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Chapter 13: Problem 9

You play the following bargaining game. Player \(A\) moves first and makes Player \(B\) an offer for the division of \(\$ 100 .\) (For example, Player \(A\) could suggest that she take \(\$ 60\) and Player \(B\) take \(\$ 40 .\) ) Player \(B\) can accept or reject the offer. If he rejects it, the amount of money available drops to \(\$ 90,\) and he then makes an offer for the division of this amount. If Player \(A\) rejects this offer, the amount of money drops to \(\$ 80\) and Player \(A\) makes an offer for its division. If Player \(B\) rejects this offer, the amount of money drops to 0 Both players are rational, fully informed, and want to maximize their payoffs. Which player will do best in this game?

Short Answer

Expert verified
Player A will do best in this game because he gets to make the first offer and the structure of the game allows him to ensure that he gets at least as much as he would in the second stage.

Step by step solution

01

Interpret Player B's Optimal Strategy

We start from the back (or the smallest bargain) to find the optimal strategy. Under the assumption that Player A will reject any offer where Player B gets more than half of the money, Player B should therefore offer Player A the smallest amount that A will accept. Assuming A will not accept less than $40 at the $80 stage, Player B should offer $40 to A and keep $50 for himself.
02

Determine Player A's Optimal Strategy at the $90 Stage

At the $90 stage, Player A will want to obtain at least as much as they would get at the $80 stage. Knowing that Player B will choose optimally in the next round and offer them $40, Player A should make an offer that Player B will accept but no more. Therefore, Player A should offer Player B $50 and keep $40 for himself.
03

Determine Player A's Optimal Strategy at the $100 Stage

At the $100 stage, Player A will want to obtain at least as much as they would get at the $90 stage. Therefore, Player A should offer Player B $60 and keep $40 for himself.

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