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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.
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.
In this section, we will use the idea of expected value to find the equilibrium mixed strategies for repeated two-person zero-sum games.
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. • William’s Oddments: A shorthand algebraic calculation for finding mixed strategies
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.
A mixed strategy Nash equilibrium involves at least one player playing a randomized strategy and no player being able to increase his or her expected payoff by playing an alternate strategy. A Nash equilibrium in which no player randomizes is called a pure strategy Nash equilibrium. Figure 16.14 Mixed strategy in matching pennies.
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.
