Ch 4: Nash Equilibrium

Static Solution Concepts

-Nash equilibrium and refinements -evolutionary stable strategy (ch 16) -supply and demand

Best Reply

-a best reply for a player i is a strategy that maximizes player i's payoff, given that the other n-1 players use strategies

Nash Equilibrium

-a strategy profile is a Nash equilibrium if each player's strategy maximizes his or her payoff, given the strategies used by the other players -players are rational -beliefs are accurate -ensure that each player is doing the best they can individually but does not ensure that the group of players are doing the best they can collectively

Pareto Dominant

-at least one person can be made better off without hurting anyone else

Classifications

-games of pure conflict: zero-sum games -mutual interest games: includes pure coordination games -some are neither or both: Chicken, BOS

Symmetric Strategy Profile

-if one player's strategy is a best reply, then all players' strategies are best replies -to solve symmetric games need a commitment device

Constant-Sum Game

-payoffs always sum to the same number -when that number happens to be zero, its called a zero-sum game

Nash Strategy Profile

-player i makes best response, given his beliefs of what players j are doing -player i has correct beliefs about players j -must specify all choices of each player -if a player has a dominant strategy, a Nash equilibrium requires that the player use it -if all players have a dominant strategy, then there is a unique Nash equilibrium in which each player uses their dominant strategy

Fictitious Play

-player i: predicts j's future probability of play based on frequencies of actions in j's past play -calculate the expected payoff of each action and use the one with the largest expected value -not thinking about the other.person thinking about them

Nash Equilibrium Assumptions

-players are rational -beliefs are accurate -believing others are rational is implied

Coordination Games

-players have a common interest in coordinating their actions -driving game -IDSDS does not work, but Nash has predictions

Symmetric Game

-players have the same strategy sets, and if you switch players' strategies, then their payoffs switch -players who choose the same strategy will get the same payoff

Mutual Interest Games

-rankings of strategy pair by their payoffs coincides for the players

Dynamic Solution Concepts

-specifying how players change their minds

Pareto-Effiiciency

-take money away from someone to make economy more efficient

Nash Equilibrium & IDSDS

-there are at least as many IDSDS solution as there are Nash equilibrium -Nash is more restrictive -we can apply IDSDS first to simplify game and not lose any Nash equilibriums -this is not the case for IDWDS

Fictitious Player

-unsophisticated players -don't know others' payoffs -playing a large number of times -still an abstraction -less concerned with the play-by-play, were interested in where it ends up (solution concepts)

Strategy Plays a Dual Role

1. action player i will take 2. beliefs of other players j -thus we must fully specify all actions even those not "on the equilibrium path" especially in sequential games