ECO 358/463 Game Theory/Topics in Game Theory Problem Set 4 Spring 2022
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ECO 358/463
Game Theory/Topics in Game Theory
Spring 2022
Problem Set 4
1. Brothers (Tadelis, Chapter 7, Problem 7.8, p. 149) [Note: No need to find mixed-strategy Nash equilibria for part (d).]
2. Consider the following “centipede game.”The game starts with player 1 choosing be- tween terminate (T) and continue (C). If player 1 chooses C, the game proceeds with player 2 choosing between terminate (t) and continue (c). The two players choose be- tween terminate and continue in turn if the other player chooses continue until the terminal nodes with (player 1’s payoff, player 2’s payoff) are reached as shown below.
Player 1 Player 2 Player 1 Player 2
C c C c
(3, 3)
T t T t
(1, 1) (0, 3) (2, 2) (1, 4)
(a) List all possible strategies of each player.
(b) Transform the game tree into a normal-form matrix representation.
(c) Find all pure-strategy Nash equilibria.
(d) Find the unique pure-strategy subgame-perfect equilibrium.
3. Consider the following “ultimatum game.”There is a pie with value normalized to one to be allocated between two players. Player 1 starts the game by choosing x ∈ [0, 1], which specifies an offer where player 1 receives x and player 2 receives 1 − x.
After seeing the offer, player 2 chooses whether to accept or reject. If player 2 accepts, the game ends with payoff u1 = x for player 1 and u2 = 1 − x for player 2; if player 2 rejects, both players receive u1 = u2 = 0.
(a) Find the unique pure-strategy subgame-perfect equilibrium. (b) Find a pure-strategy Nash equilibrium that is not subgame-perfect.
4. Playing it Safe (Tadelis, Chapter 8, Problem 8.10, p. 172) [Note: Find both pure- and mixed-strategy equilibria for part (d).]
2022-05-06