Question

Suppose we have a seller who owns an object and there are two people who are interested in buying it. Each of these two people, well call them a and b, are known to have values for the object, va and vb that were independently and randomly chosen from the range [0,1]. Only a and b know what the true values are. Assume the seller values the object at 0. the seller decides they want to try the following mechanism. They o§er the object at a ìtake- it-or-leave-itîprice of some amount x 2 [0; 1]. If a or b, but not both, say they want the object at that price, that person gets it and pays x. if both accept the price then they áip a coin to decide who gets it and the winner pays x. if neither accept the áat price then the object goes unsold. a. assume that x = :5 and that va = :7: should a be willing to accept the price of .5? b. suppose that neither of the bidders agrees to buy at the initial x chosen by the seller and the seller decides to make a descending series of ìtake-it-or-leave-itî o§ers. How will this process compare with the auction mechanisms we have studied? should the seller expect that the örst ìtake-it-or-leave-itîo§er they make will have an impact on their revenue? (positive, negative, no di§erence) c. Now assume that the way the seller conducts the auction is through using a second price sealed bid mechanism but with one catch. the seller sets a reserve price, r; such that they refuse to sell below that price. if only one bidder submits a bid above r then that bidder pays r instead of the second highest bid. if no bidder submits a bid above r the object goes unsold. assume that the bidders are not informed of the level of the reserve price prior to submitting their bids but they do know that one has been chosen. does this change the equilibrium bidding strategy from the standard second price sealed bid auction? d. should a reserve price set in this manner be expected to increase the expected revenue over the case of a second price auction in which a reserve price is not used? why or why not. explain

Answer 1

a. If va = 0.7 and the seller offers the price of 0.5, a would be willing to accept the price. b. The seller can make a descending series of 'take-it-or-leave-it' offers if the initial offer is not accepted. c. In a second price sealed bid auction with a reserve price, the equilibrium bidding strategy remains the same. d. Setting a reserve price is expected to increase the expected revenue over a second price auction.

Step-by-step explanation:

a. In this case, if va = 0.7 and the seller offers the price of 0.5, a would be willing to accept the price. Since va > x, a would agree to pay x and get the object.

b. If the initial x chosen by the seller is not accepted by either bidder, the seller can make a descending series of 'take-it-or-leave-it' offers. This process is similar to an auction mechanism called 'Dynamic English Auction.' The seller can decrease the price with each offer until one of the bidders accepts or until the price reaches a point where the bidder's value exceeds the price.

c. In a second price sealed bid auction with a reserve price, the equilibrium bidding strategy for the bidders remains the same as in the standard second price auction. The bidders do not know the exact level of the reserve price, but they know that one has been set. The bidders will still submit their bids based on their own valuations of the object.

d. Setting a reserve price in this manner is expected to increase the expected revenue over a second price auction without a reserve price. With a reserve price, the seller ensures that they will not sell the object below a certain threshold. This can lead to higher bids from the bidders who fear losing the auction to a low-valued bid. As a result, the expected revenue is likely to increase.