Game Theory Basics - Department of Mathematics

2.11. NASH EQUILIBRIUM 41⇒ Exercise 2.5 on page 54 studies weakly dominated strategies, and what can happenwhen these are eliminated. You can answer this exercise except for (d) which youshould do after having understood the concept of Nash equilibrium, treated in thenext section.2.11 Nash equilibriumNot every game is dominance solvable, as the games of chicken and the battle of sexesdemonstrate. The central concept of non-cooperative game theory is that of equilibrium,often called Nash equilibrium after John Nash, who introduced this concept in 1950 forgeneral games in strategic form (the equivalent concept for zero-sum games was consideredearlier). An equilibrium is a strategy profile where each player’s strategy is a “bestresponse” to the remaining strategies.Definition 2.5 Consider a game in strategic form and a strategy profile given by a strategys j for each player j. Then for player i, his strategy s i is a best response to the strategiesof the remaining players if no other strategy gives player i a higher payoff, when thestrategies of the other players are unchanged. An equilibrium of the game, also calledNash equilibrium, is a strategy profile where the strategy of each player is a best responseto the other strategies.In other words, a Nash equilibrium is a strategy profile so that no player can gain bychanging his strategy unilaterally, that is, with the remaining strategies kept fixed.(a)❅❅IT↓BII❅l ← r2 12 00 13 1↓(b)❅❅ITBII❅l r2 12 00 13 1→Figure 2.11Indicating best responses in the quality game with arrows (a), or by puttingbest response payoffs in boxes (b).Figure 2.11(a) shows the quality game of figure 2.10 where the best responses areindicated with arrows: The downward arrow on the left shows that if player II plays herleft strategy l, then player I has best response B; the downward arrow on the right showsthat the best response to r is also B; the leftward arrow at the top shows that player II’sbest response to T is l, and the rightward arrow at the bottom that the best response to Bis r.This works well for 2 × 2 games, and shows that a Nash equilibrium is where thearrows “flow together”. For the quality game, player I’s arrows both point downwards,