Rational Bidding under Dutch and First-Price Auctions

Rational Bidding under Dutch and First-Price Auctions

Some auction mechanisms do not present the bidder with a clear dominant strategy. One such auction is a first-price sealed-bid auction. In this case, the bidder with the highest bid wins the auction and must pay the amount he bid. Further, because it is a sealed-bid auction, bidders cannot observe the bids of other players before determining their own bid. If bidders are not allowed to observe the bids of others before placing their bids, there is not one single bid that clearly yields a higher payout no matter what all other bidders choose to do.

Bidders who bid their value are guaranteed a zero return in all cases. If their bid is not the highest, then they neither pay nor receive any payoff. If their bid is the high bid, they would pay exactly their valuation and receive the value from the object of the auction exactly offsetting the amount paid. The only way to receive a positive return from this auction is to bid something below the valuation of the object. Suppose bidder i places a bid of xi = vi − k, where k is a positive constant. In this case, if some other bidder bids xj = vi − k2 and wins the auction, player i receives a payoff of zero, neither receiving the object of the auction or paying any money. Player i could clearly have done better. By bidding something above xj and something below vi, player i could have won the auction and made a positive gain upon paying that bid. Thus, players face a tradeoff in placing bids. By increasing the bid up to the level of their valuation, players increase their probability of winning the auction and obtaining some positive value. However, as players increase their bid, they also reduce the eventual payoff for having won by increasing their eventual price paid. To discuss the optimal strategy for bidding, we need to use the tools of elementary probability and some discussion of decision under uncertainty.

Consider a random variable z that can take on real number values. We call the set of values the variable can take on the support of z. Thus, if z were the roll of a single sixsided game die, the support is given by {1, 2, 3, 4, 5, 6}. Let us suppose that the random variable displays predictable properties that allow us to discuss how likely some outcomes are relative to others. For example, when a fair, evenly weighted die is rolled hundreds of times, each number is rolled in approximately 1/6 of the rolls. The probability of an outcome is the proportion of times that outcome would occur if the random variable could be drawn a large number of times (approaching infinity). Thus, probability is a number between 0 and 1 representing the likelihood of a particular outcome. Further, the probability of a draw of the random variable being contained in the support of that random variable (obtained by adding the probability of each individual outcome in the support) must be 1. Thus the probability of rolling some integer between and including 1 to 6 is 1. The probability of any outcome outside of the support must be zero—the probability of rolling a 7 is 0.

The expectation, or mean, of a random variable can be defined as Ez =

where pi is the probability of outcome zi, and E is the expectation operator. The expectation is simply a sum of all outcomes weighted by their relative frequency in a large sample, thus representing the average or mean outcome of a draw from the distribution. When an individual decision will result in a randomly distributed payoff, we

bids multiplied by the value obtained if all other bids are lower. Maximizing the expected payout involves increasing xi until the marginal expected payout becomes zero. Bidding less than this means that bidders could increase the expected benefit of their bid, and bidding more than this would mean that they could receive a higher benefit by lowering their bid.

Finding the Nash equilibrium for such a game is difficult. As an example, let us suppose that valuations are drawn from a uniform distribution over all integers 0 to 100. The uniform distribution assigns the same probability to every individual outcome. Thus the probability that any particular bidder has a valuation of 10 is equal to 0.01, and the

probability that any particular bidder has a valuation of 100 is also equal to 0.01. Let us first examine the possibility that everyone bids their own valuation. If player i believes

expected payout is 0. If instead the bidder bids slightly more than his value but not enough to surpass the next highest valuation, the probability of winning remains the same. So long as the probability of winning remains positive, the expected payout is negative if his bid exceeds his valuation. Thus, given that others’ bid their value, it is not in the interest of any bidders to bid above their value. Alternatively, if the a player bids slightly below his value, then vi − x > 0, and so long as the probability of winning remains positive, the expected payout is positive. Thus, if all others bid their value, an individual bidder has an incentive to bid below his value, making him better off. Thus it cannot be a Nash equilibrium for everyone to bid their value because any individual player could be better off trying a different strategy given that others bid their value. So the first-price sealed-bid auction should result in a different outcome than either the English or Vickrey auctions.

Suppose instead that all other players bid a fraction α of their valuation. Thus, if my valuation were 50 and α = 0.5, I would bid 25. Then, if player i bids αvi, the probability

their value, and the next highest value is vi + 1, increasing their bid by less than α will not increase the probability of winning. Thus, bidding less than α above αvi decreases the expected payoff by decreasing the payoff in the event of a win, and the probability of winning remains the same. Increasing the bid by α results in a payoff of

immediately decreases the probability by 0.01vin − 1 − vi − 1n − 1 ≥ 0.01 because

the bid will potentially fall below others with value vi who are bidding αvi. But once bidders have lowered their bid, they do not further reduce the probability of winning until the bid has been lowered by α, and they continue to increase the payoff for winning. The Nash equilibrium occurs where α is such that bidding an amount either α less or α more would result in a lower expected value, or

and equation 5.6 yields

With any number of players, and for any value vi, these two conditions will always be met where α = n − 1n. In a more-general first-price auction, if values are drawn from a uniform distribution over all real numbers between 0, v, the Nash equilibrium bid is

Thus, the appropriate strategy depends on the number of players in the game, with the bid increasing toward the valuation as n gets large. The interested reader can find the derivation of this more general case in the Advanced Concept box at the end of this chapter.

Another common form of auction is the Dutch auction. In this auction, the auctioneer begins by calling out a very high price. The auctioneer then lowers the price by small increments until a bidder indicates he or she is willing to pay the price. This form of auction is common in the sale of large batches of flowers, as well as corporate and public bonds. Because the auction stops when the first bidder indicates his willingness to buy, no bidder has any information on others’ valuations or intended bids before the auction concludes. Thus, despite the difference in mechanism, the Dutch auction acts exactly like a first-price sealed-bid auction. The highest bidder wins and pays his own bid price, but this bid must be determined before any observed actions by others.

EXAMPLE 5.4 Dutch Auctions

James C. Cox, Bruce Roberson, and Vernon L. Smith conducted early experimental comparisons of Dutch and first-price sealed-bid auctions. The rational economic model of bidding suggests that both auction mechanisms should yield the same bids and that these bids should be somewhat below the individual valuation of the object. They found that the Dutch auction bids are regularly below the bids of those in identical first-price auctions. The bidder assigned the highest valuation wins the auction approximately 88 percent of the time in the first-price auction, compared to only 80 percent in the Dutch auction. Thus, lower-valuation bidders have a greater tendency to outbid the top-value bidder in the Dutch auction. Further, they found that both the first-price and Dutch auction bids are above the bids in the second-price auctions. The anchoring and adjustment model might suggest that bidders in a Dutch auction should overbid, as appears to be the case, by anchoring on the opening amount called and failing to adjust down sufficiently. The Dutch auction starts with the highest possible price, providing a high anchor, which should lead to higher overall bids. The first-price sealed-bid auction provides no clear anchor but the induced value of the bidder: The amount the bidder will receive should he win the auction. Hence, the anchoring and adjustment model suggests lower overall bids for the first-price than the Dutch auction, a prediction that is not borne out in laboratory experiments. In fact, both appear to be higher than the bids predicted by theory by a substantial amount.