Problem Set #6
- PlSc 349
- Due: Tuesday, 12 October 2010
- [5] Problem U7.7
- [6] Problem U7.11
- [5] Problem U8.7
- [7] Problems S8.9 and U8.8
- [8] Problem 8.11. In (a), instead of calculating the payoffs for Row 2,
show the calculation of payoffs for cells (1,1), (1,3), and (2,3) only.
- [7] [modified version from Osborne 2004] Just before the 2004 election,
Bush is barely leading
Kerry in Florida and Ohio. Bush needs both Florida and Ohio to win. Kerry needs
only one of them to win. (Ignore all other states.)
Bush has three discrete resources that he can use that
are equally strong (say, visits by himself, Laura Bush, and Condoleezza Rice--note:
not Cheney). Kerry has two discrete resources that he can use that are equally
strong (say, visits by himself and Edwards--note: not Theresa Heinz). Bush and
Kerry must allocate these resources for the last day of campaigning, and can
allocate each single resource to only one state. Bush will win a state
if (and only if) he assigns at least as many resources to that state as
Kerry does.
Formulate this situation as a strategic game and find all Nash equilibria.
You may find it convenient
to assign simple payoffs such as (1 if win, -1 if lose), and write out the game as a
matrix game. (Hint: Bush should have four actions, Kerry three.)
- First, argue that in any equilibrium, Bush never concentrates his resources all in one state.
- Then, argue that in any equilibrium, Kerry assigns probability zero
to the action of allocating one resource to each state.
- Then, argue that in any equilibrium, Kerry assigns probability 1/2
to each of his other actions.
- Finally, find Bush's equilibrium strategies.
- In an equilibrium, do Bush and Kerry concentrate all their resources
in one state, or spread them out?
- [12] [from Crawford 2009] Two risk-neutral, expected money-maximizing bargainers, U and V, must agree on
how to share $1. They bargain by making simultaneous demands; if their demands add up
to more than $1, they each get nothing; if they add up to less than or equal to $1, each
bargainer gets exactly his demand. Assume that any real number is a possible demand,
and is also a possible division of the money.
(a) Find an infinite number of mixed-strategy Nash equilibria in this game. Explain why,
in your equilibria, neither bargainer can do better by switching to any other strategy, pure
or mixed.
(b) Show how to compute the equilibrium probability of disagreement, and show that it is
always strictly positive in the mixed-strategy equilibria you identified in part (a).
(c) Are there any Pareto-efficient equilibria in this game?
(d) Now suppose that there are two plausible, but rival, notions of what it means to divide
the dollar fairly. Redo your analysis from part (a), assuming that bargainers can put
positive probability only on demands that are consistent with one or both of these notions
of fairness. Is the equilibrium identified here also an equilibrium in the original game?
(e) Give a fairly detailed real-world (but not experimental) example in which common
ideas of fairness appear to determine bargaining outcomes (and the likelihood of impasse)
as in your answer to (d).
- [for future planning] Which problem was most useful in learning
the concepts? Which problem was least useful?
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