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Theory Field Examination Game Theory (209A) Jan. 2008
Both questions carry approximately the same weight. Good luck!!!
Question 1
[1] Prove or give a counterexample: if the equilibrium payoff of a player in a strictly competitive game is Ļ, then any strategy pair that gives the player a payoff of Ļ is an equilibrium. [2] Prove or give a counterexample: every action used with positive prob- ability in some mixed strategy Nash equilibrium is rationalizable. [3] Prove or give a counterexample: in every 2 Ć 2 symmetric game there is a mixed strategy that is an evolutionary stable strategy. [4] Prove or give a counterexample: any mixed strategy Nash equilibrium in which every playerās strategy is undominated is a trembling hand perfect equilibrium. [5] Prove or give a counterexample: if a strategy profile is a trembling hand perfect equilibrium of an extensive form game it is also a trem- bling hand perfect equilibrium in its strategic form.
Question 2
[1] Let Ī£ be the set of all bargaining problems which are strictly convex and compact sets in R^2 + with nonempty intersection with R^2 ++ and nonincreasing and concave frontier. Let f u^ : Ī£ ā R^2 + be the maximizer of s 1 +s 2 (the utilitarian solution). Determine if f u^ satisfies symmetry (SY M ), weak Pareto optimality (W P O), invariance to equivalent utility representations (IN V ), and independence of irrelevant alternatives (IIA). [2] Consider a T period finite horizon alternating offer bargaining game over a pie of size v where there is a cost c < v to make an offer (only the proposer incurs the cost even if no agreement is reached). Assume player iās utility is his share of the pie minus the cost of his proposals. What is the (unique) SP E of the game? What happens as T ā ā.
Theory Field Exam 206
January 2008
Consider a principal-agent game with the following timing and assumptions. In the first stage, the agent learns his ability a ā { 0 , A}. His ability is his private information. It is, however, common knowledge that Pr{a = A} = Ļ. In the second stage, the principal offers the agent a contract on a take-it-or-leave-it basis. If the agent leaves it (rejects the offer), the game ends and the agentās utility is 0. If the agent takes it (accepts the offer), the agent may, depending on the contract, make a verifiable announcement (send a message to the principal). Next, the agent chooses an effort level e ā { 0 , 1 }. The agentās choice of effort is his private information. In the penultimate stage, a verifiable signal x ā { 0 , 1 } is realized. Assume Pr{x = 1|a, e} = (a + q)e. Finally, payoffs are realized. Assume the following with respect to the parameters:
0 < q < 1 ; 0 ā¤ A ā¤ 1 ā q ; and 0 < Ļ < 1.
Assume the following with respect to payoffs: the principal is risk neutral, she prefers paying the agent less to paying him more, and she prefers, , the agent choose effort level 1; the agentās utility is
U (w) ā Ce ,
where C > 0 and U (Ā·) is at least twice differentiable, with U 0 (Ā·) > 0 and U 00 (Ā·) < 0. Assume, too, that there exists a w ā„ āā such that
lim wāw U(w) = āā.
Assume, as well, that lim wāā U(w) = ā.
(a) Assume that A = 0 ( there is only one type). What is the optimal contract from the principalās perspective if she wishes to induce the agent to choose effort level 1?
From now on, assume that A > 0.
(a)
(b) Assume that A = 1 ā q. What is the optimal contract from the principalās perspective if she wishes to induce both types of agent to choose effort level 1? Which, if either type, earns an information rent?
From now on, assume that A < 1 ā q.
(a)
Question from Psychology and Economics (219A)
Callen, who will live forever, has a magic mango, which he can eat exactly once. Beginning
Day 0, he decides each morning whether or not to eat the mango. If he has not yet eaten the
mango, each night he can make plans and form beliefs as to when in the future he might eat
it; but he cannot commit his behavior on future dates. Callen is born and starts planning
on Day 0.
If Callen goes to bed on night t ā 1 believing that the probability that he will eat the
magic mango on Day t is qt then is utility on Day t is:
ut = t + Ī· Ā· t Ā· (1 ā qt) if he eats the mango on Day t, and
ut = āĪ» Ā· Ī· Ā· t Ā· qt if he does not eat the mango on Day t,
where Ī· > 0 and Ī» > 1 are parameters of his preferences. Notice that these are not
stationary preferences; the mango is getting bigger and bigger as time goes by.
Callenās goal is to maximize his total undiscounted utility: Each period t where he has not
yet eaten his magic mango, Callen seeks to maximize Ut^ =
Pā
Ļ =t uĻ^.^ But Callenās behavior
must accord to a personal equilibrium ā given his preferences and given that he cannot
commit, he must on all mornings that he has not yet eaten the mango be willing to follow
through on his continuation strategy.
a) Very briefly (not more than 2 sentences, and not worth a lot of points, and not a
substitute for solving the model) describe the nature of Callenās preferences.
b) When would Callen prefer to eat the mango? That is, ignoring whether a particular
plan is consistent in the sense of being a personal equilibrium, what day would Callen prefer
to eat the mango? If the question isnāt fully coherent, explain how not and why not.
c) As a function of Ī· and Ī», what are the set of personal-equilibrium pure-strategy plans
that Callen can have? That is, for what values of k (if any) will Callen, if he really believes
he will eat the mango on Day k, be willing to follow through on each day with that plan?
d) For values of of Ī· and Ī» where the set of pure-strategy PE is non-empty, which such
PE is Callenās favorite? (This is the āpreferred personal equilibrium.ā Give an intuition for
this result.
e) Using your answer to Part (d), how does Callenās utility for this representation of his
preferences depend on Ī·? How does it depend on Ī»? Give an intuition and interpretation of
your results.
Theory Field Exam 234
Consider a financial market with three types of agents: (i) noise traders; (ii) an insider;
(iii) market makers. The market is open for one period, and one risky financial asset
is traded. Denote the terminal value of the asset by v, a normally distributed random
variable with expected value zero and variance Ļ v^2.
The market operates the following way. The insider, who has zero endowment of the
risky asset, observes v and then places a market order for purchasing x shares. The
insider has constant absolute risk aversion a, so he maximizes
E [ā exp {āaW }]
where W is his terminal wealth.
Risk neutral market makers observe the total order flow x + u, where u is the demand
of noise traders, which is independent of v and normally distributed with mean zero and
variance Ļ u^2. Competition among market makers results in the market price
p = E [v|x + u].
The insider behaves strategically: in deciding his optimal strategy, he takes into account
the effect of his demand on the price p.
(a) Assuming that W is normally distributed, show that the insiderās optimization prob-
lem is equivalent to maximizing
E [W ] ā
a
Ā· var [W ]
where the expectation and variance are conditional on the insiderās information (i.e., v).
(b) Assume that the market price is a linear function of the total order flow
p = Ī» (x + u).
Express W as a function of x, v, u and Ī». Compute the mean and variance of W condi-
tional on v and solve for the optimal choice of x by maximizing the insiderās objective.
Express x as a function of Ī» and exogenous parameters. Why is demand x finite? What
is x in the special case where the insider is risk neutral?
(c) Denote the total order flow by y = x + u. Note that the price is determined as
p = E [v|y] = Ī¼ + Ī»y
where linearity of the conditional expectation follows from joint normality. Viewing
the conditional expectation as a regression, compute the parameter Ī¼ and derive an
equilibrium condition for Ī» as a function of exogenous parameters.
