If player 2 rejects player 1s initial proposal player 2 can

Question

Econ 521 - Week 9:

1. Bargaining with two-period deadline - Suppose that the players alternate proposals, one per "period", and that each player i regards the outcome in which she receives all of the pie after t periods of delay as equivalent to the outcome in which she receives the fraction δ_i^t of the pie immediately, where 0 < δ_i < 1 for i = 1, 2. That is, suppose that each player i "discounts" the future using the constant discount factor δ_i. Consider the game in which 2 periods are possible: If player 2 rejects player 1s initial proposal, player 2 can make a counterproposal which, if rejected by player 1, ends the game with payoffs 0 for each player. Find SPNE of the game using backward induction. What if δ₂ increases? What if δ₁ increases? Explain the intuition of this.

2. Two period bargaining with constant cost of delay - Find the Subgame Perfect Nash Equilibrium of the variant of the game above in which player i's payoff when she accepts the proposal in period 2 is y_i - c_i, where 0 < c_i < 1 (rather than δ_iy_i), and her payoff to any terminal history that ends in rejection is c_i (rather than 0), for i = 1, 2. (Payoffs may be negative but a proposal must still be a pair of nonnegative numbers.)

3. Three period bargaining with constant cost of delay - Find the Subgame Perfect Nash Equilibrium of the variant of the game in the exercise above in which the game may last for three periods, and the cost to each player i of each period of delay is c_i. (Treat the cases c₁ ≥ c₂ and c₁ < c₂ seperately.)