7.5. Welfare Effects of Picking Orders and Targets. The flexible deferred accep- tance algorithm follows a certain picking order of hospitals in each region when there are some doctors remaining to be tentatively matched after hospitals have kept doctors up to their target capacities. One issue is how to decide the picking order. One natural conjec- ture may be that choosing earlier (that is, having an earlier order in the flexible deferred acceptance algorithm) benefits a hospital. As we have mentioned earlier, this would be a problematic property: If choosing earlier benefits a hospital, then how to order hospitals will be a sensitive policy issue to cope with because each hospital would have incentives to be granted an early picking order. Fortunately, the conjecture is not true, as shown in the following example. The example also shows that the different choices of orders result in different stable matchings, thus the choice of an order does matter for the algorithm’s outcome.
3(a - e)/4, is greater than aggregate quantity in the Nash equilib- rium of the Cournot game, 2(a - e)/3, so the market-clearing price is lower in the Stackelberg game.. Thus, i[r]
If the stage game has a unique NE, then for any T , the finitely repeated game has a unique SPNE: the NE of the stage game is played in every stage irrespective of the histor[r]
(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
5. Bayesian Nash Equilibrium (12 points)
There are three different bills, $5, $10, and $20. Two individuals randomly receive one bill each. The (ex ante) probability of an individual receiving each bill is therefore 1/3. Each individual knows only her own bill, and is simultaneously given the option of exchanging her bill for the other individual’s bill. The bills will be exchanged if and only if both individuals wish to do so; otherwise no exchange occurs. That is, each individuals can choose either exchange (E) or not (N), and exchange occurs only when both choose E. We assume that individuals’ objective is to maximize their expected monetary payoff ($).
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.)
4. Incomplete Information (16 points, think carefully)
There are four different bills, $1, $5, $10, and $20. Two individuals randomly receive one bill each. The (ex ante) probability of an individual receiving each bill is therefore 1/4. An individual knows only her own bill, and is simultaneously given the option of exchanging her bill for the other individual’s bill. The bills will be exchanged if and only if both individuals wish to do so; otherwise no exchange occurs. That is, each individuals can choose either exchange (E) or not (N), and exchange occurs only when both choose E. We assume that individuals’ objective is to maximize their expected monetary payoff ($).