auctions, aftermarket competition, and risk attitudes...choices in risky environments depending on...

Int. J. Ind. Organ. 27 (2009) 274–285International Journal of Industrial Organization 27 (2009) 274–285

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International Journal of Industrial Organization

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Auctions, aftermarket competition, and risk attitudes☆

Maarten C.W. Janssen a,b,c, Vladimir A. Karamychev b,⁎a Department of Economics, University of Vienna, Hohenstaufengasse 9, 1010 Vienna, Austriab Department of Economics, Erasmus University, P.O. Box 1738, 3000 DR Rotterdam, The Netherlandsc Tinbergen Institute, The Netherlands

☆ We thank Jay Pil Choi, two anonymous referees,ASSET 2004, Stockholm School of Economics and Tinbvaluable comments.⁎ Corresponding author. Department of Economics, E

Burg. Oudlaan 50, NL-3062 PA Rotterdam, The NetherlaE-mail address: [email protected] (V.A. Karam

0167-7187/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.ijindorg.2008.08.005

a b s t r a c t
a r t i c l e i n f o
Article history:
With the experience of the Received 7 March 2007Received in revised form 5 August 2008Accepted 27 August 2008Available online 9 September 2008
JEL classification:D43D44D82

Keywords:AuctionsAftermarketsRisk attitudesSelection

sequence of UMTS auctions held worldwide in mind, we consider a situationwhere firms participate in license auctions to compete in an aftermarket. It is known that when a monopolyright is auctioned, auctions select the bidder that is least risk-averse. This firm will choose a higher value ofthe aftermarket strategic variable than any other firm will do, thereby implying a higher market price underprice setting behavior and a lower price due to higher quantity under quantity-setting behavior. This paperextends the analysis to oligopoly aftermarkets and analyzes whether the monopoly result carries over tooligopoly settings. We argue that with multiple licenses and demand uncertainty auctions actually performeven worse from a welfare point of view than the monopoly case would suggest. A strategic effectstrengthens the monopoly result with respect to prices, but weakens the result with respect to quantities.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

In many markets around the world, governments use auctions toallocate licenses to interested parties. A characteristic feature of theseauctions is that after the auction is held, the winning firms competewith each other in an aftermarket where the revenues and profitsdepend not only on the market strategies that the licensed firmschoose, but also on uncertainty concerning important market features,such as demand or cost. Winning a license is verymuch like winning alottery ticket where the value of the ticket is uncertain. Importantexamples that share these features are the UMTS auctions that havebeen held in the past years all around the world. Other examplesinclude the selling of radio frequencies for commercial radio and theselling of licenses to exploit gasoline stations on certain locations.

If firms were completely risk-neutral (an assumption often madefor analytical tractability), this uncertainty would not affect firms'behavior in a crucial way. However, as we will briefly argue below,there are good reasons to believe that firms are not necessarily risk-neutral and, in fact, that firms differ in their risk attitudes. In this

and audiences at ESEM 2004,ergen Institute Rotterdam for

rasmus University Rotterdam,nds.ychev).

l rights reserved.

paper, we analyze how firms' risk attitudes affect their biddingbehavior in an auction followed by an aftermarket.

Janssen and Karamychev (2007) ask a similar question in casegovernments decide to allocate only one license, thereby creating amonopoly in the aftermarket. They identifya risk attitudeeffect and showthat the least risk-averse firm has the highest certainty equivalent for theaftermarket game (lottery) and, therefore, will win the license in anystandard auction format. This firm chooses a higher value of its strategicvariable, e.g., price or quantity, than any other firmwould choose. Thus, ifthe strategic variable is price, auctions select thefirm that sets thehighestaftermarket price, leading to awelfare loss relative to any other allocationof licenses.On theotherhand, if the strategic variable is quantity, auctionsselect the firm that sets the highest quantity, resulting in a loweraftermarket price (and awelfare gain).We refer to this selection aspect ofauctions as to the “monopoly result”.

A crucial assumption in the present paper and in Janssen andKaramychev (2007) is that firms are not necessarily risk-neutral.Empirical studies in finance indicate that firms may indeed be risk-averse, or that their behavior is as if they were risk-averse. Nance et al.(1993) and Geczy et al. (1997), among others, argue that firms hedgeagainst different types of exogenous shocks such as exchange ratevolatility. In a study on the goldmining industry, Tufano (1996) arguesthat delegation of control to a risk-averse manager, whose renumera-tion is linked to the firm's performance, may cause the firm to takeactions in a risk-averse manner. As delegation of control differsbetween firms, and as the payment structure of managers differs from

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2 We compare consumer welfare for different number of licenses indirectly, i.e.,through its comparison with welfare generated by, e.g., a random allocation. In ouranalysis, the main effect of the number of licenses on consumer welfare is independentfrom firms' risk attitudes. Thus, a normative analysis of the optimal (from consumers'prospective) number of licenses does not depend on whether the licenses areauctioned or assigned randomly.

275M.C.W. Janssen, V.A. Karamychev / Int. J. Ind. Organ. 27 (2009) 274–285

firm to firm, firms may very well act as if their attitudes towards riskdiffer significantly from one to the other. On the other hand, liquidityconstraints and prospects of bankruptcy may force managers toundertake risky actions. Therefore, firms' managers can be both risk-averse and risk-seeking. In an experiment that they conducted with224 managers, Laughhunn et al. (1980) showed that there are indeedlarge individual differences in risk attitudes between managers.

It is well-known that under a general form of risk aversion, the actionprice, which is sunk at the market stage, affects firms' aftermarketbehavior. Inparticular,firmswith identical utility functionsmakedifferentchoices in risky environments depending on how wealthy they are.McMillan (1994) argues that as auctions force firms to pay considerableamounts of money for licenses, auctions may force firms to behavedifferently in the marketplace. This wealth effect, however, disappearswhen firms are characterized by constant absolute risk aversion (CARA).As wewant to concentrate on the selection aspect of auctions and not onthe wealth effect, we assume that firms have CARA utility functions.

The main question we address in this paper is whether themonopoly result discussed above carries over to a situation wheregovernments auction multiple licenses. This question is relevant inmost applications (UMTS, radio frequencies, etc.) as governmentsoften create oligopoly markets by auctioning off more than onelicense. We will say that the monopoly result generalizes to theoligopoly context if the auction has an equilibrium in which the mostrisk-seeking (least risk-averse) firms make the highest bids. We referto such an equilibrium as a “risk-seekers' equilibrium”, andwe analyzethe conditions under which it exists. To avoid complications arisingfrom the possibility that players would like to signal their typethrough their bids, we analyze this question in a private informationscenario, where firms are neither informed about each other bids, norabout the risk attitudes of each other in the aftermarket.1

This private information scenario arises in a sealed-bid uniform-price multi-unit auction where winning firms are the firms with thehighest bids, and they pay a license fee which is equal to the highestnon-winning bid. This is the auction formatwe adopt in this paper andthe main features of this format were used, for example, in the DanishUMTS auction in 2001, see National Telecommunications AgencyDenmark (2001). Qualitatively similar (although more complicated)results can, however, also be obtained for other formats, for example apay-your-bid multi-unit auction, see, e.g., Janssen and Karamychev(2005). Using this auction format, we consider both demand and costuncertainty, and distinguish between differentiated Bertrand (strate-gic complements) and Cournot (strategic substitutes) oligopolies.

The main result we obtain is as follows. Under demanduncertainty, auctions actually perform worse than what could beexpected on the basis of the monopoly result. Under differentiatedBertrand oligopoly, the aftermarket externality (due to strategicinteractions between firms) reinforces the risk attitude effect. Thisimplies that indeed the least risk-averse firms are selected and theychoose the highest prices. Thus, like in the monopoly setting,auctioning multiple licenses results in higher market prices than ifany other allocation mechanism is used. Under Cournot oligopoly, theaftermarket externality works in the opposite direction and weakensthe risk attitude effect. When the externality is sufficiently strong, arisk-seekers' equilibrium fails to exist. This implies that the least risk-averse firms do not necessarily get licenses. Thus, and contrary to themonopoly setting, auctioning multiple licenses does not necessarilyresult in higher quantities and, therefore, lower market price than ifthe licenses were allocated differently. Hence, from a consumerwelfare perspective the attractiveness of using auctions relative to

1 Signaling plays an important role in a different scenario where the bids of thewinning firms but not their types become public so that the auction stage has to beanalyzed as an N-player signaling game. Another scenario, used in Section 3 forillustration purposes, where types of all winning firms become public, does not seemto be very realistic. Nevertheless, all these scenarios can be analyzed in a similar way tothe private information scenario.

using any other allocation mechanism is higher in the case of a singlelicense than in the case of multiple licenses.2

The reason why the aftermarket externality works in the oppositedirection inCournot competition is that in a risk-seekers' equilibrium, thewinning firms compete with the least risk-averse rivals. These firmschoose the highest quantities and, therefore, are the most aggressivecompetitors. It remains true that for a given set of other players in theaftermarket, a relatively less risk-aversefirm iswilling topaymore for thelicense than a relativelymore risk-averse firm is willing to pay. However,it may well be that a set of the least risk-averse firms make less profitsand, therefore, have a smallerwillingness topay for the licenses thana setof more risk-averse firms do. That is why less risk-averse firms are notwilling to pay high auction prices and outbid more risk-averse firms.

Our second result for Cournot competition is that the aftermarketexternality and the ex-ante affiliation of firms' risk types together give riseto a strategic effect that does not only cause that a risk-seekers'equilibrium fails to exist, but that creates the conditions for another,“risk-averse players' equilibrium” to exist. In a risk-averse players'equilibrium, the most risk-averse firms (or the least risk-seeking firms—depending on the risk attitudes under consideration) submit the highestbids, win the licenses, and choose the lowest production levels, whichagain, as in the Bertrand case, results in the highest possible aftermarketprice.

Another type of uncertainty that may exist and which we analyzeas well is uncertainty about production costs. Here we show that if theuncertainty is about the level of fixed cost then the monopoly resultdoes generalize to the oligopoly situation, whereas if the uncertainty isabout marginal cost then a strategic effect appears, and the monopolyresult for both price and quantity setting does not generalize.

The paper is organized as follows. An overviewof related literature isprovided in Section 2. As the model itself and the equilibrium analysisare fairly complicated, we start off in Section 3 with an example wherewe make some simplifying assumptions. The example makes itintuitively clear why under demand uncertainty the monopoly resultcarries over under Bertrand competition, but not under Cournot. Section4 then presents the formal model where these simplifying assumptionsare not made. The main results for the general model are provided inSection 5. Section 6 derives implications of the general results in case ofdemand uncertainty, while Section 7 delves into the case of costuncertainty. Section 8 concludes and the Appendix A contains all proofs.

2. Literature review

There is a relatively large, recent literature on the possibility ofinefficient allocation of licenses in auctions due to the presence ofinterdependencies. First, there is a literature where one license isauctioned and the auction winner competes in the aftermarket withnon-winners.3 Moldovanu and Sela (2003) analyze a situation whereaftermarket competition is characterized by Bertrand competition andcost is private information at the auction stage.4When in such a situationa patent for a cost-reducing technology is auctioned amongst thecompetitors, they show that standard auction formats do not exhibitefficient equilibria where bids are increasing in the firms' efficiencyparameter. Goeree (2003) andDas Varma (2003) analyze a similar setting

3 Jehiel and Moldovanu (2006) provide an overview of existing work in this area andargue that in case firms’ aftermarket profits depend on private information in thehands of other winning firms there is an informational externality. See, also, Jehiel etal. (1996) and Jehiel and Moldovanu (2000) for related papers where an (informa-tional) externality may lead to inefficiency even in standard single-unit auctions.

4 To avoid signaling issues, they consider the case where the true production costs ofthe bidding firms are revealed after the auction.

276 M.C.W. Janssen, V.A. Karamychev / Int. J. Ind. Organ. 27 (2009) 274–285

but allow for signaling private information through the auction bid. DasVarma (2003) shows that when there is Cournot competition (strategicsubstitutes) in the aftermarket an efficient equilibriumdoes exist asfirmshave an incentive to signal their (high) efficiency. Under Bertrandcompetition (strategic complements), this is not the case as other firmswill adjust their market price downwards, the more efficient a firm isknown to be. Katzman andRhodes-Kropf (2008) showhowdifferent bid-announcement policies affect an auction's revenue and efficiency.

The above papers show that single-object auctions may result ininefficiencies, if values are strongly interdependent. We add to thisliterature by arguing that in multi-object auctions, inefficiencies mayarise even when the aftermarket externality is weak provided thebidders' types are ex-ante affiliated. Moreover, external effects offirms' attitudes towards risk have not been analyzed before. In thispaper we propose a technique that is capable of providing a fullanalysis of a multi-player signaling game for the imperfect informa-tion scenario, when firms' bids aremade public after the auction stage.

There is also a literature (see, for example, Hoppe et al., 2006 andGrimm et al., 2003) on inefficiencies created by multiple licenseauctions if players have interdependent valuations. However, thesepapers usually address issues related to collusion and/or asymmetriesbetween different participants.5

Apart from the auction literature, the present paper is intimatelyrelated to the early literature on price and quantity-setting behavior ofa risk-aversemonopolist (cf., Baron,1970,1971 and Leland,1972). Theyshow that the more risk-averse a price setting monopolist is the lowerthe price it sets. These results have recently been generalized to thecase of market competition (see Asplund, 2001).

3. An Example

Consider an example where N=3 risk-averse firms compete for n=2licenses. We first consider the case where firms winning the auctioncompete in prices in a differentiated product aftermarket and they faceuncertaindemand that is representedby the followingdemand functions:

q1 p1;p2ð Þ ¼ 1þ θ−p1 þ p2 and q2 p1; p2ð Þ ¼ 1þ θ−p2 þ p1;

where θ is a demand shock that is normally distributed with zeromean and variance DN0. In this example, we allow demand to benegative which happens when θb−1. Moreover, we assume in thisSection that risk attitudes of the two winning bidders are revealedafter the auction. With these simplifying assumptions, a closed-formsolution can be obtained for most of the analysis, so that we are betterable to explain the main intuition for the results we get. In the mainbody of the paper, we consider a private information case, where riskattitudes of the winners are not revealed after the auction so that thebidders remain uncertain about risk attitudes of each other in themarket stage, and we do not restrict the analysis to specific demandfunctions or specific forms of demand uncertainty.

Normalizing firms' production costs to zero, the profit of firm 1 isgiven by

π1≡π p1; p2; θð Þ ¼ p1dq1 p1;p2ð Þ ¼ p1 1þ θ−p1 þ p2ð Þ:

As explained in the Introduction, we assume firms having CARAutility functions throughout the paper so as to focus on the selectionaspect of auction. Thus, the utility function Ui(πi) of firm i is given by

Ui π i� �

≡U π i; ri� �

¼ 1− exp −πiri� ��

=ri ; ri ≠ 0πi ; ri ¼ 0

;

�ð1Þ

where ri measures the absolute risk aversion of firm i. We assume inthis example that firms' risk attitudes are positive, i.e., ri≥0, and are

5 Krishna (2002), Example 6.4, shows that it might happen that the bidder with thehighest type is the bidder with the lowest valuation and that standard auction formatsdo not have efficient equilibria.

weakly affiliated.6 This, together with the normally distributeddemand shock θ, implies that the expected utility W1 (of receivingan uncertain aftermarket profit) of firm 1 can be written as follows:

W1≡W p1;p2;w; r1ð Þ ¼ 1

2πDð Þ12

Z∞−∞

U1 π1−w� �

exp −12D

θ2� �

dθ

¼ 1r1

1− exp −r1 p1 1−p1 þ p2ð Þ−w−12Dr1p21

� �� ¼ U1 π p1;p2;0ð Þ−w−

12Dr1p21

� �where w is the price to be paid for the license. In other words, thecertainty equivalent of the uncertain aftermarket profit is the netprofit π(p1,p2,0)−w adjusted by the amount −Dr1p12/2 due to riskaversion. When firms get to know each others' risk types after theauction, theymaximizeWi w.r.t. pi taking the price of its competitor asgiven. Maximizing W1 w.r.t. p1 yields the first-order condition:

0 ¼ 1− 2þ Dr1ð Þp1 þ p2;

and the following (linear) reaction function for firm 1:

p1 p2ð Þ ¼ 1þ p2ð Þ= Dr1 þ 2ð Þ: ð2Þ

Nash equilibrium prices are then easily calculated to be

p⁎1 ¼ Dr2 þ 3Dr1 þ 2ð Þ Dr2 þ 2ð Þ−1 andp⁎2 ¼ Dr1 þ 3

Dr1 þ 2ð Þ Dr2 þ 2ð Þ−1 : ð3Þ

Here, one can see that these prices are decreasing in both firms'risk attitudes. As a player becomesmore risk-averse, his reaction curveshifts downwards (this is because the bad states of the world, inwhichthe firm optimally sets low prices, receive more weight in the utilityevaluation) and, because of strategic complementarity, both equili-brium prices decrease. The reduced-form certainty equivalent π ofwinning and getting the uncertain aftermarket profit for firm 1 is

~π r1; r2;wð Þ≡π p⁎1 ;p⁎2 ;0

� �−w−

Dr1 p⁎1� �22

¼ Dr2 þ 3ð Þ2 Dr1 þ 2ð Þ2 Dr1 þ 2ð Þ Dr2 þ 2ð Þ−1ð Þ2

−w ¼ v1−w;

where we define

v1≡v r1; r2ð Þ ¼ Dr2 þ 3ð Þ2 Dr1 þ 2ð Þ2 Dr1 þ 2ð Þ Dr2 þ 2ð Þ−1ð Þ2

: ð4Þ

The amount v(r1,r2) is firm 1's valuation of the license in theauction stage, had the firm known the risk attitude of the otherwinning firm. The signs of the partial derivatives are:

@v@r1

¼ −D Dr2 þ 3ð Þ2 Dr1 þ 2ð Þ Dr2 þ 2ð Þ þ 1ð Þ

2 Dr1 þ 2ð Þ Dr2 þ 2ð Þ−1ð Þ3b 0; and ð5Þ

@v@r2

¼ −D Dr1 þ 2ð Þ Dr2 þ 3ð Þ Dr1 þ 3ð Þ

Dr1 þ 2ð Þ Dr2 þ 2ð Þ−1ð Þ3b 0: ð6Þ

First, firms' valuations for licenses monotonically decrease withtheir own risk attitude. This is the risk attitude effect we noted in theIntroduction, and it is always negative. Second, firms' valuationsmonotonically decrease with risk attitudes of their rivals. This is theaftermarket externality that affects firms' bidding behavior during theauction. With an increase in a firm's risk attitude, its reaction function

6 Weak affiliation includes the case of statistical independence; see, e.g., Krishna(2002).


(2) moves downwards and, due to the strategic complementarity,firms' Nash equilibrium prices (3) decrease. This has a negative impacton the firm's profit and, consequently, on its valuation for the license.

In the auction stage of the game, firms' risk attitudes are drawnfrom a certain joint distribution, and are private information to them.A bidding strategy in the auction is, therefore, a monetary bidconditional on a firm's risk attitude, which we denote by b(r). In orderto calculate the optimal (equilibrium) bid, we make in this exampletwo additional simplifying assumptions. First, we formally assumethat the variance D is so small that we can approximate firms' reactionfunctions (2), the Nash equilibrium prices (3), and the valuationfunctions (4) by their linear approximations around D=0:

p1 ¼ 1þ p22

1−12r1D

� �; p⁎1 ¼ 1−

23r1 þ 1

3r2

� �D;

and v r1; r2ð Þ ¼ 1−16

5r1 þ 4r2ð ÞD:

Second, we assume that firms' risk attitudes are independently anduniformly distributed over the interval [0,1].

Given these additional simplifications, we can construct a risk-seekers' equilibrium, where the most risk-seeking firms will win thelicenses, i.e., where the equilibrium bidding function b(r) is adecreasing function. From the perspective of firm 1 in such anequilibrium it is the most risk-averse of the other two firms (let us callhim firm 3) that determines the auction price, i.e., w=b(r3). Theaftermarket rival, firms 2, has risk attitude r2≤r3. Hence, firm 1′sexpected utility V1 of winning the auction given type r3 of firm 3 is:

V1≡V r1; r3ð Þ ¼ E U1 v r1; r2ð Þ−b r3ð Þð Þjr2br3� �

¼ 1r3

Z r3

0

1− exp −r1 v r1; r2ð Þ−b r3ð Þð Þð Þr1

dr2

¼ 1r1r3

r3− exp −r1 1−56r1D−b r3ð Þ

� �� exp 2r1r3D=3ð Þ−1

2r1D=3

� �For small values of D we can approximate the latter exponent as

follows:

exp 2r1r3D=3ð Þ≈1þ 2r1r3D=3:

This leads to the final expression

V r1; r3ð Þ ¼ 1− exp −r1 1−5r1D=6−b r3ð Þð Þð Þr1

¼ U1 1−56Dr1−b r3ð Þ

� �: ð7Þ

This implies that the certainty equivalent of the auction game isequal to the ex-ante value for the license (1−5Dr1 /6) minus theauction price b(r3).

In a uniform-price auction,which is a generalization of a second-priceauction, winning firms pay the highest losing bid and a bidder's Nashequilibriumstrategy is tobid theirownvalue.Hence, the bidding function

b rð Þ ¼ 1−5Dr=6

constitutes the unique monotone symmetric Nash equilibrium.7 Thisillustrates that the monopoly result generalizes to oligopoly competi-tion settings with uncertain demand and strategic complements.

We now explain the more general intuition behind this result. Theresult depends on two forces that are at work. First, due to the riskattitude effect, a firm that is more risk-averse bids less because it has alower valuation for given types of its competitors. This is reflected in the

7 Note that this is indeed a decreasing function so that we have a consistent solution.In addition, it can be shown that the second-order condition is satisfied in thisexample. The general treatment of the second-order condition is in Proposition 1 inSection 5.

first derivative in Eq. (5). Second, even if firms' risk attitudes are ex-ante statistically independent, they are positively correlated amongstthe firms that win the auction ex-post. Thus, when bidding for alicense, a firm takes into account that if it wins the auction the rivalfirm is not just a firm with an average risk attitude. In particular, in amonotone decreasing equilibrium, a firm knows that if it wins theauction, it cannot be the case that the other two firms have lower riskattitudes than it has itself. This implies that a firm that is more risk-averse expects to compete (upon winning) with a firm that is morelikely to be more risk-averse.

As this point is one of the crucial points made in this paper, weillustrate it graphically. In Fig. 1, the square (r2,r3)∈ [0,1]2 representsthe ex-ante support of the joint distribution of (r2,r3). Let risk attitudeof firm 1 take a value of r1⁎. If firm 1 wins the auction, it realizes thatthe risk attitudes of firms 2 and 3 cannot be both below r1⁎:

Pr r2br⁎1 ; r3br⁎1 jFirm1wins a license

� �¼ 0:

The reason for this is that if it were that r2b r1⁎ and r3br1⁎, firm 1would not have won the license because firms 2 and 3 would have bidhigher than firm 1. Hence, the expected risk attitude of theaftermarket rival of firm 1 is the lowest of the two risk attitudes r2and r3, i.e., min(r2,r3) calculated over the remaining (shaded) area.

To see that there is a positive correlation between risk attitudes offirms that havewon licenses, we calculate the expected risk attitude ofa rival winning firm conditional on the risk attitude of firm 1 if thelatter has won a license. This expected value equals to

E min r2; r3ð Þjmax r2; r3ð ÞNr⁎1� �

¼ E r2jr3Nr⁎1 ; r3Nr2� �

¼ 13

1þ r⁎211þ r⁎1

!;

and it is easy to see that it increases for all r1⁎∈ (0,1). Hence, a firm'sown risk attitude has a second, indirect effect on its bid through (i) theex-post positive correlation, and (ii) the aftermarket externality ∂v/∂r2b0. This indirect effect is what we call the strategic effect.

In the case of demand uncertainty and price competition, thestrategic effect is negative (because the externality is negative) and,therefore, reinforces the risk attitude effect: a more risk-averse firmexpects (while bidding in the auction) to compete in the aftermarketwith a more risk-averse rival if it wins a license. Because a firm's riskattitude has a negative impact on the other firm's valuation, this furtherreduces the firm's incentive to bid high. Consequently, the monopolyresult in the present example carries over to the oligopoly case.

Fig. 1. Support (shaded) of the joint distribution of (r2,r3) conditional on r1=r1⁎.

8 This formulation does not affect the generality of the model as any change in thedistribution of θ can be modeled as a corresponding change of the profit function π(si,s−i,θ).


When firms compete in quantities a la Cournot the situation isdifferent. This can be illustrated by modifying our example so thatfirms compete in quantities and the profit of firm 1 is given by

π1≡π q1; q2; θð Þ ¼ q1 1þ θ−q1−q2ð Þ:

Under the CARA specification (1), expected utilityW1 of firm 1 canbe written as follows:

W1≡W q1; q2;w; r1ð Þ ¼ U1 π1−w−Dr1q21=2� �

:

Maximizing W1 with respect to q1 yields the following reactionfunction of firm 1:

q1 q2ð Þ ¼ 1−q2ð Þ= Dr1 þ 2ð Þ:

Nash equilibrium quantities are then given by

q⁎1 ¼ Dr2 þ 1Dr1 þ 2ð Þ Dr2 þ 2ð Þ−1 andq⁎2 ¼ Dr1 þ 1

Dr1 þ 2ð Þ Dr2 þ 2ð Þ−1 :

Cournot equilibrium quantities are decreasing in a firm's own riskattitude, but –more importantly – are increasing in the risk attitude ofthe other winning firm with whom they are competing in theaftermarket. As a player becomes more risk-averse, his reaction curveshifts downwards and, due to the fact that quantities are strategicsubstitutes, a player's own quantity choice decreases, whereas thequantity of his rival increases.

The valuation of firm 1 for the license in the auction stage caneasily be calculated to be:

v1≡v r1; r2ð Þ ¼ Dr2 þ 1ð Þ2 Dr1 þ 2ð Þ2 Dr1 þ 2ð Þ Dr2 þ 2ð Þ−1ð Þ2

:

This function has the following monotonicity properties:

@v@r1

¼ −D Dr2 þ 1ð Þ2 1þ Dr1 þ 2ð Þ Dr2 þ 2ð Þð Þ

2 Dr1 þ 2ð Þ Dr2 þ 2ð Þ−1ð Þ3b0;

and@v@r2

¼ D Dr1 þ 2ð Þ Dr2 þ 1ð Þ Dr1 þ 1ð ÞDr1 þ 2ð Þ Dr2 þ 2ð Þ−1ð Þ3

N0:

This implies that the risk attitude of a firm's rival has a positiveeffect on the firm's valuation, i.e., the aftermarket externality now ispositive. Consequently, in a risk-seekers' equilibrium, as a firm that ismore risk-averse expects (due to ex-post positive correlation) tocompete with another firm that is also more risk-averse, and becausefirms under Cournot competition prefer to compete with more risk-averse rivals, the indirect strategic effect is now positive, and it worksagainst the risk attitude effect.

The relation between these two effects determines the conditionfor the existence of amonotone symmetric bidding equilibrium.Whenthe strategic effect is small, the unique equilibrium is a risk-seekers'equilibrium in which players who are less risk-averse bid higher, andthe equilibrium bidding function is decreasing in a firm's risk attitude.When the strategic effect is larger than the risk attitude effect for somevalues of risk attitudes, a risk-seekers' equilibrium fails to exist.Finally, when the strategic effect is larger than the risk attitude effectfor all values of risk attitudes, not only the risk-seekers' equilibriumfails to exist, but another, risk-averse players' equilibrium appears. Insuch an equilibrium, the most risk-averse firms (or the least risk-seeking firms) submit the highest bids, and the bidding functionmonotonically increases in firms' risk attitude.

We now proceed to describe the general model that we consider inthe main body of the paper, and show that the above properties dohold more generally. The main difference with the example presentedhere is that when uncertainty is not normally distributed andinformation about risk attitudes is not revealed after the auction,closed-form solutions cannot be obtained. We therefore will resort to

the technique of Taylor expansions to arrive at solutions in the generalmodel.

4. The model

We consider amarketwhere n≥2 firms compete by simultaneouslychoosing a value of the strategic variable s. Depending on the market,we interpret s as either a price (strategic complements) or a quantity(strategic substitutes). The profit πi of firm i is determined by the levelof s that firm i chooses, and on the levels of s chosen by all other (n−1)firms. In addition to this, we assume that thefirms'market profitabilityis affected bya random shock θ, which can represent either uncertaintyconcerning market demand or industry-wide production cost. Thus,we write the market profit of firm i as πi=πi(s1,s2,…sn,θ).

We assume that πi is symmetric in all sj for j≠ i, and identical for alli. Hence, πi can be written as πi=π(si,s− i,θ), where s− i stands for choicesof s of the firms others than i. To shorten notation, we denote thepartial derivatives of π as follows:

πi≡@π=@si; πj≡@π=@sj for j≠i; πθ≡@π=@θ; πi;j≡@2π=@si@sj; etc:

We use similar notations for partial derivatives of other functions.The random shock θ is assumed to be uniformly distributed over

the interval [−α,α], where α∈ (0,1) is a parameter.8 If θ is a demandshock and firms compete in prices then s=p and firms fulfill theirrandom demands qi=q(pi,p− i,θ). If, to the contrary, firms compete inquantities then s=q, prices that consumers pay for these quantities arepi=p(qi,q− i,θ), and firms accept to sell their pre-determined quantitiesqi at these random prices. Finally, if θ is an industry cost's shock thenfirms' profit function π can be written as π(si,s− i,θ)=Ri(si,s− i)−C(qi(si,s− i),θ), where firms' revenues Ri(si,s− i) and outputs qi(si,s− i) aredetermined by the nature ofmarket competition (prices or quantities).

With respect to the uncertainty parameter θ we assume that thesingle-crossing property holds, i.e., πi,θ keeps its sign. This single-crossing property, which is sometimes referred to in this context as tothe Principle of IncreasedUncertainty (PIU) due to Leland (1972), statesthat the marginal profit πi≡∂π/∂si is monotonically increasing in θ.

Access to the market is limited to the firms that have obtainedlicenses to operate in the market. There are N≥ (n+1) firms, and thegovernment allocates n licenses in the multi-unit uniform-priceauction, where n highest bids win licenses, and the winners pay alicense fee equal to the highest non-winning bid. This uniform-priceauction allows us to simplify the exposition of results while keepingthe market stage competition quite general.

Firms differ in their attitude towards risk: some firms are morerisk-averse than others. We assume that firm i has CARA utilityfunction Ui(πi), given by Eq. (1). Risk types ri, i=1,…,N, are identicallydistributed over the finite support [r,r ] in accordance with the jointdistribution function F(r1,…,rN). Positive values of ri correspond torisk-averse firms whereas negative values correspond to risk-seekingfirms. A player's risk attitude is his type, and is private information inboth the auction stage and the aftermarket stage of the game.

The expected utility of a firm i that gets a license to operate in themarket at the auction price w, has a risk type ri, sets a level si of thestrategic variable, and competes with firms that set levels s− i of thestrategic variable, is denoted by Wi. It follows that

Wi≡W si; s−i;w; rið Þ ¼ 12α

Zα−α

U π si; s−i; θð Þ−w; rið Þdθ:

In order to ensure the existence, uniqueness and stability of the(Bayes–) Nash equilibrium in the aftermarket, the expected utility


function W must satisfy a stability requirement. We follow Bulowet al. (1985) and assume thatWi, i +∑ j≠ iWi , jb0 ifWi , jN0, and thatWi , i−Wi , jb0 for all j≠ i ifWi , jb0.

In the auction stage, a firm i submits a bid bi, which is based on itsrisk type ri, and we denote a monotone symmetric equilibriumbidding function by b(x), so that bi=b(ri). We define Z to be the risktype of the firm that submits the nth highest bid amongst all (n−1)firms other than firm i. Hence, if the bidding function is an increasingfunction b(+)(x), Z is the nth highest-order statistics amongst rj, j≠ i;we denote the distribution of Z conditional on ri=x by G(+)(z|x). If, onthe other hand, the bidding function b(−)(x) decreases, Z is the nthlowest-order statistics; its conditional distribution is denoted by G(−)

(z|x). Thus,

G Fð Þ zjxð Þ ¼ Pr Zbzjri ¼ xð Þ:

The corresponding distribution density function is denoted by g(±)

(z|x):

g Fð Þ zjxð Þ ¼ dG Fð Þ=dz:

In the market stage, firm i maximizes its expected utility W withrespect to its strategy si conditional on the fact that all its competitors(we index them by j) won the auction and all other firms (we indexthem by k) lost the auction. Suppose that Z takes a value z. Thus, if afirm i of type ri=x wins a license and pays an auction price w=b(z), itchooses its Nash equilibrium strategy si⁎=s(x,z) where

s x; zð Þ ¼ argmaxsi

E W si; s−i;w; rið Þjri ¼ x; b rj� �

Nb zð Þ; b rkð ÞVb zð Þ� �:

We denote by V(x,z) the reduced-form expected utility of awinning firm i:

V x; zð Þ ¼ E W s⁎i ; s⁎−i;w; ri

� �jri ¼ x; b rj

� �Nb zð Þ; b rkð ÞVb zð Þ

� �:

This expression has its counterpart in the previous section in Eq. (7).Suppose firm i of a risk type ri=x submits a bid bi=b(y) in the

auction stage. If b(y)bb(z), firm i looses the auction and gets its utilityfrom the outside option which we normalize to zero. If, on the otherhand, b(y)Nb(z), firm i gets the license at the auction pricew=b(z) andgets the expected utility V(x,z). The ex-ante expected utility of a firmwith risk type x which bids b(y) is

V x; yð Þ ¼Z

b zð Þbb yð Þ

V x; zð ÞdG Fð Þ zjxð Þ:

The firm maximizes V(x,y) with respect to y, and the conditionx ¼ argmaxy V x; yð Þ implicitly defines the equilibrium bidding func-tion b(x), which we will also denote by b(+)(x) or b(−)(x) depending onwhether it is increasing or decreasing.

We derive the exact solution to the model as an expansion powerseries with respect to the parameter α, i.e., we solve the model for

s x; zð Þ ¼ ∑∞

t¼0αts tð Þ x; zð Þ; and b xð Þ ¼ ∑

∞

t¼0αtb tð Þ xð Þ;

and analyze conditions for a risk-seekers' (with decreasing biddingfunction) and a risk-averse players' (increasing bidding function)equilibrium to exist or not to exist.9 The first non-trivial term in thoseexpansions will capture the main effects of firms' risk attitudes ontheir equilibrium bidding behavior and aftermarket strategy. It turnsout that the first such term is of the second order, and in the nextSectionwe analyze the second-order Taylor expansions of firms' exactstrategies s(x,z)=s(0)(x,z)+α2s(2)(x,z), and b(x)=b(0)(x)+α2b(2)(x).

9 We implicitly, therefore, make the standard assumption that the exact solutionexists and can be represented by its Tailor expansion with positive radius ofconvergence.

5. Analysis

Without uncertainty, the aftermarket game has a unique symmetricNash equilibrium s(x,z)=s(0), in which neither firms' risk attitudes norauctionpricew play a role. For the sake of notational simplicity, we dropthe arguments of the profit function π and its derivatives, and implicitlyassume that they are evaluated at the point (si,s− i,θ)=(s(0),s− i(0),0).The stability assumption requires that in case of strategic complementsπi,ib−(n−1)πi,jb0, and in case of strategic substitutes πi,ibπi,jb0.

Let firm i have type x and let it win a license at auction price w=b(z). By H(±)(x,z) we denote the expected risk attitude of an after-market competitor in case of an increasing and a decreasing biddingfunction b(±)(x), respectively:

H −ð Þ x; zð Þ≡E rjjri ¼ x; rjbz; rkzz� �

; andH þð Þ x; zð Þ≡E rjjri ¼ x; rjNz; rkVz� �

:

In other words, H(−) (H(+)) is an expectation of risk attitudes of anyof the (n−1) winning firms j conditional on these risk types beingbelow (above) z, risk types of all the other (N−n) loosing firms k areabove (below) z, and the risk type of firm i itself being equal to x.When risk types are affiliated (including the case of statisticalindependence), both partials of H(±)(x,z) are non-negative (see, e.g.,Krishna, 2002, pp. 272), i.e., Hx

(±)(x,z)≥0 and Hz(±)(x,z)≥0. Finally, by

continuity we have H −ð Þ x;Pr� � ¼ Pr, and H(+)(x,r)= r.

In the following proposition, we derive bidding functions andexistence conditions for a risk-seekers' equilibrium (i.e., an equili-brium where players who are less risk-averse bid more and theequilibrium bidding function is decreasing in a firm's risk attitude),and a risk-averse players' equilibrium (i.e., an equilibrium whereplayers who are more risk-averse bid more and the equilibriumbidding function is increasing in a firm's risk attitude).

Proposition 1. Let coefficients B(0), B(1), B(2), and function v(±)(x,z) bedefined as follows:

B 0ð Þ≡16

πθ;θ−n−1ð Þπjπi;θ;θ

πi;i þ πi;j n−1ð Þ� � !; B 1ð Þ≡ n−1ð Þπjπi;jπi;θπθ

3 πi;i þ n−1ð Þπi;j� �

πi;i−πi;j� �þ 1

6πθð Þ2;

B 2ð Þ≡ n−1ð Þπjπi;iπi;θπθ

3 πi;i þ n−1ð Þπi;j� �

πi;i−πi;j� � ; and v Fð Þ x; zð Þ≡π þ α2 B 0ð Þ−B 1ð Þxþ B 2ð ÞH Fð Þ x; zð Þ� �

:

The auction stage has at most one risk-seekers' equilibrium and atmost one risk-averse players' equilibrium, such that:

a) A risk-averse players' (+) or risk-seekers' (−) equilibrium exists if0b±vx(±) and 0b±(vx(±)+vz(±)) (sufficient conditions).

b) If a risk-averse players' or risk-seekers' equilibrium does exist, it isunique and given by b(±)(x)=v(±)(x,x).

c) A risk-averse players' (+) or risk-seekers' (−) equilibrium exists onlyif 0≤±vx(±)(x,x) and 0≤±(vx(±)(x,x)+vz(±)(x,x)) (necessary conditions).

In accordance with Proposition 1, there is no first-order effect offirms' risk attitudes on their bidding behavior. The reason for this isthat the concept of attitude towards risk is essentially of the second-order, i.e., it is determined by the second-order derivative of the utilityfunction. In the first-order approximation, all firms behave as if theywere all risk-neutral.

When α=0, all firms get identical market profits π and, therefore,bid bi=π in the auction. The other part α2(B(0)−B(1)x+B(2)H(±)(x,x)) ofthe bidding function is the second-order effect of firms' risk attitudes.In accordance with Proposition 1, v(±)(x,z) appears to be a valuationfunction, i.e., a certainty equivalent of themarket-stage game, of a firmi, which has a risk type x and pays an auction price determined by afirm of type z. When z=x, i.e., when both firms i and j have the samerisk type and compete for only one remaining license, they bid theirvalues v(±)(x,x). Hence, the auction price w must be equal to v(±)(x,x),which is, essentially, the first-order condition b(±)(x)=v(±)(x,x).

The first term B(0) is of least importance as it captures the effect ofaftermarket uncertainty on firms' valuations that is independent of


firms' risk attitudes. This effect disappears when firms' profits arelinear in uncertainty θ, i.e., when πθ,θ=πi,θ,θ=0.

The second term B(1)x represents the effect of the firm's riskattitude on its valuation for the license for given types of the firm'scompetitors. Its derivative with respect to x, i.e., B(1), is the direct, orrisk attitude effect. This corresponds to inequality (5) in the exampleof Section 3. If B(1)N0, a less risk-averse firm is willing to pay more forthe license in order to compete in the aftermarket with a given set ofcompetitors. If only one license were auctioned, i.e., in a monopolisticaftermarket, the direct effect is the only effect that is present and theonly equilibrium that exists is a risk-seekers' equilibrium.

The third term B(2)H(±)(x,z) represents the aftermarket externality,i.e., the effect of the expectation of competitors' risk attitudes rj on thefirm's valuation for a given type x of the firm. When B(2)b0, theexternality is negative, and it reinforces the risk attitude effect so thatonly a risk-seekers' bidding equilibrium can exist and, in fact, doesexist. This is the case in the Bertrand example in Section 3; seeinequality (6). When, on the other hand, B(2)N0, the externality ispositive and works against the direct risk attitude effect so that a risk-seekers' equilibrium may fail to exist. The expression for B(2) makes itpossible to analyze special cases where the strategic effects are absentso that πj=0, or the uncertainty does not effect marginal profit (as isthe case when the only uncertainty is with respect to a fixed costparameter) so that πi,θ=0. In these two special cases, B(2)=0 so that themonopoly result generalizes and the equilibrium is always a risk-seekers' equilibrium.

The sensitivity of the externality B(2)H(±)(x,z) with respect to its owntype x, i.e., B(2)Hx

(±)(x,z) is what we call the indirect, or strategic effect.When the externality is positive and sufficiently strong, and Hx

(±)(x,z) issufficiently large, not only does a risk-seekers' equilibrium fail to existbut also another, risk-averse equilibriummayappear (see Section6). Thestrategic effect is absent if B(2)=0, i.e., when there is no externality.Alternatively, the strategic effect is absent if firm types are statisticallyindependent, i.e., when H(±)(x,z) is independent of x and Hx

(±)(x,z)=0.Hence, risk-averse players' equilibria may only exist when multiplelicenses are auctioned andwhen firms' risk types are ex-ante correlated.

In light of the direct and strategic effects, the necessary (0≤±vx(±)(x,x)) and sufficient (0b±vx(±)) conditions in Proposition 1 can be restatedas follows. If the sum of the direct and strategic effects is alwaysstrictly positive (negative) then a risk-averse players' (risk-seekers')equilibrium exists. If a risk-averse players' (risk-seekers') equilibriumdoes exist then the sum of the direct and strategic effects inequilibrium is not negative (positive).10

6. Cournot competition under demand uncertainty

In this Section and the next, we provide illustrations of Proposition1 for regular oligopoly models when there is either demand or costuncertainty. Section 3 has already provided an illustration for the caseof differentiated Bertrand competition with three firms competing fortwo licenses and linear uncertain demand. Applying Proposition 1 to amore general setting with an arbitrary number of firms and licenses,and more general demand and cost functions, gives a similar result,namely that only a risk-seekers' equilibrium (an equilibrium whereplayers who are less risk-averse bid more, and the equilibrium biddingfunction is decreasing in a firm's risk attitude) exists, resulting in thehighest possible aftermarket prices being set.

This section proceeds with an analysis of Cournot competition as thisanalysis is richer than the analysis of the Bertrand model. We first showthat evenwith lineardemandand statistically independent risk attitudes,a risk-seekers' equilibrium may fail to exist, providing evidence that thepositive welfare result for the quantity-setting monopoly case does not

10 The other necessary (0≤±vx(±)(x,x)±vz(±)(x,x)) and sufficient (0b±(vx(±)+vz(±))) condi-tions in Proposition 1 are just necessary and sufficient conditions for the biddingfunction b(±)(x)=v(±)(x,x) to be monotonically increasing (decreasing).

automatically generalize. Next, we show by means of an example thatunder Cournot competition, one actually may get a completely reverseresult compared to the quantity-setting monopoly result.

Let firms compete in quantities, i.e., si=qi, with market (inverse)demand being given by p=1+θ−qi−∑j ≠ iqj. Normalizing productioncosts to zero, we write firms' profit function as π=pqi. We denote Nashequilibrium outputs at θ=0 by q0: q0=1/(n+1). In this set-up one canderive the following expressions for the partial derivatives:

πθ ¼ q0; πθ;θ ¼ πi;θ;θ ¼ 0; πi;θ ¼ 1; πi;i ¼ −2; πj ¼ −q0; πi;j ¼ −1:

Applying Proposition 1 leads to the following expressions:

B 0ð Þ ¼ 0; B 1ð Þ ¼ 3n−16 nþ 1ð Þ3

N0; B 2ð Þ ¼ 2 n−1ð Þ3 nþ 1ð Þ3

N0; and

v Fð Þ x; zð Þ ¼ π þ α2

6 nþ 1ð Þ3− 3n−1ð Þxþ 4 n−1ð ÞH Fð Þ x; zð Þ� �

:

As B(2)N0, the externality is positive and works against the directeffect. Intuitively, in a risk-seekers' equilibrium, firms suffer fromcompetingwithmore risk-seeking firms, and this effect may actually beso strong that a risk-seekers' equilibriumdoes not exist evenwhenfirms'types are statistically independent. Inorder to showthis,we take the risktypes to be independently distributed over an interval [r,r ] with adistribution function F rð Þ ¼ r−PrÞγ= r−PrÞγ ; γN0

��. The conditional

expectation function H(−)(x,z) for this distribution is:

H −ð Þ x; zð Þ ¼Zz

Pr

rdF rð Þ=Zz

Pr

dF rð Þ ¼ γzþPrγ þ 1

:

This implies that a risk-seekers' equilibrium does not exist if thefunction

b −ð Þ xð Þ ¼ v −ð Þ x; xð Þ ¼ π þ α2

6 nþ 1ð Þ3− 3n−1ð Þxþ 4 n−1ð ÞγxþPr

γ þ 1

� �increases in x, i.e., if 4(n−1)γN (3n−1)(γ+1). This condition is satisfiedwhen n≥4 and γN (3n−1)/(n−3).

Hence, the monopoly result does not always generalize to theoligopoly case if firms compete in quantities, evenwith linear demandand independent risk types. The effect, however, appears in thecurrent example only with more than four licenses and, therefore,with at least five competing firms in the auction. The positive (forwelfare) selection effect of auctioning monopoly rights when firmschoose quantities thus weakens and sometimes disappears whenmultiple licenses are auctioned.

6.1. When a risk-averse equilibrium exists

Until now, we focused on the question whether or not a risk-seekers' equilibrium exists. We now show that when the externality issufficiently strong, and at the same time, firms' types are stronglycorrelated, a risk-averse players' equilibrium may emerge underCournot competition. In such an equilibrium, we have the worstpossible outcome, namely that the most risk-averse firms win theauctions and they choose the lowest possible production level,resulting in the highest possible market price.

In the example we take the following distribution F⁎ of firms' riskattitudes. Let there be N=n+1 firms, and let a macroeconomicfundamental such as an interest rate, oil price, the growth rate ofthe economy, a democracy index, etc., β be distributed over theinterval [0, 1] in accordance with an arbitrary twice differentiabledistribution function Fβ(t)≡Pr(βb t), Fβ∈C2 (the support [0, 1] hasbeen chosen for exposition purposes only; the linearity of firms'equilibrium strategies guarantees that any other interval leads to thevery same equilibrium conditions). Then, for any given realization of

Fig. 2. Support of the conditional distribution Fr(x|t)=Pr(rjbx|β= t).


β, let all N firms' risk types rj be independently and uniformlydistributed over the interval Px βð Þ; x βð Þ�

, where Px βð Þ≡β−eβ 1−βð Þ;x βð Þ≡β þ eβ 1−βð Þ, and ε∈ (0,1) is a parameter, i.e., let the conditionaldistribution Fr(x|t)≡Pr(rjbx|β= t) be

Fr xjtð Þ ¼ x−Px tð Þx tð Þ−Px tð Þ for xa Px tð Þ; x tð Þ �

:

The reason why we consider this specific distribution is that forsmall values of ε, if a firm i has a type x, the distribution of types of allother firms conditional on x is concentrated on a small neighborhoodof x. Therefore, all firms that are competing in the aftermarket haveapproximately the same type, the Nash equilibrium is almostsymmetric, and an equilibrium increasing bidding function can beanalytically calculated in the limit when ε converges to zero. Fig. 2shows the support (bold line) of the conditional distribution Fr(x|t).

In the following lemma, we derive the conditional expectationfunction H(±)(x,z).

Lemma 1. Let ri be distributed in accordance with the distributionfunction F⁎. Then, H(±)(x,z) for small ε can be written as follows:

H Fð Þ x; zð Þ ¼ nxþ nþ 2ð Þz2 nþ 1ð Þ F

nx 1−xð Þnþ 1ð Þ eþ o eð Þ:

Let firms compete in the market with stochastic demand p=θ+Q−1/e

so that the stability requirement eN1/n holds. Let firms marginal costsbe constant and equal to cN0. Firms' profit function then is π=(Q−1/e+θ−c)qi. The aftermarket Nash equilibrium output q0 is then given by

q0 ¼ ne−1ð Þ= necð Þð Þe=n;

and the partials of the profit function at this output are:

πθ ¼ q0; πθ;θ ¼ πi;θ;θ ¼ 0; πi;θ ¼ 1; πj ¼ −c= ne−1ð Þ;πi;i ¼ −

2ne− 1þ eð Þð Þcneq0 ne−1ð Þ ; andπi;j ¼ −

ne− 1þ eð Þð Þcneq0 ne−1ð Þ :

This lead to the following expressions:

B 0ð Þ ¼ 0; B 1ð Þ ¼ 2 n−1ð Þ ne− 1þ eð Þð Þ þ n ne−1ð Þ6n ne−1ð Þ q20; andB 2ð Þ

¼ n−1ð Þ 2ne− 1þ eð Þð Þ3n ne−1ð Þ q20:

Using Lemma 1 and considering it in the limit when ε converges tozero yields the following expression for the valuation function:

v Fð Þ ¼ π þα2q20 − 2 n−1ð Þ ne− 1þ eð Þð Þ þ n ne−1ð Þð Þxþ 2 n−1ð Þ 2ne− 1þ eð Þð Þ nxþ nþ2ð Þz

2 nþ1ð Þ� �

6n ne−1ð Þ :

It is easy to see that vz(±)(x,z)N0. Thus, the only sufficient conditionto be satisfied for the risk-averse players' equilibrium to exist is vx(±)(x,z)N0, that is:

0b− 2 n−1ð Þ ne− 1þ eð Þð Þ þ n ne−1ð Þð Þ þ 2 n−1ð Þ 2ne− 1þ eð Þð Þ n2 nþ 1ð Þ ; or

ebe nð Þ≡1n

1þ n−1ð Þ n2 þ nþ 2� �

n−1ð Þ n2 þ n−2ð Þ þ 2n2

!:

Hence, if e∈(1/n,e(n)), then there exists an εN0 and αN0 so that theonly equilibrium in the auction stage of the game is the risk-averseplayers' equilibrium. In such an equilibrium, the most risk-averse firmssubmit the highest bids, get the licenses, and compete in the aftermarket.They choose lower outputs (due to πi,θπθN0) and, therefore, market priceis higher than if licenses were assigned by any other mechanism.

7. Cost uncertainty

We now briefly consider the implications of the general resultsobtained in Section 5 for the case where marginal costs depend on acommon uncertain component. We show that in this case themonopolyresults donot generalize for bothprice andquantity competition settings.

7.1. Differentiated Bertrand competition

Let firms compete in prices, i.e., si=pi, in differentiated Bertrandoligopoly with demand given by qi(pi,p− i)=1−pi+(∑j≠ ipj)/(n−1). Thiscase represents consumers with unit demand who only decide on fromwhich firm to buy. Let the stochastic cost function be given by (c−θ)qi. Inthis case, a firms' profit function is given by π=(pi−c+θ)qi. DenotingNash equilibriumoutputs at θ=0 by q0, we get the following expressionsfor the partials:

πθ ¼ q0; πθ;θ ¼ πi;θ;θ ¼ 0; πi;θ ¼ −1; πi;i ¼ −2; πj ¼ q0= n−1ð Þ; πi;j ¼ 1= n−1ð Þ:

This leads to the following expressions for B(1) and B(2):

B 1ð Þ ¼ 2n−36 2n−1ð Þ q

20; B 2ð Þ ¼ 2 n−1ð Þ

3 2n−1ð Þ q20:

Here the externality is positive and works against the direct effect.As in Section 6 under Cournot oligopoly, we show that the risk-seekers' equilibrium may fail to exist even when firms' types arestatistically independent. Assuming that risk types are independentlydistributed over some interval [r,r] with the distribution functionF rð Þ ¼ r−PrÞγ= r−PrÞγ ;γN0

��, we get H −ð Þ x; zð Þ ¼ γzþ PrÞ= γ þ 1ð Þ�

and,consequently,

b −ð Þ xð Þ ¼ v −ð Þ x; xð Þ ¼ π þ α2q206 2n−1ð Þ γ þ 1ð Þ 4 n−1ð Þ γxþ PrÞ− 2n−3ð Þ γ þ 1ð ÞxÞ:��

The risk-seekers' equilibrium does not exist when the abovefunction b(−)(x) does not decrease in x, i.e., γN (2n−3)/(2n−1).

Hence, the monopoly result does not generalize to the oligopolycase if firms compete in prices and have uncertain production costs.The risk-seekers' equilibrium fails to exists even for n=2 licenses anduniform (and independent) types' distribution where γ=1.

7.2. Cournot competition

Let us go back to the case of Cournot competition with market(inverse) demand given by p=1−qi−∑j≠ iqj and stochastic costs (c−θ)qi,


as in the previous example. Firms' profit function in this case isgiven by π=(p−c+θ)qi. Denoting Nash equilibrium outputs at θ=0 byq0, we get q0=(1−c)/(n+1) and the following expressions for thepartials:

πθ ¼ q0; πθ;θ ¼ πi;θ;θ ¼ 0; πi;θ ¼ 1; πi;i ¼ −2; πj ¼ −q0; πi;j ¼ −1:

One may see that these expressions exactly coincide with thosefrom the Cournot example analyzed in the previous Section. There-fore, under quantity competition setting, whether there is a demandor cost uncertainty, the monopoly result does not generalize.

Comparing the results obtained for demand and cost uncertaintyone may notice an asymmetry between different oligopoly settings.Under differentiated Bertrand competition, the sign of the aftermarketexternality is negative with demand uncertainty, and is positive withcost uncertainty. Consequently, firms prefer competing with less risk-averse competitors if market demands are uncertain, whereas theyprefer more risk-averse competitors if production costs are uncertain.To the contrary, under Cournot competition, the sign of the after-market externality is always negative. Consequently, firms alwaysprefer competing with more risk-averse competitors irrespective ofthe source of uncertainty.

The reason for this asymmetry goes back to the single-crossingproperty of the effect of uncertainty on firms' profits. Under Cournotcompetition, if the uncertainty positively affects firms' profits, it alsoaffects firms' marginal profits positively, in other words, πθπi,θN0irrespective of the source of uncertainty. Consequently, theexternality is always positive, and it always works against the riskattitude effect. Under differentiated Bertrand competition, a positivedemand shock increases firms' marginal profits, i.e., πθπi,θN0,whereas a positive cost shock decreases marginal profits of thefirms, i.e., πθπi,θb0. That is why the sign of the externality, thedirection of the strategic effect, and consequently, whether themonopoly result generalizes or not in differentiated Bertrand com-petition, depends on whether the aftermarket uncertainty affectsdemand or cost.

8. Conclusion

In this paper, we have analyzed auctions where the winning firmsget licenses to operate in an aftermarket, and play an oligopoly game.Profits in the marketplace are uncertain as there is uncertaintyconcerning future demand and/or future cost, and firms differ in theirrisk attitude. The difference between this model and a standardauction model with interdependent valuations (see, e.g., Milgrom andWeber, 1982) is that in the current model firms' valuations onlydepend on risk attitudes of the winning firms; risk types of loosingfirms only affect the distribution of risk attitudes of the firms that doplay the market stage.

We have provided necessary and sufficient condition for amonotonic equilibrium to exist in a very general set-up. We haveapplied this set-up to the case of demand and cost uncertainty. Wehave shown that in oligopoly markets with demand uncertaintyauctions perform worse compared to the monopoly result. Theinefficiency of auctions for monopoly rights due to the selection ofthe least risk-averse firms (compared to a random allocation oflicenses) carries over when the firms play a differentiated Bertrandgame. The efficiency of auctions for monopoly rights when firmschoose production levels does not generalize to an oligopolysetting (Cournot competition). The main reason for this is astrategic effect that works against the risk attitude effect: firmsprefer to compete with competitors who are more risk-averse. Wehave also shown that when a firm's choice of output has asufficient impact on price and when firms' risk types are ex-anteaffiliated, the strategic effect becomes so large that the more risk-averse firms win licenses and set lower quantities (leading again to

higher market prices) than when licenses were allocated by anyother mechanism.

Appendix A

Proof of Proposition 1. The proof consists of two parts. In part 1, wederive firm's aftermarket strategies s(x,z). In part 2, we derive firms'bidding functions and equilibrium necessary and sufficient existenceconditions.

Part 1. For α=0, the aftermarket equilibrium si=si(0) is symmetric, andfirms' expected utility function W is

W si; s−i;w; rið Þ ¼ U π si; s−i;0ð Þ−w; rið Þ:

Maximizing this expressionw.r.t. si is equivalent tomaximizing theaftermarket profit π(si,s− i,0). The first-order condition πi=0 definesthe unique symmetric Nash equilibrium strategy si=si(0) and therealized aftermarket profit π.

In the first-order approximation, si=si(0)+αsi(1) and (taking into

account πi=0):

π si; s−i; θð Þ ¼ π þ απj ∑j≠is 1ð Þj þ θπθ; πi si; s−i; θð Þ ¼ α πi;is

1ð Þi þ πi;j ∑

j≠is 1ð Þj

!þ θπi;θ; and

Uπ π si; s−i; θð Þ−w; xð Þ ¼ Uπ π−wð Þ; xð Þ 1− απj ∑j≠is 1ð Þj þ θπθ

!x

!:

Hence, the first-order condition 0=Wi(si⁎,s− i⁎ ,w) which defines si⁎,in the first-order approximation becomes:

0 ¼ Wi x; zð Þ ¼ @

@siE

12α

Zα−α

U π s⁎i ; s⁎−i; θ

� �−w; ri

� �dθjri ¼ x; b rj


0@ 1A¼ 1

2αE

Zα−α

πi s⁎i ; s⁎−i; θ

� �Uπ π s⁎i ; s

⁎−i; θ

� �−w; ri

� �dθjri ¼ x; b rj


0@ 1A¼ αUπ π−wð Þ; xð Þ πi;is

1ð Þi þ πi;jE ∑j≠i s

1ð Þj jri ¼ x; b rj


� �� :

Thus, the following first-order condition must hold:

πi;is1ð Þi ¼ −πi;j n−1ð ÞE s 1ð Þ

j jri ¼ x; b rj� �

Nb zð Þ; b rkð ÞVb zð Þ� �

: ðA:1Þ

In order to solve Eq. (A.1) for si(1), we first rewrite it for anotherfirm j:

πi;is1ð Þj ¼ −πi;j n−1ð ÞE s 1ð Þ

l jl≠j; rj;…� �

¼ −πi;j n−1ð ÞE s 1ð Þl jl≠i; rj;…

� �þ E s 1ð Þ

i jrj;…� �

−s 1ð Þj

� �and, consequently,

πi;i−πi;j� �

s 1ð Þj ¼ −πi;j n−1ð ÞE s 1ð Þ

l jl≠i; rj;…� �

−πi;jE s 1ð Þi jrj;…

� �:

Then, taking the expectation conditional on information that firm ihas yields:


E s 1ð Þj jj≠i; ri;…

� �¼ −πi;j n−1ð ÞE E s 1ð Þ

l jl≠i; rj;…� �

jj≠i; ri;…� �

−πi;jE E s 1ð Þi jrj;…

� �jj≠i; ri;…

� �:

ðA:2Þ

By the law of iterative expectations:

E E s 1ð Þl jl≠i; rj;…

� �jj≠i; ri;…

� �¼ E s 1ð Þ

j jj≠i; ri;…� �

and E E s 1ð Þi jrj;…

� �jj≠i; ri;…

� �¼ s 1ð Þ

i ;

so that Eq. (A.2) can be written as follows


E s 1ð Þj jj≠i; ri;…

� �¼ −πi;j n−1ð ÞE s 1ð Þ

j jj≠i; ri;…� �

−πi;js1ð Þi :


This equation together with Eq. (A.1) implies that

s 1ð Þi ¼ E s 1ð Þ



¼ 0:

Hence, risk attitudes have no first-order effect of on the after-market behavior.

In the next, second-order approximation, si=si(0)+α2si(2) and:

π si; s−i; θð Þ ¼ π þ θπθ þ α2πj ∑j≠is 2ð Þj þ 1

2θ2πθ;θ;

πi si; s−i; θð Þ ¼ θπi;θ þ α2 πi;is2ð Þi þ πi;j ∑

j≠is 2ð Þj

!þ 12θ2πi;θ;θ; and

Uπ π si; s−i; θð Þ−w; xð Þ ¼ Uπ π−w; xð Þ 1−θπθxð Þ:

Hence, the first-order condition 0=Wii(si⁎,s− i⁎ ,w) in the second-order

approximation becomes:

0 ¼ E12α

Zα−α

πi s⁎i ; s⁎−i; θ

� �Uπ π s⁎i ; s

⁎−i; θ

� �−w; ri

� �dθjri;…

0@ 1A¼ α2Uπ π−w; xð Þ πi;is

2ð Þi þ πi;jE ∑

j≠is 2ð Þj jri;…

!þ 16πi;θ;θ−

13πi;θπθx

!

This can be written as

6πi;is2ð Þi ¼ 2πi;θπθx−6πi;j n−1ð ÞE s 2ð Þ

j jj≠i; ri…� �

−πi;θ;θ ðA:3Þ

In order to solve Eq. (A.3) for si(2), we first rewrite it for another firm j:

6πi;is2ð Þj ¼ 2πi;θπθrj−6πi;j n−1ð ÞE s 2ð Þ

l jl≠j; rj…� �

−πi;θ;θ

¼ 2πi;θπθrj−6πi;j n−1ð ÞE s 2ð Þl jl≠i; rj…

� �þ E s 2ð Þ

i jrj…� �

−s 2ð Þj

� �−πi;θ;θ

so that

6 πi;i−πi;j� �

s 2ð Þj ¼ 2πi;θπθrj−6πi;j n−1ð ÞE s 2ð Þ

l jl≠i; rj…� �

þ E s 2ð Þi jrj…

� �� −πi;θ;θ:

In a similar way as in the first-order approximation, we write:


E s 2ð Þj jj≠i; ri…

� �¼ 2πi;θπθE rjjj≠i; ri…

� �−6πi;jE E s 2ð Þ

i jrj…� �

jj≠i; ri…� �

−6πi;j n−1ð ÞE E s 2ð Þl jl≠i; rj…

� �jj≠i; ri…

� �−πi;θ;θ


E s 2ð Þj jj≠i; ri…

� �¼ 2πi;θπθE rjjj≠i; ri…

� �−6πi;js

2ð Þi

−6πi;j n−1ð ÞE s 2ð Þj jj≠i; ri…

� �−πi;θ;θ

:

6 πi;i þ πi;j n−2ð Þ� �E s 2ð Þ


¼ 2πi;θπθE rjjj≠i; ri…� �

−6πi;js2ð Þi −πi;θ;θ

This equation together with Eq. (A.3) yields the followingexpressions:

s 2ð Þi ¼ πi;i þ πi;j n−2ð Þ� �

x−πi;j n−1ð ÞE rjjj≠i; ri…� ��

πi;θπθ


πi;i þ πi;j n−1ð Þ� � −πi;θ;θ

6 πi;i þ πi;j n−1ð Þ� �E s 2ð Þ


¼ πi;θπθ πi;iE rjjj≠i; ri…� �

−πi;jx� �


πi;i þ πi;j n−1ð Þ� � −πi;θ;θ

6 πi;i þ πi;j n−1ð Þ� �8>>><>>>: ðA:4Þ

This ends Part 1 of the proof.

Part 2. In the second-order approximation, firms' equilibrium profitsπi⁎≡π(si⁎, s− i⁎ ,θ), utilities U(πi⁎,b(z),ri) and expected utilities V(x,z) are

π⁎i ¼ π þ θπθ þ α2πj ∑j≠is 2ð Þj þ 1

2θ2πθ;θ;

U π⁎i −b zð Þ; ri� �

¼ θπθ þ α2 πj ∑j≠is 2ð Þj þ 1

2θ2 πθ;θ−ri πθð Þ2� �

−b 2ð Þ zð Þ !

;

V x; zð Þ ¼ E W s⁎i ; s⁎−i; b zð Þ

� �jri ¼ x; b rj


� �¼ α2 n−1ð ÞπjE s 2ð Þ



þ 16

πθ;θ−ri πθð Þ2� �

−b 2ð Þ zð Þ� �

Using Eq. (A.4), this can be written as follows:

V x; zð Þ ¼ α2 B 0ð Þ−B 1ð Þxþ B 2ð ÞH Fð Þ x; zð Þ−b 2ð Þ zð Þ� �

;

where B(0), B(1), and B(2) are defined as in Proposition 1.Suppose that the bidding function is decreasing, i.e., db(2)/dxb0. The

ex-ante expected utility of a firmwith risk type x which bids b(y) is

V x; yð Þ ¼Zybz

V x; zð ÞdG −ð Þ zjxð Þ:

Maximizing V(x,y) w.r.t. y yields the following first-order condition:

0 ¼ @V@y

x; xð Þ ¼ −V x; xð Þg −ð Þ xjxð Þ:

This implies that V(x,x)=0, i.e.,

b 2ð Þ xð Þ ¼ B 0ð Þ−B 1ð Þxþ B 2ð ÞH −ð Þ x; xð Þ and b −ð Þ xð Þ ¼ π þ α2b 2ð Þ xð Þ¼ v −ð Þ x; xð Þ:

This bidding function is a unique decreasing equilibrium biddingfunction if b(2)(x) is indeed decreasing. Hence, the condition 0≥vx(−)(x,x)+vz(−)(x,x) is the first necessary condition for the risk-seeking equilibrium

to exist. The second necessary condition is the second-order condition:

0z@2V@y2

x; xð Þ ¼ ddx

@V@y

x; xð Þ !

−@

@x@V@y

!x; xð Þ ¼ −

@2V@x@y

x; xð Þ

¼ Vx x; xð Þg −ð Þ xjxð Þ;

which can be written as 0≥vx(−)(x,x).Suppose now that, first 0Nvx(−)+vz(−) so that b(−)(x) is a strictly

decreasing function, and second 0Nvx(−). In equilibrium, firm i of type xbids b(−)(x) and gets expected utility

V x; xð Þ ¼Zxbz

V x; zð ÞdG −ð Þ zjxð Þ ¼Zxbz

v −ð Þ x; zð Þ−v −ð Þ z; zð Þ� �

dG −ð Þ zjxð Þ:

If, on the other hand, firm i of bids b(−)(y), it gets

V x; yð Þ ¼Zybz

V x; zð ÞdG −ð Þ zjxð Þ ¼Zybz


dG −ð Þ zjxð Þ:

This deviation yields strictly lower utility level because

V x; yð Þ−V x; xð Þ ¼Zxy


dG −ð Þ zjxð Þ

¼Zxy

Zxz

v −ð Þx t; zð Þdt

0@ 1AdG −ð Þ zjxð Þb0:

Hence, b(−)(x)=v(−)(x,x) is indeed an equilibrium bidding function.In exactly the sameway, onemay show that if a risk-averse players'

equilibrium exists, it is unique and given by b(+)(x)=v(+)(x,x). Thenecessary conditions for b(+)(x) to be a unique risk-averse players'equilibrium are 0≤vx(+)(x,x)+vz(+)(x,x) and 0≤vx(+)(x,x), and the sufficientconditions are 0bvx(+)+vz(+) and 0bvx(+). This ends the proof. ■

Proof of Lemma 1. We denote distribution functions as follows:

Fβ tð Þ≡Pr βbtð Þ; Fr x; zð Þ≡Pr ribx; rkNzð Þ; and Fr x; zjtð Þ≡Pr ribx; rkNzjβ ¼ tð Þ;

and the corresponding densities are

fβ tð Þ≡dFβ tð Þ=dx ¼ 1; fr x; zð Þ≡dFr x; zð Þ=dx; and fr x; zjtð Þ≡dFr x; zjtð Þ=dx:


Next, we define inverse functions

Pt xð Þ≡ 1þ eð Þ−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ eð Þ2−4ex

q� �= 2eð Þ and t xð Þ≡

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−eð Þ2þ4ex

q− 1−eð Þ

� �= 2eð Þ;

so that x≡Px t xð Þ� �and x≡x Pt xð Þ� �

for any x∈ [0,1].As ri e U Px βð Þ; x βð Þ� �

, it follows that for all xa Px tð Þ; x tð Þ �:

Fr x; zjtð Þ ¼ x−Px tð Þ� �x tð Þ−zð Þn−1 x tð Þ−Px tð Þ� �−n

; if za Px tð Þ; x tð Þ� �x−Px tð Þ� �

= x tð Þ−Px tð Þ� �; if za 0;Px tð Þ� ��

and, therefore,

fr x; zjtð Þ ¼ x tð Þ−zð Þn−1 x tð Þ−Px tð Þ� �−n; if za t−et 1−tð Þ; x tð Þð Þ

1= x tð Þ−Px tð Þ� �; if za 0; t−et 1−tð Þð Þ

�Hence,

fβ tjx; zð Þ≡ fr x; zjtð Þfβ tð Þfr x; zð Þ

¼x tð Þ−zð Þn−1fβ tð Þ

x tð Þ−Px tð Þ� �nfr x; zð Þ ; if ta max Pt xð Þ;Pt zð Þ� �;min t xð Þ; t zð Þ� ��

fβ tð Þx tð Þ−Px tð Þ� �

fr x; zð Þ ; if ta min t xð Þ; t zð Þ� �; t xð Þ� �

8>>><>>>:We define H (+)(x,z,t)≡E(rk|ri=x,rl=zb rk,β= t) and consider two

cases.

a) When z≥x,~H

þð Þx; z; tð Þ ¼ 1

2x tð Þ þ zð Þ for ta Pt zð Þ; t xð Þ� �

. Hence, H(+)(x,z) can be written as

H þð Þ x; zð Þ ¼ Eβ~H

þð Þx; z;βð Þjri ¼ x; rl ¼ zbrk

� �¼ P1 x; zð Þ=Q1 x; zð Þ; where

P1 x; zð Þ≡fr x; zð ÞZ t xð Þ

Pt zð Þ

12

x tð Þ þ zð Þfβ tjx; zð Þdt ¼Z t xð Þ

Pt zð Þ

x tð Þ þ zð Þ x tð Þ−zð Þn−1fβ tð Þ2 x tð Þ−Px tð Þ� �n dt;

and

Q1 x; zð Þ≡fr x; zð ÞZ t xð Þ

Pt zð Þ

fβ tjx; zð Þdt ¼Z t xð Þ

Pt zð Þ

x tð Þ−zð Þn−1fβ tð Þx tð Þ−Px tð Þ� �n dt:

b) When z≤x,~H

þð Þx; z; tð Þ ¼ 1

2x tð Þ þ zð Þ for ta Pt xð Þ; t zð ÞÞ�

and H(+)(x ,z ,t)= t for t∈ (t (z) , t (x)). Hence, H(+)(x ,z) can be written as

H þð Þ x; zð Þ ¼ Eβ~H

þð Þx; z;βð Þjri ¼ x; rl ¼ zbrk

� �¼ P2 x; zð Þ=Q2 x; zð Þ; where

P2 x; zð Þ≡Z t zð Þ

Pt xð Þ

x tð Þ þ zð Þ x tð Þ−zð Þn−1fβ tð Þ2 x tð Þ−Px tð Þ� �n dt þ

Z t xð Þ

t zð Þ

tfβ tð Þx tð Þ−Px tð Þdt

and

Q2 x; zð Þ ¼Z t zð Þ

Pt xð Þ

x tð Þ−zð Þn−1fβ tð Þx tð Þ−Px tð Þ� �n dt þ

Z t xð Þ

t zð Þ

fβ tð Þx tð Þ−Px tð Þdt:

In order to evaluate H(+)(x,z) and its partials for small values of εweuse the 3rd-order approximation

ffiffiffiffiffiffiffiffiffiffiffi1þ e

p ¼ 1þ 12e−1

8e2 þ 1

16e3 þ o e3

� �,

so that

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1−eð Þ2þ4ex

q¼ 1þ 2x−1ð Þeþ 2x 1−xð Þe2 þ 2x 1−xð Þ 1−2xð Þe3 þ o e3

� �; and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ eð Þ2−4ex

q¼ 1− 2x−1ð Þeþ 2x 1−xð Þe2−2x 1−xð Þ 1−2xð Þe3 þ o e3

� �:

Hence,

t xð Þ ¼ xþ x 1−xð Þeþ x 1−xð Þ 1−2xð Þe2 þ o e2� �

; t V xð Þ ¼ 1þ o 1ð Þ;

Pt xð Þ ¼ x−x 1−xð Þeþ x 1−xð Þ 1−2xð Þe2 þ o e2� �

; andPtV xð Þ ¼ 1þ o 1ð Þ;

For an arbitrary ε ∈ (0,1), the uniform convergence with respect toε∈ (0,ε ) of the limits

limxY0

t xð Þx

¼ limxY1

1−Pt xð Þ1−x

¼ 11−eð Þ ; and lim

xY0Pt xð Þx

¼ limxY1

1−t xð Þ1−x

¼ 11þ eð Þ ;

implies the following expressions for t and t:

t xð Þ ¼ xþ x 1−xð Þe 1þ 1−2xð Þeþ o eð Þð Þ; and Pt xð Þ¼ x−x 1−xð Þe 1− 1−2xð Þeþ o eð Þð Þ:

We consider cases z≥x and z≤x separately.

a) Let z≥x. Using the above Tailor expansions yields

Q1 x; zð Þ ¼ 1n

fβ xð Þ þ n−1nþ 1

fβ xð Þ 1−2xð Þ þ f Vβ xð Þx 1−xð Þ� �e

� �;

@Q1

@xx; xð Þ ¼ tV xð Þ fβ xð Þ þ f Vβ xð Þx 1−xð Þ−fβ xð Þ 1−2xð Þ� �

e� �

2x 1−xð Þe ;

@Q1

@zðx; xÞ ¼ −

12xð1−xÞe fβðxÞ þ n−2

nf Vβ ðxÞxð1−xÞe

� �;

P1 x; xð Þ ¼ xQ1 x; xð Þ þ fβ xð Þx 1−xð Þenþ 1

;

@P1@x

x; xð Þ ¼ tV xð Þ fβ xð Þ þ x fβ xð Þ þ f Vβ xð Þ 1−xð Þ� �e

� �2 1−xð Þe ; and

@P1@z

x; xð Þ ¼ Q1 x; xð Þ2

−1

2 1−xð Þe fβ xð Þ þ 1−xð Þn

n−1ð Þfβ xð Þ þ n−2ð Þf Vβ xð Þx� �e

� �:

Hence,

H þð Þ x; xþ 0ð Þ ¼ xþ nnþ 1

x 1−xð Þe;

H þð Þx x; xþ 0ð Þ ¼ n

2 nþ 1ð Þ ; andH þð Þz x; xþ 0ð Þ ¼ nþ 2ð Þ

2 nþ 1ð Þ :

b) Let z≤x. As P2(x,x)=P1(x,x) and Q2(x,x)=Q1(x,x), it follows that H(+)

(x,x−0)=H(+)(x,x+0). Similarly, as ∂P2/∂x=∂P1/∂x, ∂P2/∂z=∂P1/∂z,∂Q2/∂x=∂Q1/∂x, and ∂Q2/∂z=∂Q1/∂z at (x,x), it follows that Hx

(+)

(x,x−0)=Hx(+)(x,x+0) and Hz

(+)(x,x−0)=Hz(+)(x,x+0).

Thus,H(+)(x,z) is continuously differentiable it at (x,x) functionwiththe partial derivatives Hx

(+)(x,x)=n/(2(n+1))+o(1) and Hz(+)(x,x)=(n+2)/

(2(n+1))+o(1), and, therefore, can be written as

H þð Þ x; zð Þ ¼ nxþ nþ 2ð Þz2 nþ 1ð Þ þ nx 1−xð Þ

nþ 1ð Þ eþ o eð Þ:

The expression for H(−) immediately follows from H(−)(x,z)=1−H(+)

(1−x,1−z). ■

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auctions, aftermarket competition, and risk attitudes...choices in risky environments depending on...

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