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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.
In this section, we will use the idea of expected value to find the equilibrium mixed strategies for repeated two-person zero-sum games.
Mixed strategies Notation: Given a set X, we let Δ(X) denote the set of all probability distributions on X. Given a strategy space S i for player i, the mixed strategies for player i are Δ(S i). Idea: a player can randomize over pure strategies.
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.
a mixed strategy is one where a player plays (some of) the available pure strategies with certain probabilities. concept best understood in repeated games, where each player’s aim is to keep the other guessing. Examples: Rock–Scissors–Paper game, penalty kicks, tennis point, bait cars, tax audits, drug testing. etc.
Step 1a. Find the probability ( p p) and payoff ( m m) if Player 1 always plays A. If Player 1 plays pure strategy A, then she never plays B. Thus the probability she plays B is 0. Hence, p= 0. p = 0. In the case where Player 1 plays A and Player 2 plays C, what is the payoff to Player 1? This is m, m, so m =1. m = 1.
Use you graph to determine if there is a mixed strategy equilibrium point. If there is, how often should Player 1 play each strategy? What is the expected payoff to each player?
