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ECON 159 PSet April 7

Andrew HuGame Theory

April 7, 2014

Problem 1. a. Choosing prices below 1 is weakly dominated for both firms as they couldguarantee themselves a payoff of 0 (vs a negative payoff) by setting the price at 1. There areno strictly dominated strategies as the other firm could always undercut and force a payoff

of 0.b. Both player choosing price 1 is a Nash Equilibrium as neither player has incentive todeviate off. Any price choice lower than 1 cannot be a Equilibrium as that player wouldwant to choose 1. Any price combination where a player is above 1 is not an equilibriumas either that player is making no money and would want to lower price or the other playerwishes to barely undercut to make more money.

c. We take the derivative of this with respect to pLand get 200000(2pL)200000(pL1) =600000 400000pL= 0 = pL= 1.5.

d. In this case, one player makes (2000001000001.5)(1.51) = 0.5500000.5 = 12500using a price 1.5, just below the collusive price.

e. We will assume that if one firm undercuts, he will undercut exactly under 1.5 for a value 1.5. Then, he stands to make 25000. For him to decide to undercut, it must be the casethat 25000> 12500 1

1 = 1 > 1

2 = < 1

2. Thus, this individual cheats as long as

< 12

and that is the lowest such to sustain the monopoly price. The equilibrium strategyhere is to play the monopoly price forever, unless the other firm cheats, in which case to play1 forever.

Problem 2. a. By the method of best responses, the two pure nash equilibria are (C, D)and (D, C). Pursuing a mixed strategy, and assigning row probability p of playing C, tomake col indifferent, we have 5p + 2(1p) = 6p + 1p = 3p + 2 = 5p + 1 = p= 0.5.

Thus, these is a mixed strategy equilibria where each player plays Cwith probability 0.5(since the players are symmetric.

b. Repeating the single period Nash Eq. are Nash Eq in this two period game. However, weshould search for more. The following strategy for both players is a Nash Eq: Play C round1 and in round 2, play the mixed strategy if the other player played C round 1, otherwise,play D. To see that no player has any incentive to deviate, notice that deviation in round2 cannot happen because this is simply a subgame and the mixed strategy from (a) was

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ECON 159: PSet 4.7 Andrew Hu

an equilibrium. Thus, any deviation would come in round 1. Under the above strategy,the expected payoff is 5 + 3.5 = 8.5. Suppose a player played D the first round, then, theexpected payoff is 6 + 2 = 8