12. winner’s curse36 Ralph wants to purchase a family heirloom from a neighbor. The heirloom has private value to the neighbor denoted v. Neighbor knows v; Ralph only knows v is uniformly distributed between v = 0 and v = 100. Neighbor knows this about Ralph. Finally, whatever v is, the value of the heirloom to Ralph is 1.5v. Neighbor also knows this about Ralph.

(a) Who should own the heirloom, Ralph or neighbor?

(b) Suppose the trade encounter between the two individuals proceeds as follows. Ralph offers to purchase the heirloom at price P. If neighbor agrees, Ralph pays P in exchange for the heirloom.

If neighbor does not agree, the game ends, and neighbor keeps the heirloom. Now, from Ralph’s perspective, what is the expected value (to Ralph) of the heirloom, before any conversation with neighbor? Conversely, suppose Ralph offers price P > 0 and neighbor accepts the offer; what now is the expected value (to Ralph) of the heirloom? (As an aside, if Ralph pays P for the heirloom, at what price will Ralph’s accountant value the heirloom?)

(c) What is the equilibrium in this game? Why does no trade take place, despite the fact Ralph is known to value the object higher than the neighbor?