For each of these different types of auction, economists are concerned with finding the bidding strategies of rational bidders, and to determine the outcome of the auction, which involves determining who gets the item on offer and at what price. Of particular interest to the seller is the expected price offered under each type of auction. Vickrey (1961) argues that the four different auctions are equivalent, because in all cases the good ends up with the individual who values it most highly, and all four auctions offer the same expected price to the seller. I will now examine whether Vickrey’s claim is valid.
There are a number of assumptions we need to state before going on to examine the outcomes of the four different auctions. We assume there is a single item for sale, and the initial owner’s private valuation of this item is zero, implying the owner wants to sell the good rather than retain ownership. The owner is also risk neutral, so is only interested in the expected price from the sale. The bidders are also assumed to be risk neutral. There are N bidders who all have personal valuations of the good, which are independent from each other. (A bidder’s valuation of the good would be unaffected if they knew the valuations of other bidders.) We assume that nature draws these valuations from the same probability distribution. (For simplicity, we imagine this probability distribution is a uniform one ranging from zero to one.) The bidders are also said to be symmetric, seeing as they have the same attitude to risk, and their valuations are drawn from the same distribution. Given that the N valuations are drawn independently from this distribution, the expected value of the gth order statistic is: N+1-g/N+1. Finally, at the time of bidding, any particular bidder is the only one who knows his own valuation. The fact that there are N risk neutral bidders, and that each bidders valuation is drawn from the uniform distribution mentioned earlier, are common knowledge.
I will now go on to analyse the outcomes of each of the four different auctions. I will begin with the English auction. Assuming infinitesimally small increments are possible, bidder i has a dominant strategy: raise the standing bid if and only if i’s valuation is more than the bid made by bidder k. To understand why this is the case, we need to realise that as the auction progresses, bidders drop out when the standing bid reaches their valuations. Thus, when the bidder with the highest valuation submits a bid equal to the second highest valuation, the bidding process stops. Therefore, the highest bidder receives the good and pays a price equal to the second-highest valuation, which we denote Pe = v2. (where Pe denotes the equilibrium price for the English auction, and v2 is the second highest valuation.) The expected price to the seller is N-1/N+1. (Obtained from the equation for the gth order statistic.)
The English auction bears a strong resemblance to the case of Second Price Sealed Bidding. The dominant bidding strategy of a bidder in this case is to bid his true valuation, whatever the strategies of other bidders might be. To see why this is the case, suppose bidder i’s valuation is more than the highest other bid, which we will denote b. Bidder i would get the item at a price of b and receive a surplus of vi – b. Bidding more than their own valuation would make no difference as the price paid for the good would still be b. By bidding less than their own valuation, i would risk getting the item at a price above their own valuation. Thus bidding the true valuation dominates bidding less than or more than the true valuation. The dominant strategy equilibrium, which is necessarily a Nash equilibrium, involves each bidder bidding his valuation. The item will go to the highest bidder at a price equal to the second highest valuation, a result similar to the one found in English auctions. The expected price to the seller is the same: N-1/N+1.
I will now move on to analyse Dutch and First Price Sealed Bidding auctions, which I will analyse together because there is a strong equivalence between them. Remember that in FPSB auctions, bidders submit one bid before a designated deadline. The item goes to the highest bidder at a price equal to that bid. In a Dutch auction, nothing which takes place during the auction is informative, so a bidder could instruct someone as to which price they would “stop the clock.” We can interpret this as someone’s bid, and if all bidders informed the auctioneer of these bids, the auctioneer could determine the outcome without starting the clock. This would amount to FPSB.
The equivalence between the two auctions is highlighted as we examine the bidder’s actions in either case. The bidder does not have a dominant strategy. Given the bidder’s valuation of the item, his knowledge of the number of other bidders and of the common probability distribution, and his beliefs about the bidding strategy of others, a risk neutral bidder will wish to maximise expected surplus. The bidder faces a trade off: the lower the bid, the greater the surplus on winning, but the lower the probability of winning. There are a number of factors which would affect the bidders optimal bid. One of these is the individual’s own valuation of the good. The higher this is, the higher the optimal bid. Also, the greater the number of rival bidders, the higher the optimal bid.
To determine the outcome of the auctions, theorists have focused on Nash equilibria. Each bidder is employing their optimal bidding strategy given their beliefs about the bidding strategy of others, where their beliefs are confirmed in equilibrium. With N bidders, a symmetric Nash equilibrium for our uniform distribution involves: bi(vi)=N-1/N(vi), for i=1…N. since each bidders bid is increasing in their own valuation, the item goes to bidder 1 at a price N-1/N(vi). The expected price in these auctions is: N-1 / N x E(vi) = (N-1/N) x (N/N+1) = N-1/N+1. As we can see, all four auctions result in the same expected price to the seller. In general, the actual prices will not be the same for all auctions, since the equilibrium price for the Dutch and FPSB auctions does not equal the equilibrium price for the Dutch an SPSB auctions. However, the fact that the expected prices are the same is nonetheless an important result. We have thus confirmed the revenue equivalence theorem. Under the stipulated assumptions, the four primary types of auction are “equivalent” in that they would all yield the same expected price to the seller, with the item going to bidder who values it most highly. This theorem explains why it would not matter to a seller which type of auction is used.
