ECO00055M Design and Analysis of Mechanisms and Institutions

Part I: 1. Mechanism Design [50 marks]

1A Solve the following two problems (a) and (b): [20 marks]

Let N = {1, 2, 3, 4, 5, 6, 7} be a set of patients in need of kidney transplants. Each patient in N has an incompatible living donor.

Each patient only ranks the patients whose donors they are compatible with. For example, 2 is incompatible with all donors except 6’s and 7’s donors and she prefers 6’s donor to 7’s.

(a) Use the Top Trading Cycle mechanism to determine who should receive a kidney transplant from whom. Describe clearly what happens in each step.

(b) Consider the same patients and donors as before. The set of donors each patient can safely receive a kidney transplant from remains unchanged. However, suppose that each patient now has dichotomous preferences over the donors of the other patients as in Roth et al. (2005). Use the information in (a) to construct the compatibility matrix for these patient-donor pairs and draw the corresponding compatibility graph.

1B Answer the following three questions (a), (b), and (c): [30 marks]

A seller wishes to sell two disjoint sets S1 and S2 of items to n potential bidders through auction. The seller valuates every bundle at zero. Each bidder i has a private integer valuation V i (S) on every bundle S, views items in the same set as substitutes but items across the two sets as complements, and faces no budget constraint.

(a) Formulate the notions of Walrasian equilibrium and efficient assignment for the market. Show that if (π, p) is a Walrasian equilibrium, then π must be an efficient assignment.

(b) Describe the double-track dynamic auction that can find a Walrasian equilibrium in the market.

(c) Consider a simple market with three bidders and two chairs {c1, c2} and one table {t}. The valuations of bidders are given as in Table 1. Show how the above auction finds a Walrasian equilibrium by starting with the price of c1 at 2, the price of c2 at 1, and the price of t at 23.

2. Shapley-Scarf’s Swap Market [50 marks]

There are n traders in a market. Each trader holds one indivisible item and has no medium of exchange like money. Every trader has preferences over items and wants to exchange with others but demands at most one item.

(a) Formulate the concept of core allocation and explain it in detail.

(b) Describe the top trading cycle mechanism in detail.

(c) Find a core allocation and a competitive equilibrium of the following example with six traders. The traders called Alex, Ben, Carol, Dan, Eng, and Fen, and their houses are denoted by Ha, Hb, Hc, Hd, He, and Hf respectively.

(d) Prove that the allocation found by the top trading cycle mechanism must be a core allocation.

Part II:

1. Bargaining [50 marks]

Consider the following strategic game with two players, Alice and Bob, who are bargaining with each other on sharing a joint payoff of 100. In the 1st round, Alice makes a proposal which Bob then either accepts or rejects. In case of rejection, the game may break down with probability θBob ∈ [0, 1] so that Alice and Bob will end up with the disagreement payoffs, 2, and 5, respectively, and with probability 1 − θBob the game may continue to the 2nd round where Bob makes a proposal which Alice then either accepts or rejects. In case of rejection, the game may break down with probability θAlice ∈ [0, 1], where θAlice = 3 5 θBob, so that both get the disagreement alternative, i.e., Alice receives 2 while Bob receives 5, and with probability 1 − θAlice the game may continue to the 3rd round where Alice again makes a proposal which Bob then either accepts or rejects. The game proceeds in this fashion. Calculate the subgame perfect Nash equilibrium outcome of this game. What is the associated bargaining problem of this game and what is its (generalised) Nash bargaining solution.

2. Mechanisms [50 marks]

2A

Consider a society of six individuals N = {1, 2, 3, 4, 5, 6} and five alternatives X = {a, b, c, d, e}.

Answer the following three problems: [20 marks]

(a) Find the social preference relation selected by the majority rule and determine which of the following properties it satisfies: transitivity, quasitransitivity, acyclicity.

(b) Given your answer in (a), discuss whether the majority rule could be used by this society to select a Pareto efficient alternative.

(c) Find the social preference relation selected by Borda count and show that Borda count does not satisfy independence of irrelevant alternatives using the preferences of this society.

2B

Consider the paper “Efficiency and Compromise: A Bid-Offer-Counteroffer Mechanism with Two Players” (Ju, 2013). What are the requirements for a strategic mechanism to “adequately” solve the related problem? In what sense is the paper related to the Nash Programme? [30 Marks]