Lecture 3 Representation of Games

Transcription

1 ecture 3 epresentation of Games 4. Game Theory Muhamet Yildiz oad Map. Cardinal representation Expected utility theory. Quiz 3. epresentation of games in strategic and extensive forms 4. Dominance; dominant-strategy equilibrium

2 Cardinal representation definitions Z = a finite set of consequences or prizes. A lottery is a probability distribution on Z. P = the set of all lotteries. A lottery: $M $0 Cardinal representation Von Neumann-Morgenstern representation: A lottery (in P) p f q z Z Expected value of u under p u ( z ) p ( z ) z Z u ( z ) q ( z ) U(p ) U(q )

3 VNM Axioms Axiom A: is complete and transitive. VNM Axioms Axiom A (Independence): For any p,q,r P, and any a (0,], ap+ (-a)r aq+ (-a)r p q. p $ q $M.5.5 $ A trip to Florida A trip to Florida $0 3

4 VNM Axioms Axiom A3 (Continuity): For any p,q,r P, if p q r, then there exist a,b (0,) such that ap+ (-a)r q bp+ (-b) r. Theorem VNM-representation A relation on P can be represented by a VNM utility function u : Z iff satisfies Axioms A-A3. u and v represent iff v = au + b for some a > 0 and b. 4

5 Exercise Consider a relation among positive real numbers represented by VNM utility function u with u(x) = x. Can this relation be represented by VNM utility function u * (x) = x /? What about u **(x) = /x? Normal-form representation Definition (Normal form): A game is any list G = (S,K, S n ; u,k, u n ) where, for each i N = {,, K, n}, S i is the set of all strategies available to i, u i : S S is the VNM utility function of n player i. Assumption: G is common knowledge. Definition: A player i is rational iff he tries to maximize the expected value of u i given his beliefs. 5

6 Chicken (-,-) (0,) (,0) (/,/) Figures by MIT OCW. Extensive-form representation Definition: A tree is a set of nodes connected with directed arcs such that. There is an initial node;. For each other node, there is one incoming arc; 3. each node can be reached through a unique path. 6

7 A tree Non-terminal nodes Terminal Nodes Extensive form definition Definition: A game consists of a set of players a tree an allocation of each non-terminal node to a player an informational partition (to be made precise) a payoff for each player at each terminal node. 7

8 Information set An information set is a collection of nodes such that. The same player is to move at each of these nodes;. The same moves are available at each of these nodes. An informational partition is an allocation of each non-terminal node of the tree to an information set. A game (,) l r u (0,0) λ ρ Λ Ρ (,3) (3,) (3,3) (,) 8

9 Another game x T B The Same Game x T B 9

10 What is wrong? x T B Up What is wrong? x T B 3 0

11 What is wrong? A 3 B T D Strategy A strategy of a player is a complete contingent-plan, determining which action he will take at each information set he is to move (including the information sets that will not be reached according to this strategy).

12 Matching pennies with perfect information Head Tail head tail head tail (-,) (,-) (,-) (-,) Matching pennies with perfect information HH HT TH TT Head Head Tail Tail head tail head tail (-,) (,-) (,-) (-,)

13 Matching pennies with Imperfect information Head Tail Head Head Tail (-,) (,-) head tail head tail (-,) (,-) (,-) (-,) Tail (,-) (-,) A game A α a (,-5) D δ d (4,4) (5,) (3,3) 3

14 A game with nature eft (5, 0) / Head ight (, ) Nature / Tail eft (3, 3) ight (0, -5) Mixed Strategy Definition: A mixed strategy of a player is a probability distribution over the set of his strategies. Pure strategies: S i = {s i,s i,,s ik } A mixed strategy: σ i : S [0,] s.t. σ i (s i ) + σ i (s i ) + + σ i (s ik ) =. If the other players play s -i =(s,, s i-,s i+,,s n ), then the expected utility of playing σ i is σ i (s i )u i (s i,s -i ) + σ i (s i ) u i (s i,s -i ) + + σ i (s ik ) u i (s ik,s -i ). 4

15 How to play Dominance s -i =(s,, s i-,s i+,,s n ) Definition: A pure strategy s i * strictly dominates s i if and only if u i ( s *, s ) > u i ( s, s ) s i. i i i i A mixed strategy σ i strictly dominates s i iff σ i (s i )u i (s i, s i ) + +σ i (s ik )u i (s ik, s i ) > u i (s i, s i ) s i A rational player never plays a strictly dominated strategy. 5

16 Prisoners Dilemma Cooperate Defect Cooperate (5,5) (0,6) Defect (6,0) (,) U M = U T A Game UB = U T (,0) (-,) U B U T M (0,0) (0,0) 0 U M B (-,-6) (,0) p -p - 0 p 6

17 Weak Dominance Definition: A pure strategy s i * weakly dominates s i if and only if u i ( s *, s ) u i ( s, s ) s i. i i i i and at least one of the inequalities is strict. A mixed strategy σ i * weakly dominates s i iff σ i (s i )u i (s i, s i ) + +σ i (s ik )u i (s ik, s i ) u i (s i, s i ) s i and at least one of the inequalities is strict. If a player is rational and cautious (i.e., he assigns positive probability to each of his opponents strategies), then he will not play a weakly dominated strategy. Dominant-strategy equilibrium Definition: A strategy s i * is a dominant strategy iff s i * weakly dominates every other strategy s i. Definition: A strategy profile s* is a dominant-strategy equilibrium iff s i * is a dominant strategy for each player i. If there is a dominant strategy, then it will be played, so long as the players are 7

18 Prisoners Dilemma Cooperate Defect Cooperate (5,5) (0,6) Defect (6,0) (,) Second-price auction N = {,} buyers; The value of the house for buyer i is v i ; Each buyer i simultaneously bids b i ; i* with b i* = max b i gets the house and pays the second highest bid p = max j i b j. Clip art image removed for copyright reasons. 8

19 A Game T M B (,0) (0,0) (-,-6) (-,) (0,0) (,0) 9