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Unformatted text preview: Section VIII: Minmax Strategies and Infinitely Repeated Games Daniel Egel November 3, 2008 1 Minmax Strategies Minmax strategies are treated formally in your book on pages 141-147. However, this treatment is quite technical and is unlikely to be too useful for this part of the course. The treatment of minmax strategies presented here follows from lecture and draws from the notes of Matt Levy, though my presentation is significantly simplified. 1.1 Minmax Defined Minmax strategies are very useful in finding equilibrium in infinitely repeated games. Here I outline the idea of minmax strategies and talk about our assumptions: • Minmax strategies can be used as a threat in any sort of repeated game • A minmax strategies is a strategy that minimizes the payoff of the other player. 1.2 A simple way to find minmax strategies Consider a very simple game with two players: A and B. To find the payoff for B if he is being “minmax-ed”, perhaps because he is being punished by B, we: 1 1. Find the best response of B to each play by A and underline B’s payoff. This is identical to the approach for finding Nash equilibrium. 2. Identify and circle the lowest underlined payoff for B. This is player B’s minmax payoff. The strategy for player A that corresponds to this payoff is player A’s minmax strategy. A symmetric approach can be used for finding the minmax payoff for player A. 1.3 Some Examples of Minmax 1.3.1 Example 1 Let’s do an example together. Consider the following game: 1 This intuitive algorithm is based on that proposed by Matt Levy. 1 Player 1 Player 2 E F G H A 2,5 10,3 4,5 3, 9 B 3,7 1,4 6, 8 7,7 C 1, 1 0,0 0, 3 4,1 D 4, 6 5,5 8, 8 2, 6 Let’s start by finding the minmax strategy of player 1: 1. What does player 2 choose if player 1 plays A? B? C? D? 2. Which of the above strategies minimizes the payoff for player 2? (that is player 1’s minmax strategy!) 3. What is the player 2’s minmax payoff?...
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This note was uploaded on 12/25/2010 for the course PO 137 taught by Professor Power during the Fall '10 term at Berkeley.
- Fall '10