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Unformatted text preview: Microeconomic Theory II Assignment 7: Even more on Game Theory and some Externalities and Public Goods Solutions * April 4, 2007 Problem 1. Consider the following extensive form game of complete information (analogous to the partner- ship game studied by Admati and Perry which we discussed in class.) There are two players, A and B. Time is discrete, and the players alternate making moves toward the finish line of a race. (Specifically, let us say that player A moves in periods 1,3,5,... and player B moves in periods 2,4,6...) After each move, a period of time passes before the other players turn. The players have common per period discount factor (0 , 1). At the beginning of the game, player i stands at distance x i from the finish line. (We are NOT assuming x A = x B . The whole point of this exercise is to find out how the play of the game depends on the initial positions.) The first player to reach the finish line wins the race, receiving at that time a prize worth V > 0. The other player receives nothing. A player, when taking his turn, decides how far to advance toward the finish line in that period, and incurs a cost in that period which is an increasing, convex function of the distance advanced. Thus, let y it be the progress toward the finish line of player i in period t . (So, y At = 0, t = 2 , 4 , 6 ... and y At = 0, t = 1 , 3 , 5 ... in light of the alternating move assumption.) If player i stands at distance x i from the finish line, and if that player advances the distance y i on his turn, then he incurs the cost C ( y i ) in the period of his move, and stands at distance x i- y i at the start of his next turn, where C > 0 and C > 0.) The sugame perfect Nash equlibrium of this game can be characterized by an elegant argument which I want you to discover for yourselves. Here are some things that are true about equi- librium: Generically (i.e., for almost all values of parameters) at most one of the players ever advances toward the finish line. The other simply stays put. Also, assuming the prize is worth winning at all, only the first move of the winning player is affected by the presence of his rival. * Prepared by Omer Ozak (email@example.com), Department of Economics, Brown University. If you find any typos or mistakes please let me know, so that it can be fixed. 1 Microeconomic Theory II Solutions From the second move onward, the winning player advances toward the finish line in same manner as he would if there were no competiton. [Here is a hint about how to approach this problem: Use a backward induction logic similar to that used to solve the partnership game. Consider the critical distance X from the finish line, with the property that if a player were within distance X , and if it were his turn, then the player could get a positive net return by winning the race on that turn [that is, C ( X ) = V .] Clearly if one player initially stands inside of this distance and the other beyond it (that is if...
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