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Computing Profit-Maximizing Bid Shading Factors in First-Price Sealed-Bid Auctions

A Correction to this article was published on 06 January 2023

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Computational methods are used to determine a profit-maximizing shading factor by which rational bidders shade their bid in first price sealed bid auctions for a broad range of realistic scenarios when the prior is diffuse. Bidders’ valuations may have both common value and firm-specific components, and the accuracy of their estimates of the common value component may differ. In addition, we allow for a subset of “naive” rivals, defined as bidders who do not account for the Winners’ Curse. Our computations show that profit-maximizing shading is greatly impacted by asymmetries in the bidding population and, in particular, by the presence of naive bidders. Failing to account for the presence of naive bidders results in underbidding only in one case, when facing a single rival who is naive, and in overbidding in all other cases. Overbidding is particularly severe when the population of naive competitors is large.

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  1. We use rational and sophisticated as equivalent terms. A naive bidder is a type of irrational bidder whose behavior does not follow any profit maximization given all the available information. These can use basic rules of thumb or other strategies. We assign a strategy to our naive bidder later in the text.

  2. As Heumann (2019) notes, because of its tractability the model studied by Wilson (1998) is often used in empirical work. Diffuse priors are also considered in recent theoretical and empirical papers, e.g. Vives (2011), Hong et al. (2013).

  3. Wilson (1969) derived the first explicit bidding function; however it was limited to only two symmetric bidders.

  4. Hoernig and Fagandini (2018) decomposes the optimal shading as the expected value plus the dispersion index of the maximum error. While the first deals with the Winner’s Curse, the second accounts for the competitive effect.

  5. The term naive is used in the same sense by Kagel and Levin (2002), and in a similar way in Lorentziadis (2012).

  6. Thaler (1988) provides a very good review of laboratory and field studies that document the existence of the Winner’s Curse.

  7. Notable literature reviews can be found in Klemperer (2004), Krishna (2010), Milgrom (2004), Salant (2014).

  8. For example (Crawford, 1970; Smiley, 1979).

  9. Following the literature, we use the term “symmetry” when referring to the distribution from where the signals are drawn. To avoid confusion, we will use “heterogeneity” when referring to differences in the the bidders’ level of sophistication.

  10. R. Wilson and M. Rothkopf are probably among the most prolific authors that actually looked at both sides, game theory and decision sciences.

  11. For example (Friedman, 1956; Ortega Reichert, 1968), among others.

  12. E.g.Capen et al. (1971).

  13. The analysis can be extended to three or more groups. However, 2 groups keeps the comparative statistics parsimonious.

  14. Note however that, while the moments may differ, the nature (Normal or Log Normal) of the underlying distribution is the same for all bidders. This is consistent with the auction literature and reflects the fact that the distribution of the error term is related to the characteristics of the auctioned item and not to the characteristics of the bidders.

  15. An objection to a constant additive shading model is that it may appear unrealistic to apply the same additive shading factor, say 2, in cases where the signals are as different as, say 5 and 100. Furthermore, this specification does not guarantee a natural lower bound—zero—for bids. However, while we focus on additive model, it is important to bear in mind that this specification includes the log transformation of the multiplicative model, which does not suffer from any of those problems.

  16. i.e. instead of using Monte Carlo simulations. This process is briefly described in Appendix C

  17. Note the importance of the diffuse prior assumption for this result.

  18. We used as tolerance \(10^{-4}\). This choice was arbitrary. We saw no further gains in precision to the bias factors and shading factors for the distributions employed in our simulations, but this parameter can be adjusted as required.

  19. We plot convergence on iterations instead of reporting time to convergence, which depends on the particular hardware that is available to us.

  20. In a private communication with the author we were informed that he does not have those proofs anymore.

  21. Hoernig and Fagandini (2018) also find the solution for the non symmetric, but all rational, equilibrium as a linear function.

  22. Tried for several symmetric and all-rational bidders drawing signals from a standard normal distribution.

  23. On Appendix C we describe the Monte Carlo simulations and show some histograms for the estimates for a couple of symmetric bidder scenarios and also for a couple of scenarios with asymmetric bidders.

  24. Wilson divided by \(\sigma \) just to have a result that was scale-independent.

  25. For example, the analytical solution (Hoernig & Fagandini, 2018), coinciding with all our simulations, allows to compute the symmetric values for 50 (\(BF=2.25\), \(CF=0.10\), \(SF=2.34\)), 100 (\(BF=2.51\), \(CF=0.07\), \(SF=2.58\)), and even 10,000 (\(BF=3.85\), \(CF=0.02\), \(SF=3.88\)) bidders.

  26. This finding was confirmed also with Monte Carlo simulations to rule out potential coding issues.

  27. See also (Porter, 1995), and (Hendricks & Porter, 2007).

  28. Code available at .


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We are indebted to Steffen Hoernig for his insightful advice in this paper. This paper also profited from valuable interaction with Guido Maretto, Susana Peralta, Fernando Anjos, Pedro Vicente, Patrick Rey, and Robert Wilson, as well as the referees who helped to significantly improve this paper. The usual disclaimer applies.


Paulo Fagandini: Fundação para a Ciência e a Tecnologia FCT, grant SFRH/BD/105669/2015.

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Paulo Fagandini thanks the support of the Fundação para a Ciência e a Tecnologia FCT under the grant SFRH/BD/105669/2015. This is partially based on a dissertation for a PhD in Economics at Nova School of Business and Economics.

The original online version of this article was revised due to the second affiliation update for the corresponding author.


Appendix A Effect of naive competitors on the Winner’s Curse

See Fig. 8.

Fig. 8
figure 8

Bias factor for an increasing number of bidders. In one group only one of the bidders is rational (solid) and in the other, all bidders are rational (dashed)

Appendix B Proof of Lemma 1

As stated in the main text, the Shading Factors satisfy the following:

$$\begin{aligned} c_i\in \arg \max _{{\hat{c}}} \int _{\mathbb {R}} f_i(\epsilon _i) ({\hat{c}}-\epsilon _i)\left( \prod _{j\ne i} F_j(\varDelta _i-{\hat{c}}-(\varDelta _j-c_j)+\epsilon _i)\right) d\epsilon _i \end{aligned}$$

Take the derivative with respect to \(c_i\) and set it equal to zero:

$$\begin{aligned} \frac{\partial }{\partial c_i}\left[ \int _{\mathbb {R}} f_i(\epsilon _i) (c_i-\epsilon _i)\left( \prod _{j\ne i} F_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)\right) d\epsilon _i\right] =0 \end{aligned}$$

Noting that \(f_i(\epsilon _i)\) does not depend on \(c_i\), solve:

$$\begin{aligned} \int _{\mathbb {R}} f_i(\epsilon _i) \frac{\partial }{\partial c_i}\left[ (c_i-\epsilon _i)\left( \prod _{j\ne i} F_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)\right) \right] d\epsilon _i=0 \end{aligned}$$

The derivative corresponds to the addition of two terms:

$$\begin{aligned} \frac{\partial }{\partial c_i}\left[ (c_i-\epsilon _i)\right] \left( \prod _{j\ne i} F_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)\right) + \ldots \end{aligned}$$
$$\begin{aligned} (c_i-\epsilon _i)\frac{\partial }{\partial c_i}\left[ \left( \prod _{j\ne i} F_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)\right) \right] \end{aligned}$$

It is clear that (10) is equal to:

$$\begin{aligned} \left( \prod _{j\ne i} F_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)\right) \end{aligned}$$

Finally, to solve (11) let’s focus for now on the term:

$$\begin{aligned}&\frac{\partial }{\partial c_i}\left[ \prod _{j\ne i} F_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)\right] = \\&\quad -\sum _{j\ne i} \left( f_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)\prod _{k\ne j, i} F_k(\varDelta _i-c_i-(\varDelta _k-c_k)+\epsilon _i)\right) \end{aligned}$$

Multiplying and dividing by \(F_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)\) we obtain

$$\begin{aligned} -\sum _{j\ne i} \left( \frac{f_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)}{F_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)}\prod _{k\ne i} F_k(\varDelta _i-c_i-(\varDelta _k-c_k)+\epsilon _i)\right) \end{aligned}$$

Note that the product does not depend on j, so it can go out of the summation:

$$\begin{aligned} -\left( \prod _{k\ne i} F_k(\varDelta _i-c_i-(\varDelta _k-c_k)+\epsilon _i)\right) \left( \sum _{j\ne i} \frac{f_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)}{F_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)}\right) \end{aligned}$$

So the third term, replacing k with j, corresponds to:

$$\begin{aligned} -(c_i-\epsilon _i)\left( \prod _{j\ne i} F_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)\right) \left( \sum _{j\ne i} \frac{f_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)}{F_j(\varDelta _i-c_i-(\varDelta _j-c_j)+\epsilon _i)}\right) \end{aligned}$$

Finally, to obtain the full expression, apply the change of variables \(z_i = \varDelta _i-c_i-(\varDelta _j-c_j)\), to obtain:

$$\begin{aligned} \left( \prod _{j\ne i} F_j(z_{ij}+\epsilon _i)\right) -(c_i-\epsilon _i)\left( \prod _{j\ne i} F_j(z_{ij}+\epsilon _i)\right) \left( \sum _{j\ne i} \frac{f_j(z_{ij}+\epsilon _i)}{F_j(z_{ij}+\epsilon _i)}\right) \end{aligned}$$

Replacing back into the integral in (9) we have:

$$\begin{aligned} \int _{\mathbb {R}}f_i(\epsilon _i)\left( \prod _{j\ne i} F_j(z_{ij}+\epsilon _i)\right) \left( 1-(c_i-\epsilon _i)\sum _{j\ne i} \frac{f_j(z_{ij}+\epsilon _i)}{F_j(z_{ij}+\epsilon _i)}\right) d\epsilon _i = 0 \end{aligned}$$

This is the first order condition.

Appendix C Monte Carlo method

See Figures 9, 10, 11 and 12

We benchmarked the Shading Factor in some cases using Monte Carlo simulations, as set out below:

Generate signals randomly for the number of bidders, in a matrix of size \(n\times m\). Each column represents a bidder, and each row a draw. Define an interval in which to look for the equilibrium. Apply the shading to the competitors and test shading within that interval and up to 2 decimal places, and pick the one with the highest expected profit.

This procedure has limitations. First, we are taking a fixed number of shadings within a bounded and fixed interval; however, given the results obtained using the first order condition, we can use intervals easily wide enough to give reasonable assurances that it includes the optimum. Second, this procedure is inefficient, as it tries every single possibility within that set. However, this procedure also has important advantages. It can deal with any trouble embedded functions might have and it is agnostic about the curvature of the expected profits functions (does not assume differentiability or a unique maximum within the interval).

We plot the histograms of the estimates of specific simulations.Footnote 28 Each SF in the sample, consists on the equilibrium shading that maximizes expected profits using 100.000 hypothetical draws for each bidder. We do this 500 times to obtain the sample of optimal factors we include in the histograms shown below. We use a tolerance of 0.01 for each optimal factor and a maximum of iterations (over best responses) of 50 times. In all these samples, the algorithm converged (within tolerance) every time.

Fig. 9
figure 9

Two sophisticated bidders, both with signals N(0, 1). The SF is 1.77 for both

Fig. 10
figure 10

Four sophisticated bidders, with signals N(0, 1). The \(SF=1.51\) for all

Fig. 11
figure 11

Two sophisticated bidders, one with signals N(0, 1), the other with signals N(1, 1). The true corrections are \(SF=1.62\) and \(SF=2.14\) respectively

Fig. 12
figure 12

Two sophisticated bidders, one with signals N(0, 1), the other with signals N(0, 0.5). The true correction is 1.40 for both

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Fagandini, P., Dierickx, I. Computing Profit-Maximizing Bid Shading Factors in First-Price Sealed-Bid Auctions. Comput Econ (2022).

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  • Winner’s curse
  • Auctions
  • Bidding
  • Asymmetric agents
  • Naive bidders