The Winner’s Curse and Optimal Auction Bidding Strategies

Auctions are popular mechanisms for the exchange of goods and services in the marketplace. Common examples of items offered at auctions include real estate, mineral rights, construction contracts, agricultural products, United States Treasury bills, and government procurement contracts.[1] Auctions are now even more commonplace in our personal lives, where online auction sites have made it possible for individuals to bid for and sell items en masse.[2]

Auctions are particularly useful whenever there is uncertainty about the value of an item to be sold and the seller is willing to accept whatever price the market (i.e., the bidders) establishes for the item. Unfortunately, the competition amongst bidders in auctions often results in a phenomenon termed the “winner’s curse,” an outcome in which the winner prevails by submitting a bid that is not only higher than competing bids, but also higher than the true value of the item. In this article, we discuss how this phenomenon can be modeled via simulation and how the simulation results can be used to develop strategies for hedging bids, creating an optimal balance between the probability of winning and the economic gain from the transaction.



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Auction Settings

The study of auctions and bidding strategies is complicated by the great variety in types of auctions. Two of the most common types of auctions are the open auction the bidding price is publicly announced in ascending order (i.e., an English auction) or descending order (i.e., a Dutch auction) and the closed auction sealed bids are submitted simultaneously and the winner is the individual submitting the highest bid or, less frequently, the second-highest bid (i.e., a Vickery auction).[3]

Here, because of their predominance in business settings, the authors focus on bidding strategies for first-price “sealed bid” auctions, where either the highest bid wins or the lowest bid wins. The relatively limited amount of information available for bidders participating in these types of auctions makes the analysis of such bidding strategies all the more important.[4] The focus was further narrowed to sealed bid auctions where the highest bid wins the auction; however, it is straightforward to consider mirror cases where the lowest bid prevails, as would be the case with auctions for government procurement contracts, for example.

Regardless of auction type, one of two value models is appropriate:

  • Independent private values model: The value of the object at auction is different for the various bidders, for example, an antique to be purchased for a private collection.
  • Common value model: The value of the object is approximately the same for all rational bidders. This is the model most typical for business auctions, and it is also the model that gives rise to the “winner’s curse.”

The Winner’s Curse

Uncertainty exists in first-price sealed bid auctions with common item values for many reasons, including:

  • Bidders have access to different information,
  • Bidders interpret the same information differently, and
  • Valuation of items is a complicated and subjective process.

To see how such uncertainty can lead to the winner’s curse, consider the following example: Several firms are bidding for the rights to an oil exploration tract and each firm has developed its own internal estimate of its value. The firm winning this auction will typically be the one that produced the highest estimated value for the tract of land. However, the greater the submitted bid, the greater too is the likelihood that such a bid will exceed the true value of the tract, resulting in a “cursed” winner.

The winner’s curse was first identified, and then verified empirically, in the early 1970s for precisely this type of situation.[5] Since that time, economists, operations researchers, sociologists, and others have studied the phenomenon, confirming its existence in numerous empirical studies. The amount of the winner’s curse is the difference between the true value of the item being auctioned and the amount paid for it by the winning bidder.

An illustration of the curse commonly employed in the classroom setting is the auctioning of a jar of coins to student “bidders.” Almost always, the winning (i.e., the highest) bid exceeds the true value of coins contained in the jar, even though the average of all the bids is typically less than the true value, due to students’ risk aversion.[6] The winner’s curse has also been modeled mathematically, and its existence has been confirmed under very general conditions, including those cases where an object being auctioned has different values for the bidders involved and where competitors’ estimates are sometimes biased (i.e., higher or lower than the true value), for all types of auctions.[7]

Implications for Bidders

In view of the winner’s curse, how should a bidder behave in an auction to avoid or at least minimize its effect? Intuitively, it seems one should probably bid less aggressively. Analytical results for very simple auction models suggest that rational bidders in common value sealed bid auctions can generally avoid the winner’s curse if they presume that their estimate of an item’s value is the highest amongst all competitors and then bid some fraction of their original estimate.

In principle, there is no real cost associated with such a strategy as losing bidders neither gain nor lose anything; moreover, taken to the extreme, bidders can completely eliminate the threat of the winner’s curse by bidding such a small fraction of their estimate so as to never win an auction. Such a conservative strategy would clearly not be viable for bidders seeking to generate a profit; therefore, the challenge is to determine the fraction of the estimated value that should be bid to optimally balance risk and reward. In addition, common basic findings in virtually all research on auctions indicate that bidders should be aware that the winner’s curse is most severe in the following situations:

  1. Bidders have less information than their competitors,
  2. There is significant uncertainty about the true value of the item being auctioned, and
  3. There are a large number of bidders.

Under these conditions, the need for an optimal bidding strategy becomes more critical.

Developing an Optimal Bid

To illustrate how one can develop an optimal bid in an auction, consider the example of a two-bidder, first-price sealed bid auction for a tract of real estate. As is the case in most real estate auctions, we assume that the exact value of the tract of land is uncertain and that the bidders know enough about the parcel to describe its worth using a range of likely values (e.g., from analysis of “comparables”) using a probability distribution. For simplicity, we assume that the true value of the tract can be described by the normal distribution (i.e., the bell curve) with an expected value of $1 million and a standard deviation of $200,000 (i.e., the tract is worth $1 million on average). We also assume that both bidders will base their respective bids on their own proprietary estimates of the value of the tract and that both bidders have generated an unbiased estimate of that value. However, because there is great subjectivity inherent in their estimation processes, there is also significant uncertainty in the values bid. Hence, the two unbiased bids are not deterministic (i.e., certain); instead, they derive from a normal probability distribution, with a mean of $1 million and a standard deviation of $200,000. In the end, the higher of the two bids will be deemed the winner of the auction, and the winning bidder will realize a “profit” equal to the value of the tract of land acquired, minus what she or he paid for it.

Even with this simple example, an analytical estimate of the winning bidder’s expected profit would be quite difficult requiring, for example, the derivation of the probability distribution for the maximum of two random normal variables, and the integration of the functional form for this distribution to determine expected value.[8] An empirical solution, based upon the historical data collected for repeated auctions of similar tracts of land, could be used instead; however, this is usually difficult to obtain as the information from sealed bid auctions is typically not publicized. It seems that a rational bidding strategy based upon quantitative analysis is not readily available to the practitioner.

Simulation as a Tool for the Analysis of Auctions

Fortunately, commercially available simulation packages, such as @RISK or Crystal Ball, can be used to simulate repeated random samplings from realistic probability distributions for the values of the tract and the bidders’ estimates. The maximum of the two bids can be identified to determine a winner, and simple arithmetic can be used to calculate the winning bidder’s profit. If this process is repeated a large number of times, and those results are averaged, the long-run expected profit for a bidding strategy can be determined computationally. In short, one can create one’s own set of empirical data for the bidding analysis a sort of computational empirical analysis.

In Figure 1, we summarized the results from such a simulation study (for the example above) by plotting the output data for both the expected profit and the probability of winning over a variety of bidder-specific scenarios, given various opposing strategies. We define a strategy in terms of “hedging,” where the hedging percentage is the reduction in the bidder’s expected value that is used to calculate the final bid submitted in the auction.[9] For example, a 10 percent hedge indicates that a bidder would submit 90 percent of his or her original expected value as a final bid for the auction.

The simulation results shown in Figure 1 provide clear evidence of a winner’s curse; moreover, we can see the cost associated with the curse. For example, for the case where both bidders hedged their original bids by zero percent (i.e., they both bid their original estimated values), the expected profit is a loss of more than $50,000. And, as one might expect, both bidders are equally likely to “win” the auction. This can be seen at the extreme left-hand tails of the solid black and dashed black curves, respectively. While this is a desired outcome for the seller of the tract of land, it is clearly a suboptimal outcome for the bidders involved.

Assuming that sophisticated bidders are at least intuitively aware of the winner’s curse, we next consider the case where one or both bidders adopt different hedging strategies. Under these circumstances, we expect the probability of winning the auction to decline with increasing hedging percentage, holding the opposing bid constant.

This can be seen in Figure 1 by observing the following characteristics:

  1. The dashed lines, which show the probability of winning as a function of a bidder’s hedge all monotonically decrease as the magnitude of percent hedge increases, and
  2. Those dashed lines are ordered from top to bottom in decreasing amounts of the opponent’s hedge amount.

A related, and perhaps more important observation, can be made from the solid profit curves in Figure 1; one can see that there is a maximum value associated with each of the curves. At the extreme left-hand side of each of the profit curves, the bidder suffers from the winner’s curse, since she or he realized a high probability of winning the auction, but paid too high a price for the tract. Conversely, at the right-hand extreme of each curve, conservative bidding offers the potential for high profit, and yet the conservative bid almost never results in a win for the bidder.





Figure 1 Expected Profit and P(Winning) vs. Hedge % for a Two-Bidder Auction





The greatest height achieved by each profit curve in Figure 1 offers an indication of the amount of hedge that would lead to maximum profit, given a particular opposition strategy. For example, we can see that profit is maximized when the bidder’s hedge is in the range of 20 to 40 percent, depending on the particular balance or imbalance between the bidder’s hedge and the competitor’s hedge. We believe this clearly demonstrates the extraordinary value of using simulation to understand auctions and to develop optimal bidding strategies.

Formulating a Bidding Strategy with Incomplete Information

If our example bidders could agree to share information and cooperate with each other in aggressively hedging their bids, they could drive the auction price down to a small fraction of its actual value and produce a large profit for the winning bidder, assuming the seller was obliged to sell to the higher bidder. Evidence of this can also be seen in Figure 1: the profit to the winning bidder would be in excess of $200,000 if both bidders agreed to hedge their original bids by 50 percent. It is also clear from the figure, however, that there would be a strong temptation for one of the bidders to deviate from such a plan (e.g., hedging slightly less than the agreed amount), thereby increasing her or his chances of winning the auction and receiving a handsome profit. Thus, even if such cooperation amongst bidders was considered fair and ethical, it is unlikely that it would be at all common in practice.

A more realistic scenario is an auction where bidders neither act in cooperation with each other nor share information; instead, they prepare rational bids, and they expect other bidders to behave in an equally rational manner. Thus, while we have seen that simulations can be a very useful tool for determining auction outcomes, given bidder strategies, we would like to address the following question: How should a bidder behave without knowledge of the strategy her or his opponent will use?

For auctions characterized by rational bidders with incomplete information, economic game theory can be used to explore the possibility of a dominant strategy for bidding in auctions. Unfortunately, the common assumptions of game theory must be simplified radically in order to find the analytical point of equilibrium between bidders. Nevertheless, some researchers have attempted to derive the optimum bid for a two-person auction game by assuming restrictive probability distribution forms for bids or the value of an auction item.[10], [11] By using simulation models, however, we are not bound by simplifications of or restrictions to the assumptions. To illustrate, the payoff table in Figure 2 was generated using output from repeated simulation runs of the example above, and the table shows the payoffs for our two bidders over a wide range of bidding strategies. A comparable graphic could be generated for virtually any probability distribution of values or bids, not simply the normal distribution, and for very general parameter specifications.

Once generated, the payoff table below can be used to search for the equilibrium point between two bidders. In the jargon of game theory, this is referred to as a Nash equilibrium the point or points where bidders cannot unilaterally improve their expected gain by moving to a different strategy.[12]

  • If Bidder 1 hedges at zero percent, Bidder 2′s optimal strategy would be to hedge at 25 percent (green cell showing the maximum profit in column two).
  • If Bidder 2 hedges at 25 percent, Bidder 1′s optimal strategy would be to respond by changing to hedge at 25 percent (yellow cell showing the maximum profit for Bidder 1 along that row).
  • If Bidder 1 hedges at 25 percent, Bidder 2′s optimal strategy would be to respond by changing to hedge at 30 percent (orange cell).
  • Finally, if Bidder 2 hedges at 30 percent, Bidders 1′s optimal strategy would be to respond by changing to hedge at 30 percent (see the blue cell).

The last combination of strategies (i.e., Bidder 1 hedges at 30 percent; Bidder 2 hedges at 30 percent) is Nash equilibrium, because neither bidder can improve her or his profit by changing the hedge percentage. Note also that profit for Bidder 1 ($111 thousand) is very close to the profit for Bidder 2 ($110 thousand), with the slight difference being a computational artifact of numerical simulation. (Click here for Figure 2 Expected Profit Table for a Two-Bidder Auction.)

In this example, we assumed that our bidders acted rationally, and of course, bidders sometimes behave in irrational ways at auction, especially when the item up for auction holds sentimental or emotional value to a particular bidder.[13] Still, our analysis of the rational bidder case can be used to set the standard for more complex auction strategies. Also, to limit the length of our discussion, we considered a simple case with only two bidders; however, we could generalize our two-bidder example to cases where there are n bidders in an auction, in two ways. First, we could stay with a two-player model and assume that the opponent is the collective of all other bidders, who act in the same way. Alternatively, we could use the approach discussed here to rigorously model each bidder in an auction that is, we could simulate each bidder in the n-bidder game. The difficulty with the latter approach lies in the computational burden, which increases as more bidders are added. Finally, under certain assumptions, it is also possible to model the case of n bidders as sequential auctions between pairs of bidders whose bids are repeated until equilibrium bidding strategies are reached.[14]

In each of these cases, the ability to simulate a two-bidder auction serves as the basic framework, and the information required to set up the simulation analysis is simply the estimated probability distribution functions for the value estimates for both the bidder and the competitor(s).

Conclusion

Given the prevalence of auctions in business today, it is important for decision-makers (i.e., bidders) to fully understand the nature of auctions and the winner’s curse. We have shown that simulation, an analytical tool from the management science field, can be of tremendous value in generating empirical evidence about auctions when actual data does not exist. Moreover, simulation can deliver insights for the formulation of effective bidding strategies. The results from a simulation analysis of the relatively simple example discussed in this paper, for instance, allow us to make several important observations about auctions:

  1. When bidders do not hedge their value of an item in a competitive auction, they will likely pay too high a price for an item, if they win an auction. The expected loss can be quantified under very general conditions using simulation modeling.
  2. To avoid the winner’s curse, rational managers should choose a valuation model carefully and then decide on an appropriate hedge for their final bid. The expected profit, as a function of hedging percentage, has a maximum at the point where the tradeoff between risk and reward is optimal. This optimal hedge can also be found using simulation.
  3. In the usual case where competitor-bidding strategy is unknown, a range of bidding strategies, for both the bidder and the competitor(s), can be simulated and then economic game theory can be used to determine the optimal bidding strategy. While this type of analysis cannot guarantee an outcome or provide assurance that the winner’s curse will be avoided, the resulting bidding strategy can provide the bidding decision-maker with the best opportunity for success.

As mentioned earlier, we have focused on the so-called common value model, where the value of the object, and thus the profit from acquiring it, should be approximately the same for all rational bidders. An interesting extension that is beyond the scope of this paper would be to consider the independent private values model in which the value of the object at auction will be different for the various bidders to see what effect the additional uncertainty about competitor valuation would have on bidding strategy.

[1] R. McAfee and J. McMillan, “Auctions and Bidding,” Journal of Economic Literature, 25 (1987): 699 738.

[2] M. Rothkopf and S. Park, “An Elementary Introduction to Auctions,” Interfaces, 31 (2001): 83 97.

[3] M. Rothkopf and R. Harstad, “Modeling Competitive Bidding: A Critical Essay,” Management Science, 40 (1994): 364 384.

[4] W. Vickery, “Counterspeculation, Auctions, and Competitive Sealed Tenders,” Journal of Finance, 16 (1961): 8 37.

[5] E. Capen, R. Clapp, and W. Campbell, “Competitive Bidding in High-Risk Situations,” Journal of Petroleum Technology, 23 (1971): 641 653.

[6] R. Thaler, “Anomalies: The Winner’s Curse,” The Journal of Economic Perspectives, 2 (1988): 191 202.

[7] S. Oren and A. Williams, “On Competitive Bidding,” Operations Research, 23 (1975): 1072 1079.

[8] P. Milgrom and R. Weber, “A Theory of Auctions and Competitive Bidding,” Econometrica, 50 (1982): 1089 1122.

[9] M. Rothkopf, “On Multiplicative Bidding Strategies,” Operations Research, 28 (1980): 570 575.

[10] L. Friedman, “A Competitive Bidding Strategy,” Operations Research, 4 (1956): 104 112.

[11] E. Dougherty and M. Nozaki, “Determining Optimum Bid Fraction,” Journal of Petroleum Technology, 27 (1975): 349 356.

[12] B. Smith and J. Chase, “Nash Equilibria in a Sealed Bid Auction,” Management Science, 22 (1975): 487 497.

[13] M. Rothkopf, “A Model of Rational Competitive Bidding,” Management Science, 15 (1969): 362 373.

[14] S. Oren and M. Rothkopf, “Optimal Bidding in Sequential Auctions,” Operations Research, 23, no. 6 (1975): 1080 1090.

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