UNIT II: The Basic Theory Zero-sum Games Nonzero-sum Games Nash Equilibrium: Properties and Problems Bargaining Games Review Midterm3/19 3/12

Prudent v. Best-Response Strategies Problem Sets 1 & 2 Graduate Assignment Review

Opera Fight O F O F (2,1) (0,0) (0,0) (1,2) Player 1 Player 2 2, 1 0, 0 0, 0 1, 2 O F O F Battle of the Sexes Review Compare best response and prudent strategies.

Opera Fight O F O F (2,1) (0,0) (0,0) (1,2) Player 1 Player 2 2, 1 0, 0 0, 0 1, 2 O F O F Battle of the Sexes Review NE = {(1, 1); (0, 0); } Find all the NE of the game. NE = {(O,O); (F,F); } Both are correct

O F P1P1 P221P221 Battle of the Sexes Mixed Nash Equilibrium Review O F 2, 1 0, 0 0, 0 1, 2 NE (1,1) NE (0,0) 1 2 NE = {(1, 1); (0, 0); (MNE)}

O F 2, 1 0, 0 0, 0 1, 2 O F NE = {(1, 1); (0, 0); ( 2/3,1/3 )} Prudent: {1/3, 2/3)} Battle of the Sexes Review Let (p,1-p) = prob 1 (O, F ) (q,1-q) = prob 2 (O, F ) Then EP 1 (Olq) = 2q EP 1 (Flq) = 1-1q q* = 1/3 EP 2 (Olp) = 1p EP 2 (Flp) = 2-2p p* = 2/3

q=1 q=0 2, 1 0, 0 0, 0 1, 2 q Battle of the Sexes EP 1 2/3 0 2 p=1 p=0 Review NE = {(1, 1); (0, 0); ( 2/3,1/3 )} EP 1 = 2q +0(1-q) Player 1’s Expected Payoff

q=1 q=0 2, 1 0, 0 0, 0 1, 2 q Battle of the Sexes EP 1 1 2/ p=1 p=0 Review p=1 p=0 NE = {(1, 1); (0, 0); ( 2/3,1/3 )} Player 1’s Expected Payoff

q=1 q=0 2, 1 0, 0 0, 0 1, 2 q Battle of the Sexes EP 1 1 2/ p=1 p=0 Review p=1 NE = {(1, 1); (0, 0); ( 2/3,1/3 )} EP 1 = 0q+1 (1-q) Player 1’s Expected Payoff

q=1 q=0 2, 1 0, 0 0, 0 1, 2 q Battle of the Sexes EP 1 1 2/ p=1 p=0 Review Opera Fight NE = {(1, 1); (0, 0); ( 2/3,1/3 )} Player 1’s Expected Payoff

q=1 q=0 2, 1 0, 0 0, 0 1, 2 q Battle of the Sexes EP 1 1 2/ p=1 p=0 Review p=1 p=0 0<p<1 NE = {(1, 1); (0, 0); ( 2/3,1/3 )} Player 1’s Expected Payoff

q=1 q=0 2, 1 0, 0 0, 0 1, 2 q Battle of the Sexes 2020 p=1 p=0 Review p=1 p=0 p = 2/3 4/3 EP 1 2/3 1/3 If Player 1 uses her (mixed) b-r strategy (p=2/3), her expected payoff varies from 1/3 to 4/3. NE = {(1, 1); (0, 0); ( 2/3,1/3 )}

q=1 q=0 2, 1 0, 0 0, 0 1, 2 q Battle of the Sexes 2020 p=1 p=0 Review p=1 p=0 EP 1 2/3 1/3 2/3 If Player 2 uses his (mixed) b-r strategy (q=1/3), the expected payoff to Player 1 is 2/3, for all p. q = 1/3 NE = {(1, 1); (0, 0); ( 2/3,1/3 )} 4/3

O F 2, 1 0, 0 0, 0 1, 2 O F Battle of the Sexes Review Find the prudent strategy for each player. q* = 2/3 Prudent strategies: 1/3; 2/3

O F 2, 1 0, 0 0, 0 1, 2 O F Battle of the Sexes Review Let (p,1-p) = prob 1 (O, F ) (q,1-q) = prob 2 (O, F ) Then EP 1 (Olp) = 2p EP 1 (Flp) = 1-1p p* = 1/3 EP 2 (Oiq) = 1q EP 2 (Flq) = 2-2q q* = 2/3 Prudent strategies: 1/3; 2/3

O F 2, 1 0, 0 0, 0 1, 2 q Battle of the Sexes 2020 O F Review p=1 p=0 p = 2/3 4/3 If Player 1 uses her prudent strategy (p=1/3), expected payoff is 2/3 no matter what player 2 does NE = {(1, 1); (0, 0); ( 2/3,1/3 )} Prudent: {( 1/3, 2/3 )} EP 1 2/3 1/3 2/3 p = 1/3

O F 2, 1 0, 0 0, 0 1, 2 q Battle of the Sexes 2020 O F Review p=1 p=0 p = 2/3 4/3 NE = {(1, 1); (0, 0); ( 2/3,1/3 )} Prudent: {( 1/3, 2/3 )} EP 1 2/3 1/3 2/3 q = 1/3 2/3 If both players use (mixed) b-r strategies, expected payoff is 2/3 for each. p = 1/3

O F P1P1 P /3 Battle of the Sexes Review O F 2, 1 0, 0 0, 0 1, 2 NE (1,1) NE (0,0) 2/3 1 2 If both players use (mixed) b-r strategies, expected payoff is 2/3 for each. NE = {(1, 1); (0, 0); ( 2/3,1/3 )} Prudent: {( 1/3, 2/3 )}

O F P1P1 P /3 Battle of the Sexes Review O F 2, 1 0, 0 0, 0 1, 2 NE (1,1) NE (0,0) 2/3 1 2 If both players use prudent strategies, expected payoff is 2/3 for each. NE = {(1, 1); (0, 0); ( 2/3,1/3 )} Prudent: {( 1/3, 2/3 )}

O F P1P1 P /3 Battle of the Sexes Review O F 2, 1 0, 0 0, 0 1, 2 NE (1,1) NE (0,0) 2/3 1 2 NE = {(1, 1); (0, 0); ( 2/3,1/3 )} Prudent: {( 1/3, 2/3 )} Is the pair of prudent strategies an equilibrium?

O F 2, 1 0, 0 0, 0 1, 2 q Battle of the Sexes 2020 O F Review p=1 p=0 p = 2/3 4/3 Player 1’s best response to Player 2’s prudent strategy (q=2/3) is Opera (p=1). NE = {(1, 1); (0, 0); ( 2/3,1/3 )} Prudent: {( 1/3, 2/3 )} EP 1 2/3 1/3 2/3 q = 1/3 2/3 Opera p = 1/3 Therefore not an equilibrium!

Review [I]f game theory is to provide a […] solution to a game- theoretic problem then the solution must be a Nash equilibrium in the following sense. Suppose that game theory makes a unique prediction about the strategy each player will choose. In order for this prediction to be correct, it is necessary that each player be willing to choose the strategy predicted by the theory. Thus each player’s predicted strategy must be that player’s best response to the predicted strategies of the other players. Such a prediction could be called strategically stable or self- enforcing, because no single player wants to deviate from his or her predicted strategy (Gibbons: 8).

Review SADDLEPOINT v. NASH EQUILIBRIUM STABILITY:Is it self-enforcing?YES UNIQUENESS: Does it identify an unambiguous course of action? YES NO EFFICIENCY: Is it at least as good as any other outcome for all players? --- (YES) NOT ALWAYS SECURITY: Does it ensure a minimum payoff? YES NO EXISTENCE: Does a solution always exist for the class of games? YESYES

Review 1.Indeterminacy: Nash equilibria are not usually unique. 2. Inefficiency: Even when they are unique, NE are not always efficient. Problems of Nash Equilibrium

Review T 1 T 2 S 1 S 2 5,5 0,1 1,0 3,3 Multiple and Inefficient Nash Equilibria When is it advisable to play a prudent strategy in a nonzero-sum game? What do we need to know/believe about the other player? Problems of Nash Equilibrium

Review T 1 T 2 S 1 S 2 5,5 -99,1 1,-99 3,3 Multiple and Inefficient Nash Equilibria When is it advisable to play a prudent strategy in a nonzero-sum game? What do we need to know/believe about the other player? Problems of Nash Equilibrium

Review Dominant Strategy: A strategy that is best no matter what the opponent(s) choose(s). Prudent Strategy: A prudent strategy maximizes the minimum payoff a player can get from playing different strategies. Such a strategy is simply max s min t P(s,t) for player i. Mixed Strategy: A mixed strategy for player i is a probability distribution over all strategies available to player i. Best Response Str’gy: A strategy, s’, is a best response strategy iff P i (s’,t) > P i (s,t) for all s. Dominated Strategy:A strategy is dominated if it is never a best response strategy.

Review Saddlepoint: A set of prudent strategies (one for each player), s. t. (s*, t*) is a saddlepoint, iff maxmin = minmax. Nash Equilibrium: a set of best response strategies (one for each player), (s’,t’) such that s’ is a best response to t’ and t’ is a b.r. to s’. Subgame: a part (or subset) of an extensive game, starting at a singleton node (not the initial node) and continuing to payoffs. Subgame Perfect Nash Equilibrium (SPNE): a NE achieved by strategies that also constitute NE in each subgame. eliminates NE in which the players threats are not credible.