Demand reduction and preemptive bidding in multi-unit license auctions


Multi-unit ascending auctions allow for equilibria in which bidders strategically reduce their demand and split the market at low prices. At the same time, they allow for preemptive bidding by incumbent bidders in a coordinated attempt to exclude entrants from the market. We consider an environment where both demand reduction and preemptive bidding are supported as equilibrium phenomena of the ascending auction. In a series of experiments, we compare its performance to that of the discriminatory auction. Strategic demand reduction is quite prevalent in the ascending auction even when entry imposes a (large) negative externality on incumbents. As a result, the ascending auction performs worse than the discriminatory auction both in terms of revenue and efficiency, while entrants’ chances are similar across the two formats.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8


  1. 1.

    For instance, in his analysis of bidding behavior in the FCC’s AB-block auction, Weber (1997) finds evidence that the large bidders dropped out of some markets at prices far below market expectations. See also Cramton and Schwartz (2000).

  2. 2.

    Grimm et al. (2003b) study a simultaneous, ascending multi-unit setting that fits the 2nd generation GSM spectrum auction in Germany and find that in the unique equilibrium that survives the iterated elimination of dominated strategies, the two bidders immediately reduce their demand to clear the auction at the minimum bid.

  3. 3.

    Demand reduction has also been observed in the laboratory. Alsemgeest et al. (1998), for instance, compare the open ascending auction to a sealed-bid auction where the price for each unit equals the lowest accepted bid. They find evidence for demand reduction in the open ascending auction, which generally yields lower revenues than the sealed-bid format. Kagel and Levin (2001) consider environments without strategic uncertainty where a single human bidder competes against robot bidders in a sealed-bid uniform-price auction, an open ascending auction, and the Vickrey/Ausubel auction. Demand reduction occurs in both the sealed-bid uniform-price and the open ascending auction, but the level of demand reduction is more pronounced in the latter. Kagel and Levin (2005) obtain a similar finding for a setting where the single human bidder faces a tradeoff between bidding above value due to synergies and demand reduction. Engelmann and Grimm (2009) compare five auction formats: the sealed-bid uniform-price, the open ascending, the discriminatory, the Vickrey, and the Ausubel auction. They observe more demand reduction in the open ascending auction than in the sealed-bid uniform-price auction. Finally, List and Lucking-Reiley (2000) conduct a field experiment with sports cards and find evidence for demand reduction in the sealed-bid uniform-price auction, although revenues do not differ from those of a Vickrey auction because bidders bid too high on the first unit. Pooling the results of these different studies suggests that demand reduction is more pronounced in open ascending auctions than in sealed-bid uniform-price auctions.

  4. 4.

    See Jehiel et al. (1996, 1999), Jehiel and Moldovanu (2000) and Das Varma (2002, 2003) for a theoretical analysis of bidding behavior in the presence of externalities.

  5. 5.

    A similar evaluation has to be made by a seller interested in maximizing efficiency or entry.

  6. 6.

    Klemperer (2004, p. 159, footnote 27) compares Deutsche Telekom’s behavior to that of someone who waits in a queue for a long time but then quits in frustration before it is his turn.

  7. 7.

    Klemperer (2004, p. 202) criticizes the assumption made by Ewerhart and Moldovanu (2001) that there is only a single strong bidder; their model cannot explain why initially both Deutsche Telekom and Mannesman pushed up the price but then stopped doing so.

  8. 8.

    We chose to focus attention to the two auction formats that are most often observed in practice. The optimal format is unknown for our setting, see Jehiel et al. (1999).

  9. 9.

    Note that the actual number of units bought by X is irrelevant for the external effect, as long as this number is positive.

  10. 10.

    In accordance with the usual practice of license auctions, we did not use reserve prices in either auction format. In the recent 3G auctions, most countries refrained from setting a reserve price (Netherlands) or they set very low reserve prices (e.g., Germany, Austria, Switzerland, Italy). As Klemperer (2003) notes, “[But] serious reserve prices are often unpopular with politicians and bureaucrats who—even if they have the information to set them sensibly—are often reluctant to run even a tiny risk of not selling the objects, which outcome they fear would be seen as a ‘failure’.”

  11. 11.

    In all treatments, we had an upper bound on subjects’ bids. In the treatments with x=0, x=50 and x=100, the upper limit was equal to respectively 100, 125 and 150. These upper limits were never reached in the experiments (neither in the ascending nor in the discriminatory auctions).

  12. 12.

    Notice that in both auctions there was a possibility that some goods remained unsold in a period. In the ascending auctions this happened when the sum of the initially demanded quantities was less than 6. In the discriminatory auctions this would occur when in total fewer than 6 bids were submitted.

  13. 13.

    If bidders behave competitively also off the equilibrium path, bidders have no incentive to deviate from their own competitive bidding strategy. For instance, reducing demand to one or two units before one’s value is reached serves no purpose then; if a bidder wins one or two units at some price below v by doing so, she would have won three units at that price had she not deviated.

  14. 14.

    Recall that if the other bidders each demand two licenses and bidder 3 initially demands three licenses, she cannot lower her demand below two licenses since total demand cannot fall below the total supply of six licenses.

  15. 15.

    Suppose instead that B I (v)=p where v<p. As the price level approaches p, an incumbent of type v would be better off reducing demand to zero units slightly before p and incur the negative externality, rather than waiting to reduce demand to two units at p since then she also incurs the negative externality plus 2(vp)<0.

  16. 16.

    As for the case without externalities, off the equilibrium path strategies are assumed to be such that, if one of the bidders deviates before price level p is reached (by reducing demand to one or two units), the other bidders behave as if no deviation has occurred.

  17. 17.

    Suppose two bidders each submit only a single bid that applies to all three licenses. We have to show that the third bidder’s best response is to also submit a single bid. Let v denote the bidder’s value and b 1b 2b 3 her bids. The optimal b 3 is determined by trading off the profit conditional on winning, vb 3, against the probability of winning as determined by the distribution of the sixth-highest of the others’ bids. Likewise, the optimal b 2 (b 1) is determined by trading off vb 2 (vb 1) against the winning probability as determined by the distribution of the fifth-highest (fourth-highest) of others’ bids, since one (two) of the third bidder’s own bids are higher. But if the other two bidders submit only a single bid then the distributions of the sixth, fifth, and fourth highest of others’ bids are identical. Hence, b 1=b 2=b 3. Lebrun and Tremblay (2003) prove that for the case of two bidders (and no externalities), the equilibrium in which bidders submit only a single bid is the unique equilibrium of the discriminatory auction.

  18. 18.

    The equilibrium bid functions of Proposition 3 can also be derived more directly. Note that the payoff of a bidder who has value v but bids as if her value is w is given by π e(w|v)=(vB(w))(1−(1−w)2). Optimizing with respect to w and equating the result to zero at w=v yields a first-order differential equation that is solved by (7).

  19. 19.

    The demand reduction equilibrium may also vanish in some cases where the negative externality depends on the number of units bought by the entrant. If the difference in negative externality when the entrant acquires 2 licenses instead of 1 license is sufficiently large, an incumbent may want to deviate from equally sharing the licenses and try to obtain 3 units so that possibly only 1 unit remains for the entrant (when the other incumbent bids sufficiently high on the remaining 2 units).

  20. 20.

    With this in mind, one could consider allowing an incumbent to purchase four units. This would prevent a monopoly, but would still allow an incumbent to aggressively pursue preemption even if the other incumbent wanted to settle on demand reduction.

  21. 21.

    Parts 1 and 2 of the experiment resulted in statistically similar revenues for 5 out of 6 treatments; when we consider the realized revenues as a fraction of the available Nash revenues at the preemptive equilibrium, the only significant difference is obtained for the treatment disc100. Here the relative revenue provides the appropriate measure for comparison between parts 1 and 2, because we kept values constant across treatments but not across parts. As it appears the randomly drawn values of part 2 are accidentally more favorable for raising revenue. In treatment disc100, average observed revenue equals 190.5 in part 1 and 284.8 in part 2, while the predicted Nash preemptive revenues equal 217.6 and 242.4, respectively. The ratios of these observed and predicted revenues differ significantly (Mann-Whitney rank test (m=n=8, p=0.02). The test results of the other 5 treatments are far from significant, however (all p>0.28).

  22. 22.

    In all cases the predictions listed in Table 2 are based on the actual private value draws used in the experiments. These predictions may slightly differ from the ones based on the U[0,100] distribution (cf. Sect. 3). E.g., when x=0 the competitive equilibria of the two auction formats are revenue equivalent in the general model and are predicted to yield the seller 150. Yet for the particular private values that we use the competitive equilibrium in the ascending auction yields a slightly higher revenue (162.8 on average) than the equilibrium in the discriminatory auction(148.9).

  23. 23.

    The levels of demand reduction in parts 1 and 2 are of the same magnitude. For instance, 6 of the 8 groups in asc0 successfully reduced demand in part 2. This suggests that bidders reduce their demand for the ‘right’ non-cooperative reasons, and that it is not due to a repeated game effect or low stakes.

  24. 24.

    To the contrary, society as a whole may actually become strictly better off when competition is intensified. Dana and Spier (1994) and Grimm et al. (2003a) characterize the optimal government mechanism to regulate market entry in a setting where government tradeoffs both consumer surplus and producer surplus (as well as the revenue from the mechanism).

  25. 25.

    Again note that the predictions appearing in Table 5 are based on the actual private value draws. For the theoretical U[0,100] distribution entry probabilities in the preemptive equilibria of the ascending auction equal 57.3 % and 45.4 % for x=50 and x=100, respectively (cf. Sect. 3.2). In the discriminatory auction these numbers equal 61.0 % and 55.3 %.

  26. 26.

    Recall that in Sect. 3 the units were scaled down by a factor of 1/100.

  27. 27.

    Ideally, one would like to identify a subject’s loss-aversion in a different, unrelated task. We do not have such data, however.

  28. 28.

    Notice that rather counterintuitively the curve for x=50 bends downward for very high minimum private values. However, this part of the figure is based on few data only.

  29. 29.

    The row labeled PDR contains the corresponding cases for the situation without external effects. Because in that regime the preemptive motive is absent, these cases are labeled as partial demand reduction outcomes.


  1. Abbink, K., & Brandts, J. (2005). Price competition under uncertainty: a laboratory analysis. Economic Inquiry, 43, 636–648.

    Article  Google Scholar 

  2. Alsemgeest, P., Noussair, C., & Olson, M. (1998). Experimental comparisons of auctions under single- and multi-unit demand. Economic Inquiry, 36, 87–97.

    Article  Google Scholar 

  3. Ausubel, L., & Cramton, P. (1998). Demand reduction and inefficiency in multi-unit auctions. Working paper 96-07.

  4. Cramton, P. (2002). Spectrum auctions. In M. Cave, S. Majumdar, & I. Vogelsang (Eds.), Handbook of telecommunications economics (pp. 605–639). Amsterdam: Elsevier Science B.V.

    Google Scholar 

  5. Cramton, P., & Schwartz, J. (2000). Collusive bidding: lessons from the fcc spectrum. Journal of Regulatory Economics, 17, 229–252.

    Article  Google Scholar 

  6. Dana, J., & Spier, K. (1994). Designing a private industry. Government auctions with endogenous market structure. Journal of Public Economics, 53, 127–147.

    Article  Google Scholar 

  7. Das Varma, G. (2002). Standard auctions with identity-dependent externalities. Rand Journal of Economics, 33(4), 689–708.

    Article  Google Scholar 

  8. Das Varma, G. (2003). Asymmetric equilibria in the open ascending-price auction with shared allocative externalities. Working paper.

  9. Dufwenberg, M., & Gneezy, U. (2000). Price competition and market concentration: an experimental study. International Journal of Industrial Organization, 18, 7–22.

    Article  Google Scholar 

  10. Engelbrecht-Wiggans, R., & Kahn, C. (1998). Multi-unit auctions with uniform prices. Economic Theory, 12, 227–258.

    Article  Google Scholar 

  11. Engelmann, D., & Grimm, V. (2009). Bidding behavior in multi-unit auctions—an experimental investigation. Economic Journal, 119, 855–882.

    Article  Google Scholar 

  12. Ewerhart, C., & Moldovanu, B. (2001). A stylized model of the German UMTS auction. Working paper, University of Mannheim.

  13. Grimm, V., Riedel, F., & Wolfstetter, E. (2003a). Implementing efficient market structure. Review of Economic Design, 7, 443–463.

    Article  Google Scholar 

  14. Grimm, V., Riedel, F., & Wolfstetter, E. (2003b). Low price equilibrium in multi-unit auctions: the gsm spectrum auction in Germany. International Journal of Industrial Organization, 21, 1557–1569.

    Article  Google Scholar 

  15. Jehiel, P., & Moldovanu, B. (2000). Auctions with downstream interaction among buyers. Rand journal of economics, 31, 768–791.

    Article  Google Scholar 

  16. Jehiel, P., Moldovanu, B., & Stacchetti, E. (1996). How (not) to sell nuclear weapons. American Economic Review, 86, 814–829.

    Google Scholar 

  17. Jehiel, P., Moldovanu, B., & Stacchetti, E. (1999). Multidimensional mechanism design for auctions with externalities. Journal of Economic Theory, 85, 258–294.

    Article  Google Scholar 

  18. Kagel, J., & Levin, D. (2001). Behavior in multi-unit demand auctions: experiments with uniform price and dynamic auctions. Econometrica, 69(2), 413–454.

    Article  Google Scholar 

  19. Kagel, J., & Levin, D. (2005). Multi-unit demand auctions with synergies: behavior in sealed-bid versus ascending-bid uniform-price auctions. Games and Economic Behavior, 53, 170–207.

    Article  Google Scholar 

  20. Klemperer, P. (2002). How (not) to run auctions: the European 3G Telecom auctions. European Economic Review, 46, 829–845.

    Article  Google Scholar 

  21. Klemperer, P. (2003). Using and abusing economic theory—lessons from auction design. Journal of the European Economic Association, 1, 272–300.

    Article  Google Scholar 

  22. Klemperer, P. (2004). Auctions: theory and practice. Princeton: Princeton University Press.

    Google Scholar 

  23. Lebrun, B. & Tremblay, M.-C. (2003). Multi-unit pay-your-bid actions with onedimensional multi-unit demands. International Economic Review, 44, 1135–1172.

    Article  Google Scholar 

  24. List, J., & Lucking-Reiley, D. (2000). Demand reduction in multiunit auctions: evidence from a sportscard field experiment. American Economic Review, 90(4), 961–972.

    Article  Google Scholar 

  25. Milgrom, P. (2004). Putting auction theory to work. Cambridge: Cambridge University Press.

    Book  Google Scholar 

  26. Noussair, C. (1995). Equilibria in a multi-object uniform-price sealed bid auction with multi-unit demands. Economic Theory, 5, 337–351.

    Article  Google Scholar 

  27. Spulber, D. (1995). Bertrand competition when rivals’ costs are unknown. Journal of Industrial Economics, 43, 1–11.

    Article  Google Scholar 

  28. Weber, R. (1997). Making more from less: strategic demand reduction in the fcc spectrum auctions. Journal of Economics and Management Strategy, 6, 529–548.

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Randolph Sloof.

Additional information

A former editor of this journal, Timothy Cason, acted as a guest editor on this submission. His constructive remarks and those of two anonymous referees improved the presentation of the material. We gratefully acknowledge financial support from the Dutch Royal Academy of Sciences (KNAW) and the European Research Council (ERC Advanced Investigator Grant, ESEI-249433). We thank CREED programmer Jos Theelen for programming the experiment.

Electronic Supplementary Material

Below is the link to the electronic supplementary material.

(PDF 420 kB)

Appendix B

Appendix B

In this Appendix we provide the analytic solution for the incumbent’s equilibrium bid function in the ascending auction as determined by (3). To solve this first-order condition it will prove useful to consider a related differential equation:

$$ zG^{\prime}(z)=2\bigl(1+G(z)-z^{2}G(z)^{2}\bigr), $$

with general solution

$$ G_{\alpha_{p,x}}(z)=\frac{1}{z}\frac{I_{1}(2z)+\alpha_{p,x}K_{1}(2z)}{I_{2}(2z)-\alpha_{p,x}K_{2}(2z)} $$

for z≥0. Here I n (K n ) is the n th modified Bessel function of the first (second) kind and α p,x is a constant chosen such that the boundary condition B I (p)=p is satisfied. In the proof of Theorem 1 below we show that this boundary condition is met if \(\alpha_{p,x}=\alpha (\frac{1-p}{x/3} )\) where we define

$$ \alpha(z)=\frac{I_{2}(2z)-(1-1/z)I_{1}(2z)}{K_{2}(2z)+(1-1/z)K_{1}(2z)} $$

Lemma 1

\(G_{\alpha_{p,x}}(\cdot)\) and α(⋅) satisfy the following properties:

  1. (i)

    α(z) is strictly increasing in z with α(0)=0.

  2. (ii)

    \(G_{\alpha_{p,x}}(z)\) has an asymptote at z=z where z solves I 2(2z )/K 2(2z )=α p,x .

  3. (iii)

    The inverse \(G_{\alpha_{p,x}}^{(-1)}(z)\) is well defined for \(z\in\mathcal{D}\) where

    $$\mathcal{D}=\left \{ \begin{array}{l@{\quad}l} \lbrack\frac{x/3}{1-x/3},\frac{x/3}{1-x/3-p}] & \mbox{\emph{if}}\ p<1-x/3\\[2mm] \lbrack\frac{x/3}{1-x/3},\infty)\cup(-\infty,\frac{x/3}{1-x/3-p}] & \mbox{\emph{if}}\ p>1-x/3 \end{array} \right . $$

    an d \(G_{\alpha_{p,x}}^{(-1)}(z)\) for \(z\in\mathcal{D}\) is minimized at \(z=\frac{x/3}{1-x/3-p}\) with \(G_{\alpha _{p,x}}^{(-1)}(\frac{x/3}{1-x/3-p})=\frac{1-p}{x/3}\).

  4. (iv)

    For \(z\in\mathit{Int}({\mathcal{D}})\) we have

    $$ \frac{1}{z} \biggl(\frac{1}{z}+1 \biggr)<G_{\alpha_{p,x}}^{(-1)}(z)^{2}< \biggl(\frac{1}{z}+1 \biggr)^{2} $$


Properties (i) and (ii) can be verified by plotting the right side of (13) and the ratio of modified Bessel functions I 2(2z)/K 2(2z). To establish (iii) note that if \(G_{\alpha_{p,x}}(z)\geq0\) then (12) implies that \(G_{\alpha_{p,x}}(z)\geq G_{0}(z)\equiv I_{1}(2z)/(zI_{2}(2z))\) and using standard properties of the modified Bessel functions we have

$$\frac{1}{G_{0}(z)} \biggl(\frac{1}{G_{0}(z)}+1 \biggr)= \biggl( \frac {I_{0}(2z)I_{2}(2z)}{I_{1}(2z)I_{1}(2z)} \biggr)z^{2}<z^{2}$$

Since \(G_{\alpha_{p,x}}(z)\geq G_{0}(z)\) we conclude that

$$ \frac{1}{G_{\alpha_{p,x}}(z)} \biggl(\frac{1}{G_{\alpha_{p,x}}(z)}+1 \biggr)<z^{2}$$

when \(G_{\alpha_{p,x}}(z)\geq0\). Moreover, inequality (15) is (trivially) satisfied when \(G_{\alpha_{p,x}}(z)\leq-1\) since then the left side is non-positive. Combined with (11) inequality (15) implies that \(G_{\alpha_{p,x}}(z)\) is strictly decreasing, and, hence, invertible when \(G_{\alpha_{p,x}}(z)\leq-1\) or \(G_{\alpha_{p}}(z)\geq0\). This proves that \(G^{(-1)}_{\alpha_{p,x}}(z)\) is well defined for \(z\in\mathcal{D}\) since \(\mathcal{D}\subseteq[0,\infty)\cup(-\infty,-1]\) for all 0≤p≤1 and x>0. If p<1−x/3 then \(G^{(-1)}_{\alpha_{p,x}}(z)\) is strictly decreasing on \([\frac{x/3}{1-x/3},\frac{x/3}{1-x/3-p}]\) and is thus minimized at \(z=\frac{x/3}{1-x/3-p}\). If p>1−x/3 then \(G^{(-1)}_{\alpha_{p,x}}(z)\) declines on \([\frac{x/3}{1-x/3},\infty)\) and it declines on \((-\infty ,\frac{x/3}{1-x/3-p}]\) with \(G^{(-1)}_{\alpha_{p,x}}(\pm\infty)=z^{*}\), so the minimum is again attained at \(z=\frac{x/3}{1-x/3-p}\). A direct computation verifies that \(G_{\alpha_{p,x}}(\frac{1-p}{x/3})=\frac{x/3}{1-x/3-p}\), or, equivalently \(G^{(-1)}_{\alpha_{p,x}}(\frac{x/3}{1-x/3-p})=\frac{1-p}{x/3}\). Finally, the left inequality of (iv) follows from (15) and the fact that \(G^{(-1)}_{\alpha_{p,x}}(z)\) is well defined for \(z\in\mathcal{D}\). To show the right inequality of (iv), note that \(G_{\alpha_{p,x}}^{(-1)}(z)<1/z+1\) is equivalent to \(z<1/G_{\alpha_{p,x}}(z)+1\), which using (12) can be written as

$$1-\frac{1}{z}<\frac{I_{2}(2z)-\alpha_{p,x} K_{2}(2z)}{I_{1}(2z)+\alpha_{p,x} K_{1}(2z)}$$

Using \(\alpha_{p,x}=\alpha (\frac{1-p}{x/3} )\) and the definition of α(⋅) in (13) this can be rewritten as \(\alpha (z)>\alpha (\frac{1-p}{x/3} )\). Since α(⋅) is increasing, the right inequality in (iv) follows if \(z> (\frac{1-p}{x/3} )\) for all z such that \(G_{\alpha_{p,x}}(z)\in\mbox{Int}({\mathcal{D}})\). Since \(\frac{1-p}{x/3}=G^{(-1)}_{\alpha_{p,x}}(\frac{x/3}{1-x/3-p})\) this is true if \(G^{(-1)}_{\alpha_{p,x}}(z)>G^{(-1)}_{\alpha_{p,x}}(\frac{x/3}{1-x/3-p})\) for all \(z\in\mbox{Int}({\mathcal{D}})\), which holds since \(G^{(-1)}_{\alpha_{p,x}}(z)\) is minimized at \(z=\frac{x/3}{1-x/3-p}\). □

With Lemma 1 we are able to characterize B I (v) in Proposition 2.

Theorem 1

The incumbents’ equilibrium bid function in the ascending auction is given by:

$$ B_{I}(v)=1-\frac{(x/3)^{2}}{1-v}G_{\alpha_{p,x}}^{(-1)} \biggl( \frac {x/3}{1-x/3-v} \biggr)^{2} $$

B I (v) is strictly increasing for 0≤v<p with B I (p)=p, satisfies v<B I (v)<v+x/3 for 0≤v<p, and lim x→0 B I (v)=v.


Note that for 0≤vp and 0≤p≤1, the ratio (x/3)/(1−x/3−v) lies in the set (−∞,−1]∪[0,∞) on which \(G_{\alpha_{p,x}}^{(-1)}\) is well defined (see proof of Lemma 1). We first verify the necessary (first-order and boundary) conditions. Differentiating (16) with respect to v yields

Using (16) we can rewrite this as

$$B_{I}^{\prime}(v)=-\frac{1-B_{I}(v)}{1-v} \biggl(1+ \frac{(1-v)(x/3)}{(1-x/3-v)^{2}}\frac{2}{G_{\alpha_{p,x}}^{(-1)} (\frac{x/3}{1-x/3-v})G_{\alpha_{p,x}}^{\prime} (G_{\alpha_{p,x}}^{(-1)}(\frac {x/3}{1-x/3-v}) )} \biggr) $$

which can be rewritten using (11)

which, using (16), further simplifies to

$$B_{I}^{\prime}(v)=-\frac{1-B_{I}(v)}{1-v} \biggl(1+ \frac{(1-v)(x/3)}{(1-x/3-v)^{2}}\frac{1}{1+\frac{(x/3)}{(1-x/3-v)}-\frac{(1-v)(1-B_{I}(v))}{(1-x/3-v)^{2}}} \biggr) $$

and, after rearranging terms


which is the first-order condition in (3). To verify the boundary condition, note that B I (p)=p can be rewritten as

$$G_{\alpha_{p,x}}^{(-1)} \biggl(\frac{x/3}{1-x/3-p} \biggr)= \frac{1-p}{x/3}$$

or, equivalently,

$$\frac{x/3}{1-x/3-p}=G_{\alpha_{p,x}} \biggl(\frac{1-p}{x/3} \biggr) $$

which yields a linear equation in α p,x , see (12), that can readily be solved to yield \(\alpha_{p,x}=\alpha (\frac{1-p}{x/3} )\), with α(⋅) defined in (13).

To show that B I (v)<v+x/3, use the left inequality of (14) in property (iv) of Lemma 1

$$B_{I}(v)<1-\frac{(x/3)^{2}}{1-v}\frac{1}{\frac{x/3}{1-x/3-v}}\biggl( \frac{1}{\frac{x/3}{1-x/3-v}}+1 \biggr)=v+x/3 $$

Similarly, the right inequality in (14) implies

$$B_{I}(v)>1-\frac{(x/3)^{2}}{1-v} \biggl(\frac{1}{\frac{x/3}{1-x/3-v}}+1 \biggr)^{2}=v $$

Note that v<B I (v)<v+x/3 for 0≤v<p implies that \(B_{I}^{\prime }(v)>0\) for all 0≤v<p, see the first-order condition (3).

Finally, the limiting property of B I (v) follows from the fact that \(G_{\alpha_{p,x}}^{(-1)}(z)\sim1/z\) for z small. Hence, as x tends to zero, B I (v) limits to

$$\lim_{x\rightarrow0}B_{I}(v)=\lim_{x\rightarrow0}1-\frac{(x/3)^{2}}{1-v} \biggl(\frac{1-x/3-v}{x/3} \biggr)^{2}=1-(1-v)=v $$


Rights and permissions

Reprints and Permissions

About this article

Cite this article

Goeree, J.K., Offerman, T. & Sloof, R. Demand reduction and preemptive bidding in multi-unit license auctions. Exp Econ 16, 52–87 (2013).

Download citation


  • Multi-license auctions
  • Demand reduction
  • External effects
  • Preemption

JEL Classification

  • D44
  • D45
  • C91