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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 strategy in strategic-form games of mixed-strategy Nash equilibrium , while Chapter 6 deals with mixed strategies in dynamic 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.
Solution: The best responses of the players are. There are two NE in pure strategies: (A, D) and (B, C). We look for a NE in mixed strategies of the form. σ1 = xA + (1 − x)B σ2 = yC + (1 − y)D. The expected utilities of the players are. u1(A, σ2) = −50y + 100(1 − y) = 100 − 150y u1(B, σ2) = 0 × y + 0 × (1 − y) = 0. and. u2(σ1, C) u2(σ1, D)
We can use the graphical method to find the maximin and minimax mixed strategies for repeated two-person zero-sum games. Using the same game matrix as above: \(\begin{bmatrix}1 & 0 \\-1 & 2 \end{bmatrix}\)
Let's look at some examples and use our lesson to nd the mixed-strategy NE. Example 1 Battle of the Sexes. 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. Suppose that there was another equilibrium in which the row mixed on both A and B.
We can use the graphical method to find the maximin and minimax mixed strategies for repeated two-person zero-sum games. Using the same game matrix as above: \begin{equation*} \left[\begin{matrix}1\amp 0\\ -1\amp 2 \end{matrix} \right], \end{equation*}
