# (Optional ) Consider the following ultimatum bargaining game, which has been studied in laboratory..

(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.

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(Optional ) Consider the following ultimatum bargaining game, which has been studied in laboratory..
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(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.