Homework

cl. Show -:iirectl:,F [without resorting to a theorem] that the matching produced in parts a and b are stable. e. Show that there are no other stable matchmgs. 1'. Suppose qt is a social choice function that always selects a stable matching, given any {strict} preference prole. If :35 chooses or given the above preference prole i, show that to; can gain by [unilaterally] misrepresenting her preference. g. In part f, suppose e5 chooses , given the above preference prole i. Show that there is an agent who can gain by (unilaterally) misrepresenting hisfher preference. h. Carefull}r formulate a conclusion from the above ndings. cl. Show -:iirectl:,F [without resorting to a theorem] that the matching produced in parts a and b are stable. e. Show that there are no other stable matchmgs. 1'. Suppose qt is a social choice function that always selects a stable matching, given any {strict} preference prole. If :35 chooses or given the above preference prole i, show that to; can gain by [unilaterally] misrepresenting her preference. g. In part f, suppose e5 chooses , given the above preference prole i. Show that there is an agent who can gain by (unilaterally) misrepresenting hisfher preference. h. Carefull}r formulate a conclusion from the above ndings. In the remaining parts of the problem, assume that E initially does not lniov.r the can- didate's type and believes it is H with probability A, where if? c: A c: 1. e. Draw a game tree representing the interaction between E and the job candidate, f. In the game described above, how many pure strategies does the employer E have? Give an example of one. g. In the game described above, how many pure strategies does the job candidate have? Give an example of one. h. Show that there is a weak sequential equilibrium (WEE, also called WPBE} in which neither type candidate gets training. What can be said about the plausibility of E's beliefs in such an equilibrium? i. Show that there is a pure WEE in which both candidate types get training. Compare the efciency of the allocation for type L in this WEE to that in the SPE of part d. Use the comparison to describe the han of E's uncertainty about the candidate's type. 4. A monopoly produces one unit of a perishable good per period at D cost to itself. The only potential buyer of the good1 attaches value v1 = 'v 2:: U to consuming one unit of it in period 1 and the monopoly knows it. In period 1, the monopoly and the consiuner believe that the consumer will attach value to to consuming one unit of the good in period 21 where 1:2 is uniformly distributed over the interval [1: src + s]1 with [l i: s E c. The monopoly offers one unit of the good to the consumer at the price pl 3 U in period 1 and the consumer accepts or rejects the oer. If the consumer rejects the offer? then the unit offered disappears and the monopoly offers the unit of the good produced in period 2 at price pa in period 2. The consumer learns the value of 112 in ['[J' sj 'U-I-S] and accepts or rejects the offer. This interaction can be treated as a game of common knowledge in which the players maximise their expected payoffs. The monopoly gets payoff olpl + (1ng and the consumer gets :11 {c1 p1) + 52ch Iii-'43}? where at equals 1 or U depending on whether the consumer accepts or rejects the offer made in period t. Note that the consumer might buy the good in both periods and the agents do not discount period '2 payoffs. a. Give a complete description of a pure strategyr for the monopoly. b. Under what {if any} conditions on the parameters does the period 2 price offer satisfy Pg 33 c in a subgame perfect equilibrium {SPE} of the game? c. Under what [if any] conditions on the parameters does the monopoly charge a price pg that it believes will be accepted with probability 1 in SPE? d. Find a pure SPE of the game} assuming 1: