History and Notes

explanations for why the Dutch and first-price auctions might produce realized prices that are above those in second-price auctions. In the derivation of the optimal bids in this chapter we have assumed that people maximize the expected payout. This assumption eliminates the possibility that people might value less-risky choices over riskier choices, called risk aversion. For example, a strategy that has a 50 percent chance of resulting in a payoff of five cents and a 50 percent chance of resulting in a loss of five cents has an expected payout of $0. Suppose an alternative strategy were available that offered a 50 percent chance of obtaining $1,000,000 and a 50 percent chance of losing $1,000,000. This also has an expected payout of $0, but many might be more inclined to choose the lower-stakes strategy because it offers less risk. More generally, risk preferences are often represented by a utility of money function, uπ, so that people maximize the expected utility of their payout rather than the expected payout itself. If the utility function is increasing and displays diminishing marginal utility of payout (in other words, it is a concave function), then the consumer prefers lower levels of risk given the same expected payout (this is discussed in greater detail in Chapter 6).

In the Vickrey and English auctions, the dominant strategy still prevails if bidders maximize their expected utility. In these auctions, bidding your valuation is always the optimal bid no matter what the behavior of others, and it thus involves no uncertainty. Alternatively, in the Dutch and first-price auctions, there is no dominant strategy and each possible bid involves some level of uncertainty. Although both the Dutch and English auctions should result in the same behavior, risk aversion might lead to higher bids. In essence, higher bids reduce the payout in the event of winning the auction, while increasing the probability of a win, making the bid safer. On the other hand, no matter what the bid, the payout in the event of a loss is 0. Thus, bidding higher than the riskneutral (or expected payout-maximizing) bid can reduce the risk involved in bidding at a cost of some reduction in expected payout. Aversion to risk can induce bids that are somewhat higher than the bids in the Nash equilibrium we have discussed. Further, the risk-neutral bids in the first-price and Dutch auctions generally result in the same expected price for the item being auctioned as in the Vickrey and English auctions. Thus, risk aversion can lead to higher prices in the Dutch and first-price sealed-bid auctions in general, as has been observed. Despite this, many properties of the observed bids are inconsistent with risk aversion or any sort of Nash equilibrium strategy.

Auction with Uncertainty

Consider a different type of uncertainty in auctions. Suppose, for example, that the bidder wishes to bid on an object so that he can sell it to a third party at a later date. He may be uncertain of the price he will be able to obtain for the item in the future. Thus, it may be that the bidder is uncertain about his own valuation of the item. In addition, some objects have some intrinsic value that is common to all involved. For example, when bidding on an oil well, generally all bidders may be uncertain regarding the actual amount of oil in the well, but all are agreed that it is the value of the oil in the market that determines the value of the well. In this case, valuation is not subjective because the object will not be consumed by the bidder but will be used to derive some monetary value (e.g., through sale on a market). This is called a common-value auction. In a common-value auction, different bidders might have differing estimates or beliefs

regarding the eventual value of the item they are bidding on and must form a bid without knowing for certain the value they will obtain. In this case, the estimate of the value plays a significant role. Auctions that are commonly thought of as common-value auctions include oil lease contracts, bidding for free agents in professional sports, or bidding for the rights to publish highly anticipated books (e.g., biographies of recent presidents or their counselors).

Rational Bidding under Uncertainty

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 pv, where v must be in the set v − k2, , v + k2, k is some positive even integer, and the average value of the estimates is Ev = v. Then each bidder must formulate a bid to solve

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 signal vi, this value could be as much as k2 above the true valuation or as much as k2 below the true value. The probability that vi is k2 above v is pv + k2, and the probability that vi is k2 − 1 greater than v is pv + k2 − 1, and so on. Thus, the bidder receiving the signal vi, might view the probability function as representing their beliefs regarding the placement of the true value of winning the auction. If the probability that vi is k2 above v is pv + k2, then to the bidder, the probability that v = vi − k2 is equal to pv + k2. 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 pv + k2 pv − k2. To see this, note that given the probability function specified, the only way vj could fall k below vi is if vi = v + k2 and vj = v − k2. If we assume that each estimate is an independent draw from the distribution, then the probability of drawing vi = v + k2 and vj = v − k2 is just the product of the probabilities of each individual draw. On the other hand, the proba-

where the first 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 probability 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 bv2 = v2 − 1, so that b − 1x = x + 1 = v. Then, bidding 9 in this case would yield Eπ = 19, bidding 10 would yield Eπ = − 19, and bidding 11 would yield Eπ = − 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 possibility of a win and thus results in a lower expected payout. Thus, the optimal bid when faced with an opponent who bids according to bvj = vj − 1 is to bid bv1 = v1 − 1. The same can be shown for any possible vi, 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 valuation 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

Further, when his estimate of the value is high, the expected profit for bidding might become low relative to the Nash equilibrium expected return, or it might even become negative. This phenomenon is called the winner’s curse. Because the highest bidder is the most likely to have overbid the true underlying valuation of the good, if he fails to account for the intricacies of optimal bidding in a Nash equilibrium he is highly likely to lose money for having won the auction.

In describing behavioral theories of why this might happen, two points are important. First, although market prices are the result of the behavior of many market participants, auction prices tend to be the result of just a couple of bidders—and in the case of the firstprice auction one bidder. Thus, in an auction with 100 participants, one irrational bidder who happens to draw a high estimate of the valuation could lead to the behavioral anomaly. Anomalous market prices would require irrational behavior by nearly all participants—an unlikely event. Thus, markets are very stable and robust. Auctions are very volatile, with the outcome depending on each individual participant. Second, the Nash equilibrium under the common-value first-price auction is highly complex. Even in the very simplified version under which I have derived the Nash equilibrium in this chapter, I would guess (from previous experience in teaching) that whereas students understand the concept that bids should be below the estimated valuation, many students of economic theory fail to fully understand how the strategy is derived. I have omitted the derivation of the more complicated general Nash strategy for bidding because it is lengthy enough to be omitted from nearly all game-theory texts, as well as academic papers discussing optimal bidding in the common-value auction. It is then a very strange idea that people who have good training in mathematics and theory have trouble understanding this strategy even when they are concentrating significant cognitive effort on it, yet somehow this same strategy comes naturally to untrained persons who are participating in auctions.

The typical behavioral model of bidding under uncertainty is based on the anchoring and adjustment model. Bidders recognize the error in their estimate and the need to make profit. Thus, they anchor on their estimated value and adjust downward to obtain a bid that is closer to the Nash equilibrium bid. However, because their anchor is high, they fail to adjust downward fully, and thus the bid is too close to the anchor. One result of this model is that the larger the number of bidders, the greater the probability of the winner’s curse. When each participant has an independently drawn estimate of the underlying value, the higher the number of draws, the higher the probability that the highest estimate of value is significantly over the true value. Suppose the estimate of value that is required in order to estimate at or above the true value is v, and the probability of drawing v > v is p. Then the probability that no one draws an estimate greater than v with n bidders is 1 − pn, which is clearly decreasing as n increases. Thus, we would expect to see many more cases of the winner’s curse with many auction participants rather than with fewer auction participants.

An alternative behavioral model supposes that people underappreciate the impact of information differences between bidders regarding individual bidding behavior. Thus, whereas bidders might correctly anticipate the bids of their competitors, they might not fully attribute the differences in bids to the differences in estimates of the underlying value. Rather they might believe the difference is due to poor strategy on the part of others. This possibility has been implicitly used to explain both the

winner’s curse and several other anomalies in the trade of goods with common yet unknown values.

Most notably, Erik Eyster and Matthew Rabin propose an alternative to the Nash equilibrium in such games. In the Nash equilibrium, people are assumed to know the distribution of estimates of value and to employ the strategy that anticipates the strategies of other players based on this distribution. Alternatively, Eyster and Rabin suppose that people know the true distribution of actions taken by others, but they suppose that these actions have no particular relationship to the underlying estimates of values. Thus, the distribution of actions by others is seen as not representing strategic behavior but just as some static environmental variables. Then the fully cursed equilibrium is given where each player is maximizing the expected payout given the distribution of actions by other players. More generally, the χ -cursed equilibrium assumes that each player assigns probability χ to the event that all other players’ actions are not related to their underlying estimate of valuation (or type), and the remaining 1 − χ probability is assigned to the event that players are fully strategic in responding to their estimate of valuation.

Let us consider the fully cursed equilibrium of the simple first-price auction considered in the previous section. There are three possible values of the estimate v, v − 1, v, v + 1, with a uniform probability distribution, so that pv = 13 for each possible value of v, and n 2. Suppose again that player 1 draws v1 = 10. As derived previously, the probability distribution of the other player’s estimate of valuation is given by

This would result in the probability distribution of the other player’s action z given by

Further, let μ = Ev =vpvv be the expected value of the item being auctioned given the signal received by the player in question. This expected value in the auctions we have described is just the value of the signal itself. In this case μ = v1 = 10. Given this distribution, player 1 must solve

where Eχ denotes expectation given cursed beliefs, and the probabilities pbv2 are drawn from equation 5.15. The key difference between the Nash and the cursed equilibrium is in how the player evaluates the potential payout from a bid. Here, the player multiplies the overall expected benefit should he win the auction, 10 − x, by the probability of winning should he bid, x. However, the player treats winning the auction and the payout he receives as if they are unrelated. He doesn’t recognize that he has a higher probability of winning when he has overbid. When the player does not consider the strategic nature of others’ bids, the relationship between the possible true value and the bids of other players is not considered.

Compare this to the equation 5.12, where the player considers the impact of the true underlying valuation on the impact of others’ bids. For example, if the true valuation is low (in this case, 9), the probability of the other player drawing an 8 increases from 0 to 1/3. Equation 5.12 takes this possibility into account when determining the expected payout, whereas the fully cursed equilibrium does not. Here the player only considers the average value given the signal he received and then considers his chances of winning based on that value. The player behaves as if winning the auction is independent of the underlying value.

Recall that the Nash equilibrium strategy was to bid one below the estimated valuation. If player 2 plays bv = v − 1, then consider the payout for bidding 9 when your drawn estimate is 10:

Alternatively, bidding just slightly more than 9, say 9.1, would yield

which is much greater than 1/3. Thus, bidding 1 less than your estimated value cannot be the fully cursed equilibrium strategy. In fact, the only strategy that will produce a probability distribution such that the player is not tempted to increase his bid is bv = v, yielding Eχ π = 0. Further, the true expected payout is found by multiplying the probability of winning given a particular true underlying value by the value of winning in that instance. For simplicity, assume that the bidder in question loses if there is a tie bid. Thus, the probability that the underlying value is 9 is 1/3, and the probability of winning when bidding 10 if the value is 8 is 2/3 (the probability of the other player drawing a 8 or a 9). The value of winning in this case is − 1. The probability that the value is 10 is 1/3, and the probability of winning in this case is 1/3 (the probability the other player draws a 9). The value of winning in this case would be 0. The probability that the true value is 11 is 1/3. The probability of winning in this case is 0. Thus, the true expected profit from winning the auction with a bid of 10 given v = 10 is

Thus, the winner would on average lose money. Hence, a rational bidder would never choose to follow this strategy.

More generally, Eyster and Rabin show that in Vickrey auctions with large numbers of bidders, the true expected payout of bidders will be negative, whereas with smaller numbers this might not be the case. They show evidence of such behavior in several experiments involving a wide variety of games with uncertainty. The cursed equilibrium is closely related to the previously mentioned literature on the level of rationality, or degree to which people anticipate the actions of others. In the cursed equilibrium, the player does not believe the others are reacting to his or her private information rather, the player just takes their actions as given.

EXAMPLE 5.7 Experimental Evidence

John Kagel and Dan Levin (and others at the University of Houston) have run several sets of economics experiments to test for the winner’s curse in a well-controlled setting. Subjects were placed in a series of common-value sealed-bid first-price auctions. In each auction, the true value, v, of the item was drawn at random from a uniform distribution on a known interval, v, v, that was unknown to the bidders. Each bidder was given a private estimate of the value, v, drawn from a uniform distribution on an interval of known width v − ε, v + ε. The support of the true value ranged from as low as $12 in some auctions to as high as $500 in others. Additionally, they varied the number of bidders in the auctions from as low as three to as many as seven. In all auctions with more than five bidders, the average realized profits were negative, providing substantial evidence of the winner’s curse. In nearly all treatments the winner took away strictly less than the amount implied by Nash-equilibrium bidding. Perhaps more puzzling, the winner was often not the participant given the highest estimate of value. In fact, this phenomenon was more prevalent when there were more bidders. This might suggest that some people respond to the competitiveness of the situation with larger bids in excess of their estimated valuation. Over the course of many rounds of auctions, bidders who had won previous auctions eventually showed some learning and convergence toward the Nash equilibrium behavior.

William Samuelson and Max Bazerman used M.B.A. students to conduct a hypothetical choice experiment describing a mergers and acquisitions problem. With some regularity, it has been shown that when one company decides to purchase another, the purchasing company pays more than the acquired company is worth and receives a negative return on the investment. Many have linked this to the winner’s curse, where the private company might be using their private information and ignoring the publicly available information regarding the company’s value embodied in publicly traded stock prices. For example:

Place yourself in the position of a CEO of a major pharmaceuticals firm who is considering the purchase of a smaller firm in the same field. Should you decide to acquire the smaller firm, your firm will pay cash and purchase all of the firms outstanding stock. The smaller firm is heavily invested in a project that could produce a cure for common form of cancer. In the best case

scenario, if the project pays off, the company will be worth $100/share. If the project does not yield such a cure, the company could be worth as little as $0/ share. You consider all share values between $0 and $100 to be equally likely to occur. Because the companies perform similar tasks, there are some potential cost savings should you decide to acquire the smaller firm. Thus, the smaller firm is worth approximately 1.5 times as much to your firm as it would be worth on the market generally. The smaller firm already knows whether their project will pay off or not, and will only accept an offer that exceeds the anticipated fair price per share of their firm. What would you bid?

The correct answer to this question would consider the amount of value obtained if the bid is accepted. Samuelson and Bazerman conducted an experiment asking participants to make a similar decision. If any bid of $x per share is accepted, then the fair value can be at most $x per share, otherwise this bid would be rejected. With a uniform distribution of possible values between $0 and $x per share, the expected value per share given the bid is accepted is Ex = $x2. In this case, the expected value of purchasing the smaller firm per share is $1.5Ex− $x = $3x4 − $x = − $x4. Thus, if the firm is willing to accept your bid, you do not want the company. The optimal bid is zero. However, if, as in the cursed equilibrium, the bidder does not consider the strategic action of the firm but only the expected gain, then the bidder will use the expected value of the smaller firm, not conditioned on whether the bid is accepted, to calculate the potential gain. This expected value is $50 per share. The expected benefit of acquiring the smaller firm would be 1.5 times this expected value, $75, and thus the bidder would choose some value between $50 and $75.

Samuelson and Bazerman’s experiment provided monetary incentives to some participants by paying them a fraction of the value of the acquisition. In fact, bidders who were given monetary incentives, as well as those who were not, all bid positive amounts for the firm, contrary to the optimal strategy. In both cases, more than half of the bids fell between $50 and $75, demonstrating something that closely resembles cursed equilibrium behavior.

EXAMPLE 5.8 Building Contractors

Construction firms are often required to bid on larger projects that could be worth substantial amounts in terms of revenue and profits. However, in this case, the firm bids their payment for construction, and then they must construct the building project for some unknown cost. The uncertainties involved in construction projects lead costs to be substantially volatile. Given this need to participate in regular auctions, one might expect construction firms to face the winner’s curse, leading to eroding profits and potential failure. In laboratory experiments, building contractors, in fact, fall prey to the winner’s curse. But there is much less evidence of the winner’s curse in real-life construction bidding. Douglas Dyer and John Kagel used interviews with building contractors as well as some publicly available bidding data to learn why there might be a difference