IBM Ponder This #119 · March 2008

S, has a good, G, which he is considering selling to B. The values of G to S and B are independent uniformly distributed random variables between 0,1. S and B know this and know their own valuations but not the valuation of the other party.

1. Suppose B makes a single offer which S accepts or rejects depending on whether or not the offer exceeds the value S places on the item. What should B offer to maximize his expected gain (the difference when a sale occurs between B's valuation of G and the sales price, 0 when no sale occurs) as a function of B's valuation of G? What is B's expected gain? What is S's?

2. Suppose there are 2 buyers B1, B2 (each with a valuation independently uniformly distributed between 0 and 1). Suppose each makes a single bid for G and S accepts the larger if and only if it exceeds S's valuation. Find a bidding strategy for B1 and B2 which is optimal in that if one adopts it the other can do no better than to adopt it also. What will be the expected gains of B1, B2 and S?

Clarifications: Give the expected gains as an overall average not as a function of the buyer's or seller's valuation. The seller's gain is the excess of the sales price over the seller's valuation (or 0 if no sale occurs). In part 2 each buyer bids without knowing the other buyer's bid or valuation.

Solution

Best opened after a real attempt

To be added.