29. Game Applications - Solutions

29. Game Applications - Solutions - Chapter 29 NAME Game...

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Chapter 29 NAME Game Applications Introduction. As we have seen, some games do not have a “Nash equi- librium in pure strategies.” But if we allow for the possibility of “Nash equilibrium in mixed strategies,” virtually every game of the sort we are interested in will have a Nash equilibrium. The key to solving for such equilibria is to observe that if a player is indiFerent between two strategies, then he is also willing to choose ran- domly between them. This observation will generally give us an equation that determines the equilibrium. Example: In the game of baseball, a pitcher throws a ball towards a batter who tries to hit it. In our simpli±ed version of the game, the pitcher can pitch high or pitch low, and the batter can swing high or swing low. The ball moves so fast that the batter has to commit to swinging high or swinging low before the ball is released. Let us suppose that if the pitcher throws high and batter swings low, or the pitcher throws low and the batter swings high, the batter misses the ball, so the pitcher wins. If the pitcher throws high and the batter swings high, the batter always connects. If the pitcher throws low and the batter swings low, the batter will connect only half the time. This story leads us to the following payoF matrix, where if the batter hits the ball he gets a payoF of 1 and the pitcher gets 0, and if the batter misses, the pitcher gets a payoF of 1 and the batter gets 0. Simplifed Baseball Pitcher Batter Swing Low Swing High Pitch High 1 , 0 0 , 1 Pitch Low . 5 ,. 5 1 , 0 This game has no Nash equilibrium in pure strategies. There is no combination of actions taken with certainty such that each is making the best response to the other’s action. The batter always wants to swing the same place the pitcher throws, and the pitcher always wants to throw to the opposite place. What we can ±nd is a pair of equilibrium mixed strategies.
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358 GAME APPLICATIONS (Ch. 29) In a mixed strategy equilibrium each player’s strategy is chosen at random. The batter will be willing to choose a random strategy only if the expected payoF to swinging high is the same as the expected payoF to swinging low. The payoFs from swinging high or swinging low depend on what the pitcher does. Let π P be the probability that the pitcher throws high and 1 π P be the probability that he throws low. The batter realizes that if he swings high, he will get a payoF of 0 if the pitcher throws low and 1 if the pitcher throws high. The expected payoF to the batter is therefore π P . If the pitcher throws low, then the only way the batter can score is if pitcher pitches low, which happens with probability 1 π P .Ev enthen the batter only connects half the time. So the expected payoF to the batter from swinging low is . 5(1 π P
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29. Game Applications - Solutions - Chapter 29 NAME Game...

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