Game Theory and its Application博弈论及其应用

• If you bid 110
• Someone bids more than 110: you lose anyway • Someone bids less than 100: you win anyway and pay second-price • Someone bids 105: you win; but you pay 105, i.e., 5 more than what you value
• Total cost = 3/4
• Game theory solution (selfish routing)
• Each bit will be transmitted using the lower link • Not optimal: total cost = 1
• Price of anarchy is, therefore, 4/3
Prisoner 2 Confess Prisoner 1 Confess -3,-3 Not Confess 0,-9
Not Confess
-9,0
-1,-1
Mixed strategies
• A probability distribution over the pure strategies of the game • Rock-paper-scissors game
Do high-speed links always help?
C(x) = x C(x) = 1 C(x) = 1 C(x) = x C(x) = x C(x) = 1 C(x) = 0 C(x) = 1 C(x) = x
• ½ of the data will take route s-u-t, and ½ s-v-t • Total delay is 3/2 • Add another zero-delay link from u to v • All data will now switch to s-u-v-t route • Total delay now becomes 2 • Adding the link actually makes situation worse
1
N (0,0)
Economic applications of game theory
• The study of oligopolies (industries containing only a few firms) • The study of cartels, e.g., OPEC • The study of externalities, e.g., using a common resource such as a fishery • The study of military strategies • The study of international negotiations • Bargaining
• Each player simultaneously forms his or her hand into the shape of either a rock, a piece of paper, or a pair of scissors • Rule: rock beats (breaks) scissors, scissors beats (cuts) paper, and paper beats (covers) rock
Second-price auction
• Suppose you value an item at 100 • You should bid 100 for the item • If you bid 90
• Someone bids more than 100: you lose anyway • Someone bids less than 90: you win anyway and pay second-price • Someone bids 95: you lose; you could have won by paying 95
Games
• Normal Form representation – Payoff Matrix
Husband
Movie Movie
Wife
Cricket 0,0 1,2
Βιβλιοθήκη Baidu
2,1 0,0
Cricket
Nash equilibrium
• Each player’s predicted strategy is the best response to the predicted strategies of other players • No incentive to deviate unilaterally • Strategically stable or self-enforcing
Games
• Normal Form representation – Payoff Matrix
Prisoner 2
Confess
Not Confess
Confess
Prisoner 1
-3,-3 -9,0
0,-9 -1,-1
Not Confess
Battle of Sexes
• A couple deciding how to spend the evening • Wife would like to go for a movie • Husband would like to go for a cricket match • Both however want to spend the time together • Scope for strategic interaction
• No social choice mechanism is desirable
• Akin to algorithms in computer science
Inefficiency of Nash equilibrium
• Can we quantify the inefficiency? • Does restriction of player behaviors help? • Distributed systems
• Does centralized servers help much?
• Price of anarchy
• Ratio of payoff of optimal outcome to that of worst possible Nash equilibrium
• In the Prisoner’s Dilemma example, it is 3
• Second Price Sealed Bid Auction
• Same rules • Exception – Winner pays the second highest bid and gets the good • Nash equilibrium: Each person exactly bids the good’s valuation
Auctions
• Games of incomplete information • First Price Sealed Bid Auction
• Buyers simultaneously submit their bids • Buyers’ valuations of the good unknown to each other • Highest Bidder wins and gets the good at the amount he bid • Nash Equilibrium: Each person would bid less than what the good is worth to you
Game Theory and its Applications
XIUJU SONG
Prisoner’s Dilemma
• Two suspects arrested for a crime • Prisoners decide whether to confess or not to confess • If both confess, both sentenced to 3 months of jail • If both do not confess, then both will be sentenced to 1 month of jail • If one confesses and the other does not, then the confessor gets freed (0 months of jail) and the non-confessor sentenced to 9 months of jail • What should each prisoner do?
• No pure strategy Nash equilibrium • One mixed strategy Nash equilibrium – each player plays rock, paper and scissors each with 1/3 probability
Nash’s Theorem
Mechanism design
• How to set up a game to achieve a certain outcome?
• Structure of the game • Payoffs • Players may have private information
• Example
Network example
• Delay (or cost) of transmission is C(x)
C(x) = 1 C(x) = x
• Simple network from s to t with two links
• Total amount of data to be transmitted is 1 • Optimal: ½ is sent through lower link
• Existence • Any finite game will have at least one Nash equilibrium possibly involving mixed strategies • Finding a Nash equilibrium is not easy • Not efficient from an algorithmic point of view
Dynamic games
• Sequential moves
• One player moves • Second player observes and then moves
• Examples
• Industrial Organization – a new entering firm in the market versus an incumbent firm; a leader-follower game in quantity competition • Sequential bargaining game - two players bargain over the division of a pie of size 1 ; the players alternate in making offers • Game Tree
Game tree example: Bargaining
(x1,1-x1) 1 x1 B A N B x2 A N Y 0 Period 1: A offers x1. B responds. 0 (x2,1-x2) 0 Period 3: A offers x3. B responds. A Y Period 2: B offers x2. A responds. (x3,1-x3) 1 x3 B Y
• To design an efficient trade, i.e., an item is sold only when buyer values it as least as seller
• Second-price (or second-bid) auction
• Arrow’s impossibility theorem