This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 14.12: Problem Set 3 Solution Ruitian Lang October 27, 2010 1 There are three subgames: the whole game, the game following L , and the game following R . The subgame following L is the following normal form: a b x 1,0 0,1 y 0,1 1,0 The only Nash equilibrium of this game is ( 1 2 x + 1 2 y, 1 2 a + 1 2 b ) , and Player 1 gets payoff 1 2 . The subgame following R has the following normal form: A B X 1,01,1 Y1,1 0,1 This game has three Nash equilibria: ( X,A ), ( Y,B ), and ( 2 3 X + 1 3 Y, 1 3 A + 2 3 B ) , and Player 1’s payoffs are 1 , , and 1 3 , respectively. Finally, Player 1 chooses L when his payoff following R is greater than 1. Therefore, there are three subgameperfect equilibria: 1. Player 1 plays R at the beginning, 1 2 x + 1 2 y following L , and X following R . Player 2 plays 1 2 a + 1 2 b following L and A following R . 2. Player 1 plays L at the beginning, 1 2 x + 1 2 y following L , and Y following R . Player 2 plays 1 2 a + 1 2 b following L and B following R . 3. Player 1 plays L at the beginning, 1 2 x + 1 2 y following L , and 2 3 X + 1 3 Y following R . Player 2 plays 1 2 a + 1 2 b following L and 1 3 A + 2 3 B following R . 2 (a) There are two subgames: the whole game and the Stage 2 Cournot competition. In the Stage 2 Cournot competition, firm i ’s payoff is u i ( q ) = q i 1 n X j =1 q j  F. 1 (We set q j = 0 if j is not in the market.) Therefore, the best response is BR i ( q i ) = 1 ∑ j 6 = i q j 2 , if firm i is in the market, (1) where q i means quantities chosen by other firms: q i = ( q 1 ,...,q i 1 ,q i +1 ,...,q n ). The unique Nash equilib rium is that q i = 1 / ( m + 1) for all i , and each firm in the market gets payoff 1 ( m +1) 2 F . Now consider the entry game. A firm gets payoff ( m + 1) 2 F by entering the market if m firms enter and 0 by staying out. Let n * = max m ≤ n : 1 ( m + 1) 2 > F , which is the maximum number of firms that can earn positive profits. If F > 1 4 , it is not profitable to enter even for a monopoly, so the only equilibrium involves no entry. Assume that F ≤ 1 4 . Pick any set S of n * firms, the following is a subgame perfect equilibrium: firm i enters if and only if i ∈ S , and firms who enter the market choose q = 1 / ( m + 1). The reason is that there will be n * firms in the market and each earns positive profit; a firm not in S does not have incentive to enter since if it did there would be n * + 1 firms and it could not earn positive profit. In addition, there are equilibria in mixed strategies where more than n * firms enter with positive probability. Pick any set S of n firms where n > n * , and assume that every firm i ∈ S enters with probability...
View
Full
Document
This note was uploaded on 10/31/2011 for the course 18 18.445 taught by Professor Liewang during the Spring '11 term at MIT.
 Spring '11
 LieWang

Click to edit the document details