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Mixed Strategy Nash Equilibrium. Definitions. Example 1: Public Goods Provision. Example 2: Defense against Terrorism. Choice Under Uncertainty. So far we have been talking about preferences over certain alternatives. Let’s think about preferences over what might be called “risky” alternatives.
Example 3.4.1 : Matching Pennies Game. Consider the game in which each player can choose HEADS (H) or TAILS (T); if the two players match, Player 1 wins; if the two players differ, Player 2 wins. What strategy should each player play?
Mixed strategies. Mixed strategy Nash equilibrium. Existence of Nash equilibrium. Examples. Discussion of Nash equilibrium Notation: Given a set X, we let Δ(X) denote the set of all probability distributions on X. Given a strategy space S for player i, mixed strategies for player i are Δ(S the. i) Idea: a player can randomize over pure strategies.
This game has two pure strategy equilibria: (Swerve, Don’t) and (Don’t, Swerve). In addition, it has a mixed strategy. Suppose that Column swerves with probability p.Then Row gets 0p + –1(1 – p) from swerving, 1p + (–4)(1 – p) from not swerving, and Row will randomize if these are equal, which requires p = ¾. That is, the probability that Column swerves in a mixed strategy ...
Example taken from Game Theory and Strategy (Straffin 1993) p.16. Three Methods for Solving Mixed Strategy Solutions. Method of Equalizing Expectation: A mixed strategy for the column player must result in equivalent payoff in active row strategies. Results in solving simultaneous linear equations.
Solutions. Departament of Economics. Problem 1: For each one of the following normal form games find. The pure-strategy Nash equilibria and their payofs; The mixed-strategy Nash equilibria and their payofs; Are there any equilibria in dominant strategies? If yes, which ones? A. B. Solution: The best responses of the players are.
Finding Mixed-Strategy Nash Equilibria. Let’s look at some examples and use our lesson to nd the mixed-strategy NE. Example 1 Battle of the Sexes a b A 2;1 0;0 B 0;0 1;2 In this game, we know that there are two pure-strategy NE at (A;a) and (B;b). Let’s see if there are any other mixed-strategy NE.
