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Theory Field Examination Game Theory (209A) Jan 2010
Good luck!!!
Question 1 (duopoly games with imperfect information)
Consider a duopoly game in which the inverse demand function is linear where it is positive
P (Q) =
α − Q if Q ≤ α 0 if Q > α
where Q = q 1 + q 2 and qi is firm i’s output (both firms produce the same good). Firm i’s profit is given by
πi(q 1 , q 2 ) = qiP (Q) − Ci(qi).
where Ci(qi) is the cost of producing qi units. Assume that unit cost is constant Ci(qi) = ciqi for all qi and that each firm has probability μ of having unit cost cL and (1 − μ) of having cost cH > cL. Each firm knows only its unit cost. Each firm only knows its unit cost.
Solve for the Bayesian Nash equilibrium of the Cournot game, and for the sequential equilibrium of the Stackelberg’s game (firm 1 moves at the start of the game).
Question 2 (trembling hand perfection)
Consider the following three-player game (van Damme, 1983) in which player 1 chooses rows, player 2 chooses columns, and player 3 chooses boxes l L R T 1 , 1 , 1 1 , 0 , 1 B 1 , 1 , 1 0 , 0 , 1
r L R T 1 , 1 , 0 0 , 0 , 0 B 0 , 1 , 0 1 , 0 , 0
Show that the game has a Nash equilibrium in which no action is weakly dominated but it is not a trembling hand perfect equilibrium.
Best wishes for 2010!
Psychology and Economics (219A)
Part I a) Sami has two possible career moves (he must choose exactly one): He could take a job as a forklift operator, and earn $ 163. Or he could become a spoonlift operator, and earn $0 – no money. (These wages come from a strange labour-market equilibrium in Sami’s coun- try. And if you don’t know what these two jobs are, they are irrelevant to the problem.) Sami’s utility is given by u = w + p^2 , where w is his income and p is his probabilistic beliefs (after retiring from his job) that he is a good spoonlift operator. (He either knows or doesn’t care how good a forklift operator he is.) Sami (and all other characters in this problem) wishes to maximize the expected value of this utility. Sami has (rational) priors p ∈ [0, 1] that he is good with spoonlift operating. If he takes the job as forklift operator, he never updates those beliefs. If he takes the spoonlift job, he will learn one way or another for sure whether he is good at it. Write down (and circle) clearly Sami’s expected utility for each of the two jobs.
b) For what values of p ∈ [0, 1] will Sami take the job as a spoonlift operator? Give a brief intuition for what you find. (And, as always, you can ‘cheat’ by double checking your math answers by using your intuition!)
c) Hamlet is also choosing between these same two careers. But he has a different utility function, and his labour market is different. He, like Sami, only cares about money and how good a spoonlift operator he is, but his utility is u = w + ϕp, where ϕ might be positive, negative, or zero. (He might hate the idea of being a good spoonlift operator.) Like Sami, Hamlet learns whether he is good at spoonlifting if and only if he takes the spoonlifting job. Hamlet’s market is such that: if he takes the forklifting job, he earns wf. If he takes the job spoonlifting, he earns wsg if he is good at spoonlifting, but wbs < wgs if he is bad at it. As a function of ϕ, wf , wbs, and wgs , for what values of prior beliefs p ∈ [0, 1] will Hamlet take a job as a spoonlift operator? Give a succinct intuition for your answer and how it depends on the various variables. (And if you cannot solve the math, also give a succinct intuition for what you’d expect.)
d) Without doing math (I’m not giving you enough to solve it anyhow!), give a brief intuition for how your answer might change if Hamlet’s performance as a spoonlifter provides only a noisy signal (to be both Hamlet and those paying him...) of whether he good at spoonlifting. Don’t spend much time on this part.
Theory Field Exam
January
Two parties are interested in cooperating with each other. Let the payoff to party i in a given period from choosing level of cooperation (or effort) qi in that period be β(qi, qj ) − c(qi), where qj is the level of effort chosen by the other party (i.e., j ̸= i) in that period, β : R^2 + → R, and c : R+ → R+. Assume all functions are at least twice continuously differentiable in all their arguments. Assume, further, that
- Cost and marginal cost are both increasing in q; that is, c(·) is increasing and strictly convex. Assume c′(0) = 0.
- β(0, 0) = 0.
- The benefit function is symmetric; that is, for any q and q′, β(q, q′) = β(q′, q).
- ∂β(qi, qj )/∂qi > 0.
- The function β is strictly concave.
- There exists a qM < ∞ such that
∂β(q, q) ∂q
− c′(q) < 0
for all q > qM.
(a) Prove that, for any{ x > 0, the welfare-maximizing element from the set (q 1 , q 2 ) q 1 + q 2 = x
is (x/ 2 , x/2).
Consider the following contracting technology (call it the formal system): Each period, the parties can write a contract that fixes their levels of cooperation (i.e., q 1 and q 2 ). If one party breaches by choosing a level less than the agreed upon level, the law permits the victim of that breach to claim expectation damages— the difference between the benefit it would have received had the other party honored the contract and the benefit it actually receives. A difficulty is that the court system is imperfect: Although it can perfectly determine that a breach has occurred, it correctly identifies the breaching party with probability θ ∈ [1/ 2 , 1). Whoever is identified by the court as being the breaching party must pay the other party the expectation damages. The parameter θ is common knowledge between the parties at the time they contract. Assume for questions (b) and (c) that the parties do not take into account any future interactions they may have when deciding on their contract in any given period.
(b) Prove that the first-best level of cooperation is not attainable under this contracting technology.
(c) Prove that welfare is increasing in θ.
Now consider a different contracting technology (call it the informal system). The only contracts that are feasible are relational contracts. Absent a contract, the parties choose the levels of cooperation that constitute a Nash equilibrium. Let δ ∈ (0, 1) be the common per-period discount factor. Assume the game is infinitely repeated.
(d) Prove that there exists a δ∗^ < 1 such that the first-best level of cooperation is attainable every period under a relational contract provided δ ≥ δ∗. Be sure to identify clearly the punishment for breach.
(e) Suppose that β(q 1 , q 2 ) = b(q 1 )+b(q 2 ), where b(·) is such that all previously assumed properties of the function β continue to hold. Prove that for all δ > 0, the parties are able to sustain a level of cooperation via relational contracts greater than the level they would play in the Nash equilibrium of a one-period game.
Now suppose that both the formal system and informal system exist (call this the alternatives system). In any given period, the parties decide whether they are employing the formal or informal system (they cannot employ both).
(f) Prove there exists a δ∗(θ) < 1 such that the first-best level of cooperation is attainable every period under a relational contract provided δ ≥ δ∗. Prove that δ∗(·) is an increasing function. Interpret.
(g) Suppose that β(q 1 , q 2 ) = b(q 1 )+b(q 2 ), where b(·) is such that all previously assumed properties of the function β continue to hold. Prove that, for all δ ∈ (0, 1), welfare is decreasing in θ in some neighborhood of 1/2 (i.e., for θ ∈ [1/ 2 , t), t > 1 /2). Interpret.
Finally, suppose that both the formal system and informal system exist. More- over, the parties can employ both if they wish (i.e., have a formal contract and exploit their ongoing relationship). Call this the joint system.
(h) Prove in this regime that any level of cooperation that could have been sustained in equilibrium under the alternatives system can be sustained in equilibrium in the joint system. Interpret.
