Final Exam Answers Economics 502 Prof. Steven Williams May 9, 2011 1:30-4:30

1. (25 points) Consider in this problem a marriage market with 3 men and 4 women. The strict preferences of the men and women are as follows: P (m1): w1, w2, w3, w4 P (w1): m2, m3, m1

P (m2): w2, w1, w4, w3 P (w2): m2, m3

P (m3): w1, w2, w3 P (w3): m3, m2

P (w4): m1, m3, m2

(a) (10 points) Apply the deferred acceptance (or Gale-Shapley) algo- rithm to determine the M-optimal stable matching: Men make the proposals: stage 1: P (m1): w2, w3, w4 w1 w2 w3 w4 P (m2): w1, w4, w3 m3 m2 P (m3): w2, w3 stage 2: P (m1): w3, w4 w1 w2 w3 w4 P (m2): w1, w4, w3 m3 m2 P (m3): w2, w3 stage 3: P (m1): w4 w1 w2 w3 w4 P (m2): w1, w4, w3 m3 m2 P (m3): w2, w3 stage 4: P(m1): w1 w2 w3 w4 P(m2): w1, w4, w3 m3 m2 m1 P(m3): w2, w3 (b) (10 points) Apply the deferred acceptance (or Gale-Shapley) algo- rithm to determine the W-optimal stable matching: Women make the proposals: P (m1): w1, w2, w3, w4 P(w1): m2, m3, m1

P (m2): w2, w1, w4, w3 P(w2): m2, m3

P (m3): w1, w2, w3 P(w3): m3, m2

P(w4): m1, m3, m2 stage 1:

1 P (w1): m3, m1

w4 w2 w3 P (w2): m3 m1 m2 m3 P (w3): m2

P (w4): m3, m2 stage 2: P (w1): m1

w4 w2 w1 P (w2): m3 m1 m2 m3 P (w3): m2

P (w4): m3, m2 stage 3: P (w1): m1

w4 w2 w1 P (w2): m3 m1 m2 m3 P (w3):

P (w4): m3, m2 (c) (5 points) Explain using your answer to a)-b) how you know that there is at least one woman who never gets married in any possible stable matching. w3 is not matched in the W-optimal matching, which means that being single is her favorite achievable match. If she were married to man m′ in some stable matching, then she would strictly prefer being single to man m′. This contradicts stability. Alternatively, some of you noticed that the M-optimal and W-optimal stable matchings are the same. These are the best and the worst stable matchings for each side of the market. Consequently, this is the only stable matching in this problem, and woman w3 is unmarried.

2. (25 points) There are two agents, each of whom owns an item. The two items are identical. Each agent i privately knows his value vi and regards the value of the other agent as uniformly distributed on [0, 1]. The utility of each agent is quasilinear in the return from acquiring or giving up an item and money: agent i’s utility is

vi δ + xi

where xi is any monetary transfer he receives, δ = 1 if he acquires the item of the other agent, δ = −1 if he sells his item to the other agent, and δ = 0 if neither agent acquires the item of the other agent. Gains from trade exist whenever vi = vj: the item of the agent with the lower value can be transferred to the agent with the higher value at a price such that each agent makes a positive proﬁt. It is eﬃcient that the item of the

2 lower-value agent be transferred to the agent with the higher value. If trade is eﬃcient, then the ex ante expected gain from trade equals

1 1 1 v1 1 (v1 − v2) dv2dv1 + (v2 − v1) dv2dv1 = . 0 0 0 3 v1 I’ve calculated this for you and you need not derive it for yourself.

(a) (10 points) Deﬁne the basic VCG mechanism in this problem.

Given the reports v1∗, v2∗, agent 1 gives his item to agent 2 if v2∗ > v1∗ and agent 2 gives his item to agent 1 if v1∗ > v2∗. No item is transferred if v1∗ = v2∗. The transfer of agent i is as follows:

−v∗ i if vi∗ > v∗ i − − xi (v1∗, v2∗) = 0 if vi∗ = v∗ i .  − v∗ i if vi∗ < v∗ i  − − (b) (5 points) What is the ex ante budget deficit incurred by the basic VCG mechanism? The answer is clear from the gains from trade: the deficit equals 1/3. (c) (10 points) Does there exist an efficient, Bayesian incentive compatible, ex ante budget-balanced and interim individually rational mechanism in this problem? Explain your answer.

If such a mechanism exists, then one exists in the family of VCG mecha- nisms with constant individualized taxes. We therefore restrict our search to this family. Let’s assume that each agent i pays a constant ki regardless of the reports of the two agents. Ex ante budget balance requires 1 k1 + k2 = 3 We next determine the interim expected utility of the "worst-oﬀ" type of each agent in the basic VCG mechanism. This will bound above the individualized taxes. Agent 1’s interim expected utility is

v1 1 U1(v1) = (v1 − v2) dv2 + (v2 − v1) dv2 0 v1 2 v1 2 1 v2 v2 = v1v2 − + − v1v2 2 0 2 v1 2 2 v1 1 v1 2 = + − v 1 − + v1 2 2 2 1 2 = − v1 + v1 2

3 We have dU1 = −1 + 2v1, dv1 which changes from negative to positive at v1 = 1/2. We therefore have 1 1 min U1 (v1) = U1 = . v1 2 4 It is therefore possible to choose k1, k2 as large as 1/4 while still satisfying interim individual rationality. Choose any k1, k2 ≤ 1/4 such that k1+k2 = 1/3. The VCG mechanism with these individualized taxes demonstrates the existence of an incentive compatible, eﬃcient, ex ante budget balanced and interim individually rational revelation mechanism. 3. (25 points) Consider the following two player game of incomplete infor- mation: 1/2 L R 1 1 T θ1, θ2 2 , 3 1 B 0, − 3 1,−θ2

Each player i privately observes his own type θi and regards the type θ i of his opponent as uniformly distributed on [0, 1]. − (a) (5 points) Player 2 has a dominant strategy. State it. 1 R if θ2 < 3 σ2 (θ2) = 1 . L if θ2 ≥ 3 (b) (10 points) Determine a Bayesian-Nash equilibrium of this game. We’ll construct an equilibrium in which player 2 uses his dominant strategy. Player 1 receives the following expected payoffs from each of his pure strategies: 2 1 1 2 1 T : θ1 + = θ1 + 3 3 2 3 6 1 1 B : 1 = 3 3 The expected payoff from T exceeds the expected payoff from B if and only if 2 1 1 θ1 + ≥ 3 6 3 2 1 θ1 ≥ 3 6 1 θ1 ≥ 4 Player 1’s best response strategy is therefore 1 B if θ1 < 4 σ1 (θ1) = 1 . T if θ1 ≥ 4

4 (c) (10 points) Apply the revelation principle to construct a revelation game in which honest revelation is a Bayesian-Nash equilibrium that achieves the same outcome as the equilibrium in b. The following square depicts the payoﬀs for each pair (θ1, θ2) of types and each pair of reported types (θ1∗, θ2∗):

Grading: I deducted points if you failed to distinguish between a player’s report and his true type. I also deducted points if you did not discuss a mechanism in which players report their types.

4. A monopolist can produce a quantity q of his output at a total cost of θq, where θ is a parameter that he privately observes. The inverse demand function for the output is p(q) = 1 − q. His proﬁt function is therefore

π (q, θ) = p(q)q − θq = q (1 − q − θ) .

A regulator can select the price 1 − q(θ∗) and output q(θ∗) based upon a reported value θ∗. The selection of the function q() is a regulatory policy. The regulator regards θ as distributed according to the cumulative distribution F on [0, 1], whose density is denoted as f.

(a) (5 points) Given a regulatory policy q(θ∗), what is the monopolist’s proﬁt as a function of his true cost parameter θ and his reported cost parameter θ∗?

π (θ, θ∗) = q (θ∗) (1 − q (θ∗) − θ)

(b) (10 points) A regulatory policy q() is incentive compatible if honest reporting of his cost parameter is a dominant strategy for the

5 monopolist. Assuming that q() is differentiable, apply the enve- lope theorem to derive a formula for the monopolist’s profit π (θ) as a function of his cost parameter and his profit when θ = 1 in an incentive compatible regulatory policy q (θ).

dπ (θ) ∂π = (q, θ, θ∗ = θ) = −q (θ) dθ ∂θ 1 π (θ) = π (1) + q (θ) dθ. θ (c) (10 points) Use your answer to b. to determine an equation that an incentive compatible regulatory policy q(θ) must satisfy. Do not try to solve this equation for q(θ).

1 q (θ) (1 − q (θ) − θ) = π (θ) = π (1) + q (θ) dθ θ