Game Theory. Professor Peter Cramton Economics 300

Transcription

1 Game Theory Professor Peter Cramton Economics 300

2 Definition Game theory is the study of mathematical models of conflict and cooperation between intelligent and rational decision makers. Rational: each individual maximizes her expected utility Intelligent: individual understands situation, including fact that others are intelligent rational decision makers

3 Game Theory Game theory lets us study multi-person decision problems Use game theory to model: Trading process (auctions, bargaining, markets) Competition among firms Sporting events Voting Military decisions Competition or collusion among countries in choosing tariffs, trade policies, environmental standards, etc.

4 A set of rules specifying: What is a game? Players Alternatives (actions players choose from) Order of play Outcomes and payoffs

5 Normal form of a game Players: I = {1, 2,, N} Action sets: A 1, A 2,, A N Players simultaneously choose their actions a 1, a 2,, a N Payoffs are realized U i (a 1, a 2,, a N ) for each player i

6 Example: The Prisoner s Dilemma Two suspects are arrested by the police If both stay mum they get a light 1-year sentence If one finks and other stays mum, the finker goes free and the other gets 9 years If both fink on the other, they both get 6 years What should the prisoner s do?

7 Example: The Prisoner s Dilemma Players: Prisoner 1 and Prisoner 2 Actions: A 1 = A 2 = {Mum, Fink} Payoffs: u ( Mum, Mum) 1 u ( Mum, Mum) 1 2 u ( Fink, Fink) 6 u ( Fink, Fink) 1 2 u ( Mum, Fink) 9 u ( Fink, Mum) 1 2 u ( Fink, Mum) 0 u ( Mum, Fink) 1 2

8 Example: The Prisoner s Dilemma Mum P2 Fink P1 Mum Fink -1, -1-9, 0 0, -9-6, -6

9 Mum Fink Solution Concepts Mum Fink -1, -1-9, 0 0, -9-6, -6 Best Response Mapping: Each player wants to make an optimal (i.e., the best) action for himself, given actions of others. Example: Prisoner s Dilemma BR ( Mum) Fink (since 0 1) 1 BR ( Fink) Fink (since - 6-9) 1 Fink is the dominant strategy for prisoner 1.

10 Definition: A dominant strategy is the best choice for a player regardless of what the others are doing (i.e., Best Response is always the same). If each player has a dominant strategy, then we call this strategy profile a dominant strategy equilibrium (DSE).

11 Mum The Prisoner s Dilemma Mum Fink -1, -1-9, 0 Fink 0, -9-6, -6 BR ( Mum) Fink (since 0 1) 1 BR ( Fink) Fink (since - 6-9) 1 BR ( Mum) Fink (since 0 1) 2 BR ( Fink) Fink (since - 6-9) 2 Hence, (Fink, Fink) is a unique DSE

12 P1 U D L Example Two P2 M R 1, 0 1, 2 3, 1 0, -9 0, 3 1, 2 BR 1 (L) U BR 1 ( M ) U U is dominant strategy for P1 BR 1 (R) U BR2 ( U ) M (2 0 and 2 1) M is dominant strategy for P2 BR2 ( D) M (3 9 and 3 2) (U, M) is DSE.

13 Example: U P1 D P2 L M R 1, 0 1, 2 0, 1 0, 3 0, 1 2, 0 BR 1 (L) U BR 1 ( M ) U BR 1 (R) D BR 2 (U ) M BR 2 (D) L No dominant strategy No DSE

14 Solution Concept 2: Iterated Elimination of Never a Best Response Strategies Rational players do not play strategies that are never a best response So remove strategies that are never a best response

15 Example: U P1 D P2 L M R 1, 0 1, 2 0, 1 0, 3 0, 1 2, 0 BR 1 (L) U BR 1 ( M ) U BR 1 (R) D BR 2 (U ) M BR 2 (D) L For P2, R is never a best response, eliminate R BR ( L) 1 1 BR ( M ) BR ( U) 2 U U M For P1, D is never a best response, eliminate D Hence, (U,M) is the Iterated Elimination of Never a Best Response Strategies

16 Example: Battle of the Sexes Football W Ballet M Football Ballet 2, 1 0, 0 0, 0 1, 2 BR (B) B (1 0) M BR (F) F (2 0) M BR ( B) B (2 0) W BR ( F) F (1 0) No dominant strategy & No IENS W

17 Nash Equilibrium is a set of mutual best responses: Each player is playing a best response to what the others are doing. With two players: Solution Concept 3: Nash Equilibrium (Pure Strategies) (a, b) is NE if BR 1 (b)=a and BR 2 (a)=b

18 Example: Battle of the Sexes Football W Ballet M Football Ballet 2, 1 0, 0 0, 0 1, 2 BR (B) B (1 0) M BR (F) F (2 0) M (F,F) and (B,B) are NE BR ( B) B (2 0) W BR ( F) F (1 0) W

19 P1 Solution Concept 3: Nash Eq. in Pure Strategies a b L P2 M R 1, 0 2, 1 3, 2 2, 1 0, 0 2, 0 BR (L) b (2 1) 1 BR (a) R BR ( M ) a (2 0) 2 1 BR (b) L BR (R) a (3 2) 2 1 (a, R) and (b, L) are Nash Eq.

20 Nash Eq. in pure strategies don t always exist Example: Matching Pennies P1 H T H P2 T -1, 1 1, -1 1, -1-1, 1 BR (H) T 1 BR (T ) H 1 BR ( H ) 2 BR ( T ) 2 H T No Nash Eq. in pure strategies

21 Solution Concept: Nash Eq in Mixed Strategies Example: Matching Pennies P2 P1 H (q) T (1-q) H (r) T (1-r) -1, 1 1, -1 1, -1-1, 1 A mixed strategy is a probability distribution over a player s pure strategies.

22 Solution Concept: Nash Eq in Mixed Strategies Example: Matching Pennies P2 P1 H (q) T (1-q) H (r) T (1-r) -1, 1 1, -1 1, -1-1, 1 P1 plays P2 plays H with probability q T with probability 1-q H with probability r T with probability 1-r, where r, q [0,1]

23 Why mixed strategies? Mathematical point of view Needed if pure strategy equilibria do not exist May coexist with pure strategy NE Practical point of view Be unpredictable Tennis service Penalty kick in football Poker War

24 Defining the expected payoffs from mixed strategy play for P1 H (r) P2 T (1-r) EU 1 P1 H T -1, 1 1, -1 1, -1-1, 1 r(-1)+(1-r)1 r(1)+(1-r)(-1) 1 1 EU H, r,1 r r 1 1 r 1 1 2r EU T, r,1 r r 1 1 r 1 2r 1 P1 will only randomize if indifferent! 1 2r 2r 1 r 1/ 2

25 Defining the expected payoffs from mixed strategy play for P2 P1 H (q) T (1-q) 2 2 H P2 T -1, 1 1, -1 1, -1-1, 1 EU 2 : q(1)+(1-q)(-1) q(-1)+(1-q)(1) EU ( q,1 q), H q 1 1 q ( 1) 2q 1 EU ( q,1 q), T q 1 1 q 1 1 2q P2 will only randomize if indifferent! 2q 1 1 2q q 1/ 2

26 Each player randomizes to make other indifferent P1 H 1/2 T 1/2 H (1/2) P2-1, 1 1, -1 1, -1-1, 1 T (1/2) EU 1 0 = 0 EU 2 : 0 = 0

27 Nash Existence Theorem (Nash, 1950) Theorem: Every finite game has at least one Nash equilibrium (when mixed strategies are permitted). Remark: If, in a mixed-strategy equilibrium, player i places positive probability on each of two strategies then player i must be indifferent between these two strategies i.e., they yield player i the same expected payoff.

28 Mixed strategy in battle of the sexes F (r) P2 B (1-r) EU 1 P1 F B 2, 1 0, 0 0, 0 1, 2 r(2)+(1-r)0 r(0)+(1-r)(1) 1 1 EU F, r,1 r r 2 1 r 0 2r EU B, r,1 r r 0 1 r 1 1 r P1 will only randomize if indifferent! 2r 1 r r 1/ 3

29 P1 Mixed strategy in battle of the sexes P2 F (q) B (1-q) 2 2 F (r) B (1-r) 2, 1 0, 0 0, 0 1, 2 EU 2 : q(1)+(1-q)(0) q(0)+(1-q)(2) (,1 ), 1 1 (0) EU q q F q q q EU ( q,1 q), B q 0 1 q 2 2 2q P2 will only randomize if indifferent! q 2 2q q 2 / 3

30 Each player randomizes to make other indifferent P1 F (2/3) B (1/3) EU 2 : F (1/3) P2 2, 1 0, 0 0, 0 1, 2 2/3 = 2/3 B (2/3) EU 1 2/3 = 2/3