I 0 I Note that the expected payout is a weighted average of payouts 1 1 1 1 1 I What is the expected payoff of ( 3 ; 3 ; 3 ) against ( 2 ; 0; 2 )? Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 S -1, 1 1, -1 0, 0 I 0 I Note that the expected payout is a weighted average of payouts Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 S -1, 1 1, -1 0, 0 1 1 1 1 1 I What is the expected payoff of ( 3 ; 3 ; 3 ) against ( 2 ; 0; 2 )? I Note that the expected payout is a weighted average of payouts Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 S -1, 1 1, -1 0, 0 1 1 1 1 1 I What is the expected payoff of ( 3 ; 3 ; 3 ) against ( 2 ; 0; 2 )? I 0 Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 S -1, 1 1, -1 0, 0 1 1 1 1 1 I What is the expected payoff of ( 3 ; 3 ; 3 ) against ( 2 ; 0; 2 )? I 0 I Note that the expected payout is a weighted average of payouts 1 1 1 I u((1; 0; 0); ( 2 ; 0; 2 )) = 2 1 1 I u((0; 1; 0); ( 2 ; 0; 2 )) = 0 1 1 1 I u((0; 0; 1); ( 2 ; 0; 2 )) = − 2 to play rock I By removing pure strategies that lower the average 1 1 I Best strategy against ( 2 ; 0; 2 ) is I A best response to a strategy will consist of pure strategies that have the same (high) expected payout 1 1 I How can we raise the expected payout against ( 2 ; 0; 2 )? Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 S -1, 1 1, -1 0, 0 1 1 1 I u((1; 0; 0); ( 2 ; 0; 2 )) = 2 1 1 I u((0; 1; 0); ( 2 ; 0; 2 )) = 0 1 1 1 I u((0; 0; 1); ( 2 ; 0; 2 )) = − 2 to play rock I A best response to a strategy will consist of pure strategies that have the same (high) expected payout I By removing pure strategies that lower the average 1 1 I Best strategy against ( 2 ; 0; 2 ) is Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 1 1 I How can we raise theS expected -1, 1 1, payout-1 0, 0 against ( 2 ; 0; 2 )? to play rock I A best response to a strategy will consist of pure strategies that have the same (high) expected payout 1 1 1 I u((1; 0; 0); ( 2 ; 0; 2 )) = 2 1 1 I u((0; 1; 0); ( 2 ; 0; 2 )) = 0 1 1 1 I u((0; 0; 1); ( 2 ; 0; 2 )) = − 2 1 1 I Best strategy against ( 2 ; 0; 2 ) is Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 1 1 I How can we raise theS expected -1, 1 1, payout-1 0, 0 against ( 2 ; 0; 2 )? I By removing pure strategies that lower the average to play rock I A best response to a strategy will consist of pure strategies that have the same (high) expected payout 1 1 I u((0; 1; 0); ( 2 ; 0; 2 )) = 0 1 1 1 I u((0; 0; 1); ( 2 ; 0; 2 )) = − 2 1 1 I Best strategy against ( 2 ; 0; 2 ) is Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 1 1 I How can we raise theS expected -1, 1 1, payout-1 0, 0 against ( 2 ; 0; 2 )? I By removing pure strategies that lower the average 1 1 1 I u((1; 0; 0); ( 2 ; 0; 2 )) = 2 to play rock I A best response to a strategy will consist of pure strategies that have the same (high) expected payout 1 1 1 I u((0; 0; 1); ( 2 ; 0; 2 )) = − 2 1 1 I Best strategy against ( 2 ; 0; 2 ) is Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 1 1 I How can we raise theS expected -1, 1 1, payout-1 0, 0 against ( 2 ; 0; 2 )? I By removing pure strategies that lower the average 1 1 1 I u((1; 0; 0); ( 2 ; 0; 2 )) = 2 1 1 I u((0; 1; 0); ( 2 ; 0; 2 )) = 0 to play rock I A best response to a strategy will consist of pure strategies that have the same (high) expected payout 1 1 I Best strategy against ( 2 ; 0; 2 ) is Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 1 1 I How can we raise theS expected -1, 1 1, payout-1 0, 0 against ( 2 ; 0; 2 )? I By removing pure strategies that lower the average 1 1 1 I u((1; 0; 0); ( 2 ; 0; 2 )) = 2 1 1 I u((0; 1; 0); ( 2 ; 0; 2 )) = 0 1 1 1 I u((0; 0; 1); ( 2 ; 0; 2 )) = − 2 I A best response to a strategy will consist of pure strategies that have the same (high) expected payout to play rock Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 1 1 I How can we raise theS expected -1, 1 1, payout-1 0, 0 against ( 2 ; 0; 2 )? I By removing pure strategies that lower the average 1 1 1 I u((1; 0; 0); ( 2 ; 0; 2 )) = 2 1 1 I u((0; 1; 0); ( 2 ; 0; 2 )) = 0 1 1 1 I u((0; 0; 1); ( 2 ; 0; 2 )) = − 2 1 1 I Best strategy against ( 2 ; 0; 2 ) is I A best response to a strategy will consist of pure strategies that have the same (high) expected payout Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 1 1 I How can we raise theS expected -1, 1 1, payout-1 0, 0 against ( 2 ; 0; 2 )? I By removing pure strategies that lower the average 1 1 1 I u((1; 0; 0); ( 2 ; 0; 2 )) = 2 1 1 I u((0; 1; 0); ( 2 ; 0; 2 )) = 0 1 1 1 I u((0; 0; 1); ( 2 ; 0; 2 )) = − 2 1 1 I Best strategy against ( 2 ; 0; 2 ) is to play rock Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 1 1 I How can we raise theS expected -1, 1 1, payout-1 0, 0 against ( 2 ; 0; 2 )? I By removing pure strategies that lower the average 1 1 1 I u((1; 0; 0); ( 2 ; 0; 2 )) = 2 1 1 I u((0; 1; 0); ( 2 ; 0; 2 )) = 0 1 1 1 I u((0; 0; 1); ( 2 ; 0; 2 )) = − 2 1 1 I Best strategy against ( 2 ; 0; 2 ) is to play rock I A best response to a strategy will consist of pure strategies that have the same (high) expected payout I Each player asks \if the other players stuck with their strategies, am I better off mixing the ratio of strategies?" I If pi is a best response to p−i , the payouts of the pure strategies in pi are equal I Note that pure Nash equilibria are still Nash equilibria Nash Equilibrium I Mixed strategies (p1;:::; pn) are a Nash equilibrium if pi is a best response to p−i I If pi is a best response to p−i , the payouts of the pure strategies in pi are equal I Note that pure Nash equilibria are still Nash equilibria Nash Equilibrium I Mixed strategies (p1;:::; pn) are a Nash equilibrium if pi is a best response to p−i I Each player asks \if the other players stuck with their strategies, am I better off mixing the ratio of strategies?" I Note that pure Nash equilibria are still Nash equilibria Nash Equilibrium I Mixed strategies (p1;:::; pn) are a Nash equilibrium if pi is a best response to p−i I Each player asks \if the other players stuck with their strategies, am I better off mixing the ratio of strategies?" I If pi is a best response to p−i , the payouts of the pure strategies in pi are equal Nash Equilibrium I Mixed strategies (p1;:::; pn) are a Nash equilibrium if pi is a best response to p−i I Each player asks \if the other players stuck with their strategies, am I better off mixing the ratio of strategies?" I If pi is a best response to p−i , the payouts of the pure strategies in pi are equal I Note that pure Nash equilibria are still Nash equilibria I 0 1 1 1 I When both players use ( 3 ; 3 ; 3 ) I Test this: what is the payoff of a pure strategy against 1 1 1 ( 3 ; 3 ; 3 )? I What should the (unique) Nash equilibrium be? I Note that the expected payoff for each player is 0 (the game is fair) Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 S -1, 1 1, -1 0, 0 I 0 1 1 1 I When both players use ( 3 ; 3 ; 3 ) I Test this: what is the payoff of a pure strategy against 1 1 1 ( 3 ; 3 ; 3 )? I Note that the expected payoff for each player is 0 (the game is fair) Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 S -1, 1 1, -1 0, 0 I What should the (unique) Nash equilibrium be? I 0 I Test this: what is the payoff of a pure strategy against 1 1 1 ( 3 ; 3 ; 3 )? I Note that the expected payoff for each player is 0 (the game is fair) Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 S -1, 1 1, -1 0, 0 I What should the (unique) Nash equilibrium be? 1 1 1 I When both players use ( 3 ; 3 ; 3 ) I 0 I Note that the expected payoff for each player is 0 (the game is fair) Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 S -1, 1 1, -1 0, 0 I What should the (unique) Nash equilibrium be? 1 1 1 I When both players use ( 3 ; 3 ; 3 ) I Test this: what is the payoff of a pure strategy against 1 1 1 ( 3 ; 3 ; 3 )? I Note that the expected payoff for each player is 0 (the game is fair) Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 S -1, 1 1, -1 0, 0 I What should the (unique) Nash equilibrium be? 1 1 1 I When both players use ( 3 ; 3 ; 3 ) I Test this: what is the payoff of a pure strategy against 1 1 1 ( 3 ; 3 ; 3 )? I 0 Rock Paper Scissors I Consider Rock Paper Scissors: R P S R 0, 0 -1, 1 1, -1 P 1, -1 0, 0 -1, 1 S -1, 1 1, -1 0, 0 I What should the (unique) Nash equilibrium be? 1 1 1 I When both players use ( 3 ; 3 ; 3 ) I Test this: what is the payoff of a pure strategy against 1 1 1 ( 3 ; 3 ; 3 )? I 0 I Note that the expected payoff for each player is 0 (the game is fair) Theorem (Nash) Suppose that: I a game has finitely many players I each player has finitely many pure strategies I we allow for mixed strategies Then the game admits a Nash equilibrium I Can we guarantee a mixed Nash equilibrium? Nash Equilibrium I We saw that there are not always pure Nash equilibrium Theorem (Nash)