Lecture notes 5 - Economics 112 Lecture Notes Lecture 5....

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Economics 112 Lecture Notes Lecture 5. N-person Games So far, most of the games we have studied have been greatly simplified by restricting the story to two players. But most strategic situations involve more than two players, sometimes many more. This can add great complexity to game theory models. For example, Nash equilibrium requires every player to choose a best response to the choices of all the other players. You can imagine that as the number of players gets large it rapidly becomes impossible to consider each possible combination of strategies and test whether any player can benefit from switching strategies. Happily, in many interesting cases other simplifications besides limiting the number of players will also permit straightforward analysis. In general we speak of "N player" games , where N stands for any integer, 2, 3,4,. ..to some potentially very large (but finite) number. Key terms: N-person game; state variable; representative agent; proportional game 1. The traffic game Twenty commuters can either take the train or drive their cars. All they care about is the length of time it takes to get to work. The train takes 40 minutes no matter how many passengers are riding, but the time to drive depends on how many commuters choose to drive. Assume that payoffs are equal to the negative of the commuting time. Thus the payoff for the train is -40. The payoff for driving is given in the following table: Cars Payoff Cars Payoff 1 -22.5 11 -47.5 2 -25 12 -50 3 -27.5 13 -52.5 4 -30 14 -55 5 -32.5 15 -57.5 6 -35 16 -60 7 -37.5 17 -62.5 8 -40 18 -65 9 -42.5 19 -67.5 10 -45 20 -70 A simple way to find an equilibrium is to test some allocations between driving and riding the train to see if any player would wish to switch behavior. Imagine that two players drove. Is this a Nash Equilibrium? No. Although neither driver would wish to change to riding the train, each of the train riders would prefer to be the third car, with a payoff of -27.5 than stay as a train rider getting -40.
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It is a Nash Equilibrium for eight players to drive. Neither drivers nor train riders can unilaterally improve their payoff by switching to the other mode of travel. Inefficiency
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This note was uploaded on 01/28/2011 for the course ECON 112 taught by Professor Notsure during the Winter '08 term at University of Victoria.

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Lecture notes 5 - Economics 112 Lecture Notes Lecture 5....

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