Strategic Form of a Game.The individual most closely associated with the creation of the theory of games is John von Neumann,one of the greatest mathematicians of this century.Although others preceded him in formulating a theory of games-notably´Emile Borel-it was von Neumann who published in1928the paper that laid the foundation for the theory of two-person zero-sum games.Von Neumann’s work culminated in a fundamental book on game theory written in collaboration with Oskar Morgenstern entitled Theory of Games and Economic Behavior,1944.Other more current books on the theory of games may be found in the text book,Game Theory by Guillermo Owen,2nd edition,Academic Press,1982,and the expository book,Game Theory and Strategy by Philip D.Straﬃn,published by the Mathematical Association of America,1993.The theory of von Neumann and Morgenstern is most complete for the class of games called two-person zero-sum games,i.e.games with only two players in which one player wins what the other player loses.In Part II,we restrict attention to such games.We will refer to the players as Player I and Player II.1.1Strategic Form.The simplest mathematical description of a game is the strate-gic form,mentioned in the introduction.For a two-person zero-sum game,the payoﬀfunction of Player II is the negative of the payoﬀof Player I,so we may restrict attention to the single payoﬀfunction of Player I,which we call here L.Deﬁnition1.The strategic form,or normal form,of a two-person zero-sum game is given by a triplet(X,Y,A),where(1)X is a nonempty set,the set of strategies of Player I(2)Y is a nonempty set,the set of strategies of Player II(3)A is a real-valued function deﬁned on X×Y.(Thus,A(x,y)is a real number for every x∈X and every y∈Y.)The interpretation is as follows.Simultaneously,Player I chooses x∈X and Player II chooses y∈Y,each unaware of the choice of the other.Then their choices are made known and I wins the amount A(x,y)from II.Depending on the monetary unit involved, A(x,y)will be cents,dollars,pesos,beads,etc.If A is negative,I pays the absolute value of this amount to II.Thus,A(x,y)represents the winnings of I and the losses of II.This is a very simple deﬁnition of a game;yet it is broad enough to encompass the ﬁnite combinatorial games and games such as tic-tac-toe and chess.This is done by being suﬃciently broadminded about the deﬁnition of a strategy.A strategy for a game of chess,for example,is a complete description of how to play the game,of what move to make in every possible situation that could occur.It is rather time-consuming to write down even one strategy,good or bad,for the game of chess.However,several diﬀerent programs for instructing a machine to play chess well have been written.Each program constitutes one strategy.The program Deep Blue,that beat then world chess champion Gary Kasparov in a match in1997,represents one strategy.The set of all such strategies for Player I is denoted by X.Naturally,in the game of chess it is physically impossible to describe all possible strategies since there are too many;in fact,there are more strategies than there are atoms in the known universe.On the other hand,the number of games of tic-tac-toe is rather small,so that it is possible to study all strategies andﬁnd an optimal strategy for each ter,when we study the extensive form of a game,we will see that many other types of games may be modeled and described in strategic form.To illustrate the notions involved in games,let us consider the simplest non-trivial case when both X and Y consist of two elements.As an example,take the game called Odd-or-Even.1.2Example:Odd or Even.Players I and II simultaneously call out one of the numbers one or two.Player I’s name is Odd;he wins if the sum of the numbers if odd. Player II’s name is Even;she wins if the sum of the numbers is even.The amount paid to the winner by the loser is always the sum of the numbers in dollars.To put this game in strategic form we must specify X,Y and A.Here we may choose X={1,2},Y={1,2}, and A as given in the following table.II(even)yI(odd)x12 1−2+3 2+3−4A(x,y)=I’s winnings=II’s losses.It turns out that one of the players has a distinct advantage in this game.Can you tell which one it is?Let us analyze this game from Player I’s point of view.Suppose he calls‘one’3/5ths of the time and‘two’2/5ths of the time at random.In this case,1.If II calls‘one’,I loses2dollars3/5ths of the time and wins3dollars2/5ths of the time;on the average,he wins−2(3/5)+3(2/5)=0(he breaks even in the long run).2.If II call‘two’,I wins3dollars3/5ths of the time and loses4dollars2/5ths of the time; on the average he wins3(3/5)−4(2/5)=1/5.That is,if I mixes his choices in the given way,the game is even every time II calls ‘one’,but I wins20/c on the average every time II calls‘two’.By employing this simple strategy,I is assured of at least breaking even on the average no matter what II does.Can Player Iﬁx it so that he wins a positive amount no matter what II calls?Let p denote the proportion of times that Player I calls‘one’.Let us try to choose p so that Player I wins the same amount on the average whether II calls‘one’or‘two’.Then since I’s average winnings when II calls‘one’is−2p+3(1−p),and his average winnings when II calls‘two’is3p−4(1−p)Player I should choose p so that−2p+3(1−p)=3p−4(1−p)3−5p=7p−412p=7p=7/12.Hence,I should call‘one’with probability7/12,and‘two’with probability5/12.On theaverage,I wins−2(7/12)+3(5/12)=1/12,or813cents every time he plays the game,nomatter what II does.Such a strategy that produces the same average winnings no matter what the opponent does is called an equalizing strategy.Therefore,the game is clearly in I’s favor.Can he do better than813cents per gameon the average?The answer is:Not if II plays properly.In fact,II could use the same procedure:call‘one’with probability7/12call‘two’with probability5/12.If I calls‘one’,II’s average loss is−2(7/12)+3(5/12)=1/12.If I calls‘two’,II’s average loss is3(7/12)−4(5/12)=1/12.Hence,I has a procedure that guarantees him at least1/12on the average,and II has a procedure that keeps her average loss to at most1/12.1/12is called the value of the game,and the procedure each uses to insure this return is called an optimal strategy or a minimax strategy.If instead of playing the game,the players agree to call in an arbitrator to settle thisconﬂict,it seems reasonable that the arbitrator should require II to pay813cents to I.ForI could argue that he should receive at least813cents since his optimal strategy guaranteeshim that much on the average no matter what II does.On the other hand II could arguethat he should not have to pay more than813cents since she has a strategy that keeps heraverage loss to at most that amount no matter what I does.1.3Pure Strategies and Mixed Strategies.It is useful to make a distinction between a pure strategy and a mixed strategy.We refer to elements of X or Y as pure strategies.The more complex entity that chooses among the pure strategies at random in various proportions is called a mixed strategy.Thus,I’s optimal strategy in the game of Odd-or-Even is a mixed strategy;it mixes the pure strategies one and two with probabilities 7/12and5/12respectively.Of course every pure strategy,x∈X,can be considered as the mixed strategy that chooses the pure strategy x with probability1.In our analysis,we made a rather subtle assumption.We assumed that when a player uses a mixed strategy,he is only interested in his average return.He does not care about hismaximum possible winnings or losses—only the average.This is actually a rather drastic assumption.We are evidently assuming that a player is indiﬀerent between receiving5 million dollars outright,and receiving10million dollars with probability1/2and nothing with probability1/2.I think nearly everyone would prefer the$5,000,000outright.This is because the utility of having10megabucks is not twice the utility of having5megabucks.The main justiﬁcation for this assumption comes from utility theory and is treated in Appendix1.The basic premise of utility theory is that one should evaluate a payoﬀby its utility to the player rather than on its numerical monetary value.Generally a player’s utility of money will not be linear in the amount.The main theorem of utility theory states that under certain reasonable assumptions,a player’s preferences among outcomes are consistent with the existence of a utility function and the player judges an outcome only on the basis of the average utility of the outcome.However,utilizing utility theory to justify the above assumption raises a new diﬃculty. Namely,the two players may have diﬀerent utility functions.The same outcome may be perceived in quite diﬀerent ways.This means that the game is no longer zero-sum.We need an assumption that says the utility functions of two players are the same(up to change of location and scale).This is a rather strong assumption,but for moderate to small monetary amounts,we believe it is a reasonable one.A mixed strategy may be implemented with the aid of a suitable outside random mechanism,such as tossing a coin,rolling dice,drawing a number out of a hat and so on.The seconds indicator of a watch provides a simple personal method of randomization provided it is not used too frequently.For example,Player I of Odd-or-Even wants an outside random event with probability7/12to implement his optimal strategy.Since 7/12=35/60,he could take a quick glance at his watch;if the seconds indicator showed a number between0and35,he would call‘one’,while if it were between35and60,he would call‘two’.1.4The Minimax Theorem.A two-person zero-sum game(X,Y,A)is said to be aﬁnite game if both strategy sets X and Y areﬁnite sets.The fundamental theorem of game theory due to von Neumann states that the situation encountered in the game of Odd-or-Even holds for allﬁnite two-person zero-sum games.Speciﬁcally,The Minimax Theorem.For everyﬁnite two-person zero-sum game,(1)there is a number V,called the value of the game,(2)there is a mixed strategy for Player I such that I’s average gain is at least V no matter what II does,and(3)there is a mixed strategy for Player II such that II’s average loss is at most V no matter what I does.This is one form of the minimax theorem to be stated more precisely and discussed in greater depth later.If V is zero we say the game is fair.If V is positive,we say the game favors Player I,while if V is negative,we say the game favors Player II.1.5Exercises.1.Consider the game of Odd-or-Even with the sole change that the loser pays the winner the product,rather than the sum,of the numbers chosen(who wins still depends on the sum).Find the table for the payoﬀfunction A,and analyze the game toﬁnd the value and optimal strategies of the players.Is the game fair?2.Player I holds a black Ace and a red8.Player II holds a red2and a black7.The players simultaneously choose a card to play.If the chosen cards are of the same color, Player I wins.Player II wins if the cards are of diﬀerent colors.The amount won is a number of dollars equal to the number on the winner’s card(Ace counts as1.)Set up the payoﬀfunction,ﬁnd the value of the game and the optimal mixed strategies of the players.3.Sherlock Holmes boards the train from London to Dover in an eﬀort to reach the continent and so escape from Professor Moriarty.Moriarty can take an express train and catch Holmes at Dover.However,there is an intermediate station at Canterbury at which Holmes may detrain to avoid such a disaster.But of course,Moriarty is aware of this too and may himself stop instead at Canterbury.Von Neumann and Morgenstern(loc.cit.) estimate the value to Moriarty of these four possibilities to be given in the following matrix (in some unspeciﬁed units).HolmesMoriartyCanterbury Dover Canterbury100−50 Dover0100What are the optimal strategies for Holmes and Moriarty,and what is the value?(His-torically,as related by Dr.Watson in“The Final Problem”in Arthur Conan Doyle’s The Memoires of Sherlock Holmes,Holmes detrained at Canterbury and Moriarty went on to Dover.)4.The entertaining book The Compleat Strategyst by John Williams contains many simple examples and informative discussion of strategic form games.Here is one of his problems.“I know a good game,”says Alex.“We pointﬁngers at each other;either oneﬁnger or twoﬁngers.If we match with oneﬁnger,you buy me one Daiquiri,If we match with twoﬁngers,you buy me two Daiquiris.If we don’t match I letyou oﬀwith a payment of a dime.It’ll help pass the time.”Olaf appears quite unmoved.“That sounds like a very dull game—at least in its early stages.”His eyes glaze on the ceiling for a moment and his lipsﬂutterbrieﬂy;he returns to the conversation with:“Now if you’d care to pay me42cents before each game,as a partial compensation for all those55-cent drinks I’llhave to buy you,then I’d be happy to pass the time with you.Olaf could see that the game was inherently unfair to him so he insisted on a side payment as compensation.Does this side payment make the game fair?What are the optimal strategies and the value of the game?2.Matrix Games —DominationA ﬁnite two-person zero-sum game in strategic form,(X,Y,A ),is sometimes called a matrix game because the payoﬀfunction A can be represented by a matrix.If X ={x 1,...,x m }and Y ={y 1,...,y n },then by the game matrix or payoﬀmatrix we mean the matrix A =⎛⎝a 11···a 1n ......a m 1···a mn⎞⎠where a ij =A (x i ,y j ),In this form,Player I chooses a row,Player II chooses a column,and II pays I the entry in the chosen row and column.Note that the entries of the matrix are the winnings of the row chooser and losses of the column chooser.A mixed strategy for Player I may be represented by an m -tuple,p =(p 1,p 2,...,p m )of probabilities that add to 1.If I uses the mixed strategy p =(p 1,p 2,...,p m )and II chooses column j ,then the (average)payoﬀto I is m i =1p i a ij .Similarly,a mixed strategy for Player II is an n -tuple q =(q 1,q 2,...,q n ).If II uses q and I uses row i the payoﬀto I is n j =1a ij q j .More generally,if I uses the mixed strategy p and II uses the mixed strategy q ,the (average)payoﬀto I is p T Aq = m i =1 n j =1p i a ij q j .Note that the pure strategy for Player I of choosing row i may be represented as the mixed strategy e i ,the unit vector with a 1in the i th position and 0’s elsewhere.Similarly,the pure strategy for II of choosing the j th column may be represented by e j .In the following,we shall be attempting to ‘solve’games.This means ﬁnding the value,and at least one optimal strategy for each player.Occasionally,we shall be interested in ﬁnding all optimal strategies for a player.2.1Saddle points.Occasionally it is easy to solve the game.If some entry a ij of the matrix A has the property that(1)a ij is the minimum of the i th row,and(2)a ij is the maximum of the j th column,then we say a ij is a saddle point.If a ij is a saddle point,then Player I can then win at least a ij by choosing row i ,and Player II can keep her loss to at most a ij by choosing column j .Hence a ij is the value of the game.Example 1.A =⎛⎝41−3325016⎞⎠The central entry,2,is a saddle point,since it is a minimum of its row and maximum of its column.Thus it is optimal for I to choose the second row,and for II to choose the second column.The value of the game is 2,and (0,1,0)is an optimal mixed strategy for both players.For large m ×n matrices it is tedious to check each entry of the matrix to see if it has the saddle point property.It is easier to compute the minimum of each row and the maximum of each column to see if there is a match.Here is an example of the method.row min A =⎛⎜⎝3210012010213122⎞⎟⎠0001col max 3222row min B =⎛⎜⎝3110012010213122⎞⎟⎠0001col max 3122In matrix A ,no row minimum is equal to any column maximum,so there is no saddle point.However,if the 2in position a 12were changed to a 1,then we have matrix B .Here,the minimum of the fourth row is equal to the maximum of the second column;so b 42is a saddle point.2.2Solution of All 2by 2Matrix Games.Consider the general 2×2game matrix A = a b d c.To solve this game (i.e.to ﬁnd the value and at least one optimal strategy for each player)we proceed as follows.1.Test for a saddle point.2.If there is no saddle point,solve by ﬁnding equalizing strategies.We now prove the method of ﬁnding equalizing strategies of Section 1.2works when-ever there is no saddle point by deriving the value and the optimal strategies.Assume there is no saddle point.If a ≥b ,then b <c ,as otherwise b is a saddle point.Since b <c ,we must have c >d ,as otherwise c is a saddle point.Continuing thus,we see that d <a and a >b .In other words,if a ≥b ,then a >b <c >d <a .By symmetry,if a ≤b ,then a <b >c <d >a .This shows thatIf there is no saddle point,then either a >b ,b <c ,c >d and d <a ,or a <b ,b >c ,c <d and d >a .In equations (1),(2)and (3)below,we develop formulas for the optimal strategies and value of the general 2×2game.If I chooses the ﬁrst row with probability p (es the mixed strategy (p,1−p )),we equate his average return when II uses columns 1and 2.ap +d (1−p )=bp +c (1−p ).Solving for p ,we ﬁndp =c −d (a −b )+(c −d ).(1)Since there is no saddle point,(a−b)and(c−d)are either both positive or both negative; hence,0<p<1.Player I’s average return using this strategy isv=ap+d(1−p)=ac−bda−b+c−d.If II chooses theﬁrst column with probability q(es the strategy(q,1−q)),we equate his average losses when I uses rows1and2.aq+b(1−q)=dq+c(1−q)Hence,q=c−ba−b+c−d.(2)Again,since there is no saddle point,0<q<1.Player II’s average loss using this strategyisaq+b(1−q)=ac−bda−b+c−d=v,(3)the same value achievable by I.This shows that the game has a value,and that the players have optimal strategies.(something the minimax theorem says holds for allﬁnite games). Example2.A=−233−4p=−4−3−2−3−4−3=7/12q=samev=8−9−2−3−4−3=1/12Example3.A=0−1012p=2−10+10+2−1=1/11q=2+100+10+2−1=12/11.But q must be between zero and one.What happened?The trouble is we“forgot to test this matrix for a saddle point,so of course it has one”.(J.D.Williams The Compleat Strategyst Revised Edition,1966,McGraw-Hill,page56.)The lower left corner is a saddle point.So p=0and q=1are optimal strategies,and the value is v=1.2.3Removing Dominated Strategies.Sometimes,large matrix games may be reduced in size(hopefully to the2×2case)by deleting rows and columns that are obviously bad for the player who uses them.Deﬁnition.We say the i th row of a matrix A=(a ij)dominates the k th row if a ij≥a kj for all j.We say the i th row of A strictly dominates the k th row if a ij>a kj for all j.Similarly,the j th column of A dominates(strictly dominates)the k th column if a ij≤a ik(resp.a ij<a ik)for all i.Anything Player I can achieve using a dominated row can be achieved at least as well using the row that dominates it.Hence dominated rows may be deleted from the matrix.A similar argument shows that dominated columns may be removed.To be more precise,removal of a dominated row or column does not change the value of a game .However,there may exist an optimal strategy that uses a dominated row or column (see Exercise 9).If so,removal of that row or column will also remove the use of that optimal strategy (although there will still be at least one optimal strategy left).However,in the case of removal of a strictly dominated row or column,the set of optimal strategies does not change.We may iterate this procedure and successively remove several rows and columns.As an example,consider the matrix,A .The last column is dominated by the middle column.Deleting the last column we obtain:A =⎛⎝204123412⎞⎠Now the top row is dominated by the bottomrow.(Note this is not the case in the original matrix).Deleting the top row we obtain:⎛⎝201241⎞⎠This 2×2matrix does not have a saddle point,so p =3/4,q =1/4and v =7/4.I’s optimal strategy in the original game is(0,3/4,1/4);II’s is (1/4,3/4,0).1241 A row (column)may also be removed if it is dominated by a probability combination of other rows (columns).If for some 0<p <1,pa i 1j +(1−p )a i 2j ≥a kj for all j ,then the k th row is dominated by the mixed strategy that chooses row i 1with probability p and row i 2with probability 1−p .Player I can do at least as well using this mixed strategy instead of choosing row k .(In addition,any mixed strategy choosing row k with probability p k may be replaced by the one in which k ’s probability is split between i 1and i 2.That is,i 1’s probability is increased by pp k and i 2’s probability is increased by (1−p )p k .)A similar argument may be used for columns.Consider the matrix A =⎛⎝046574963⎞⎠.The middle column is dominated by the outside columns taken with probability 1/2each.With the central column deleted,the middle row is dominated by the combination of the top row with probability 1/3and the bottom row with probability 2/3.The reducedmatrix, 0693,is easily solved.The value is V =54/12=9/2.Of course,mixtures of more than two rows (columns)may be used to dominate and remove other rows (columns).For example,the mixture of columns one two and threewith probabilities 1/3each in matrix B =⎛⎝135340223735⎞⎠dominates the last column,and so the last column may be removed.Not all games may be reduced by dominance.In fact,even if the matrix has a saddle point,there may not be any dominated rows or columns.The 3×3game with a saddle point found in Example 1demonstrates this.2.4Solving 2×n and m ×2games.Games with matrices of size 2×n or m ×2may be solved with the aid of a graphical interpretation.Take the following example.p 1−p 23154160Suppose Player I chooses the ﬁrst row with probability p and the second row with proba-bility 1−p .If II chooses Column 1,I’s average payoﬀis 2p +4(1−p ).Similarly,choices of Columns 2,3and 4result in average payoﬀs of 3p +(1−p ),p +6(1−p ),and 5p respectively.We graph these four linear functions of p for 0≤p ≤1.For a ﬁxed value of p ,Player I can be sure that his average winnings is at least the minimum of these four functions evaluated at p .This is known as the lower envelope of these functions.Since I wants to maximize his guaranteed average winnings,he wants to ﬁnd p that achieves the maximum of this lower envelope.According to the drawing,this should occur at the intersection of the lines for Columns 2and 3.This essentially,involves solving the game in which II is restrictedto Columns 2and 3.The value of the game 3116is v =17/7,I’s optimal strategy is (5/7,2/7),and II’s optimal strategy is (5/7,2/7).Subject to the accuracy of the drawing,we conclude therefore that in the original game I’s optimal strategy is (5/7,2/7),II’s is (0,5/7,2/7,0)and the value is 17/7.Fig 2.10123456col.3col.1col.2col.4015/7pThe accuracy of the drawing may be checked:Given any guess at a solution to a game,there is a sure-ﬁre test to see if the guess is correct ,as follows.If I uses the strategy (5/7,2/7),his average payoﬀif II uses Columns 1,2,3and 4,is 18/7,17/7,17/7,and 25/7respectively.Thus his average payoﬀis at least17/7no matter what II does.Similarly, if II uses(0,5/7,2/7,0),her average loss is(at most)17/7.Thus,17/7is the value,and these strategies are optimal.We note that the line for Column1plays no role in the lower envelope(that is,the lower envelope would be unchanged if the line for Column1were removed from the graph). This is a test for domination.Column1is,in fact,dominated by Columns2and3taken with probability1/2each.The line for Column4does appear in the lower envelope,and hence Column4cannot be dominated.As an example of a m×2game,consider the matrix associated with Figure2.2.If q is the probability that II chooses Column1,then II’s average loss for I’s three possible choices of rows is given in the accompanying graph.Here,Player II looks at the largest of her average losses for a given q.This is the upper envelope of the function.II wants toﬁnd q that minimizes this upper envelope.From the graph,we see that any value of q between1/4and1/3inclusive achieves this minimum.The value of the game is4,and I has an optimal pure strategy:row2.Fig2.2⎛⎝q1−q154462⎞⎠123456row1row2row3011/41/2qThese techniques work just as well for2×∞and∞×2games.2.5Latin Square Games.A Latin square is an n×n array of n diﬀerent letters such that each letter occurs once and only once in each row and each column.The5×5 array at the right is an example.If in a Latin square each letter is assigned a numerical value,the resulting matrix is the matrix of a Latin square game.Such games have simple solutions.The value is the average of the numbers in a row,and the strategy that chooses each pure strategy with equal probability1/n is optimal for both players.The reason is not very deep.The conditions for optimality are satisﬁed.⎛⎜⎜⎜⎝a b c d eb e acd c a de b d c e b ae d b a c ⎞⎟⎟⎟⎠a =1,b =2,c =d =3,e =6⎛⎜⎜⎜⎝1233626133313623362163213⎞⎟⎟⎟⎠In the example above,the value is V =(1+2+3+3+6)/5=3,and the mixed strategy p =q =(1/5,1/5,1/5,1/5,1/5)is optimal for both players.The game of matching pennies is a Latin square game.Its value is zero and (1/2,1/2)is optimal for both players.2.6Exercises.1.Solve the game with matrix−1−3−22 ,that is ﬁnd the value and an optimal (mixed)strategy for both players.2.Solve the game with matrix 02t 1for an arbitrary real number t .(Don’t forget to check for a saddle point!)Draw the graph of v (t ),the value of the game,as a function of t ,for −∞<t <∞.3.Show that if a game with m ×n matrix has two saddle points,then they have equal values.4.Reduce by dominance to 2×2games and solve.(a)⎛⎜⎝5410432−10−1431−212⎞⎟⎠(b)⎛⎝1007126476335⎞⎠.5.(a)Solve the game with matrix 3240−21−45 .(b)Reduce by dominance to a 3×2matrix game and solve:⎛⎝08584612−43⎞⎠.6.Players I and II choose integers i and j respectively from the set {1,2,...,n }for some n ≥2.Player I wins 1if |i −j |=1.Otherwise there is no payoﬀ.If n =7,for example,the game matrix is⎛⎜⎜⎜⎜⎜⎜⎜⎝0100000101000001010000010100000101000001010000010⎞⎟⎟⎟⎟⎟⎟⎟⎠。

学习数学的英语作文

The Importance of Learning MathematicsMathematics, often regarded as a challenging and abstract subject, plays a crucial role in our lives. It is not just a collection of formulas and equations but a language that helps us understand and solve problems in various fields.First and foremost, mathematics is the foundation of many scientific and technological advancements. It has been the driving force behind numerous discoveries in physics, engineering, computer science, and beyond. The principles of mathematics have enabled us to invent complex machines, build skyscrapers, and explore the vastness of the universe.Moreover, mathematical thinking improves our logical reasoning and problem-solving skills. It trains us to approach problems systematically, break them down into smaller parts, and find patterns and relationships. These skills are invaluable in everyday life, whether we are making financial decisions, managing projects, or simply trying to make sense of the world.Additionally, mathematics fosters creativity. While it may seem counterintuitive, mathematics actually provides a platform for innovation and exploration. Mathematicians are constantly pushing the boundaries of knowledge, discovering new theorems and concepts that have the potential torevolutionize entire fields.Finally, mathematics is a universal language that transcends cultural and linguistic barriers. It is a powerful tool for communication and collaboration, allowing mathematicians from different parts of the world to share ideas and work together on complex problems.In conclusion, learning mathematics is not just about solving equations or passing exams. It is about acquiring a powerful toolkit of skills and perspectives that will enhance our understanding of the world and empower us to make a positive impact in every aspect of life.。

未来我将成为一名数学老师英语作文

未来我将成为一名数学老师英语作文全文共3篇示例，供读者参考篇1My Dream to Become a Math TeacherAs I sit here in my final year of university, I can't help but reflect on the winding path that has led me to this point. From a young age, I've always had a deep fascination with numbers and their intricate patterns, an insatiable curiosity that has only grown stronger with each passing year. It's this very passion that has fueled my dream of becoming a mathematics teacher – a dream that I'm now closer to realizing than ever before.Growing up, math was always my favorite subject. I remember being that kid who would eagerly raise their hand in class, eager to solve the most complex equations on the board. While my classmates would often groan at the mere mention of math homework, I relished the opportunity to immerse myself in a world of numbers, algorithms, and logical puzzles.It wasn't just the subject matter that captivated me; it was the way math seemed to unlock a deeper understanding of the world around us. From the intricate patterns found in nature tothe complex calculations required for space exploration, math was like a universal language that helped make sense of it all.As I progressed through my academic journey, my love for mathematics only intensified. I reveled in the satisfaction of solving a particularly challenging problem, the thrill of discovering new theorems and concepts that expanded my understanding of the subject.However, it wasn't until my high school years that I truly recognized the power of a great teacher. Mr. Thompson, my calculus instructor, had a way of making even the most abstract concepts feel approachable and engaging. His infectious enthusiasm and unwavering dedication to his students inspired me in ways I never could have imagined.It was then that I realized that teaching mathematics wasn't just about imparting knowledge; it was about igniting a passion for learning, about helping students see the beauty and relevance of math in their everyday lives. From that moment on, I knew that becoming a math teacher was my true calling.Throughout my university years, I've had the privilege of studying under some of the most brilliant minds in the field of mathematics. Each lecture, each seminar, has only reinforced my conviction that teaching is the path I was meant to take.I've witnessed firsthand the transformative power of a great educator, someone who can take complex concepts and break them down into digestible, engaging lessons. I've seen the look of understanding dawn on a student's face, that moment when everything clicks into place, and the sense of accomplishment that follows.As I approach the end of my academic journey, I'm filled with a sense of excitement and purpose. I know that the road ahead won't be easy; teaching is a demanding profession that requires unwavering dedication, patience, and a genuine love for what you do. But it's a challenge that I embrace wholeheartedly, for I can think of no greater calling than to share my passion for mathematics with the next generation of learners.In my future classroom, I envision an environment where curiosity is nurtured, and mistakes are celebrated as opportunities for growth. I want to create a space where students feel empowered to ask questions, to explore new ideas, and to develop a deep appreciation for the elegance and practicality of mathematics.I'll strive to be the kind of teacher who not only imparts knowledge but also inspires a lifelong love of learning. I'll encourage my students to think critically, to challengeassumptions, and to approach problems from multiple angles. After all, mathematics is more than just a set of formulas and equations; it's a way of thinking, a tool for understanding the world around us.And beyond the confines of the classroom, I hope to make a lasting impact on my students' lives. I want them to see math not as a dry, disconnected subject, but as a powerful tool that can open doors to countless careers and opportunities. Whether they aspire to be engineers, scientists, economists, or anything in between, a solid foundation in mathematics will serve them well in their future endeavors.Of course, teaching is about more than just imparting knowledge; it's also about fostering a love for learning, cultivating resilience, and helping students develop the skills they need to navigate an ever-changing world. In my classroom, I'll strive to create an environment that nurtures creativity, problem-solving abilities, and a growth mindset.I'll encourage my students to embrace challenges, to persevere through setbacks, and to understand that failure is not a destination but merely a stepping stone on the path to success. By instilling these values, I hope to equip them with the toolsthey need to thrive not just in their academic pursuits, but in all aspects of their lives.As I look towards the future, I can't help but feel a sense of excitement and anticipation. The journey ahead will undoubtedly be filled with challenges, but I'm ready to tackle them head-on. I'm prepared to dedicate myself fully to the art of teaching, to constantly refine my craft, and to adapt to the ever-changing landscape of education.I know that there will be days when I'll feel discouraged, when the weight of responsibility will seem overwhelming. But in those moments, I'll draw strength from the knowledge that I'm playing a vital role in shaping the minds of tomorrow's leaders, innovators, and problem-solvers.And when I see that spark of understanding in a student's eyes, when I witness their confidence grow as they master a new concept, I'll know that every effort, every late night spent perfecting a lesson plan, was worth it.For me, becoming a math teacher is more than just a career choice; it's a calling, a chance to make a lasting impact on the lives of countless students. It's an opportunity to share my passion for a subject that has brought me so much joy andfulfillment, and to inspire the next generation to embrace the beauty and power of mathematics.So, as I stand on the precipice of this new chapter in my life, I'm filled with a sense of purpose and determination. I know that the road ahead won't be easy, but I'm ready to face whatever challenges come my way. For in my heart, I know that this is my true calling, and I couldn't imagine a more rewarding path to take.篇2In the Future I Will Become a Math TeacherEver since I was a young child, I have been fascinated by the world of numbers. The way they interact and intertwine, forming beautiful patterns and relationships, has always captivated my mind. As I grew older, this fascination only deepened, and I found myself drawn to the subject of mathematics like a moth to a flame.My love affair with math truly blossomed in high school. It was during those formative years that I was fortunate enough to have a teacher who not only possessed a profound understanding of the subject but also had a gift for imparting that knowledge in a way that made it accessible and engaging.Mr. Thompson, with his infectious enthusiasm and unwavering patience, ignited a spark within me that would eventually grow into an all-consuming passion.As I sat in his classroom, meticulously taking notes and soaking up every word, I found myself in awe of the elegance and power of mathematical concepts. From the simple arithmetic of my early years to the more complex realms of algebra, geometry, and calculus, each new revelation felt like a door opening onto a world of endless possibilities.It was during those hours spent poring over equations and theorems that I began to understand the true beauty of mathematics. It wasn't just a collection of dry formulas and abstract ideas; it was a language that described the very fabric of the universe. Through mathematics, we could quantify the movements of celestial bodies, predict the behavior of subatomic particles, and even capture the intricate patterns found in nature.As my proficiency in the subject grew, so too did my desire to share this knowledge with others. I began to see teaching not merely as a profession but as a calling, a sacred duty to ignite the same flames of curiosity and wonder in the minds of future generations.Upon graduating from high school, I knew that my path was set. I would dedicate my life to the pursuit of mathematical knowledge, and in doing so, I would strive to become the kind of teacher that had so profoundly impacted my own life.The journey ahead was not an easy one. College presented its own set of challenges, with advanced coursework that pushed the boundaries of my understanding and tested the limits of my determination. Yet, with each obstacle I overcame, my resolve only grew stronger.During those years of intense study, I had the privilege of working alongside some of the most brilliant minds in the field of mathematics. Their passion and dedication were contagious, and I found myself drawn deeper and deeper into the intricate tapestry of mathematical theory.As I approached the end of my academic journey, the prospect of stepping into the role of a teacher filled me with a sense of both excitement and trepidation. I knew that the responsibility of shaping young minds was a heavy one, but I also understood the profound impact that a great teacher could have on a student's life.In the years that followed, I embarked on a career that would challenge me in ways I had never imagined. Standing before aclassroom full of eager faces, I quickly learned that teaching was an art form unto itself. It required not only a deep understanding of the subject matter but also the ability to communicate complex ideas in a way that resonated with each individual student.Through trial and error, I honed my craft, developing techniques and strategies that made mathematics not just accessible but also enjoyable. I learned to adapt my teaching style to the unique needs of each student, recognizing that every mind is wired differently and that what works for one might not work for another.As the years passed, I witnessed the transformative power of mathematics firsthand. I saw students who had once struggled with the subject blossom into confident problem-solvers, their eyes alight with the thrill of discovery. I watched as the seeds of understanding took root, sprouting into a lifelong love of learning and exploration.In those moments, when a student's face lit up with the joy of comprehension, I knew that I had found my true calling. The challenges, the long hours, and the occasional frustrations all faded into insignificance in the face of that singular, transcendent experience.Through my journey as a teacher, I have come to realize that mathematics is more than just a collection of formulas and equations. It is a way of thinking, a method of approaching problems with logic and precision. By instilling these skills in my students, I am not merely teaching them how to solve mathematical problems; I am equipping them with the tools they need to navigate the complexities of life itself.As I look towards the future, I am filled with a sense of gratitude and optimism. The path that lies ahead is one of continuous learning and growth, both for myself and for the countless students whose lives I will have the privilege of shaping.I know that there will be challenges to overcome, new concepts to grasp, and ever-evolving pedagogical methodologies to embrace. But I also know that with every obstacle I face, I will be fortified by the knowledge that I am doing work that truly matters – work that has the power to change lives and shape the course of generations to come.In the end, my goal as a math teacher is not merely to impart knowledge but to inspire a love of learning that will endure long after my students have left the classroom. I want to be the catalyst that ignites their curiosity, the guide that helps themnavigate the wonders of the mathematical universe, and the mentor who instills in them the confidence to pursue their dreams, no matter how lofty or ambitious they may be.For it is in those moments of epiphany, when a student's eyes widen with understanding, that I find my true purpose. It is in witnessing the spark of comprehension grow into a blazing fire of passion that I experience the greatest joy and fulfillment.And so, as I stand before each new class, chalk in hand and a world of knowledge to impart, I am reminded of the journey that has brought me to this point – a journey that began with a simple fascination with numbers and has blossomed into a lifelong devotion to the art of teaching.In the years to come, I will continue to walk this path, ever mindful of the profound responsibility I bear and the countless lives I have the honor of shaping. For in the end, it is not just mathematics that I teach, but the very essence of human potential – a potential that knows no bounds, limited only by the depths of our curiosity and the heights of our ambition.篇3In the Future, I Will Become a Math TeacherEver since I was a young child, I have been fascinated by the world of numbers. The way they interact and fit together like an intricate puzzle has captivated my mind and sparked my curiosity. As I grew older, my love for mathematics only intensified, and it became clear to me that this was the path I wanted to pursue, not just as a student but as a future educator.My journey with mathematics began in elementary school, where I was introduced to the basics of arithmetic. At first, it seemed like a daunting task, but with the guidance of my exceptional teachers, I slowly unraveled the mysteries of addition, subtraction, multiplication, and division. Their patience and dedication were instrumental in helping me develop a strong foundation in mathematics, and their passion for the subject was contagious.As I progressed through middle school and high school, my appreciation for mathematics deepened. I marveled at the way it could be applied to solve real-world problems, from calculating the trajectory of a rocket to predicting stock market trends. Each new concept I encountered, whether it was algebra, geometry, or calculus, unveiled a new layer of complexity and beauty, further fueling my enthusiasm for the subject.However, my journey was not without its challenges. There were times when I struggled to grasp certain concepts, and the frustration of not understanding would threaten to diminish my love for mathematics. But it was during these moments that I realized the true value of perseverance and the importance of seeking help. My teachers were always there to support me, offering alternative explanations and encouraging me to keep pushing forward.Through their guidance and my unwavering determination, I overcame these obstacles and emerged with a deeper appreciation for the subject. It was then that I realized the profound impact a dedicated teacher could have on a student's life, and I knew that I wanted to be that guiding light for future generations of math enthusiasts.As I stand on the cusp of pursuing higher education, I am filled with excitement and a sense of purpose. I envision myself as a math teacher, inspiring young minds to embrace the beauty and practicality of mathematics. I dream of creating an engaging and nurturing environment where students feel empowered to ask questions, explore their curiosities, and develop a lifelong love for learning.In my classroom, I will strive to make mathematics relatable and accessible to all students, regardless of their background or initial aptitude. I will employ innovative teaching methods, incorporating technology and hands-on activities to cater to different learning styles. My goal will be to cultivate a space where students feel comfortable making mistakes and learning from them, for it is through these experiences that true understanding and growth occur.Furthermore, I will emphasize the importance of perseverance and resilience in the face of challenges. Mathematics is a subject that requires dedication and consistent effort, and I will encourage my students to embrace these qualities. I will share my own struggles and triumphs, serving as a living example of how determination and a positive mindset can overcome even the most daunting obstacles.Beyond imparting knowledge, I aspire to be a mentor and a role model for my students. I will strive to foster an environment of mutual respect, where diverse perspectives are valued and celebrated. By creating a supportive and inclusive classroom culture, I hope to empower students to reach their full potential and develop a sense of confidence in their abilities.Moreover, I will encourage my students to explore the vast applications of mathematics in various fields, from science and technology to economics and social sciences. By exposing them to the interdisciplinary nature of mathematics, I aim to broaden their horizons and ignite their curiosity, inspiring them to pursue careers that align with their passions and contribute to the advancement of society.As a future math teacher, I understand the weight of responsibility that comes with shaping young minds. It is a role that demands not only a deep understanding of the subject matter but also a genuine passion for teaching, a commitment to continuous learning, and a dedication to fostering a love for mathematics in each and every student.I am prepared to embrace this challenge with open arms, for I firmly believe that education has the power to transform lives and shape the future. By igniting the spark of curiosity and nurturing a strong foundation in mathematics, I hope to equip my students with the tools they need to navigate the complexities of the modern world and contribute to the advancement of knowledge.In the years to come, I envision a classroom filled with lively discussions, where students eagerly engage with mathematicalconcepts, sharing their ideas and supporting one another's growth. I see myself as a facilitator of learning, guiding my students through their journeys of discovery and celebrating their achievements along the way.Beyond the walls of the classroom, I aspire to foster a lifelong love for learning in my students, encouraging them to continue exploring the vast realms of mathematics and its applications throughout their lives. For it is through this continuous pursuit of knowledge that we can unlock the secrets of the universe and push the boundaries of human understanding.As I stand on the precipice of this exciting journey, I am filled with a sense of determination and purpose. I know that the road ahead will be challenging, but it is a challenge that I embrace wholeheartedly. For in the end, the true reward lies not only in imparting knowledge but in igniting the flames of curiosity and inspiring generations of students to embrace the beauty and power of mathematics.。

我想当一名数学家英语作文

My Dream of Becoming a MathematicianEver since I was a young child, I have been fascinated by the beauty and power of mathematics. The precision and elegance of mathematical equations have always captivated me, and I find myself constantly seeking out new mathematical challenges to conquer. As I grew older, my interest in mathematics deepened, and I realized that I wanted to pursue a career in this field.Mathematics is the language of the universe, and it is the tool we use to understand the world. Mathematicians are the architects of knowledge, building bridges between different fields of science and engineering. They are the innovators who push the boundaries of human understanding, discovering new theorems and concepts that change the way we view the world.Becoming a mathematician is not an easy feat. It requires years of hard work, dedication, and perseverance. However, I am willing to make the necessary sacrifices to achieve my dream. I know that I will need to develop a strong foundation in mathematics, and I am committed to studying and practicing constantly. I also understand thatI will need to be patient and persistent, as mathematical breakthroughs often come after years of struggle and failure.Despite the challenges, I am excited about the opportunities that lie ahead. I look forward to working with other mathematicians, sharing ideas and collaborating on projects that will push the boundaries of mathematical knowledge. I envision myself contributing to the field of mathematics, making discoveries that will have a lasting impact on science and society.In conclusion, my dream of becoming a mathematician is not just a pipe dream; it is a goal that I am actively pursuing. I am committed to the hard work and dedication it will require to achieve my dream, and I am excited about the opportunities it will bring. I believe that with hard work and perseverance, I can make a significantcontribution to the field of mathematics and leave alasting impact on the world.**我的数学家之梦**自我孩提时代起，我就被数学的美丽和力量深深吸引。

你喜欢不喜欢的学科英语作文

你喜欢不喜欢的学科英语作文英文回答：The subject that I both love and hate is Mathematics. Math has always been a fascination to me since childhood, with its abstract concepts and logical puzzles that tantalized my curiosity. The beauty of discovering patterns, solving complex equations, and unraveling theorems filled me with immense satisfaction and a sense of accomplishment.However, as I progressed further into my mathematical studies, my initial enthusiasm began to dwindle. The sheer volume of formulas, algorithms, and proofs became overwhelming, and the pace at which new concepts were introduced left me struggling to keep up. The once-intriguing problems now seemed like insurmountable obstacles, and the joy of discovery turned into a burden of frustration.The subject that I both love and hate is Mathematics.Math has always been a fascination to me since childhood, with its abstract concepts and logical puzzles that tantalized my curiosity. The beauty of discovering patterns, solving complex equations, and unraveling theorems filled me with immense satisfaction and a sense of accomplishment.However, as I progressed further into my mathematical studies, my initial enthusiasm began to dwindle. The sheer volume of formulas, algorithms, and proofs became overwhelming, and the pace at which new concepts were introduced left me struggling to keep up. The once-intriguing problems now seemed like insurmountable obstacles, and the joy of discovery turned into a burden of frustration.The frustrating aspect of Math is its unforgiving nature. Unlike other subjects where creativity orsubjective opinions play a role, Math demands precision and accuracy. A single misplaced decimal point or a forgotten minus sign can lead to disastrous consequences, leaving me with a haunting feeling of inadequacy. The constant need to memorize formulas and theorems, coupled with the relentlesspressure to perform well in exams, created a sense of anxiety and self-doubt.Despite the challenges, Math continues to hold a strange allure for me. The elegance of its proofs, the interconnectedness of its concepts, and the universality of its language still fascinate me. Even when I struggle with a particularly difficult problem, I find a grudging respect for the beauty of the mathematics behind it.Ultimately, my feelings towards Math are a complex mix of love and hate. It is a subject that both challenges and inspires me, that brings me both joy and frustration. I may not always enjoy the journey, but I cannot deny the allure of its destination.中文回答：我既爱又恨的科目是数学。

我的梦想成为数学家英语作文

我的梦想成为数学家英语作文English:Becoming a mathematician has been my lifelong dream, fueled by a deep passion for unraveling the mysteries of numbers and patterns. From a young age, I found solace and fascination in the elegance of mathematical equations, seeing them as a language that speaks the truths of the universe. My journey to pursue this dream has been marked by countless hours of study, problem-solving, and exploration of mathematical concepts. I am drawn to the beauty of pure mathematics, where abstract ideas manifest into concrete solutions, and the satisfaction of discovering new theorems or proofs is unparalleled. Moreover, the versatility of mathematics intrigues me; its applications span across various fields, from physics to computer science, demonstrating its indispensability in understanding and shaping the world around us. As I delve deeper into the realm of mathematics, I am continuously inspired by the profound impact it has on both theoretical knowledge and practical advancements. Through dedication and perseverance, I aspire to contribute to the ever-expanding landscape of mathematical understanding, pushingthe boundaries of human knowledge and leaving a lasting legacy in the field.中文翻译:成为一名数学家是我一生的梦想，这个梦想源于对数字和模式解密的深刻热爱。

我想当一名数学家英语作文

我想当一名数学家英语作文Ever since I was a child, I have been fascinated by the beauty of mathematics. The way numbers and equations can elegantly solve complex problems has always intrigued me. As I grew older, my passion for mathematics only intensified, leading me to dream of becoming a mathematician.The allure of mathematics lies in its universality and precision. It is a language that transcends cultural and linguistic barriers, speaking a truth that is both absolute and unchanging. As a mathematician, I would have theprivilege of exploring this language in its purest form, delving into the depths of abstract concepts and discovering new theorems that could potentially change the world.My journey to become a mathematician is not without its challenges. The path is fraught with complex theories and demanding calculations that require a high level of dedication and perseverance. However, the rewards of such a pursuit are immeasurable. The satisfaction of solving a difficult problem, the thrill of making a new discovery, and the opportunity to contribute to the field of mathematics are all compelling reasons to pursue this career.In order to achieve my goal, I am committed to a rigorous course of study. I am currently excelling in my mathematics classes and have taken additional courses to further my understanding of advanced topics. I also participate in mathcompetitions, which not only sharpen my skills but also expose me to a community of like-minded individuals who share my passion for mathematics.Moreover, I believe that becoming a mathematician is not just about personal achievement; it is about contributing to the collective knowledge of humanity. Mathematics has been the foundation of many scientific breakthroughs and technological advancements. By becoming a mathematician, I hope to play a part in the ongoing quest for knowledge and to inspire future generations to appreciate the elegance and power of mathematics.In conclusion, my aspiration to become a mathematician stems from a deep-seated love for the subject and a desire to make meaningful contributions to the field. I am prepared to face the challenges that lie ahead and am excited about the prospect of a future where I can immerse myself in the world of mathematics and help unlock its many secrets.。

零和游戏理论英语作文

零和游戏理论英语作文Zero-Sum Game Theory。

In game theory, a zero-sum game is a situation in which one participant's gain is exactly balanced by another participant's loss. This means that the total amount of wealth or resources in the system remains constant. Examples of zero-sum games include poker, chess, and many sports.In a zero-sum game, each participant's strategy is based on anticipating the actions of the other participants and responding in a way that maximizes their own gain while minimizing the other participants' gain. This often leads to a situation of conflict, as each participant tries to outmaneuver the others.One of the key concepts in zero-sum game theory is the Nash equilibrium, named after the mathematician John Nash. The Nash equilibrium is a state in which no participant canimprove their outcome by changing their strategy, given the strategies of the other participants. This means that the Nash equilibrium represents a stable state of the system, where each participant is acting optimally given the actions of the others.In many real-world situations, however, the assumptions of zero-sum game theory do not hold. For example, in a business negotiation, the two parties may be able to find a mutually beneficial outcome that creates value for both sides. This is known as a positive-sum game, where thetotal amount of wealth or resources in the system can increase.Despite its limitations, zero-sum game theory remains a useful tool for analyzing conflicts and competition in many different fields, from economics to politics to international relations. By understanding the strategies and motivations of different participants, we can better predict and manage the outcomes of complex systems.。

我想要当一名数学家英语作文

我想要当一名数学家英语作文Becoming a mathematician has always been my dream ever since I was a child. I am fascinated by the beauty and logic of numbers, equations, and patterns. 数学家一直是我童年时代的梦想。

我被数字、方程式和模式的美感和逻辑深深吸引着。

Mathematics is not just a subject or a career to me, it is a passionand a way of life. I love the challenge of solving complex problems and the thrill of discovering new theorems. 数学对我来说不仅仅是一门学科或一种职业，它更是一种热情和生活方式。

我喜欢解决复杂问题的挑战，也享受发现新定理的刺激。

One of the reasons why I want to become a mathematician is the endless possibilities and opportunities for exploration and discoveryin this field. 数学家的一个吸引我的原因是在这个领域中无限的探索和发现机会。

I am determined to contribute to the field of mathematics and makea difference in the world through my work. 我决心通过我的工作为数学领域做出贡献，并在世界上产生影响。

As a mathematician, I hope to inspire and mentor the next generation of mathematicians, encouraging them to pursue their passion for numbers and problem-solving. 作为一名数学家，我希望能激励和指导下一代的数学家，鼓励他们追求对数字和问题解决的热情。

英语第一章阅读 game theory 原文及翻译

原则上，在有限的一系列动作之后结束的任何连续游戏都可以完全“解决”。我们通过展望每一个可能的结果来确定每个玩家的最佳策略。简单的游戏，如井字游戏，可以用这种方式解决，因此不具有挑战性。对于许多其他游戏，如国际象棋，计算过于复杂，无法在实践中执行 - 即使使用计算机。因此，球员们会看到前进的几步，并尝试根据经验评估所得到的位置。
The essence of a game is the interdependence of player strategies. There are two distinct types of strategic interdependence: sequential and simultaneous. In the former the players move in sequence, each aware of the others’ previous actions. In the latter the players act at the same time, each ignorant of the others’ actions.
Game theory was pioneered by Princeton mathematician john von Neumann. In the early years the emphasis was on games of pure conflict (zero-sum games). Other games were considered in a cooperative form. That is, the participants were supposed to choose and implement their actions jointly. Recent research has focused on games that are neither zero sum nor purely cooperative. In these games the players choose their actions separately, but their links to others involve elements of both competition and cooperation.

1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。

Discovering Theorems in Game Theory:Two-Person Games with Unique NashEquilibriaFangzhen Lin and Pingzhong TangDepartment of Computer ScienceHong Kong University of Science and TechnologyClear Water Bay,Kowloon,Hong KongAbstractIn this paper we provide a logical framework for using com-puters to discover theorems in(two-personﬁnite)games in strategic form,and apply it to discover classes of games that have unique(pure-strategy)Nash equilibria.We consider all possible classes of games that can be expressed by a conjunc-tion of two binary clauses,and our program re-discovered Kats and Thisse’s class of weakly unilaterally competitive two-person games,and came up with several other classes of games that have unique Nash equilibria.It also came up with new classes of strict games that have unique Nash equilibria, where a game is strict if for each player different proﬁles have different payoffs.Partly motivated by theseﬁndings,we also (manually)prove a result that says that a strict game has a unique Nash equilibrium iff it is best-response equivalent toa strictly competitive game.IntroductionIn game theory,a key notion is that of Nash equilibria.A game in strategic form can have one,more than one,or zero Nash equilibria(see,e.g.[Osborne and Rubinstein,1994]), and there has been extensive work on classes of games that always have Nash equilibria,such as potential games[Mon-derer and Shapley,1996]and supermodular games[Topkis, 1998],as well as classes of games with unique Nash equi-libria,such as strictly competitive games[Moulin,1976; Friedman,1983]In this paper,as part of our project on using computers to discover theorems in game theory,we consider the pos-sibility of using computers to discover new classes of two-person games that have unique(pure)Nash equilibria.Our starting point is the class of two-person strictly competitive games.Weﬁrst formulate the notions of games,strictly competitive games and Nash equilibria inﬁrst-order logic. Under our formulation,a class of games corresponds to a ﬁrst-order sentence.In particular,the sentence that corre-sponds to the class of strictly competitive games is a con-junction of two binary clauses with all variables universally quantiﬁed.So we implemented a program that examine all these universally quantiﬁed conjunctions of binary clauses to see if there is another such condition that also captures a Copyright c 2007,American Association for Artiﬁcial Intelli-gence().All rights reserved.class of games with unique Nash equilibria.We did not ex-pect much as these conditions are rather simple,but to our surprise,our program returned a condition that is more gen-eral than the strict competitiveness condition.As it turned out,it exactly corresponds to Kats and Thisse’s[1992]class of weakly unilaterally competitive two-person games.Our program also returned some other conditions.Two of them capture a class of“unfair”games where one player has ad-vantage over the other.The remaining ones capture games where everyone gets what he wants-each receives his max-imum payoff in every equilibrium state,thus there is no real competition among the players.Thus one conclusion that we can draw from this experiment is that among all classes of games that can be expressed by a conjunction of two binary clauses,the class of weakly unilaterally competitive games is the most general class of“competitive”and“fair”games that have unique Nash equilibria.Of course,this does not mean that the other conditions are not worth investigating. For instance,sometimes one may be forced to play an unfair game.For the same set of conditions,we also consider strict two-person games where different proﬁles have different payoffs for each player.Among the results returned by our pro-gram,two of them are exactly the two conjuncts in Kats and Thisse’s weakly unilaterally competitive condition,but the others all turn out to be special cases of games with domi-nant strategies.Motivated by these results,we consider cer-tain equivalent classes of games,and show that a strict game has a unique Nash equilibrium iff it is best-response equiva-lent[Rosenthal,1974]to a strictly competitive game.The rest of the paper is organized as follows.Weﬁrst review some basic concepts in two-person games in strate-gic form,and then reformulate them inﬁrst-order logic.We then show that for a class of conditions,whether any of them entails the uniqueness of Nash equilibria needs only to be checked on games up to certain size.We then describe a computer program based on this result,and report our ex-perimental results.Two-person gamesA(two-person)game(in strategic form)is a tuple(A,B,≤1 ,≤2),where A and B are sets of strategies of players1and 2,respectively,and≤1and≤2are total orders on A×B called preference relations for players1and2,respectively.Instead of two preference relations,a two-person game can also be speciﬁed by two payoff functions,one for each player,which map proﬁles to numbers.The relationship be-tween these two formulations are as follows:for any proﬁles s and s ,s≤i s iff u i(s)≤u i(s ),where u i is the payoff function for player i.In the following,we shall use these two formulations interchangeably.For each b∈B,we deﬁne B1(b)to be the set of best responses by player1to the strategy b by player2:B1(b)={a|a∈A,and for all a ∈A,(a ,b)≤1(a,b)}. Similarly,for each a∈A,the set of best responses by player 2is:B2(a)={b|b∈B,and for all b ∈B,(a,b )≤2(a,b)}.A proﬁle(a,b)∈A×B is a Nash equilibrium if both a∈B1(b)and b∈B2(a).Notice that we consider only pure-strategy Nash equilibria.A game can have exactly one, more than one,or no Nash equilibria.We say that a game has unique Nash equilibria if for each player,all Nash equilibria of the game have the same payoffs,that is,whenever(a1,b1) and(a2,b2)are two Nash equilibria of the game,then (x1,y1)≤i(x2,y2)∧(x2,y2)≤i(x1,y1),for i=1,2.Notice that according to this deﬁnition,if a game has no Nash equilibria,then it is also a game with unique Nash equilibria.Thus games with unique Nash equi-libria are really games that have at most one equivalent class of Nash equilibria.One interesting class of two-person games is that of strictly competitive games.A game is strictly competitive [Moulin,1976;Friedman,1983]if for every pair of proﬁless1and s2in A×B,we have that s1≤1s2iff s2≤2s1. Thus in strictly competitive games,the two players’prefer-ences are exactly opposite.Strictly competitive games have many nice properties.If (a,b)and(a ,b )are both Nash equilibria of a strictly com-petitive game,then(1)they are equivalent in the sense that (a,b)≤i(a ,b )and(a ,b )≤i(a,b)for both i=1,2;(2) they are interchangeable in the sense that(a ,b)and(a,b ) are also Nash equilibria.Thus if a strictly competitive game has Nash equilibria,then they are unique.Furthermore they can be computed using the minmax procedure.Another class of games that we shall consider in this pa-per is that of strict games.A game is strict if for both play-ers,different proﬁles have different payoffs,that is,(a,b)= (a ,b )whenever(a,b)≤i(a ,b )and(a ,b )≤i(a,b), where i=1,2.As we shall see,strict games have some nice properties that general games do not have. Formulating two-person games inﬁrst-orderlogicWe consider aﬁrst-order language with two sortsαandβ, equality,and two predicates≤1and≤2.We use“∧”for con-junction,“∨”for disjunction,“¬”for negation,“⊃”for im-plication,and“≡”for equivalence.Negation has the highest precedence,followed by conjunction and disjunction,impli-cation,and then equivalence.The rule of precedence can be overridden by a new line.For instance,the following ex-pressionp⊃q∧q⊃pstands for the sentence(p⊃q)∧(q⊃p).In our language,sortαis for player1’s strategies,andβfor player2’s strategies.In the following,we use variables x,x1,x2,..to range overα,and y,y1,y2,...to range over β.The two predicates represent the two players’preference relations.In the following,as we have already done above, we write≤i(x1,y1,x2,y2)in inﬁx notation as(x1,y1)≤i (x2,y2),i=1,2,and(x1,y1) i(x2,y2)as a shorthand for(x1,y1)≤i(x2,y2)∧(x2,y2)≤i(x1,y1), where i=1,2.We also write(x1,y1)<i(x2,y2)as a shorthand for(x1,y1)≤i(x2,y2)∧¬(x2,y2)≤i(x1,y1).The two relations need to be total orders(in the rest of the paper,unless otherwise stated,all free variables in a displayed formula are assumed to be universally quantiﬁed from outside):(x,y)≤i(x,y),(1)(x1,y1)≤i(x2,y2)∨(x2,y2)≤i(x1,y1),(2)(x1,y1)≤i(x2,y2)∧(x2,y2)≤i(x3,y3)⊃(x1,y1)≤i(x3,y3),(3)where i=1,2.In the following,we denote byΣthe set of the above sentences.Thus two-person games correspond to ﬁrst-order models ofΣ,and two-personﬁnite games corre-spond toﬁrst-orderﬁnite models ofΣ.This correspondence extends to other type of games as well.For instance,letΣs be the union ofΣwith the following two axioms:(x1,y1) 1(x2,y2)⊃(x1=x2∧y1=y2),(x1,y1) 2(x2,y2)⊃(x1=x2∧y1=y2).Then strict games and models ofΣs are isomorphic.We now show how some other notions in game theory can be formulated inﬁrst-order logic.The condition for a proﬁle (ξ,ζ)to be a Nash equilibrium is captured by the following formula:∀x.(x,ζ)≤1(ξ,ζ)∧∀y.(ξ,y)≤2(ξ,ζ)(4)In the following,we shall denote the above formula by NE(ξ,ζ).The following sentence expresses the uniqueness of Nash equilibria in a game:NE(x1,y1)∧NE(x2,y2)⊃(x1,y1) 1(x2,y2)∧(x1,y1) 2(x2,y2)(5)A game is strictly competitive if it satisﬁes the following property:(x1,y1)≤1(x2,y2)≡(x2,y2)≤2(x1,y1).(6)Thus it should follow thatΣ|=(6)⊃(5).(7) Notice that we have assumed that all free variables in a displayed formula are universally quantiﬁed from outside. Thus(6)is a sentence of the form∀x1,x2,y1,y2ϕ.Simi-larly for(5).Theorems like(7)can actually be generated automatically using the following theorem.Theorem1Suppose Q is a formula without quantiﬁers, x1 and x2tuples of variables of sortα,and y1and y2tuples of variables of sortβ.We have that1.Σ|=∃ x1∃ y1∀ x2∀ y2Q⊃(5)iff for all model G ofΣsuch that|A|≤| x1|+2and |B|≤| y1|+2,we have thatG|=∃ x1∃ y1∀ x2∀ y2Q⊃(5),where A is the domain of G for sortα,and B the domain of G for sortβ.2.Σ|=∃ x1∃ y1∀ x2∀ y2Q⊃¬∃x,y.NE(x,y)iff for all model G ofΣsuch that|A|≤| x1|+1and |B|≤| y1|+1we have thatG|=∃ x1∃ y1∀ x2∀ y2Q⊃¬∃x,y.NE(x,y),where A is the domain of G for sortα,and B the domain of G for sortβ.Proof:“Only if”is trivial.To show“if”,suppose there is a game G(model ofΣ)such that it satisﬁes the con-dition∃ x1∃ y1∀ x2∀ y2Q but has two non-equivalent Nash equilibria,(a,b)and(a ,b ):either(a,b) 1(a ,b )or (a,b) 2(a ,b ).Thus there is a tuple a1of elements from A and a tuple b1of elements from B such that| a1|=| x1|, | b1|=| y1|,and G satisﬁes(∀ x2∀ y2Q)|x1/ a1, y1/ b1,which is obtained from(∀ x2∀ y2Q)by replacing in it every free occurrence of each variable in x1by its corresponding element in a1,and every free occurrence of each variable in y1by its corresponding element in b1.Now construct a new game G =(A ,B ,≤ 1,≤ 2)as follows:•A ={a,a }∪ a1and B ={b,b }∪ b1.•≤ 1is the restriction of≤1on A ,and≤ 2is the restriction of≤2on B .Notice that this game is well-deﬁned as≤ 1and≤ 2are both total orders,i.e.G |=Σ.Clearly,the size of G is smaller or equal to(| x1|+2)×(| y1|+2),both(a,b)and(a ,b )are still non-equivalent Nash equilibria of G ,and the formula ∃ x1∃ y1∀ x2∀ y2Q is still true in G .In other words,to prove that a sentence of the form ∃ x1∃ y1∀ x2∀ y2Q is a sufﬁcient condition for the uniqueness of Nash equilibria,it sufﬁces to verify that this is the case for all games of sizes up to(| x1|+2)×(| y1|+2),and to prove that it is a sufﬁcient condition for the non-existence of Nash equilibria,it sufﬁces to verify this for games of sizes up to(| x1|+1)×(| y1|+1).Theorem1holds for many specialized games as well.For instance,it holds for strict games as well.Theorem2Theorem1holds whenΣis replaced byΣs.In fact,Theorem1holds whenΣis replaced by any set of universally quantiﬁed sentences.Theorem discoveringSince p≡q is logically equivalent to(¬p∨q)∧(p∨¬q),the condition(6)for strictly competitive games can be written as a conjunction of two binary clauses:(l1∨l2)∧(l3∨l4),(8) where each l i,1≤i≤4,is a literal,i.e.either an atom or the negation of an atom.As we mentioned,we want to know if there are other sentences of the form(8)that also capture classes of games with unique Nash equilibria.In the follow-ing,we say that a conditionϕis a uniqueness condition if whenever a game satisﬁes this condition,it has unique Nash equilibria,that is,ifΣ|=ϕ⊃(5).Based on Theorem1,a straightforward way of discov-ering uniqueness conditions of the form(8)is as follows: For each condition of the form(8),check that if a2×2 game does not have unique Nash equilibria,then it does not satisfy this condition.There are810,000such conditions, 1950non-isomorphic2×2two-person games,and among them709games that do not have unique Nash equilibria. Thus this strategy can be implemented on a modern com-puter even by brute-force search.The search space can also be pruned by noticing that the conditions of the form(8)are not independent.For instance, condition(x1,y1)≤1(x2,y2)entails(is stronger than)condition(x1,y1)≤1(x1,y2).Once we know that a condition C is a uniqueness condition, those that entail C are no longer interesting as they become special cases of C,thus can be pruned.However,checking logical entailment is in general not de-cidable forﬁrst-order logic.But as a strategy for pruning search space,we can use a weaker notion called subsump-tion on conditions of the form(8):C subsumes C if there is a substitutionσsuch that Cσ=C .For our language, subsumption can be checked efﬁciently,and the search tree can be designed in such a way that the condition associated with a node always subsumes the conditions associated with the ancestors of the node.Thus once a condition is found to be a uniqueness condition,the entire sub-tree under this condition can be pruned.However,we still need a way to check for complete logi-cal entailment underΣfor conditions of the form(8).This is because we want every condition returned by our program to be a most general,“weakest”uniqueness condition in the sense that it does not entail any other uniqueness condition of the form(8).Fortunately,this can be done using the fol-lowing proposition.Proposition1To check whether condition∀ x1 y1Q1entails condition∀ x2 y2Q2for all two-person games,it sufﬁces to check this for all games up to max{| x2|,1}×max{| y2|,1}, where Q1and Q2are formulas without quantiﬁers.This result holds for strict games as well.Notice that what we have described applies to the task of discovering uniqueness conditions of the form(8)for strict two-person games as well.We now report our experimental results,ﬁrst for general two-person games,and then for strict two-person games.General gamesFor two-person general games,our program returns the fol-lowing seven uniqueness conditions for2x2games.(x1,y)≤1(x2,y)⊃(x2,y)≤2(x1,y)∧(x,y1)≤2(x,y2)⊃(x,y2)≤1(x,y1)(9) (x1,y)≤1(x2,y)⊃(x1,y)≤2(x2,y)∧(x,y1)≤2(x,y2)⊃(x,y2)≤1(x,y1)(10) (x1,y)≤1(x2,y)⊃(x2,y)≤2(x1,y)∧(x,y1)≤2(x,y2)⊃(x,y1)≤1(x,y2)(11) (x1,y1)≤1(x2,y1)⊃(x1,y2)≤2(x2,y2)∧(x,y1)≤2(x,y2)⊃(x,y1)≤1(x,y2)(12) (x1,y)≤1(x2,y)⊃(x1,y)≤2(x2,y)∧(x1,y1)≤2(x1,y2)⊃(x2,y1)≤1(x2,y2)(13) (x1,y1)≤1(x2,y2)⊃(x1,y1)≤2(x2,y1)∧(x1,y1)≤2(x2,y2)⊃(x2,y1)≤1(x2,y2)(14) (x1,y1)≤1(x2,y2)⊃(x1,y2)≤2(x2,y2)∧(x1,y1)≤2(x2,y2)⊃(x1,y1)≤1(x1,y2).(15) By Theorem1,these are also uniqueness conditions for all two-person games.Furthermore,since these are the only conditions returned by our program,for any sentence C of the form(8),if it is a uniqueness condition,then it must entail one of the above conditions underΣ.In other words, the above seven conditions are the weakest(most general) uniqueness conditions of the form(8).Notice that condition(10)and condition(11)are symmet-ric in the sense that one can be obtained from the other by swapping the roles of the two players.So are(12)and(13), and(14)and(15).On the other hand,(9)is symmetric to itself.It is easy to see that if two conditions are symmetric, then one is a uniqueness condition iff the other is. Condition(9)looks like condition(6)for strictly compet-itive games,except that the strategy of one of the players isﬁxed in each implication.As it turned out,it captures exactly the class of two-person games that are weakly uni-laterally competitive[Kats and Thisse,1992]:“a game belongs to this class if a unilateral move by one player which results in an increase in that player’s payoff also causes a(weak)decline in the payoffs of all other players.Furthermore,if that move causes no change in the mover’s payoff then all other players’payoffs remain unchanged.”Clearly,if a game is strictly competitive,then it is also weakly unilaterally competitive,but the converse is not true in general.Kats and Thisse[1992]showed that if a game is weakly unilaterally competitive,then it has at most one Nash equilibrium.For us,for two-person games,this fol-lows directly from our computer output and Theorem1.Condition(10)can be given a similar interpretation:A two-person game satisﬁes this condition if a unilat-eral move by player1which results in a(weak)increase in his payoff also causes a(weak)increase in the pay-off of player2,but a unilateral move by player2which results in a(weak)increase in his payoff will causes a (weak)decline in the payoff of player1.Thus in this class of games,the two players are not equal, and it clearly favors player2.The game may be competitive for player1,but not for player2.Proposition2Given a game that satisﬁes(10),if player2’s payoff is maximal at(a,b),i.e.(a ,b )≤2(a,b)for all a ,b ,then there is a strategy a∗such that(a∗,b)is a Nash equilibrium and(a∗,b) 2(a,b).Thus for the class of games that satisfy condition(10), the optimal strategy for player2is to do the strategy for which there is a strategy by the other player that will give him the maximum payoff.The following is an example of such games(as usual,player1is the row player,and player 2the column player;theﬁrst number in a cell is the payoff of the row player,the second the column player):3,64,55,12,31,46,2It has a unique equilibrium(3,6).As we mentioned,condition(11)is symmetric to condi-tion(10),with the roles of the two players swapped.For the classes of games corresponding to the other conditions,(12) -(15),both players can obtain their maximal payoffs. Proposition3Given a game that satisﬁes one of the con-ditions(12)-(15),if player1’s(player2’s)payoff at(a,b) maximal,then there is a strategy b∗(a∗)such that(a,b∗) ((a∗,b))is a Nash equilibrium where both players receive the maximum payoffs.Thus,from these two propositions,we see that the classes of games represented by the conditions(10)-(15)are not re-ally“competitive”games.We can then conclude that among the classes of games that can be represented by a conjunc-tion(8)of two binary clauses,the class of weakly unilater-ally competitive games is the most general class of“com-petitive”and“fair”games that have unique Nash equilibria. As we mentioned above,by this we do not mean that other types of games are not interesting.In real life,unfair games like those described by(10)may well arise.Strict gamesWe now describe our experimental results for strict games. Recall that these are games where for each player,different proﬁles have different payoffs.Games with dominant strategiesWeﬁrst consider conditions that mention only≤1:s1≤1s2∨s3≤1s4.For this class of conditions,our program outputs the follow-ing six uniqueness conditions on2x2strict games:(x1,y1)≤1(x2,y1)∨(x2,y1)≤1(x1,y2),(x1,y1)≤1(x2,y1)∨(x2,y2)≤1(x1,y1),(x1,y1)≤1(x2,y1)∨(x2,y2)≤1(x1,y2),(x1,y1)≤1(x2,y2)∨(x2,y1)≤1(x1,y1),(x1,y1)≤1(x2,y2)∨(x2,y2)≤1(x1,y2).By Theorem2,these are also uniqueness conditions for all strict two-person games.Notice that these conditions do not mention≤2.This means that if player1’s preference rela-tion satisﬁes any of the above conditions,then the game has a unique Nash equilibrium,no matter what the other player’s preference relation is.For instance,theﬁrst condition can be written as¬(x1,y1)≤1(x2,y1)⊃(x2,y1)≤1(x1,y2).For strict games,this is equivalent to(x2,y1)<1(x1,y1)⊃(x2,y1)≤1(x1,y2)as¬(x1,y1)≤1(x2,y1)iff(x2,y1)<1(x1,y1).It is not hard to see that the above condition implies the following condition:∃x∀x ,y.(x ,y)≤1(x,y),meaning that no matter what player2does,the best response for player1is always the same.For strict games,this means that player1has a strictly dominant strategy[Shor,web ac-cessed January2007]:a strategy x is a strictly dominant strategy if for all other strategy x of player1,and any strat-egy y of player2,(x ,y)<1(x,y).As it turned out,this is also the case for the otherﬁve conditions above,as the following proposition shows.Proposition4A strict game G=(A,B,≤1,≤2)has a strictly dominant strategy for player1if and only if for any preference relation≤ 2for player2,the game G = (A,B,≤1,≤ 2)has exactly one Nash equilibrium.Given this result,there is no need to consider any condition of the form(8)that mentions only one player’s preference relation.It is interesting to note that for the prisoner’s dilemma4,40,55,01,1each player has a strictly dominant strategy,thus should play this strategy.The dilemma is that each player can get a higher payoff by a unilateral move away from his dominant strategy.Weakly unilaterally competitive games for individual playersFor other conditions of the form(8),our program returns16 uniqueness conditions for strict games.However,each of them has a symmetric one when the roles of the two players are swapped.Thus there are really only eight such condi-tions,given below:(x1,y)≤1(x2,y)∨(x1,y)≤2(x2,y),(16)(x1,y1)≤1(x1,y2)∨(x1,y2)≤2(x2,y1),(17)(x1,y1)≤1(x1,y2)∨(x2,y2)≤2(x1,y1),(18)(x1,y1)≤1(x1,y2)∨(x2,y2)≤2(x2,y1),(19)(x1,y1)≤1(x2,y2)∨(x1,y2)≤2(x1,y1),(20)(x1,y1)≤1(x2,y2)∨(x2,y2)≤2(x1,y2),(21)(x1,y1)≤1(x1,y2)∨(x1,y1)≤2(x2,y1),(22)(x1,y1)≤1(x2,y1)∨(x2,y2)≤2(x2,y1).(23) In particular,we found that for strict games,a conjunction C1∧C2of two binary clauses is a uniqueness condition iff either C1or C2is a uniqueness condition.Theﬁrst condition is equivalent to(x2,y)≤1(x1,y)⊃(x1,y)≤2(x2,y)(24) as in strict games,s1≤1s2iff s1<1s2∨s1=s2.This is exactly one of the two conjuncts in the condition(9)for weakly unilaterally competitive games.Now swap the roles of the two players in(24),we get the following condition(x,y1)≤2(x,y2)⊃(x,y2)≤1(x,y1),(25) which is exactly the other conjunct in the condition(9).In the following,we call a game that satisﬁes(24)a weakly unilaterally competitive for player1,and a game that satisﬁes(25)a weakly unilaterally competitive for player 2.Thus a game is weakly unilaterally competitive if it is weakly unilaterally competitive for both players.The fol-lowing example shows that a game can be weakly unilater-ally competitive for player1but not for player2.2,13,41,24,3This example also shows that a weakly unilaterally competi-tive game for player1may not be almost strictly competitive [Aumann,1962]:a game is almost strictly competitive if1.the set of payoff vectors of the Nash equilibria is the sameas the set of payoff vectors of the twisted equilibria;and2.there is a Nash equilibrium that is also a twisted equilib-rium,where(a,b)is a twisted equilibrium if no player can de-crease the payoff of the other player by a unilateral change of his own strategy:for every a ∈A(b ∈B),(a,b)≤2(a ,b) ((a,b)≤1(a,b )).For this example,it is easy to see that the only equilibrium of the game,(4,3),is not a twisted equi-librium.As it turns out,(24)and(25)are the only non-trivial con-ditions.The last two conditions(22)and(23)can never be satisﬁed by games larger or equal to3x3.The remaining ﬁve conditions(17)-(21)are games with strictly dominant strategies.Proposition5If G is a strict game and satisﬁes one of the conditions(17)-(21),then one of the players has a strictly dominant strategy in G.Strictly competitive game classesTo summarize,for strict games,the only interesting unique-ness conditions that can be expressed by a conjunction of two binary clauses and include games that do not have dom-inant strategies are weakly unilaterally competitive condi-tions for individual players,(24)and(25).This led us to wonder if these two conditions are also necessary conditions for a strict game to have a unique Nash equilibrium.How-ever,it is easy to see that this is not the case.In fact,a universal condition like(8)can never be both a necessary and a sufﬁcient condition for a game to have unique Nash equilibria.This is because for any given game,no matter how many Nash equilibria it has,we can always extend it by one more strategy for each player,and make it into a game with a unique Nash equilibrium by assigning payoffs large enough to a proﬁle made of the two new strategies.How-ever,if a universal condition is satisﬁed by a game,it is also satisﬁed by any of its sub-games.This led us to consider not individual games,but classes of games under certain equivalence relation.Two games G1=(A,B,≤1,≤2)and G2=(A ,B ,≤ 1 ,≤ 2)are unilaterally order equivalent1if•A=A ,and B=B .•For every a∈A,b,b ∈B,(a,b)≤2(a,b )iff(a,b)≤ 2 (a,b ).•For every b∈B,a,a ∈A,(a,b)≤1(a ,b)iff(a,b)≤ 1 (a ,b).They are best-response equivalent[Rosenthal,1974]if for all a∈A,B2(a)in G1and G2are the same,and for all b∈B,B1(b)in G1and G2are the same.Clearly,if G1and G2are unilaterally order equivalent,then they are also best-response equivalent,but the converse is not true in general. Both notions of equivalence preserve Nash equilibria.We have the following result.Theorem3A strict game has at most one Nash equilib-rium iff it is best-response equivalent to a strictly compet-itive game.To prove this theorem,for any given game G=(A,B,≤1 ,≤2),we associate the following relation with it:R={((a,b),(a ,b))|a ∈B1(b),a∈B1(b)}∪{((a,b),(a,b ))|b∈B2(a),b ∈B2(a)}.The theorem then follows from the following two lemmas about R.Notice that relation R can also be viewed as a directed graph:if(s,s )∈R,then there is an arc from s to s .Lemma1G is best-response equivalent to a strictly com-petitive game iff R has no cycle.Lemma2If G is a strict game and R has a cycle,then G has more than one Nash equilibria.1We call it unilaterally order equivalence to distinguish it from order equivalence[Rosenthal,1974]that requires both the row and column orders in the two games to be the same for both players.Theorem3does not hold if we replace best-response equivalence by unilaterally order equivalence.The follow-ing strict game1,16,28,32,65,59,43,94,87,7has a unique equilibrium(3,9)but is not unilaterally order equivalent to any strictly competitive games.Theorem3does not hold either for general two-person games.For instance,the following game1,12,22,21,1has a unique equilibrium(2,2)but is not best-response equivalent to any strictly competitive games.Concluding remarksWe have provided a logical framework for doing computer-aided theorem discovery in two-person game theory,and applied it successfully to the task of discovering classes of two-person games with unique Nash equilibria.The gen-eral methodology used here is similar to the one used by Lin [2004]for discovering state invariants in planning.There are many directions for future work.An obvious one is to see how interesting theorems can be discovered using Theorem1on classes of games that do not have any Nash equilibrium.ReferencesR.J.Aumann.Almost strictly competitive games.Journalof the SIAM,9,1962.J.Friedman.On characterizing equilibrium points in two-person strictly competitive games.International Journal of Game Theory,12:245–247,1983.A.Kats and J.Thisse.Unilaterally competitive games.In-ternational Journal of Game Theory,21:291–299,1992.F.Lin.Discovering state invariants.In Proceedings of KR’04,pages536–544,2004.D.Monderer and L.S.Shapley.Potential games.Games and Economic Behavior,14:124–143,1996.H.Moulin.Cooperation in mixed equilibrium.Mathemat-ics of Operations Research,1:273–286,1976.M.J.Osborne and A.Rubinstein.A Course in Game The-ory.MIT Press,1994.R.W.Rosenthal.Correlated equilibria in some classes of two-person games.International Journal of Game Theory, 3:119–128,1974.M.Shor.Dominant strategy.In Dictionaryof Game Theory Terms,Game . /dictionary/DominantStrategy.html, web accessed:January2007.D.Topkis.Supermodularity and Complementarity.Prince-ton University Press,New Jersey,1998.。