La expansión del comercio

Suppose n bidders are engaged in a first-price auction for an item of common value v that is unknown to each individual bidder. Further, suppose that each bidder i obtains a different estimate of v, which we will label vi. For example, the general manager of a major league baseball team might use prior performance to guess the value of a baseball player approaching free agency. The true valuation is not known, but let us suppose that each estimated valuation is drawn at random from a distribution that can be described by a probability function p v , where v must be in the set v − k 2, , v + k 2 , k is some positive even integer, and the average value of the estimates is E v = v. Then each bidder must formulate a bid to solve

max x E π , 5 10 where πi= v − p if xi= p 0 if xi< p, 5 11

subject to the strategy of all other bidders.

The problem here is that bidders need to know how to use their imperfect signal of the underlying value in compiling the optimal bid. Given that one has received the signalvi, this value could be as much as k 2 above the true valuation or as much as k 2 below the true value. The probability thatviis k 2 above v is p v + k 2 , and the probability that vi is k 2 − 1 greater than v is p v + k 2 − 1 , and so on. Thus, the bidder receiving the signalvi, might view the probability function as representing their beliefs regarding the placement of the true value of winning the auction. If the probability thatviis k 2 above v is p v + k 2 , then to the bidder, the probability that v = vi− k 2 is equal to p v + k 2 . Thus, the signal produces a probability distribution over possible valuations.

This distribution also tells the bidder about the possible estimates received by others in the auction. For example, if bidder i receives estimate vi, then the probability that bidder j receives valuation vj= vi− k is equal to p v + k 2 p v − k 2 . To see this, note that given the probability function specified, the only way v_jcould fall k below viis if vi= v + k 2 and vj= v − k 2. If we assume that each estimate is an independent draw from the distribution, then the probability of drawingvi= v + k 2 and vj= v − k 2 is just the product of the probabilities of each individual draw. On the other hand, the proba- bility that vj= vi− k + 1 is p v + k 2 p v − k 2 + 1 + p v + k 2 − 1 p v − k 2 . In this case, there are two possible ways to obtain a difference of k − 1 between the

estimates. Either vi is the highest possible estimate and vj is one above the lowest possible estimate (with a probability of p v + k 2 p v − k 2 + 1 ), or viis one below the highest possible estimate andvjis the lowest p v + k 2 − 1 p v − k 2 . The probability that either one of these events happens is the sum of the probability of each event.

As a simple example, suppose that there are three possible estimates, v − 1, v, v + 1 , with a uniform probability distribution, so that p v = 1 3 that v takes on any of these three values, and that n = 2. Suppose bidder 1 draws an estimate of the value of the good,v1= 10. Then, the true value is either 9, 10, or 11, with the probability of each outcome being 1/3. Further, the possible estimates that could be drawn by bidder 2 are 8, 9, 10, 11, or 12. Tofind the Nash equilibrium, suppose that bidder 1 believes that the other bidder will form his bid according to some rule x2= b v2 . Given a true val- uation of v, the probability that a particular bid of x is higher than bidder 2’s bid is

b− 1x

j = v − 1p j =

b− 1x

j = v − 11 3, where v − 1 is the lowest possible estimate an opponent

could draw given v, and b− 1 x is the valuation of an opponent that would result in a bid of x. In this event, the expected payout from bidding x is given by

E π = 1 3 9 − x t =max b −1_{x , 10} t =8 1 3 + 1 3 10−x t = max b−1x , 11 t = 9 1 3 + 1 3 11−x t = max b_{t = 10} −1x , 12 1 3 , 5 12

where thefirst term on the right side of the equation is the benefit to the bidder given the possible true value, v = 9, multiplied by the probability that v = 9, (1/3), and the prob- ability that the bid x will win. The second and third terms are constructed similarly for the case where v = 10 and v = 11, respectively. Suppose that b v2 = v2− 1, so that b− 1 x = x + 1 = v. Then, bidding 9 in this case would yield E π = 1 9, bidding 10 would yield E π = − 1 9, and bidding 11 would yield E π = − 1. Further, bidding slightly less than 9 yields a lower expected payout by eliminating the possibility of winning if v > 9, and resulting in only a small (less than 1) payout in the event of v = 9. Thus, the expected value for bidding less than 9 would be smaller than 1/9. Bidding slightly above 9 increases the potential price of winning without increasing the possi- bility of a win and thus results in a lower expected payout. Thus, the optimal bid when faced with an opponent who bids according to b vj = vj− 1 is to bid b v1 = v1− 1. The same can be shown for any possiblevi, and thus bidding one below the estimated value constitutes a Nash equilibrium in this case.

This strategy eliminates the possibility of overbidding. By subtracting one from the received estimate, the bid is below the estimate by the maximum possible error in the bid. By receiving an estimate of 10, one immediately knows that the lowest possible valu- ation is 9. Thus, one guarantees oneself a nonnegative payoff by bidding 9. If both bidders bid 1 below their valuation, the bidder who draws the higher valuation wins the auction.

Suppose that the true value of the object, v, is drawn from a uniform distribution over the set v, v and that the estimates of valuation are drawn from a continuous

uniform distribution, with the support of v being given by v − ε, v + ε . With n bidders participating in the auction, the Nash equilibrium bid in the common-value auction is given by1

b v = v − ε + 2ε N + 1e

− N 2ε v − v + ε _{. 5 13}

Thus, the rational model predicts that people bid their value, minus a correction term that reduces the probability of overbidding, plus another correction term that adjusts the bid up if the estimated valuation is improbably low. The second correction term becomes very small as the valuation estimate moves abovev + ε. Because this term is likely to be very small for the bidder obtaining the highest estimate of value, the highest bidder’s expected payoff is v − b v = v − v − ε + ε − 2ε

N + 1e−

N

2ε v − v + ε ₂ε n, which must be

positive. Also, both correction terms become smaller as the number of bidders increases, thus raising bids closer to the estimated valuation. The work of Robert Wilson has shown that if the information about the valuation of the good is drawn independently by each bidder from the same distribution and that this distribution is actually a nontrivial function of the true value of the good, the highest bid is very close to the true value of the item. Each bidder must expect nonnegative profits on average in the Nash equilibrium, otherwise it would be in their best interest to place a lower bid. In fact, the highest bid that conforms to equation 5.13 occurs when the highest valuation estimate is drawn, v = v + ε, and the number of bidders, N, tends toward infinity. In this case the bid converges to v. Thus, no one should ever bid above the true underlying value under this framework.

1_{This formula must be modified somewhat if the signal is in ε of v or v. I ignore these special cases in the text. See}

John Kagel and Jean-Francois Richard for more details. EXAMPLE 5.5 Oil Lease Contracts

The right to drill for oil offshore is typically auctioned off by the government using either first-price or second-price sealed-bid auctions. Oil companies bid for the right to drill, basing their bids on estimates of the amount of oil in the particular region. These esti- mates are produced by geologists working for the individual company considering a bid. In 1971, three physicists who had been working in the petroleum industry noted the extremely low profitability of oil lease contracts in the Gulf of Mexico. They noted that the returns from 1950 to the late 1960s were about that of a credit union—extremely low. They thought this occurred because in an auction for a good with an uncertain value, the highest bidder is the bidder with the highest probability of having overestimated the value of the good. Thus, this highest bidder is the one most likely to have overbid and to be turning a loss on the lease. In fact, Capen, Clapp, and Campbell found that often the highest bid is more than four times the second-highest bid, suggesting that the highest bidder was in fact overestimating the value of the oil in the well.