Lecture SlidesDixit and Skeath

Chapter 4

Simultaneous-Move Games with Discrete Strategies

Pure Strategies vs. Mixed Strategies

Game Matrix or Payoff Matrix

Zero Sum Games - Interests of two players are exactly opposite each other. We can simplify the matrix by denoting only one value since other value is same with negative sign.

Fig. 4.2

Nash EquilibriumA list of strategies, one for each player, such that no player can get a better payoff by switching to some other strategy while other players adhere to the strategies specified for them in the list.

Since game is simultaneous, don’t know actual choices, but can instead consider “beliefs” about how others will behave.

Fig. 4.3

Finding Nash Equilibria

•Cell-By-Cell Inspection

•Dominance: Both Players Have Dominant Strategy or Just One Player Has Dominant Strategy

•Successive Elimination of Dominated Strategies

•Best-Response Analysis

•Minimax Method for Zero-Sum Games

Fig. 4.4

Both Have Dominant Strategies

Fig. 4.5

Congress Has Dominant Strategy, Fed Does Not

Need to Be Careful When Eliminating Weakly Dominated Strategies

Fig. 4.8

Fig. 4.9

Fig. 4.10

Multiple Equilibria

Pure Coordination: Doesn’t matter which action they choose, only that they coordinate. Focal Points.

Assurance: One outcome preferred. Still need to have enough certainty that the other is choosing the correct action.

Need to have convergence of expectations about what the other person will choose.

Fig. 4.11

Fig. 4.12

Fig. 4.13

Asymmetric Payoffs

Resolving Conflicting Preferences

Acting “Tough” or “Nice” may not lead to choosing same café.

Coordination could achieved by negotiation.

For example, in a repeated game, could have agreement to alternate locations.

Fig. 4.14

No Equilibrium in Pure Strategies

Games often have this outcome.

In this case, we can definitely say something about what you should not do. But can not say what you should do.

Reasonable approach is to act unsystematically by choosing each strategy part of the time. But don’t choose in same pattern. Need to mix it up: Equilibrium in Mixed Strategies.

Fig. 4.15