Eden and Kevin are supposed to bake an apple pie together. If both contribute, the size of the pie is 14. If only one shows up, the size is 6, while if neither shows up there will be no pie. The cost of showing up and contributing to the 'manufacturing process' is 2 for each. Having just learned the Rubinstein bargaining game, they agree in advance to play it when splitting the pie. They decide that they will play the 2-period version of the alternating offer game and Eden won the toss so he will be the first to propose a division. This is known before they decide whether to show up to bake the pie. As time goes by, the pie gets stale, so they both prefer eating the pie as soon as possible. This is captured by their (common) discount factor 8 € (0,1). For what values of 8 is it a subgame perfect Nash equilibrium that both of them contribute to the pie? Make sure that you specify the interval for which both players have a dominant strategy to contribute, as well as the intervals over which each player will contribute as long as the other player contributes. [Hint: Think of this as a two-stage game: in the first stage they decide simultaneously whether or not to contribute to making the pie, and in stage two they play the 2-period Stahl-Rubinstein bargaining game to split the pie.]

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.5P

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Strategic bargaining - game theory!

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Eden and Kevin are supposed to bake an apple pie together. If both contribute, the size of the pie is 14.
If only one shows up, the size is 6, while if neither shows up there will be no pie. The cost of showing up
and contributing to the 'manufacturing process' is 2 for each. Having just learned the Rubinstein bargaining
game, they agree in advance to play it when splitting the pie. They decide that they will play the 2-period
version of the alternating offer game and Eden won the toss so he will be the first to propose a division.
This is known before they decide whether to show up to bake the pie. As time goes by, the pie gets stale,
so they both prefer eating the pie as soon as possible. This is captured by their (common) discount factor
8 e (0, 1). For what values of d is it a subgame perfect Nash equilibrium that both of them contribute to the
pie? Make sure that you specify the interval for which both players have a dominant strategy to contribute,
as well as the intervals over which each player will contribute as long as the other player contributes. [Hint:
Think of this as a two-stage game: in the first stage they decide simultaneously whether or not to contribute
to making the pie, and in stage two they play the 2-period Stahl-Rubinstein bargaining game to split the
pie.)

Definition Definition Economic decision-making framework that analyzes how different players optimize their outcomes with given pay-offs corresponding to their available strategies. The assumptions of game theory are: The number of players or competitors are finite/limited; All participants are rational The rules of the game are known to every player; and Each participant has a finite set of actions. A common way in which game theory is presented as an ( m x n ) matrix in which m represents the strategies of row player and n represents the strategies of column player. There are several limitations of game theory: Many businesses have a large number of competitors, and selecting an optimal strategy in these circumstances is difficult; Real-life business operating environments are full of uncertainty and difficult to model as a game theory; and The assumption of rationality and utility maximization isn't always an accurate portrayal of human nature.

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