• Abdulkadiroğlu, A., Pathak, P. A., Roth, A. E., and Sönmez, T. 
“Changing the Boston School Choice Mechanism: Strategy-proofness as Equal Access,” mimeo.
• Abdulkadiroğlu, A. and Sönmez, T.  “Matching Markets: Theory
5. Bayesian Nash Equilibrium (12 points)
There are three different bills, $5, $10, and $20. Two individuals randomly receive one bill each. The (ex ante) probability of an individual receiving each bill is therefore 1/3. Each individual knows only her own bill, and is simultaneously given the option of exchanging her bill for the other individual’s bill. The bills will be exchanged if and only if both individuals wish to do so; otherwise no exchange occurs. That is, each individuals can choose either exchange (E) or not (N), and exchange occurs only when both choose E. We assume that individuals’ objective is to maximize their expected monetary payoff ($).
payoff) while M gives 1 irrespective of player 1’s strategy.
Therefore, M is eliminated by mixing L and R .
After eliminating M , we can further eliminate D (step 2) and L
(step 3), eventually picks up ( U , R ) as a unique outcome.
(a) If an agent is risk averse, her risk premium is ALWAYS positive.
(b) When every player has a (strictly) dominant strategy, the strategy profile that consists of each player’s dominant strategy MUST be a Nash equilibrium. (c) If there are two Nash equilibria in pure-strategy, they can ALWAYS be Pareto
e z . The prices of the three goods are given by (p, q, 1) and the consumer’s wealth is given by ω.
(a) Formulate the utility maximization problem of this consumer.
(b) Note that this consumer’s preference can be expressed in the form of U (x, y, z) = V (x, y) + z. Derive V (x, y).
(a) The intersection of any pair of open sets is an open set.
(b) The union of any (possibly infinite) collection of open sets is open.
(c) The intersection of any (possibly infinite) collection of closed sets is closed. (You can use (b) and De Morgan’s Law without proofs.)