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Lotteries Expected Utility Mixed Strategy Nash Equilibrium. Definitions Example 1: Public Goods Provision Example 2: Defense against Terrorism. 4. 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. One of the significant drawbacks of the graphical solution from the previous sections is that it can only solve \(2 \times 2\) matrix games.
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 ...
Solutions. Problem 1: For each one of the following normal form games find (a)The pure-strategy Nash equilibria and their payoffs; (b)The mixed-strategy Nash equilibria and their payoffs; (c)Are there any equilibria in dominant strategies? If yes, which ones? (a) CD A −50,−50 100,0 B 0,100 0,0 Solution: The best responses of the players are CD
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.
In this section we will learn a method for finding the maximin solution for a repeated game using a graph. Let's continue to consider the game given in Example 3.1.1 by. [ 1 0 − 1 2] In order to make our analysis easier, let's name the row and column strategies as in Table 3.2.1.
