Soluts2

# Soluts2 - Solution to Problem Set 2 ECON414 Game Theory Spring 2010 1 Question 1 Let's take one firm say firm A as an example If no firms enter

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Solution to Problem Set 2 ECON414: Game Theory Spring 2010 1 Question 1 Let’s take one ﬁrm, say ﬁrm A, as an example. If no ﬁrms enter, ﬁrm A should produce and earn a (net) proﬁt of 700. So ”no ﬁrms enter” is not a Nash Equilibrium. Then suppose there’s one ﬁrm producing. If ﬁrm A enters, it can make a positive proﬁt instead of zero, so only one ﬁrm producing is not a Nash Equilibrium either. Now consider when there’re two ﬁrms producing. Each incumbent ﬁrm earns a positive proﬁt but the potential entrants will suffer a loss if they enter as net proﬁt is negative. Therefore to achieve a Nash Equilibrium in this game, there should be only 2 ﬁrms producing. As there’re 5 different ﬁrms in this game, we will have ( 5 2 ) = 10 different combinations, hence there will be 10 Nash Equilibria. 2 Question 2 Suppose initially there are 1 m n - 1 thieves (interior solution). In order for this to be a Nash equilibrium, it needs to satisfy BOTH best reply arguments: 1. For the thieves, they do not have incentives to abide the law, i.e. m - 1 m W + 1 m Z 0 (2a) 2. For the residents abiding the law, they do not have incentives to steal, i.e. m m + 1 W + 1 m + 1 Z 0 (2b) Given Z < 0 < W , then m m +1 W > m - 1 m W and 1 m +1 Z > 1 m Z (as Z is negative). Therefore, if equation (2a) holds, (2b) cannot hold because the left hand side of (2b) is strictly greater than (2a) and hence greater than 0. We have a contradiction here so there’re no interior Nash Equilibria for 1 m n - 1 . 1

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Now consider the corners. If m = 0 , if a resident deviates and steals, she will get a payoff of Z (which is negative) because the probability that she will be caught is one. Therefore, no residents have incentives to deviate and it is a Nash Equilibrium. If m = n , it will be a Nash Equilibrium if and only if n - 1 n W + 1 n Z 0 (2c) To conclude, there could be one or two Nash Equilibria at the corners depending on whether (2c) holds or not. 3 Question 3
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## This note was uploaded on 04/30/2010 for the course ECON 412 taught by Professor Gribbin,j during the Spring '08 term at UMBC.

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Soluts2 - Solution to Problem Set 2 ECON414 Game Theory Spring 2010 1 Question 1 Let's take one firm say firm A as an example If no firms enter

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