In the Memo, you will be recommending how to best structure an auction for a painting, and whether increasing advertising will boost the auction houses bottom line based on your bidding calculations.

We have been the premiere auction house in the upper New York region for over 45 years. Lately, we have been concerned about our auction format. While everyone enjoys the traditional fast-paced open outcry format, we worry that we may not be getting the most revenue. For example, over in Holland, they auction their tulips by starting high and gradually lowering their price until someone buys. Right now, we are selling a high end painting that is valued anywhere from $400,000 to $800,000 by professional art dealers. However, the clientele we usually attract with at least 100 bidders will value the painting along the range of $10,000 to $600,000. We do have the opportunity of inviting 3 professional art dealers to the auction, but it would cost us $10,000 in advertising, auction format development, VIP status updates to induce them to come. I have two questions for you. First, which auction format would generate the most revenue for us? Recall that we earn 20% commission on all sales. Second, is it worth spending the extra money to attract the art dealers?

P.S. I have calculated that if the range of values is distributed v~U[L, H], then the expected value of first highest valuation is L+(H-L)(n)/(n+1) and the expected value of second highest bid is L+(HL)(n-1)/(n+1).

Workout: Since all formats have the same expected revenue, we will just focus on the English auction format which results in the second highest valuation as the expected price. Using the formulas given, For 3 art dealers: L+(H-L)(n-1)/(n+1) = 400+(800-400)*(3-1)/(3+1) = 600,000 For regular clientele: L+(H-L)(n-1)/(n+1) = 10+(500-10)*(100-1)/(100+1) = 490,297 The difference in commissions from each group is 120,000-98,059 = 21,941, which is greater than the $10,000 promotion cost, so definitely invest. Note that for 4 art dealers: L+(H-L)(n-1)/(n+1) = 400+(800-400)*(4-1)/(4+1) = 640,000, an increase of $40,000, or $8,000 in commission.