Dynamic Game Theory and Bargaining Problem 7. Consider an ultimatum game between
ID: 1135733 • Letter: D
Question
Dynamic Game Theory and Bargaining Problem 7. Consider an ultimatum game between Firm 1 and Firm 2. In a typical ultimatum game, Firm 1 makes a take-it-or-leave-it offer to Firm 2, which then decides whether to accept or reject the offer. This problem asks you to consider the following situation. Before Firm 1 makes its offer to split the surplus of a joint venture, Firm 2 decides on an investment level that will affect the size of the surplus. If Firm 2 chooses a low level of investment (L), then the surplus is small and equal to SL. If Firm 2 chooses a high level of investment (H), then the size of the surplus is large and equal to SH. The cost to Firm 2 of choosing L is cL, while the cost of choosing H is cH. To keep up with the story, assume that SH > SL > 0, cH > cL > 0, and further Sh - CH > SL - ci. Note that, before receiving an offer, Firm 2 has only two optionsit cannot decide not to invest at all. Note also that Firm 1 observes the level of investment before making an offer. 2 points] [3 points] (c) Write down the subgame perfect equilibrium that supports this outcome. Is the SPNE 3 points] [2 points] (a) Write down the extensive form representation (game tree) of this game. (b) Find the unique equilibrium outcome using backward induction. unique? (d) Can you find any other NE of this game?Explanation / Answer
1 Sequential Bargaining
A classic economic question is how people will bargain over a pie of a certain
size. One approach, associated with Nash (1950), is to specify a set of
axioms that a “reasonable” or “fair” division should satisfy, and identify the
division with these properties. For example, if two identical agents need to
divide a pie of size one, one might argue that a reasonable division would
be (1/2, 1/2).
The other approach, associated with Nash’s 1951 paper, is to write down
an explicit game involving offers and counter-offers and look for the equi-
librium. Even a few minutes’ reflection suggests this will be difficult. Bar-
gaining can involve bluffing and posturing, and there is no certainty that
an agreement will be reached. The Nash demand game demonstrates that
a sensible bargaining protocol might have many equilibria. A remarkable
paper by Rubinstein (1982), however, showed that there was a fairly rea-
sonable dynamic specification of bargaining that yielded a unique subgame
perfect equilibrium. It is this model of sequential bargaining that we now
consider.
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