Practice Exercises 2
Spring 2021
微观经济学代写 Players’ repeated game payoffffs are given by the average discounted sum of payoffffs, where δ ∈ (0, 1) is the common discount factor.
1.Consider the following strategic form game: 微观经济学代写
(a) Find all Nash equilibria of the static game.
(b) Suppose that the above described stage-game is played twice (i.e., repeated once), so that before playing the second stage the players observe each other’s action choices for the fifirst stage. A player’s payoffff is the sum of the stage-game payoffffs (with no discounting). Find all subgame perfect Nash equilibria.
(c) Suppose that the players discount their stage-two payoffffs relative to stage-one payoffffs with discount factor δ < 1. For which values of δ does an equilibrium exist where (C, c) is played?
2.Consider a two-stage game with observed actions, where in the fifirst stage players choose simultaneously U1 or D1 (player 1) and L1 or R1 (player 2), and in the second stage players choose simultaneously U2 or D2 (player 1) and L2 or R2 (player 2). The payoffffs of the stage games are shown in the matrices below:
The players maximize the sum of their stage-game payoffffs, without discounting. Find the subgame perfect equilibria of this game.
3.Consider the infifinitely repeated game with the following (Prisoners’ Dilemma) stage game.
Players’ repeated game payoffffs are given by the average discounted sum of payoffffs, where δ ∈ (0, 1) is the common discount factor.
(a) Draw the set of feasible, strictly individually rational payoffffs.
(b) What is the most effiffifficient subgame perfect equilibrium (SPE) payoffff vector in which player 1 obtains a (repeated game) payoffff of 1.5? (In other words, what is the highest payoffff player 2 can achieve in a subgame perfect equilibrium when player 1 obtains a payoffff of 1.5?) 微观经济学代写
(c) For a given ε > 0, describe a grim-trigger strategy profifile that generates payoffffs that are within ε of the payoffffs you found in the previous part.
(d) For what values of δ do the strategies you described in the previous part form a subgame perfect equilibrium?
4.This question is designed to demonstrate that there is no corresponding “one-shot deviation principle”for Nash equilibrium. That is, you might conjecture that if a strategy profifile σ has no profifitable one-stage deviations from on-path histories, it must be a Nash Equilibrium of the repeated game. This conjecture is not correct.
Consider the infifinitely repeated prisoners’ dilemma, with payoffffs given as follows:
The following strategy profifile is called tit-for-tat:
- each player i chooses C at the initial history, h 0 = ∅;
- at every subsequent history h t , t > 0, player i chooses si(h t ) =
.
.That is, player i chooses the action that the other player, j, chose in the previous period.
(a) Show that for δ high enough, there are no profifitable one-stage deviations from histories that are on the induced outcome path (i.e., those that are reached if everyone has always followed the strategy above). 微观经济学代写
(b) Now suppose player 2 follows the tit-for-tat strategy above. Show that, for δ low enough, there is a profifitable multi-stage deviation from a history that is on the induced outcome path, in which player 1 deviates to play D in the current period and at every history in future periods. That is, for such values of δ, tit-for-tat is not a Nash equilibrium of the infifinitely repeated prisoners’ dilemma.
(c) Compare the conditions on δ in parts (a) and (b) to conclude that there are values of δ for which tit-for-tat is not a Nash Equilibrium, even though there are no profifitable one-stage deviations from the induced outcome path.
5.Consider the following 3-stage game with perfectly observed actions. In the fifirst two stages, the players play a Prisoners’ Dilemma game with the following payoffffs: 微观经济学代写
Then, in the third stage, they play a ‘Battle of the Sexes’ game with payoffffs:
Suppose players discount future payoffffs by δ ∈ (0, 1].
(a) Find all of the Nash Equilibria (NE) for the ‘Battle of the Sexes’ stage game played in the third stage.
(b) Suppose δ = 1. Show that there does not exist a subgame perfect equilibrium (SPE) in which the players choose (C, C) in each of the fifirst two stages.
(c) Nevertheless, show that there is an SPE in which the equilibrium payoffffs are at least 16 for both players.
(d) Now suppose δ < 1. For what values of δ does the SPE you identifified in part (c) survive?
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