(Optional ) Consider the following ultimatum bargaining game, which has been studied in laboratory experiments. The Proposer moves first, and proposes a split of $10 between himself and the Responder. Any whole-dollar split may be proposed. For example, the Proposer may offer to keep the whole $10 for himself, he may propose to keep $9 for himself and give $1 to the Responder, $8 to himself and $2 to the Responder, and so on. (Note that the Proposer therefore has eleven possible choices.) After seeing the split, the Responder can choose to accept the split or reject it. If the Responder accepts, both players get the proposed amounts. If she rejects, both players get $0.
(a) Write out the game tree for this game.
(b) How many complete strategies does each player have?
(c) What is the rollback equilibrium to this game, assuming the players care only about their cash payoffs?
(d) Suppose Rachel the Responder would accept any offer of $3 or more, and reject any offer of $2 or less. Suppose Pete the Proposer knows Rachel’s strategy, and he wants to maximize his cash payoff. What strategy should he use?
(e) Rachel’s true payoff (her “utility”) might not be the same as her cash payoff. What other aspects of the game might she care about? Given your answer, propose a set of payoffs for Rachel that would make her strategy optimal.
(f) In laboratory experiments, players typically do not play the rollback equilibrium. Proposers typically offer an amount between $2 and $5 to the Responder. Responders often reject offers of $3, $2, and especially $1. Explain why you think this might occur.