GAME THEORY: Problems VI

1. Consider the bargaining game under incomplete information.

(a) Prove that the equilibrium bargaining strategies are increasing in the respective reservation prices. (b) Calculate the expected payoﬀ for the seller and the buyer when they play the equilibrium strategies (Fs and Fb are uniform distributions on [0, 1]).

2. Consider the following Cournot model with incomplete information. Let ﬁrm i0 proﬁt be of the form:

ri(q1, q2) = qi ∗ (2 − qi − q2) −qi ∗ θi , | {z } |{z} the price of the good the unit cost

where qi is the quantity chosen by firm i. It is common knowledge that for firm 5 1 θ1 = 1 (firm 1 has only one potential type). Firm 1 believes that θ2 = 4 with 1 3 1 probability 2 and θ2 = 4 with probability 2 . Thus, firm 2 has two potential types: 5 3 ”high-cost type” (θ2 = 4 ) and ”low-cost type” (θ2 = 4 ). Find the pure strategy Nash (Bayesian) equilibrium in this game. Calculate the expected payoffs for players when they play the equilibrium strategies.

The Payoﬀ Matrix for the Prisoners’ Dilemma Game:

Q C Q 3,3 0,5 Q = keep quiet, C = confess. C 5,0 1,1

3. Consider the repeated Prisoner’s Dilemma game with two stages using the full pure strategy set S = {QQ, QC, CQ, CC}. Show that both players confessing in each stage is the unique Nash equilibrium.

4. Consider the iterated Prisoners’ Dilemma with pure strategy set

S1 = S2 = {sQ, sC , sT , sA}.

The strategy sT is a so-called Tit-For-Tat: begin by being quiet and then do whatever the other player did in the previous stage. The strategy sA is a cautious version of Tit-For-Tat: begin by confessing and then do whatever the other player did in the previous stage. What condition does the discount factor have to satisfy in order for (sT , sT ) to be a Nash equilibrium? 5. A game, in which the stage game with the payoff table is given below, is repeated an infinite number of times and payoffs are discounted by a factor δ ∈ (0, 1).

A B A 1,2 3,1 B 0,5 2,3

Assume that the players are limited to selecting pure strategies from the following three options:

sA : Play A in every stage game.

sB : Play B in every stage game.

sC : Begin by playing B and continue to play B until your opponent plays A. Once your opponent played A, play A forever afterwards.

Find the condition on δ such that (sC , sC ) is a Nash equilibrium. 6. Consider the iterated Prisoners’ Dilemma. Show that both players using the strategy 2 Tit-For-Tat is not a subgame perfect Nash equilibrium if the discount factor δ > 3 . 7. Consider an iterated Prisoners’ Dilemma with the following payoﬀ matrix:

Q C Q 4,4 0,5 Q = keep quiet, C = confess. C 5,0 1,1

Let sP be a strategy: confess if only one player confessed in the previous stage (regardless of which player it was); be quiet if either both players kept quiet or both players confessed in the previous stage. Use the one-stage deviation prnciple to ﬁnd a condition for (sP , sP ) to be a subgame perfect Nash equilibrium.