# What is the repeated-game Nash equilibrium considering the duopoly's enforcement mechanisms?

As part of a fractious breakup, Dana and Blair have recently sold their joint possession, an electric keyboard, for $200. They now are arguing about how to split the proceeds of the sale. In order to reach a conclusion, they have enlisted the services of Mordecai the Mediator. Mordecai proposed the following solution: "I want eash of you, seperately and independently, to write on a slip of paper the number of dollars that you would like to keep for yourself (X_D and X_B). If X_D + X_B is less than or equal to $200, then you can each keep the figure you name and I'll take the rest [ie. $200 - X_D + X_B] as my mediation fee. If X_D + X_B > $200, then I will keep the entire $200."

a) Identify the Nash equilibrium (or equilibria) for this one-shot simultaneous-move game. Explain your reasoning.

b) What do you think would be the most likely outcome of this game? Briefly explain.

Firms I and II are duopoly producers of differentiated products. In each period, the profits for each firm depend on the pricing decisions of both, as given in the following table.

Firm II

Price = "Low" Price = "Medium" Price = "High"

Firm I Low 200,200 300,100 500,0

Medium 100,300 300,300 600,0

High 0,500 0,600 400,400

a) Identify the one-shot, simultaneous-move pure-strategy Nash equilibrium or equilibria for this game.

b) Assume that each firm [i=I,II] maximizes the geometrically-discounted present value of the infinite sum of its expected current and future payoffs. Assume also that both firms follow strategies incorporating a "tough trigger" enforcement mechanism, defined by "Charge the 'High' price as long as both firms charged a 'High' price in the previous period; otherwise charge the 'Low' price forever after." For what values will these strategies specify a repeated-game Nash equilibrium?

c) Repeat your results from part c) if the firms instead adopt strategies incorporating a "mild trigger" enforcement mechanis, defined by "Charge the 'High' price as long as both firms charged a 'High' price in the previous period; otherwise charge a 'Medium' price forever after." Is it more likely that Firm I and Firm II will be able to sustain cooperation with this more lenient enforcement strategy? Briefly explain.

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Question 3

Section a

There are infinite Nash equlibria in this game. Let's start with an example. If Dana expects Blair to play Xb = $40, what would be her best strategy? Clearly, she should play Xd = $160. If she chose a higher Xd, then the sum will be higher than $200, and she will get $0. Therefore, given Xb = $40, the best choice for Dana is to play Xd = $160. Furthermore, if Blair expects Dana to play $160, what should he play? Cleary, he should play Xb = $40, for the same reasons. We have concluded then that [160, 40] (I'll always assume that the first value is for Dana and the second one is for Blair) is a Nash equilibrium, in which both players rationally expect the other player to play what they actually play, and respond in a rational way. Another way to see that this is a Nash equilibrium is to note that neither player has an incentive do unilaterally deviate from that point: if Dana changes to 161, she will get $0 (as the sum will be higher than 200); and if she changes to 159, she will get $159, which is less than the $160 she was getting.

Obviously, the same reasoning can be made for any pair of numbers where Xd + Xb = 200 (such as [170, 30], [1, 199], etc). Thus we have infinite Nash equilibria, which occur when Dana and Blair play Xd and Xb such that Xd + Xb = 200 (with Xd and Xb <= 200)

There are more Nash equilibria of a different form. These are when both players choose their X's to be higher than or equal to $200. For example, let's say Blair expects Dana to choose Xd=$250. Now Blair is indifferent among all of his choices, because he will get $0 for certain (and so will Dana), so he can ...

#### Solution Summary

This solution looks at enforcement strategy in game theory in 1205 words.