A Primer on Strategic Games Krzysztof R. Apt (so not Krzystof and - PowerPoint PPT Presentation

A Primer on Strategic Games Krzysztof R. Apt (so not Krzystof and definitely not Krystof) CWI, Amsterdam, the Netherlands , University of Amsterdam A Primer on Strategic Games – p. 1/6

Overview Best response, Nash equilibrium, Weak/strict dominance, Iterated elimination of strategies, Mixed strategies, Variations on the definition, Pre-Bayesian games, Mechanism design: implementation in dominant strategies. A Primer on Strategic Games – p. 2/6

Strategic Games: Definition Strategic game for n ≥ 2 players: (possibly infinite) set S i of strategies, payoff function p i : S 1 × . . . × S n → R , for each player i . Basic assumptions: players choose their strategies simultaneously, each player is rational: his objective is to maximize his payoff, players have common knowledge of the game and of each others’ rationality. A Primer on Strategic Games – p. 3/6

Three Examples Prisoner’s Dilemma C D C 2 , 2 0 , 3 D 3 , 0 1 , 1 The Battle of the Sexes F B F 2 , 1 0 , 0 B 0 , 0 1 , 2 Matching Pennies H T H 1 , − 1 − 1 , 1 T − 1 , 1 1 , − 1 A Primer on Strategic Games – p. 4/6

Three Main Concepts Notation: s i , s ′ i ∈ S i , s, s ′ , ( s i , s − i ) ∈ S 1 × . . . × S n . s i is a best response to s − i if ∀ s ′ i ∈ S i p i ( s i , s − i ) ≥ p i ( s ′ i , s − i ) . s is a Nash equilibrium if ∀ i s i is a best response to s − i : ∀ i ∈ { 1 , . . ., n } ∀ s ′ i ∈ S i p i ( s i , s − i ) ≥ p i ( s ′ i , s − i ) . Intuition: In a Nash equilibrium no player can gain by unilaterally switching to another strategy. s is Pareto efficient if for no s ′ ∀ i ∈ { 1 , . . ., n } p i ( s ′ ) ≥ p i ( s ) , ∃ i ∈ { 1 , . . ., n } p i ( s ′ ) > p i ( s ) . A Primer on Strategic Games – p. 5/6

Nash Equlibrium Prisoner’s Dilemma: 1 Nash equlibrium C D C 2 , 2 0 , 3 D 3 , 0 1 , 1 The Battle of the Sexes: 2 Nash equlibria F B F 2 , 1 0 , 0 B 0 , 0 1 , 2 Matching Pennies: no Nash equlibrium H T H 1 , − 1 − 1 , 1 T − 1 , 1 1 , − 1 A Primer on Strategic Games – p. 6/6

Dominance s ′ i is strictly dominated by s i if ∀ s − i ∈ S − i p i ( s i , s − i ) > p i ( s ′ i , s − i ) , s ′ i is weakly dominated by s i if ∀ s − i ∈ S − i p i ( s i , s − i ) ≥ p i ( s ′ i , s − i ) , ∃ s − i ∈ S − i p i ( s i , s − i ) > p i ( s ′ i , s − i ) . A Primer on Strategic Games – p. 7/6

Prisoner’s Dilemma C D C 2 , 2 0 , 3 D 3 , 0 1 , 1 Why a dilemma? ( D, D ) is the unique Nash equilibrium, For each player C is strictly dominated by D , ( C, C ) is a Pareto efficient outcome in which each player has a > payoff than in ( D, D ) . A Primer on Strategic Games – p. 8/6

Prisoner’s Dilemma for n Players Assume k i ( n − 1) > l i > 0 for all i . � k i | s − i ( C ) | + l i if s i = D p i ( s ) := k i | s − i ( C ) | if s i = C. For n = 2 , k i = 2 and l i = 1 we get the original Prisoner’s Dilemma game. p i ( C n ) = k i ( n − 1) > l i = p i ( D n ) , so for all players C n yields a > payoff than D n . For all players strategy C is strictly dominated by D : p i ( D, s − i ) − p i ( C, s − i ) = l i > 0 . A Primer on Strategic Games – p. 9/6

Quiz H T E H 1 , − 1 − 1 , 1 − 1 , − 1 T − 1 , 1 1 , − 1 − 1 , − 1 E − 1 , − 1 − 1 , − 1 − 1 , − 1 What are the Nash equilibria of this game? A Primer on Strategic Games – p. 10/6

Answer H T E H 1 , − 1 − 1 , 1 − 1 , − 1 T − 1 , 1 1 , − 1 − 1 , − 1 E − 1 , − 1 − 1 , − 1 − 1 , − 1 ( E, E ) is the only Nash equilibrium. It is a Nash equilibrium in weakly dominated strategies. A Primer on Strategic Games – p. 11/6

IESDS: Example 1 L M R T 3 , 0 2 , 1 1 , 0 C 2 , 1 1 , 1 1 , 0 B 0 , 1 0 , 1 0 , 0 B is strictly dominated by T , R is strictly dominated by M . By eliminating them we get: L M T 3 , 0 2 , 1 C 2 , 1 1 , 1 A Primer on Strategic Games – p. 12/6

IESDS, Example 1ctd L M T 3 , 0 2 , 1 C 2 , 1 1 , 1 Now C is strictly dominated by T , so we get: L M T 3 , 0 2 , 1 Now L is strictly dominated by M , so we get: M T 2 , 1 We solved the game by IESDS. A Primer on Strategic Games – p. 13/6

IESDS Theorem If G ′ is an outcome of IESDS starting from a finite G , then s is a Nash equilibrium of G ′ iff it is a Nash equilibrium of G . If G is finite and is solved by IESDS, then the resulting joint strategy is a unique Nash equilibrium of G . (Gilboa, Kalai, Zemel, ’90) Outcome of IESDS is unique (order independence). A Primer on Strategic Games – p. 14/6

IESDS: Example Location game (Hotelling ’29) 2 companies decide simultaneously their location, customers choose the closest vendor. Example: Two bakeries, one (discrete) street. For instance: 8 3 Then baker 1 (3 , 8) = 5 , baker 2 (3 , 8) = 6 . Where do I put my bakery? A Primer on Strategic Games – p. 15/6

Answer 6 Then: baker 1 (6 , 6) = 5 . 5 , baker 2 (6 , 6) = 5 . 5 . (6 , 6) is the outcome of IESDS. Hence (6 , 6) is a unique Nash equilibrium. A Primer on Strategic Games – p. 16/6

IEWDS Theorem If G ′ is an outcome of IEWDS starting from a finite G and s is a Nash equilibrium of G ′ , then s is a Nash equilibrium of G . If G is finite and is solved by IEWDS, then the resulting joint strategy is a Nash equilibrium of G . Outcome of IEWDS does not need to be unique (no order independence). A Primer on Strategic Games – p. 17/6

IEWDS: Example 1 Beauty-contest game (Moulin, ’86) each set of strategies = { 1 , . . ., 100 } , payoff to each player: 1 is split equally between the players whose submitted number is closest to 2 3 of the average. Example submissions: 29 , 32 , 29 ; average: 30 , payoffs: 1 2 , 0 , 1 2 . This game is solved by IEWDS. Hence it has a Nash equilibrium, namely (1 , . . ., 1) . A Primer on Strategic Games – p. 18/6

IEWDS: Example 2 The following game has two Nash equilibria: X Y Z A 2 , 1 0 , 1 1 , 0 B 0 , 1 2 , 1 1 , 0 C 1 , 1 1 , 0 0 , 0 D 1 , 0 0 , 1 0 , 0 D is weakly dominated by A , Z is weakly dominated by X . By eliminating them we get: X Y A 2 , 1 0 , 1 B 0 , 1 2 , 1 C 1 , 1 1 , 0 A Primer on Strategic Games – p. 19/6

Example 2, ctd X Y A 2 , 1 0 , 1 B 0 , 1 2 , 1 C 1 , 1 1 , 0 Next, we get X A 2 , 1 B 0 , 1 C 1 , 1 and finally X A 2 , 1 A Primer on Strategic Games – p. 20/6

IEWDS: Example 3 L R T 1 , 1 1 , 1 B 1 , 1 0 , 0 can be reduced both to L R T 1 , 1 1 , 1 and to L T 1 , 1 B 1 , 1 A Primer on Strategic Games – p. 21/6

Infinite Games Consider the game with S i := N , p i ( s ) := s i . Here every strategy is strictly dominated, in one step we can eliminate all strategies, all � = 0 strategies, one strategy per player. A Primer on Strategic Games – p. 22/6

Infinite Games (2) Conclusions For infinite games IESDS is not order independent, definition of order independence has to be modified. A Primer on Strategic Games – p. 23/6

IENBR: Example 1 X Y A 2 , 1 0 , 0 B 0 , 1 2 , 0 C 1 , 1 1 , 2 No strategy strictly or weakly dominates another one. C is never a best response. Eliminating it we get X Y A 2 , 1 0 , 0 B 0 , 1 2 , 0 from which in two steps we get X A 2 , 1 A Primer on Strategic Games – p. 24/6

IENBR Theorem If G ′ is an outcome of IENBR starting from a finite G , then s is a Nash equilibrium of G ′ iff it is a Nash equilibrium of G . If G is finite and is solved by IENBR, then the resulting joint strategy is a unique Nash equilibrium of G . (Apt, ’05) Outcome of IENBR is unique (order independence). A Primer on Strategic Games – p. 25/6

IENBR: Example 2 Location game on the open real interval (0 , 100) . s i + s 3 − i − s i  if s i < s 3 − i  2    100 − s i + s i − s 3 − i p i ( s i , s 3 − i ) := if s i > s 3 − i 2    if s i = s 3 − i 50  No strategy strictly or weakly dominates another one. Only 50 is a best response to some strategy (namely 50). So this game is solved by IENBR, in one step. A Primer on Strategic Games – p. 26/6

Mixed Extension of a Finite Game Probability distribution over a finite non-empty set A : π : A → [0 , 1] such that � a ∈ A π ( a ) = 1 . Notation: ∆ A . Fix a finite strategic game G := ( S 1 , . . ., S n , p 1 , . . ., p n ) . Mixed strategy of player i in G : m i ∈ ∆ S i . Joint mixed strategy: m = ( m 1 , . . ., m n ) . A Primer on Strategic Games – p. 27/6

Mixed Extension of a Finite Game (2) Mixed extension of G : (∆ S 1 , . . ., ∆ S n , p 1 , . . ., p n ) , where m ( s ) := m 1 ( s 1 ) · . . . · m n ( s n ) and � p i ( m ) := m ( s ) · p i ( s ) . s ∈ S Theorem (Nash ’50) Every mixed extension of a finite strategic game has a Nash equilibrium. A Primer on Strategic Games – p. 28/6

Kakutani’s Fixed Point Theorem Theorem (Kakutani ’41) Suppose A is a compact and convex subset of R n and Φ : A → P ( A ) is such that Φ( x ) is non-empty and convex for all x ∈ A , for all sequences ( x i , y i ) converging to ( x, y ) y i ∈ Φ( x i ) for all i ≥ 0 , implies that y ∈ Φ( x ) . Then x ∗ ∈ A exists such that x ∗ ∈ Φ( x ∗ ) . A Primer on Strategic Games – p. 29/6