Chap.2 Normal –Form 博弈论英文版教学课件

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研究生课程博弈论英文课件3

3. Since player 2 is intelligent, he will predict the reaction of player 1 and ﬁgure out the expected payoﬀ m2 (a2 ) from playing a2 : m2 (a2 ) = = ∀ a 1 ∈ A1 .
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a1 ∈A1 a1 ∈A1
min u1 (a1 , a2 ) min {u2 (a3 , a1 ), u2 (a2 , a2 ), u2 (a1 , a3 )}, 1 2 1 2 1 2
1\2 a1 1 a2 1 a3 1 m2 (a2 )
a1 2 0, 0 4, −4 9, −9 −9
3. Since player 1 is intelligent, he will predict the reaction of player 2 and ﬁgure out the expected payoﬀ m1 (a1 ) from playing a1 : m1 (a1 ) = = ∀ a 1 ∈ A1 .
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First argument of player 1: Player 2 moves second
1. Since player 2 is intelligent, he will predict any action a1 ∈ A1 that player 1 may choose. 2. Since player 2 is rational, he will choose the action a2 that maximizes his payoﬀ (or, equivalently, minimizes the payoﬀ of player 2). P1 : P1 : P1 : a1 1 a2 1 3 a1 ⇒ ⇒ ⇒ P2 : P2 : P2 : a1 2 a2 2 2 a2 ⇒ ⇒ ⇒ u1min u1min u1min = 0 = 2 = 0

博弈论英文课件 (7)

Player 2’s expected payoff of playing s21: EU2(s21, (r, 1-r))=r×u2(s11, s21)+(1-r)×u2(s12, s21) Player 2’s expected payoff of playing s22: EU2(s22, (r, 1-r))= r×u2(s11, s22)+(1-r)×u2(s12, s22)
Static (or SimultaneousMove) Games of Complete Information
Matching pennies
Player 2 Head Player 1 Head Tail Tail
-1 ,
1 , -1
1
-1 ,
1 , -1 1
n Head is Player 1’s best response to Player 2’s strategy Tail n Tail is Player 2’s best response to Player 1’s strategy Tail n Tail is Player 1’s best response to Player 2’s strategy Head n Head is Player 2’s best response to Player 1’s strategy Head
n Player 1 plays a mixed strategy (r, 1- r ). Player 2 plays
a mixed strategy ( q, 1- q ).
Ø Ø
n Player 2’s expected payoff from her mixed strategy: v2((r, 1-r),(q, 1-q))=qEU2(s21, (r, 1-r))+(1-q)EU2(s22, (r, 1-r))

博弈论课件 1

1Lecture 1Introduction to Game Theory,Extensive &Normal Form,Mixed Strategies and Beliefs 22Readings•Watson: Strategy_ An introduction to game theory–Ch 1‐5:1rd ed p.1-40; 3rd ed p.1‐46.•Introduction;The Extensive Form;Strategies &the Normal Form;Beliefs,Mixed Strategies and Expecte d Payoffs;General Assumptions and Methodology.•Appendix A:Review of Mathematics p.409‐420.32Outline•Introduction.•Extensive form representation.•Strategies.•Examples.•Normal form representation.•Mixed strategies.•Beliefs.•Expected payoffs.•General assumptions.42Game Theory•Mathematical models of strategic situations :–Each agent’s behavior affects the well-being of other agents.•Perfect competition and monopoly are not strategic situations.–In perfect competition no agent considers the specific action of any other individual agent.–In monopoly the monopolist doesn’t worry about specific consumer’s actions.(Chooses price/quantity based on overall demand.)•Duopoly (2firms producing)is strategic.–Each firm considers the other’s action when deciding on its own.5Ch2:The Extensive Form162Example 1•A family is on vacation is San Diego.Their young children decide whether to behave or misbehave duri ng breakfast.•After breakfast the parents decide whether the family goes to Legoland(乐高) or sits quietly in their hotel room for the rest of the day.•This is a strategic situation.–Each party considers what the other is likely to do (or has already done)when making their own decision.72Formal Definition of a Game •A game is (formally):–A list of players ;–Specification of all possible actions each player can take and when;–The players’knowledge •(what each player knows when he acts);–How actions lead to outcomes ;–A specification of preferences over outcomes.•A game can be non-cooperative or cooperative .–Non-cooperative:All decisions are made individually82Example 1•Players:–The children and their parents.•Actions:–Children –{Behave,Misbehave}–Parents –{Legoland after the kids behave, Hotel after the kids behave, Legoland after the kids misbehave, Hotel after the kids misbehave}•Knowledge:–We typically assume the players know the game and that each player is rational.–Children –Don’t know anything else.–Parents –Knows whether the children have behaved or misbehaved.•Outcomes &Preferences:–We still need to specify these.We’ll do so later.92Extensive Form Representation •One way to represent games is with the extensive form.–Nodes:Where players choose actions (or the game ends).–Branches:Specific actions.–Labels: •Player making a decision (for decision nodes);•or actions (for branches).–Payoffs:Represent preferences over outcomes.–Information sets:Reveal what a player knows when he or she makes a decision.10Example1:Extensive Form2,,112Information Sets•An information set of player i is a collection of player i ’s nodes among which i cannot distinguish.–Perfect information : all information sets in the tree have just one node.–Imperfect information : not perfect information.122Example 1:Information Sets •The children have one information set.–The initial node.•Their parents have two information sets.–When the parents make their decision they knows if the children behaved or misbehaved.•This is why their actions are labeled differently.•For this example each information set is a single node.•Each decision is associated with a single information set.132Example 1(b)•The children decide to behave or misbehave.•Their parents decide whether to take the family to Legoland or sit quietly in their hotel room.–But they don’t observe whether the children behaved or misbehaved.•This time the parents only have one information set.–Their decision cannot be contingent on anything.14Example1(b):Extensive Form2152Example 1(b):Information Sets •The children have one information set.–The initial node.•The parents also have one information set.–When the parents make their decision they don’t know if the children behaved or misbehaved.•They don’t know which node they’re at when they makes their decision.•This is why their actions are labeled the same at both nodes.•Each decision is still associated with a single information set.162Strategies•Strategy:A complete contingent plan for a player in a game.–Prescribes an action for each of this player’ information sets.•Example 1–strategy example:–Children:Behave.–Parents:Sit quietly in the hotel room if the children behave,and go to Legoland if the children misbehave.•Example 1(b)–strategy example:–Children: Behave–Parents: ?17Notation2182Example 1•S Children ={Behave,Misbehave},or for short {B ,M }•S Parents ={Legoland if behave and Legoland if misbehave,Legoland if behave and Hotel if misbehave,Hotel if behave and Legoland if misbehave,Hotel if behave and Hotel if misbeha ve}–We can simplify this notation to {LL ’, LH’, HL ’, HH ’}•One specific s is (B , LL ’)192Example 1(b)•S Children ={Behave,Misbehave}•S Parents ={Legoland,Hotel}•One specific s is (Behave,Legoland)202Example 2•A pedestrian and a car are approaching a crosswalk.–First the pedestrian decides to cross or wait.–The driver observes the decision and then decides to proceed through the crosswalk or delay.•Warning :This is a simplified example.When we eventually solve this problem do not use the results to determine how to proceed through crosswalks.21Example2:Extensive FormPedestrianDriverCrossWaitProceedProceed’DelayDelay’‐100, ‐7510, 55, 100, 0222Example 2:Strategy Sets •S Ped ={C ,W }•S Driver ={PP ’,PD ’,DP ’,DD ’}–Each specific strategy for the driver tells either to proceed or delay if the pedestrian crosses and either to proceed’ or delay’ if the pedestrian waits.232•It’s possible people can make a decision from a continuous action space.–Ie.,S1=[0,100]2's strategy needs to cover every contingency.ie., Yes for a ≥ 72.3 and no for a <72.30,012a Yes100–a,aNo242•We also can have information sets and incomplete information.–Ie.,S1=[0,100]0,012a Yes100–a,aNo2's strategy can be a single action.ie.,Yes.25The ultimatum game (最后通谍博弈)•Players：the two players；•Timing：player 1 proposes a division (x1,x2) of a pie, where x1+x2=1. If 2 accepts this division, she receives x2and player 1 receives x1; if she rejects it, neither player receives any pie.•Preferences：Each person’s preferences are represented by payoffs equal to the division of pie she receives.26Figure: The ultimatum game x 12,x x 120,0Y N 272Example 3•Centipede game:–Order of events is as follows:•Player 1can continue or quit.–If she quits she gets $1and player 2gets $0.•If she continues player 2can stay or go.–If he goes player 1gets $0and he gets $2.•If he stays player 1can agree or disagree –If she disagrees she gets $2and player 2gets $1.–If she agrees she gets $1and player 2gets $3–Graph this game in extensive form;describe each player’s information set and strategy set282Centipede Game Player 1has two information sets:The initial node,The node contingent on C ,S .Player 2has one information set:The node contingent on C .Strategies:S 1={CA ,CD ,QA ,QD }S 2={S ,G }112CQ G S D A 1, 00, 22,11, 329Ch3:Strategies and the Normal Form2302Normal Form Representation •Note that any specific strategy profile will result in a unique outcome.ie.,(CA,G)•(Sometimes)we can use this idea to express much of the essential information for the game in a more compact, matrix form called the normal form.312NormalForm32Back to Example2•A pedestrian and a car are approaching a crosswalk.–First the pedestrian decides to cross or wait.–The driver observes the decision and then decides toproceed through the crosswalk or delay.233Example2:Extensive Form234Example2:Normal Form2352Normal Form•For games with two players and a finite number of strategies,the normal form can be written as a table with appropriate labels.–Given any strategy profile the normal form tells us what the outcome is.u i :S –But we can’t determine the order of the decisions.–It’s possible to have games with different extensive forms that have identical normal forms.36237382Normal Form•There is some debate about whether the normal form contains all the relevant information about the game.•The normal form always contains all the relevant information when the players move simultaneously and independently.–Neither observes any actions chosen by the other player before making own decision.–There are several classic normal form games that fit this description.39Classic Normal Form Games240Classic Normal Form Games241Ch4:Beliefs,Mixed Strategies and Expected Payoffs2422•In strategic situations beliefs are important.•When one player chooses a strategy she will consider what strategies the other player is likely to choose.•Mathematically, a belief of player i is aprobability distribution over the strategies of the other players.432•Formally ΔS i is the set of probability distributions over S i .•In a two-player game, the belief of player i about the behavior of player j is a function θj ϵΔS j such that, for each strategy s j ϵS j of player j , θj (s j ) is interpreted as the probability that player i thinks player j will play s j .•θj has the property that θj (s j ) ≥ 0 for each s j ϵS j , and ()1j jj j s S s θ∈=∑442A specific belief of player 1 about player 2 is denoted by θ2ϵΔS2. ie.,θ2= (2/5, 3/5) over (St2, R2).452Players may choose their strategies by randomizing.ie., Player 1 may play St 1with probability ⅓ and R 1with ⅔.Formally ΔS i is the set of probability distributions over S i .A specific mixed strategy for player i is denoted by σi ϵΔS i .ie., ΔS 1= (p , 1-p ) where 0≤p ≤1, σ1= (⅓, ⅔).462•The strategies we looked at earlier are called pure strategies .–A player chooses one specific strategy with certainty.•A mixed strategy assigns probabilities on a player’s pu re strategies.–Technically a pure strategy is a mixed strategy with a probability of 1 on that pure strategy and a probability of 0 on all other pure strategies.•Beliefs look a lot like mixed strategies (for the other player)because both of them are chosen from the set of probability distributions over that player’s strategies.47Expected Payoffs•If a player uses a mixed strategy and/ or assigns positive probability to multiple strategies of the other player, the this player cannot expect to get a particular payoff for sure.2•We can extend the definition of a payoff function to mixed strategies and beliefs by using the concept of expected value.482When i uses mixed strategy, σi and the others play (pure) s -i When i uses pure strategy,s i and has a belief θ-i about the strategies of the others,When i uses mixed strategy, σi , and has a belief θ-i about the strategies of the others,()()(),,i ii i i i i i i i s S u s s u s s σσ--∈=∑()()(),,i i i i i i i i i is S u s s u s s θθ-----∈=∑()()()(),,,i i i i i i i i i i i i i i s S s S u s s u s s σθσθ-----∈∈=∑49250。

博弈论英文课件 (5)

whether he is happy or not. n If he is altruistic then he prefers to mum and he thinks that
“confess” is equivalent to additional “four months in jail”. n Prisoner 1 can not know exactly whether prisoner 2 is rational
u1(q1, q2)=q1(a-(q1+q2)-c), u2(q1, q2)=q2(a-(q1+q2)-c)
n All these information is common knowledge
10
Cournot duopoly model of incomplete information
n A homogeneous product is produced by only two firms: firm 1 and firm 2. The quantities are denoted by q1 and q2, respectively.
18
Cournot duopoly model of incomplete information (version one) cont’d
q2* (cH
)

1 3
(a

2cH
c)
1
n (Confess if rational, Mum if altruistic) is prisoner 2’s best response to prisoner 1’s Confess
n A Nash equilibrium called Bayesian Nash equilibrium

博弈论英文课件 (3)

Ø If Player 2 chooses Head, r-(1-r)=2r-1 Ø If Player 2 chooses Tail, -r+(1-r)=1-2r
Solving matching pennies
Player 2
Head
Tail
Expected payoffs
Head Player 1
Static (or SimultaneousMove) Games of Complete Information
Matching pennies
Player 2
Head
Tail
Player 1
Head Tail
-1 , 1 1 , -1 1 , -1 -1 , 1
n Head is Player 1’s best response to Player 2’s strategy Tail n Tail is Player 2’s best response to Player 1’s strategy Tail
( 1(H)=0.5, 1(T)=0.5 ) is a Mixed strategy. That is, player 1 plays H and T with probabilities 0.5 and 0.5, respectively.
( 1(H)=0.3, 1(T)=0.7 ) is another Mixed strategy. That is, player 1 plays H and T with probabilities 0.3 and 0.7, respectively.
Ø Player 2’s expected payoff of playing s22:
EU2(s22, (r, 1-r))= r×u2(s11, s22)+(1-r)×u2(s12, s22)

博弈论最全完整-讲解PPT课件

王则柯、李杰编著，《博弈论教程》，中国人民大学出版社，2004年版。
艾里克.拉斯缪森（Eric Rasmusen）著，《博弈与信息：博弈论概论》，北京大学出版社，2003年版。
因内思·马可-斯达德勒,J.大卫·佩雷斯-卡斯特里罗著，《信息经济学引论：激励与合约》,上海财经大学出版社，2004年版。
常和博弈也是利益对抗程度最高的博弈。非常和（变和）博弈蕴含双赢或多赢。
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32
导论
四、主要参考文献
.
33
张维迎著，《博弈论与信息经济学》，上海三联书店、上海人民出版社，1996年版。
Roger B. Myerson著：Game Theory（原文版、译文版），中国经济出版社，2001年版。
是关于动态博弈进行过程之中面临决策或者行动的参与人对于博弈进行迄今的历史是否清楚的一种刻划。
如果在博弈进行过程中的每一时刻，面临决策或者行动的参与人，对于博弈进行到这个时刻为止所有参与人曾经采取的决策或者行动完全清楚，则称为完美信息博弈；否则位不完美信息。
.
30
零和博弈与非零和博弈
了解自己行动的限制和约束，然后以精心策划的方式选择自己的行为，按照自己的标准做到最好。 • 博弈论对理性的行为又从新的角度赋予其新的含义— —与其他同样具有理性的决策者进行相互作用。 • 博弈论是关于相互作用情况下的理性行为的科学。
.
4
如何在博弈中获胜？
…… 真的能在博弈中（总是）获胜吗？
对手和你一样聪明！许多博弈相当复杂，博弈论并不
施锡铨编著，《博弈论》上海财大出版社，2000年版。
谢识予编著，《经济博弈论》，复旦大学出版社， 2002年版。
谢识予主编，《经济博弈论习题指南》，复旦大学出版社，2003年版。

lecture3(博弈论讲义(Carnegie Mellon University))

* * ui ( s1 ,..., si*1, si , si*1,..., sn )
for all si Si . That is, si* solves Maximize Subject to
* * * * ui ( s1 ,..., si 1, si , si 1,..., sn ) si Si
May 21, 2003 73-347 Game Theory--Lecture 3 2
Today’s Agenda
Review of previous classes Nash equilibrium Best response function
Use best response function to find Nash
(Confess, Confess) is a Nash equilibrium. Prisoner 1
May 21, 2003
Prisoner 2
Mum
Mum Confess -1 , 0 , -1 -9
Confess
-9 , -6 , 0 -6
10
73-347 Game Theory--Lecture 3
equilibria Examples
May 21, 2003
73-347 Game Theory--Lecture 3
3
Review
The normal-form (or strategic-form) representation of a
game G specifies: A finite set of players {1, 2, ..., n}, players’ strategy spaces S1 S2 ... Sn and their payoff functions u1 u2 ... un where ui : S1 × S2 × ...× Sn→R.

博弈论第三讲

x1
x2
x4
x3
x5
x6
x7
an edge connecting nodes x1 and x5
x8
7
Game tree
A path is a sequence of distinct
nodes y1, y2, y3, ..., yn-1, yn such that yi and yi+1 are adjacent, for i=1, 2, ..., n-1. We say that this path is from y1 to yn. We can also use the sequence of edges induced by these nodes to denote the path.
13
Entry game
Challenger’s strategies
In Out Incumbent’s strategies Accommodate (if challenger plays In) Fight (if challenger plays In) Payoffs Normal-form representation
Dynamic Games of Complete Information
Entry game
An incumbent monopolist faces the possibility of entry by a
challenger. The challenger may choose to enter or stay out. If the challenger enters, the incumbent can choose either to accommodate or to fight. The payoffs are common knowledge.

博弈论Chapter 2 Normal Form Game III

1. The Advantage of Mixed Strategies
• Note that each player cannot be taken advantage of.
1. The Advantage of Mixed Strategies
• Definition of Mixed Strategy: • Let G be a game with strategy spaces S1,S2,..,SI .
• 1. Function (measure): σ1 (H) = 0.5 and σ1(T) of Mixed Strategies
• 2. Vector: If the pure strategies are si1,..siNi write (σi (si1) ; …; σi (siNi)) • e.g. (0.5, 0.5)
• β=2/3
3. Find the mixed strategy NE
u2 (L; σ1* ) = u2 (R; σ1* )
E.g.. α+2(1-α)=4 α+(1-α) α=1/4
3. Find the mixed strategy NE
• Therefore, the mixed strategy NE is:
0, 0 1, -1 -1, 1
-1, 1 0, 0 1, -1
1, -1 -1, 1 0, 0
P
S
1. The Advantage of Mixed Strategies
• This game has no pure-strategy Nash equilibrium. Whatever pure strategy player 1 chooses, player 2 can beat him.

gametheory2博弈论英文精品PPT课件

Review
Elimination
Nash Equilibrium
Summary
Games
Course topics:
• Games of complete and perfect information • Static Games (Nash Equilibrium) • Dynamic Games (Backward Induction)
6 / 43
Review
Elimination
Nash Equilibrium
Summary
Games
• Consider the following game:
• Two players • Each player chooses between two actions: A and B • Payoff for all outcomes is in the table below:
1
2
A
B
your ID keyword
A 50,50 0,200
B 200,0 80,80
1
2
A
B
A 50,50 0,200
B 200,0 80,80
7 / 43
Review
Elimination
Nash Equilibrium
Summary
Games
• Game Participation:
• you can win up to 200 CZK • send SMS with your action • phone numbers + IDs will be strictly protected • participant will be matched randomly in pairs, one pair will be

博弈论第四讲

Prisoner 2 can be rational (selfish) or altruistic, depending on
whether he is happy or not. If he is altruistic then he prefers to mum and he thinks that “confess” is equivalent to additional “four months in jail”. Prisoner 1 can not know exactly whether prisoner 2 is rational or altruistic, but he believes that prisoner 2 is rational with probability 0.8, and altruistic with probability 0.2.
(Confess, (Confess if rational, Mum if altruistic))
Confess is prisoner 1’s best response to prisoner 2’s
choice (Confess if rational, Mum if altruistic). (Confess if rational, Mum if altruistic) is prisoner 2’s best response to prisoner 1’s Confess A Nash equilibrium called Bayesian Nash equilibrium
Max s .t . q 2 [ a ( q1 q 2 ) c H ] q2 0
q 2 (c H ) 1 2 ( a q1 c H )

国外博弈论课件lecture(1)_OK

• Pat’s expected payoff of playing Prize Fight: 2(1-r)
• Pat’s best response B2(r):
➢ Prize Fight (q=0) if r<2/3
➢ Opera (q=1) if r>2/3
➢ Any mixed strategy (0q1) if r=2/3,
• Player 1 plays a mixed strategy (r, 1- r ). Player 2 plays a mixed strategy ( q, 1- q ).
➢ Player 2’s expected payoff of playing s21: EU2(s21, (r, 1-r))=r×u2(s11, s21)+(1-r)×u2(s12, s21)
➢ EU1(T, (0.5, 0.5)) = 0.5×1 + 0.5×(-1)=0
➢ v1((0.5, 0.5), (0.5, 0.5))=0.50+0.50=0
• Player 2:
➢ EU2(H, (0.5, 0.5)) = 0.5×1+0.5×(-1) =0
➢ EU2(T, (0.5, 0.5)) = 0.5×(-1)+0.5×1 = 0
Static (or Simultaneous-Mov e) Games of Complete Infor
mationMixed Strategy Nash Equilibrium
2021/9/6
1
Outline of Static Games of Compl ete Information
• Introduction to games • Normal-form (or strategic-form) representation • Iterated elimination of strictly dominated strategies • Nash equilibrium • Review of concave functions, optimization • Applications of Nash equilibrium • Mixed strategy Nash equilibrium

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

博弈论Chapter 2 Normal Form Game I

• (2) Battle of the Sexes This game is interesting because it is a coordination game with some elements of conflict. The idea is that a couple want to spend the evening together.
1. Ingredients of a normal form game
• In our example, ui = 1 if both agents choose the same action, and 0 otherwise.
1. Ingredients of a normal form game
• Think of dividing up a pie. The size of the pie doesn't change. • it's all about redistribution of the pieces between the players. • tax policy is a good example.
1. Ingredients of a normal form game
• 2. Each player i can choose actions from a strategy set Si . To continue our example, each of the players has the option to go the Empire State building or meet at the old oak tree in Central Park. So the strategy sets of both players are S1 = S2 = {E;C}.

博弈论介绍PPT课件

2
2
Game of Chicken
driver who steers away looses What should drivers do?
Goal is to prescribe how conflicts can be resolved
Applications of Game Theory
But it can provide intuitions, suggestions and partial prescriptions
best mathematical tool we currently have
What is a Game?
A Game consists of
at least two players a set of strategies for each player a preference relation over possible outcomes
Normal form (strategic) game a finite set N of players
a set strategies A i for each player iN
payoff function u i ( s ) for each player iN • where sAjNAj is the set of strategies
Outline
Introduction to game theory Typical games in networking A sample game in Cognitive Radio Networks
What is Game Theory About?
Analysis of situations where conflict of interests are present

博弈论完整课件[浙江大学]__GAME_Cha

＊Two suspects are arrested and charged with
a crime. The police lack sufficient evidence to
convict the suspects, unless aห้องสมุดไป่ตู้ least one
confesses.The police hold the suspects in
players receive payoffs that depend on the
combination of actions just chosen. Within the
class of such static (or simultaneous-move)
games,we restrict attention to games of
Chapter 1 完全信息静态博弈
Static Games of Complete Information
In this chapter we consider games of the
following simple form: first, the players
simultaneously choose actions; then, the
可编辑ppt
4
招认囚徒１
沉默
囚徒２招认沉默 –5, -5 0, -8
-8, 0 -1 , -1
囚徒的困境
可编辑ppt
5
We now turn to the general case. The normal-form representation of a game specifies: (1)the players in the game;(2)the strategies available to each player;(3)the payoff received by each player for each combination of strategies that could be chosen by the players.

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Given that each firm knows the market demand, Investment of New Product is a static game of complete information in which tow firms act simultaneously, or a dynamic game of complete information in which two firm act successively.
Information
In game-theoretic analysis, Information is just all the knowledge that one player owns, such as the other Players actions/strategy, the other players payoff, and so on.
In Investment of New Product , the firms’ actions are “Invest” and “No Invest”, e.g. ai=a (Invest) or ai=b ( No Invest) .
Player i’s action set, Ai={ai} , is the entire set of actions available to him.
We will occasionally write si S i to indicate that the strategy si is a member of the set of strategies Si.
A strategy combination is an ordered set s=(s1 ,… sn) consisting of one strategy for each of the n players in the game.
But each firm still confront the following uncertainty:
1) whether he/she knows the market demand when he/she makes decision ?
2) whether he/she knows the other firm’s decision when he/she makes decision ?
A1={a, b}
A2={a, b}
An action combination is an ordered set a=(a1 ,… an) consisting of one action for each of the n players in the game. For example, in Investment of New Product , action combination (a, b) means that firm 1’s action is a and firm 2’s action is b.
when the market demand is low, each
player loses 4 million yuan if they all
invest, i.e.
, u 1 ( a ,a ) 4 0 0 , u 2 ( a ,a ) 4 0 0
or player 1 gets 2 million yuan and player 2 gets nothing if 1 invests and 2
2.2 Normal –Form Representation of Games
Normal-form representation of games which is also called strategyform representation of games or strategic form game is the basic form of describing game problem.
Given that any firm dose not know the market demand, Investment of New Product is a static game of incomplete information in which tow firms act simultaneously, or a dynamic game of incomplete information in which two firm act successively.
Chap.2
Normal –Form Representation of Games
Content
2.1 Basic Concepts 2.2 Normal –Form Representation of
Games
2.1 Basic Concepts
We will introduce some basic Concepts In our study by illustrating the following game---”Investment in New Product”.
Player
Players are individuals who make decision. Each player’s goal is to maximize his/her utility cost by choice of actions.
In Investment of New Product , the two firms are specified the players of the game.
Payoff
Player i’s payoff, denoted ui, is what player i gets from the outcome of the game.
It usually means the expected utility the player i receives as a function of the strategies chosen by himself and the other players, i.e.
Strategy
Player i’s strategy si , is a rule that tells him which action to choose at each instant of the game.
Player i’s strategy set or strategy space, Si={si} , is the entire set of strategies availl occasionally write
to
indicate that the action ai is a maiemAiber
of the set of actions Ai.
In Investment of New Product , the firm
i’s action set is
ui ui (s1,s2, ,sn)
ui (si,si )
s i (s 1 , ,s i 1 ,s i 1 , ,s n d)enotes
the
ordered set consisting of one strategy
for each player in the game except
player i.
Any player is denoted as i , and the set of players as
{1,2, ,n}
In Investment of New Product , the set of players is
{1,2}
Action
An action or move by player i, denoted ai, is a choice he can make.
A payoff combination is an ordered set u=(ui), of one for each of the n players in the game.
In Investment of New Product , the firm’s benefit is its payoff.
Information is the basis of our gametheoretic analysis, and there exists different information assumption according to the game problem that we may encounter.
does not, i.e. . u 1 (a ,b ) 2 0 0 ,u 2 (a ,b ) 0
Information
In Investment of New Product , it is a basic information assumption that every player’s payoff is common knowledge. And, given that if each player knows the market demand, it means that complete information is a basic information assumption in our study, otherwise incomplete information is a basic information assumption.
Definition 2.1: The normal-form representation of a game specifies:
(1) the players in the game;
(2) the strategies available to each player;