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stat155hw4soln

# stat155hw4soln - Stat 155 Fall 2009 Solutions to Homework...

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Stat 155 Fall 2009: Solutions to Homework 3 (was due October 1, 2009) 1. The payoff matrix is shown below, with player I’s choices along the rows, and player II’s choices along the columns. Black Red Black 1 - 2 Red - 7 8 Using equalizing strategies, as discussed in class, we will solve for the optimal p to find the value of the game where p is the probability that Player I chooses the Black Ace, and 1 - p is the probability that Player I chooses the Red 8. Equating Player I’s expected payoffs under the two cases of Player II playing Black or Red, we get: 1 · p - 7(1 - p ) = - 2 p + 8(1 - p ) We solve this to get p = 5 / 6. This gives the value of the game to be - 1 / 3. If Player II’s optimal strategy is ( q, 1 - q ), where q is the probability that Player II chooses Black, then we can use the same method to solve for q , getting q = 5 / 9. Thus, Player I’s optimal strategy: (5 / 6 , 1 / 6), Player II’s optimal strategy: (5 / 9 , 4 / 9), and Value of the game = - 1 / 3. 2. Payoff matrix: Rock Paper Scissors Rock 0 - 1 1 Paper
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