Question

Consider two players who bargain over a surplus initially equal to a whole number amount V, using alternating offers. That is, Player 1 makes an offer in round 1; if Player 2 rejects this offer, she makes an offer in round 2; if Player 1 rejects this offer, she makes an offer in round 3; and so on. Suppose that the available surplus decays by a constant value of c = 1 each round. For example, if the players reach agreement in round 2, they divide a surplus of V − 1; if they reach agreement in round 5, they divide a surplus of V − 4. Let’s start with a simple version. What is the rollback equilibrium when V = 4? In which period will the players reach agreement? What payoff x will Player 1 receive, and what payoff y will Player 2 receive?

Answer 1

In the given scenario, when V=4, the players will reach an agreement in round 3. Player 1 will receive a payoff of 3, and Player 2 will receive a payoff of 0.

Step-by-step explanation:

In the given scenario, two players are bargaining over a surplus initially equal to a whole number amount V. The surplus decreases by a constant value of c = 1 each round. We need to determine the rollback equilibrium when V = 4, the period at which the players reach an agreement, and the payoffs for Player 1 and Player 2.

Since the surplus decreases by 1 each round, in round 1 the surplus will be V-1, in round 2 it will be V-2, and so on. In this case, when V = 4, the surplus in round 1 will be 4-1 = 3, in round 2 it will be 4-2 = 2, and in round 3 it will be 4-3 = 1. Therefore, the players will reach an agreement in round 3.

In round 1, Player 1 will make an offer. To maximize their payoff, Player 1 would make a small offer close to the surplus value. Let's assume Player 1 offers 2. If Player 2 rejects this offer, she will make an offer in round 2. To maximize her payoff, Player 2 would make an offer close to the remaining surplus value (2). Let's assume Player 2 offers 1.

Since Player 2 accepts the offer made by Player 1 in round 2, the agreement is reached in round 2. Player 1 receives the entire surplus (V-1) = 4-1 = 3, and Player 2 receives 0 since she accepted Player 1's offer.