Question

Two people use the following procedure to split 1 dollar. Person 1 offers person 2 an amount of money up to 1 dollar. If 2 accepts the offer, the person 1 receives the reminder. If person 2 rejects it, both players receive 0. Suppose person 1’s preferences are represented by the payoff function given by u1(x1, x2) = x1−1/4 |x1−x2|, while person 2’s are represented by u2(x1, x2) = x2−|x1−x2|, where xi is the amount of money that person i receives. Assume that the amount of money person 1 offers can be any number between 0 and 1, not necessarily an integer number of cents.

1. Find all subgame perfect equilibria in pure strategies (hint: remember that x1 + x2 = 1 and the properties of the absolute value).

Answer 1

To find all subgame perfect equilibria in pure strategies, it is necessary to consider the possible offers that Player 1 can make to Player 2. By analyzing Player 1's preferences and Player 2's preferences, we can calculate the possible subgame perfect equilibria.

Step-by-step explanation:

In order to find all subgame perfect equilibria in pure strategies, we need to consider the possible offers that Player 1 can make to Player 2. The total amount of money to be split is 1 dollar, so let's consider the possible values of x1 (the amount of money that Player 1 receives) and calculate the corresponding values for x2 (the amount of money that Player 2 receives).

We know that Player 1's preferences are represented by the payoff function u1(x1, x2) = x1 - 1/4 |x1 - x2|, and Player 2's preferences are represented by u2(x1, x2) = x2 - |x1 - x2|. We can use these functions to calculate the payoffs for different offers.

There are a few key points to consider when analyzing the subgame perfect equilibria:

1. Player 2 will reject any offer that gives them a negative payoff, so x2 must be greater than or equal to 0.
2. Player 1 will want to maximize their own payoff, so they will try to make an offer that minimizes x1 - 1/4 |x1 - x2|.
3. Player 2 will want to maximize their own payoff, so they will try to make an offer that maximizes x2 - |x1 - x2|.

Using these principles, we can calculate the possible subgame perfect equilibria in pure strategies.