Question

4. Consider the roommate problem where they both buy a coffee maker for $1000. For each of the follow- ing mechanisms, indicate whether it is incentive compatible for Gary, incentive compatible for Frank, budget balanced and/or Pareto efficient.

   1. (a) First, Gary reports how much he is willing to pay (vˆG). Then, after hearing

      Bob’s report, Frank announces how much he is willing to pay (vˆF ). They F

      buy the machine if and only if vˆG+ vˆF≥ 1000. If they but the machine,

      Frank pays 1000 − vˆG, and Gary pays vˆG

   2. (b) Frank and Gary simultaneously report their values. Each pays 500, regardless

      of the reports, and regardless of whether they buy the machine. They buy

      the machine if and only if vˆF+ vˆG≥ 1000.

   3. (c) They buy the machine and Frank pays 1000, if Frank’s value is greater or equal than 1000. Otherwise, they do not buy the machine and nobody pays anything.

      (d) Frank and Gary simultaneously announce their values. They buy the machine if vˆF . vˆG ≥ 250000. If they buy the machine, Gary pays 250, 000/vˆF and

      Frankpays 250,000/vˆG

      (e) Frank and Gary simultaneously announce their values. They buy the machine if (vˆF− 800)2 + (vˆG− 800)2 ≤ 40000. If they buy the machine, each pays

      ti =800−?40000−(y−800)2 [hint: draw a picture of this mechanism]

Answer 1

Frank and Gray simultaneously report their values.Each pays 500 regardless

of the reports,and regardless of whether they buy the machine. They buy

the machine if and only if V^F+V^G_>1000.