Hugo and Kohei are bargaining over how to divide £3,960. They are both strategic, but Hugo is more patient than Kohei. While Hugo values £1 tomorrow at £0.75 today; Kohei values £1 at £0.5. They agree to play a three-stage bargaining game. In the first stage, Hugo proposes a division of the £3,960 and Kohei decides whether to accept or reject the proposal. If he accepts, the game ends. If Kohei rejects the proposal, he then proposes a division. If Hugo accepts Kohei’s proposal, the game ends, otherwise Hugo makes a third proposal, which Kohei may choose to accept. If Kohei rejects this last proposal, both get nothing.

a. Represent this game in extensive form, and write down the unique subgame perfect Nash Equilibrium of the game. What are the equilibrium payoffs?

b. In a new version of the game, the first two stages of the game are as in part (a). However, if the game reaches the third stage, it is Kohei who makes the third proposal (that is, Kohei makes offers in period 2 and 3, if the game gets to a third stage) and the game ends after Hugo decides whether to accept or reject this third offer. How does this change the equilibrium payoffs? Which version of the game does Kohei prefer? Explain how the change in the structure of the game has led to Kohei’s preference.

c. In a third version of the game, Hugo and Kohei make alternating offers as in part (a), but, now, there is no limit to the number of stages. That is, they agree to alternate offers potentially infinitely. What are the payoffs in the subgame perfect Nash Equilibrium of this version of the game?

d. Consider instead that Kohei and Hugo make simultaneous choices about whether to play the game in parts (a) or (b). They need to coordinate to obtain positive payoffs. If they both choose to play the same game, they play the game on which they have coordinated, and obtain the payoffs arising in that game. If they make a different decision, they both get zero payoff. Write down the normal form representation of this game and find the Nash Equilibrium/a of this game. For each equilibrium/a, is the outcome Pareto efficient? Explain why or why not.