TitleExposure Problem in Multi-Unit Auct.

Transcription

1 TitleExosure Problem in Multi-Unit Auct Author(s) MENG, Xin; GUNAY, Hikmet Citation Issue Date Tye Technical Reort Text Version ublisher URL htt://hdl.handle.net/10086/28356 Right Hitotsubashi University Reository

2 HIAS-E-43 Exosure Problem in Multi-Unit Auctions Xin Meng Dongbei University of Finance and Economics Hikmet Gunay Hitotsubashi Institute for Advanced Study, Hitotsubashi University The University of Manitoba February 2017 Hitotsubashi Institute for Advanced Study, Hitotsubashi University 2-1, Naka, Kunitachi, Tokyo , Jaan tel: htt://hias.ad.hit-u.ac.j/ HIAS discussion aers can be downloaded without charge from: htt://hdl.handle.net/10086/27202 htts://ideas.reec.org/s/hit/hiasd.html All rights reserved.

3 EXPOSURE PROBLEM IN MULTI-UNIT AUCTIONS Xin Meng Dongbei University of Finance and Economics and Hikmet Gunay Hitotsubashi Institute for Advanced Study, Hitotsubashi University and the University of Manitoba February 10, 2017 Abstract We characterize the otimal bidding strategies of local and global bidders for two heterogeneous licenses in a multi-unit simultaneous ascending auction (SAA) like the one used in the 2008 Canadian Advanced Wireless Sectrum license auction. The global bidder wants to win both licenses to enjoy synergies; therefore, she bids more than her stand-alone valuation of a license. This exoses her to the risk of losing money even when she wins all licenses. We determine the otimal bidding strategies in the resence of an exosure roblem. By using simulation methods, first, we show that the robability of ine cient allocations in the simultaneous ascending auction can be u to 9 er cent. Second, we show that the global bidder can end u with a loss with 6 er cent robability deending on the distribution. We also investigate the relation between ine cient allocation and the revenue of SAA and VCG auctions. JEL Codes: D44, D82 Keywords: Multi-Unit Auctions, Vickrey Clarke Groves (VCG) mechanism, Exosure Problem, Synergies, Comlementarity, Sectrum License Auction Authors contributed equally to this aer. Xin Meng acknowledges the suort of NSFC(National Natural Science Foundation of China under Grant No ), SSHRC and the suort of DUFE-CIBO(Subject Research Project Fund under Grant No.LNTX ). H. Gunay thanks Osaka University-ISER for hosting him while doing this research, SSHRC for an indirect suort, and the University of Manitoba for roviding internal grants. We thank M. Aoyagi, D. Mishra, P. Murto, M. Pak, L. Ulku, J. Valimaki, R. Wang, and anonymous reviewers for their comments on the (earlier versions of the) aer. The usual disclaimer alies. This aer has been resented at Aalto University, Florida State University, Hitotsubashi University, ISI-Delhi, IIM-Bangalore, METU, Kyoto University-KIER,Osaka University-ISER, SWUFE-RIEM, Shanghai Jiao Tong University, TOBB-ETU, 2009 Iowa Alumni Worksho, 5th CIREQ PhD Students Conference, 43rd CEA meetings held in Toronto, 2009 Midwest Economic Theory Conference held at Penn State University, ESAM 2011, and 2014 AMES held in Academia Sinica. We thank the articiants for their feedback. 1

4 1 Introduction In the recent Canadian Advanced Wireless Sectrum (AWS) license auction, hundreds of (heterogenous) licenses were sold to firms simultaneously. Each of these licences gave the sectrum usage right of a geograhical area to the winning firm. Rogers sent almost $ CAD 1 billion to buy sectrum rights in each rovince and territory. However, Manitoba Telecom Services only bought licenses for Manitoba, and SaskTel only bought licenses for Saskatchewan. 1 Firms such as Manitoba Telecom Services, and SaskTel are called local firms since they are interested in winning only secific licenses in order to serve local markets while firms such as Rogers are called global firms since they are interested in winning licenses across the nation. 2 The global firms enjoy synergies if they win all the licenses; this gives them an incentive to bid more than their stand-alone valuations for some licenses. As a result, this bidding has the risk of incurring losses. Therefore, global bidders may lower their bids. This is known as the exosure roblem. In a model simlifying the recent Canadian Advanced Wireless Sectrum license auctions, 3 we derive the otimal bidding strategies of local and global firms in a simultaneous ascending auction (SAA) of two heterogeneous licenses when there is the ossibility of ex-ost loss. Although one of the main concerns of the olicy makers is the e ciency of the sectrum auction (Cramton and Schwartz, 2000), we are not aware of any aer in the literature that calculates the robability of the auction resulting in an ine cient outcome. We show that this robability is u to 9 er cent for some distributions in our main model. In order to study the e ect of the ine cient outcomes on revenue, we divide the ine cient allocations into two grous. In the first grou, the global bidder wins one license or all licenses with an (ex-ost) loss. We show that the SAA auction has a higher revenue than the e VCG auction for these kind of ine cient cient outcomes (ex-ost); the seller has a higher revenue 1 Information on Canadian AWS auction is mainly taken from Industry Canada Website. 2 Firms must deosit money before the auction. The amount of money shows their intention whether they will bid for all licenses or only select ones. Also, given the firms revious serving areas, one can exect whether a firm is local or global. 3 Some recent US license auctions also follow a similar format. 2

5 at the exense of the global bidder. Our simulations showed that the global bidder might end u with a loss 6 er cent of the time for some distribution functions. In the second grou, the global bidder does not bid high enough from the e ciency oint of view, and loses all licenses or wins one license with a rofit (where as the e cient outcome requires the global bidder to win both). Since the VCG auction corrects this ine ciency roblem, it results in a higher revenue than the SAA auction (ex-ost). We believe that this is a suort for the existence of exosure roblem since the global bidder does not bid high enough. The current literature on multi-unit auctions generally assumes that global bidders have either equal valuations (Englmaier et. al (2009), Kagel and Levin (2005), Katok and Roth (2004), Albano et. al. (2001), Rosenthal and Wang (1996), and Krishna and Rosenthal (1996)), very large synergies (Albano et al. (2006)), or that the marginal value of an additional license deends only on the number of licenses already held (Goerre and Lien (2014)). The sectrum licenses for di erent geograhic areas are not homogenous objects; hence, the equal valuation assumtion or equal marginal valuation of licenses assumtion does not fit the Canadian AWS (and US) sectrum license auction. Secifically, we model situations in which one sectrum license for, say, Toronto is more valuable than the sectrum license for Manitoba. 4 We do this by drawing two valuations from a distribution function and assign the maximum value for one license (say for Toronto license) and the minimum value for the other license. This modelling assumtion ensures that the auction for the weak location concludes before the strong location. This simlifies finding the equilibrium. Since there may be other auctions that this assumtion might not hold, we also study a revised model where the valuations of the local bidders are indeendently drawn. In the revised model, there is a ositive robability that auctions in any location might end first. Our main contribution to the literature is the result of our simulations, and characterizing the equilibrium for heterogenous licenses when synergies are moderate. By using the equilibrium and simulations, we calculate the robability of ine cient allocations, exected average revenue for SAA and VCG auctions, and exected welfare loss from the SAA auction by using ex-ost 4 The winning bids on the Industry Canada website show that Toronto licenses are more valuable than the Manitoba licenses. 3

6 valuations, and show the link between ine cient allocations and revenue. 2 The Model Suose there are 2 licenses for sale: license A and B. 5 There is one global bidder who demands both licenses and m local bidders who demand only license A, andanotherdi erent m local bidders who demand only license B. To determine the valuations of global bidder, we draw two rivate valuations, X 0 and Y 0 from the commonly known distribution function F (.). The global bidder s stand alone valuation for license A is = Max{X 0,Y 0 },andhis stand alone valuation for license B is v 0B = Min{X 0,Y 0 }.Wewanttomodelheterogeneous licenses, so with this modeling we make sure that license A is more valuable (since it is a license for a big city like Toronto) than license B (since it is a license for a medium size city like Winnieg) for the global bidder. The global bidder s total valuation, given that it wins two licenses is, V 0 = + v 0B +, wherethesynergyterm >0isassumedto be rivate knowledge, and it has a continuous density function on [0,a)wherea 2 (0, 1]. 6 To determine the valuations for local bidders, we draw two rivate valuations X i and Y i for i =1, 2,...,m,fromthesamedistributionfunctionF (.). The valuation of local bidder ia for license A is v ia = Max{X i,y i },andthevaluationoflocalbidderib for license B is v ib = Min{X i,y i }. The distribution function F (.) hasasuorton[0, 1] and robability density function f(.) whichisositiveeverywherewiththeonlyexcetionthatf(0) allowed. The bidders tye, global or local, is ublicly known. The valuations of global and local bidders are rivate information. These valuations result in Max{viA : i =1, 2,...,m} Max{viB : i =1, 2,...,m}. 7 0is Later, we add a model where the local bidders valuations are drawn indeendently, and hence, the license B might be more valuable than license A. 5 We use two licenses like Albano et. al. (2001 and 2006), Brusco and Loomo (2002), and Menucicci (2003). 6 It will be clear later that the distribution of does not matter as long as we have one global bidder. 7 With this formulation, it is ossible that a few local bidders B may have a high valuation than some other local bidder A due to them being very cost e ective. Also, note that we may assume di erent number of local bidders on each license and qualitative results will not change, as long as the number of local bidders A is higher than the local bidders B. This will guarantee that max value for license A is greater than max value for license B. We relax this assumtion later. 4

7 We consider a setting where the licenses are auctioned o simultaneously through an ascending multi-unit auction. Each license is auctioned o at a di erent auction (like Krishna and Rosenthal (1996) but unlike Kagel and Levin (2005)) at the same time. Prices start from zero for both licenses and increase simultaneously and continuously at the same rate. Bidders choose when to dro out. When only one bidder is left on a given license, the clock stos for that license, and the sole remaining bidder wins the license at the rice at which the last bidder dros out. If there is more than one bidder remaining on the other license, its rice will continue to increase. If m bidders dro out at the same rice and nobody is left in the auction, then each one of them will win the license with robability 1 m.thisisa zero measure event given the valuations are drawn from a continuous distribution function. The dro-out decision is irreversible. Once a bidder dros out of bidding for a given license, he cannot bid for this license at a later time. 8 The number of active bidders and the dro-out rices are ublicly known. We also assume that there is no budget constraints for the bidders. Local bidders have a weakly dominant strategy to stay in the auction until the rice reaches their valuation. We assume all bidders will use their weakly dominant strategy in equilibrium. We derive a symmetric Bayesian-Nash equilibrium in what follows. To understand the global bidder s strategy, consider a subgame in which every local bidder dros out of license B auction, and hence, the global bidder wins license B at the rice B = max{v 1B,...,v mb } since local bidders use their weakly dominant strategy and bid u to their valuation. The global bidder knows that, at a given clock rice c < 1, it will win license A at the rice A = max{v 1A,...,v ma }. The distribution of each v ia is [F (v ia c )] 2 given the clock rice c. 9 Then the conditional distribution of A is the highest order statistic R A f A (v)dv which we will denote as G where G( A c,m A )=[F( A c,m A )] 2m A =( c R 1 c f A(v)dv )2m A,where 8 In the real-world auctions, there is an activity rule: if the bidders do not have enough highest standing bids, then the number of licenses they may bid on is decreased (in the next rounds). Hence, when there are two licenses, this translates into an irreversible dro-out. 9 If the local bidder s valuations were indeendent, the conditional distribution of v ia given the clock rice c would be F (v ia c ). Since the valuations comes from a max distribution, each v ia is distributed as [F (v ia c )] 2. 5

8 m A is the number of active local A bidders. Then to determine its otimal dro out rice A for license A, the global bidder will maximize its continuation ayo.10 The ayo is max A The first order condition is Z A v 0B B + ( + A )dg( A c,m A ) c ( + A)dG( A c,m A )=0 ) A = + It is easy to check the second order condition which shows that this is a unique maximizer. 11 The result is intuitive, the global bidder bids until the rice reaches its marginal value of license A. The number of active local bidders does not a ect the decision of the global bidder. We would also like to note that the otimal dro out rice can be found by comaring ayo s from two cases. CASE 1 is the ayo from droing out of license A auction at the clock rice c (without winning license A), and this ayo is v 0B B. CASE 2 is the ayo from continuing on license A auction and winning at rice c (with robability 1). This ayo is + v 0B + B c which is monotonically decreasing in the rice c. Clearly as long as the ayo from CASE 2 is higher than the ayo from CASE 1, the global bidder will continue to stay in license A auction. The udated otimal dro-out rice, A, is found by equating these two equations: + v 0B + B A = v 0B B ) A = + (1) Similarly, if the global bidder wins license A, thenitstaysinlicenseb auction until the rice reaches v 0B +, butthiscannothaeninequilibriumsince A > B and local bidders use weakly dominant strategy With a slight abuse of notation, we treat A as a variable and the oint of otimal dro out rice. 11 The second order condition is [( + A )dg( A c,m A )] dg( A c,m A ) but the first term in closed brackets is zero at A = A from the first order condition, and dg( A c,m A ) < 0 for all c < 1. The function is strictly concave at its unique critical oint. 12 In the revised model, this can be an equilibrium. 6

9 If the global bidder s average valuation, +v 0B + 2,exceeds1,itsequilibriumstrategyis bidding anything from 1 u to his average valuation. As a result, the global bidder outbids his rivals. If is large enough, this condition is always satisfied. To calculate the otimal dro-out rice for the global bidder for license B when his average valuation is less than 1, one should maximize the global bidder s exected ayo. The exected ayo for the global bidder, in which denotes the dro out rice for license B, is: max " Z h Z min{ +,1} c + ( + A + v 0B B )dg( A B,m A ) (2) B Z 1 i + (v 0B B )dg( A B,m A ) dh( B ) (3) Z ( A )dg( A B,m A ) # dh( B ) (4) B min{ +,1} Z max{, v0a } In this equation H( B )=[1 (1 F (.) 2 ] m B since each valuation of local bidder B has acdfof[1 (1 F (.) 2 )], and the maximum of these when there are m B local bidders B is H( B ). 13 Readers will see that, in the roofs, H( B )hasnodirectroleincalculating the otimal dro out rice but it has a role in calculating the second order condition. We exlain this in footnote 16. The exlanation for the exected ayo equation, given the sequential rationality of the global bidder, is as follows. The first line calculates the exected ayo for the global bidder from winning both license B and (then) license A. The outer integral has an uer limit since B must be distributed between c and for the global bidder to win license B. After winning license B, globalbidderwillstayinlicensea auction until + by the sequential rationality (or will win license A as long as it stays until the rice reaches 1.). The second line shows the exected ayo from winning license B but losing A. In order to lose license A after winning license B, thecondition A > + must hold given the 13 Recall that each local bidder B s valuation is a minimum random variable; hence, the cdf of minimum of two indeendently drawn random variables from F is [1 (1 F (.) 2 )]. 7

10 sequential rationality of the global bidder. The third line shows the exected ayo from winning license A only. For this to haen, B >must hold (to lose license B) andthen since < B < A,wehavethelowerlimitoftheintegralas B. The uer limit of the integral shows the sequential rationality of the global bidder. The global bidder who loses license B becomes just like a local bidder and will bid until. When + <1, the first derivative of this maximization roblem is: FOC = Z + Z 1 ( + v 0B + A )g( A, m A )d A + [< ] Z v0a + (v 0B )g( A, m A )d A (5) ( A )g( A, m A )d A h() (6) We can write this as FOC = J(, m A )h(), where J(.) isthetermintheclosedbrackets. First, note that the FOC is continuous at =. 14 Second, any 0 that solves J(, m A )=0 also solves FOC =0. Intheroofs,wewillshowthat,this 0 is indeed the unique solution to the FOC roblem. Third, note that when FOC( 0)=0,h() iscancelledoutfromfoc. Let us discuss the first order condition of this maximization roblem after cancelling h(). The global bidder must comare the ayo s for two cases at each otential dro out rice as the clock is running: Case 1 is the ayo from droing out from license B auction at rice (without winning) and otimally continuing to bid on the license A auction (or otimally dro out from license A auction at the same, if> ). The ayo is: E 1 0(, m A )= [<v0a ] Z v0a ( A )g( A, m A )d A (7) This is equation 6 above when + <1 after cancelling h(). Case 2 is the ayo from winning license B at rice (with robability 1 so this is the highest rice the global bidder would ay) and otimally continuing on the license A auction. E 2 0(, m A )= Z Min{v0A +,1} ( + v 0B + A )g( A, m A )d A Z 1 + (v 0B )g( A, m A )d A (8) Min{ +,1} 14 As aroaches from the right, the value of Equation 6 is always zero. As it aroaches from the left, the value of Equation 6 aroaches to zero. 8

11 This is equation 5 above (when + <1) after cancelling h(). Hence, as long as the ayo of Case 2 is higher, the global bidder will stay in license B auction. In the roofs, we show that J(, m A )isdecreasingin. Then, we will show that J( = v 0B,m A ) > 0 and J( = +, m A ) < 0. Hence, if J(,m A ) < 0, there is a unique root (solution) 0 2 (v 0B, ). If not, we will have a unique root (solution) 0 2 [, min{ +, 1}). We show that this is the otimal dro-out rice, 0,whichisthesolutiontothefirstorder condition of the global bidder s exected ayo maximization roblem. 15 In Proosition 1 below, we characterize the global bidder s equilibrium bids. Note that these ayo s are changing as local bidders bidding for A are droing out; that is, m A is changing. Therefore, the lemma below gives the global bidder s (udated) equilibrium droout rice as the local bidders dro out. In the roof of Proosition 1, we show that this udated rice increases as local bidders dro out of license A auction. 16 Proosition 1 Suose that the average valuation of the global bidder is less than 1 and there are m A local bidders bidding on license A where m A local bidder B. 1 and there is at least one active The global bidder will maximize its ayo by droing out of the license B auction at the unique otimal dro-out rice 0(m A ) that satisfies J(, m A )=0. Moreover, a) If + <1, and J(,m A )= R + G( A,m A )d A +(v 0B ) < 0, then 0(m A ) < and the global bidder will stay in the license A auction until (after droing out from the license B auction). b) If + <1, and J(,m A )= R + G( A,m A )d A +(v 0B ) 0, then 0(m A ) and the global bidder will also dro out of the license A auction at 0(m A ). c) If + 1, and J(,m A )= R 1 G( A,m A )d A +(v 0B + 1) < 0, then 0(m A ) < and the global bidder will stay in the license A auction until (after droing 15 The comaring ayo s method is used by Albano et. al. (2001) for identical licenses. 16 Note that when v 0B <, local bidders B droing out of auction do not a ect the global bidder s dro out rice. This seemingly surrising result arises because the global bidder s equilibrium incentives are conditioned on obtaining license B at current rice so that the remaining number of local bidders B is irrelevant. The number of local bidders A is imortant since it a ects the rice at which the global bidder can obtain that license. We are grateful to a referee for making this oint. 9

12 out from the license B auction). 17 d) If + 1, and J(,m A )= R 1 G( A,m A )d A +(v 0B + 1) 0, then 0(m A ) and the global bidder will also dro out of the license A auction at 0(m A ). e) As m A decreases, 0 increases. Proof. See Aendix. The inequality conditions on R 1 G( A,m A )d A +(v 0B + 1) in art c and d just make the first order condition at = less than or greater than zero for the + >1 case. We also note that J(, m A )=0isequivalenttoE 1 0(, m A )=E 2 0(, m A )forallcases from art a to d. We are ready to summarize our Bayesian-Nash equilibrium. Proosition 2 (Bayesian-Nash Equilibrium) a) A local bidder of each license will stay in the auction j until the rice reaches her valuation v ij where j = {A, B}, i = {1, 2, 3,..m}. b) A global bidder active only on license j will bid v 0j +, if he won the other license. He will bid v 0j if he droed out of the other license. c) When the average valuation is less than one, the global bidder who is active on both licenses and facing m A active local bidders on license A will behave as described in roosition 1. d) If the average valuation is greater than one, the global bidder will stay in both auctions until rice reaches his average valuation. 3 Simulations; Ine cient Allocations And The Revenue Comarison of SAA and VCG Auctions Policy makers want an e outcome is ine cient auction outcome (see Cramton and Schwartz, 2000). If the cient, the winning firms may choose not to use the scarce sectrum after 17 While J(, m A ) takes di erent forms for di erent cases deending on + <1 or not, we kee using the same notation since it is clear which case we are referring to. 10

13 the auction, and hence, the society cannot benefit from the auction fully. 18 clear olicy objective, the robability of ine Desite this cient SAA outcome has not been calculated in the literature, to our best knowledge. We believe that the reason was that an analytic calculation is not ossible, and one has to use simulations for this. 19 Our simulation code is written in MATLAB. It starts with drawing two indeendent valuations for the global firm from the same distribution function. 20 The global bidder s valuations for license A and B become the maximum and the minimum of these two valuations, resectively. Then, we draw two indeendent valuations from the same distribution function. We use one local bidder for each license. The local bidder A s and B s valuations take the maximum and the minimum of these valuations, resectively. One set of valuations corresonds to one auction. Knowing these valuations, by using our theoretical model, we calculate the otimal dro-out rice(s) for the global bidder for all ossible di erent cases. We count how many times, ex-ost, the outcome will be ine cient, and what the revenue would be for four di erent synergy values of 0.2, 0.4, 0.6, 0.8. Since our results converge when we run for auctions, we divide the number of ine cient outcomes with 10000, and derive the robability of an ine 3.1 Ine cient Outcome cient outcome. In order to classify an outcome as e cient or ine cient, we use the following lemma and discussions which are summarized in Table 1. Lemma 3 If the global bidder has an (ex-ost) loss, then 0 > B >v 0B must hold and the outcome is always ine cient. This lemma shows that a necessary condition for the global bidder to make a loss is that it must win license B with an (initial) loss. This initial loss may turn into a rofit if the 18 One can give an examle of ine cient allocation of AWS auction. Quebecor won 10 Mhz of sectrum for Toronto area but had not used it until According to The Globe and Mail article ublished on Setember 19, 2012, Mr. Pruneau, chief financial o cer of Quebecor, confirmed Wednesday that Quebecor has no lans to build a mobile network in Toronto with that 10 MHz of sectrum. 19 Calculating ex-ante robability might only be ossible with numerical methods but this will be more di cult than the method of using ex-ost valuations that we use in this aer. 20 This code is more than 300 ages long as a word document. Codes can be requested from the authors. 11

14 global bidder wins license A but it may also end u in an ex-ost loss. The latter will be an ine cient outcome. The (ex-ost) revenue with SAA auction will be higher for such cases as we will rove later. This lemma does not give a su cient condition. We have cases where the global bidder makes a rofit but the outcome may be e cient or ine cient (line 3 and 4 of Table 1). Also, note that a case where the global bidder wins only license A and makes a loss is not ossible (line 5 of Table 1). The reason is that when global bidder loses license B, itbehavesjust like a local bidder and will bid u to. We summarize all ossible outcomes in Table 1. License A won by License B won by Global bidder makes Allocation is Revenue SAA 1. Global Bidder Global Bidder Profit E cient A + B 2. Global Bidder Global Bidder Loss Ine cient A + B 3. Global Bidder Local Bidder B Profit E cient A Global Bidder Local Bidder B Profit Ine cient A Global Bidder Local Bidder B Loss (not ossible) N/A N/A 6. Local Bidder A Global Bidder Loss Ine cient + + B 7. Local Bidder A Global Bidder Profit E cient + + B 8. Local Bidder A Local Bidder B Zero Profit E cient Local Bidder A Local Bidder B Zero Profit Ine cient Local Bidder A Local Bidder B Zero Profit E cient Local Bidder A Local Bidder B Zero Profit Ine cient 2 0 Table 1: All e cient and ine cient outcomes. As written in Table 1, there are cases in which the local bidder may win both licenses and the outcome may be e cient or ine cient. For such ine cient cases, the global bidder may dro from both licenses at 0 (line 10 and 11) or may dro from license B at 0 and from license A at (line 8 and 9). The global bidder does not bid high enough due to the risk of ex-ost loss. As we will show later, VCG mechanism corrects this ine ciency, and also increases the revenue (ex-ost). These cases are the basis of the exosure roblem. By using Table 1, we can count the number of e cient and ine cient outcomes in our simulations for each of our auctions. Our simulation results are summarized in Figure 1and2. TheEx-ostLossProbabilityisthesummationofcases/auctionsthatfallunder row 2 and 6 in Table 1. Probability of Ine ciency is the summation of cases/auctions that 12

15 Figure 1: Beta Distribution =1, = Plot of Probability of Inefficiency Probability of Inefficiency Ex ost Loss Probability Plot of Revenue Comarison robability Average Revenue theta (Synergy) 0.4 Average Revenue for Our Model Average Revenue for efficient VCG Model theta (Synergy) 13

16 Figure 2: To: Uniform Distribution. Bottom: Beta Distribution with = 5, = Plot of Probability of Inefficiency Probability of Inefficiency Ex ost Loss Probability Plot of Revenue Comarison robability Average Revenue theta (Synergy) 0.8 Average Revenue for Our Model Average Revenue for efficient VCG Model theta (Synergy) Plot of Probability of Inefficiency Probability of Inefficiency Ex ost Loss Probability Plot of Revenue Comarison robability Average Revenue theta (Synergy) 0.93 Average Revenue for Our Model Average Revenue for efficient VCG Model theta (Synergy) 14

17 fall under row 2, 4, 6, 9, and 11. When the synergy level is 0 (no externality) or when it is greater than 2, the outcome is always e cient with the SAA auction. However, for in between values, there are ine cient allocations according to the figures; hence, there is not a monotonic relation between synergy levels and the robability of ine cient allocations. This is what we observe in the uniform distribution figure. 21 Our main result is that the ine cient outcomes can be as high as 9 er cent. The cases where the global bidder makes a loss accounts for aroximately 6 er cent of the total outcome in some simulations (see Figure 1). There are cases where local bidders winning ine ciently. This imlies that the global bidder is not bidding as high as it should from an e ciency oint of view since it is well aware of the ossibility of making a loss. In Table 2, we reort the otimal dro out rices for a samle of global bidder s valuations. The table shows that the global bidder exoses herself to more risk by bidding well-over her stand alone valuation of license B when her synergy level is higher and/or when she exects local bidder A s valuation to be lower. However, as the synergy increases by 0.2 unit, the otimal dro out rice increases less than 0.2 unit. The imlication of these on the revenue of the SAA and the VCG auction is an interesting one. When the local bidders win one or both licenses ine ciently, the revenue of the VCG auction is always greater than the revenue of the SAA auction ex-ost. When the global bidder wins licenses ine ciently (with an ex-ost loss), the revenue of SAA auction is always greater than the revenue of the VCG auction ex-ost. This is not surrising since the global bidder bids over its stand alone valuation, it ends u with an ex-ost loss but this increases the seller s revenue. We summarize the discussion above with the following roositions which exlain the imact of ine cient allocations on revenue. Proosition 4 If the global bidder wins licenses with an ex-ost loss then the revenue of the 21 In the other figures, the eak of the grah must be below 0.2. The figures (x-axis) start from 0.2 and hence show only a decreasing art. 15

18 Table 2: Global bidder s otimal dro-out rice for a samle of valuations under various distributions and synergy levels Global Global Uniform Beta Beta Bidder s Bidder s Distr. Distr. Distr. Valuation Valuation with =5 with =1 for License A for License B and =5 and =3 v 0B Synergy= Synergy= Synergy= Synergy=

19 VCG auction is lower than that of the simultaneous ascending auction (SAA). Proosition 5 a) If the local bidder wins both licenses and the allocation is ine cient, then the revenue of the VCG auction is greater than the revenue of the SAA. b) If a local bidder wins one license and the global bidder wins the other license without loss, and the allocation is ine cient, then the revenue of the VCG auction is greater than the revenue of the SAA auction. 22 Unfortunately, the revenue comarison is unclear when the allocation is e cient with the SAA auction. There are cases in which the SAA auction gives higher or lower revenue than the VCG auction in e cient allocations. Finally, we calculate the exected welfare loss that is created by the SAA auction. We calculated the welfare with the SAA and VCG auction with our code (for draws). Then, we find the ercentage di erence between the two auctions welfare. The results are summarized in Table 3. The exected welfare loss is less than one er cent in all distributions we have used. While this may seem low, the revenue of the 2008 Canadian AWS auction was more than 4 billion Canadian dollars. Even a small ercentage decrease in revenue (say 0.5 er cent) might have a relatively big magnitude e ect (more than 20 million dollars). The same logic would aly for the welfare. Table 3: The ercentage shortfall in welfare in the SAA relative to the VCG auction under various distributions Synergy Uniform Distr. Beta Distr.with =5 Beta Distr.with =1 and =5 and = % % % % % % % % % % % % 22 We ski the roof of art b since it is similar to art a. 17

20 4 Revised Model In the main model, auction for license B always ends first in equilibrium. We have also studied a revised model in which either auction might end first with a ositive robability. 23 In this model, we assume one local bidder bidding on each license. 24 To determine the valuations for local bidders A and B, wedrawtworivatevaluationsv 1A and v 1B indeendently from the distribution function F (.). As in the main model, the global bidder s valuations are drawn in airs from F.Themaximumandtheminimumofthesedrawsbecometheglobalbidder s license A and license B valuations, resectively. 25 Everything else in the main model also alies to the revised model. Since local bidder s valuations are indeendently drawn, local bidder B s valuation can be the highest among all bidders. This might haen if local bidder Bisextremelycoste cientintheregionitoerates(afterwinningthelicense). The main di erence between the main model and the revised model is as follows. Since, the local bidder A has a maximum valuation in the main model, the global bidder is more likely to lose license A after winning license B. As a result, in the revised model, the global bidder faces less ex-ost loss desite using a higher otimal dro out rices. Hence, the welfare shortfall in the SAA relative to the VCG auction in the revised model is less than the welfare loss in the SAA relative to the VCG auction in the main model. 5 Conclusion and Discussion In this aer, we showed the otimal bidding strategies of global bidders when there are moderate synergies and the licenses are heterogeneous. We also extensively analyzed the ine cient allocation, exosure roblem and their e ects on revenue. One natural question to ask is whether there is an otimal mechanism for these tyes of auctions. Unfortunately, this is a di cult and oen roblem in the economics and comuter 23 The revised model is a technical extension of the main model; hence, we admit that we cannot rovide much intuition. The revised model can be requested from the authors. 24 When we used more than one local bidder, we could not rove that the otimal dro out rice of the global bidder will increase as the local bidders dro out. Hence, we could not extend the roof for more than one local bidder case. 25 This makes the global bidder s license valuations heterogeneous. 18

21 science literature since the standard otimal mechanism design techniques cannot be alied when the valuations are multi-dimensional. In our model, the global bidder s valuation is three-dimensional; one valuation for license A, oneforlicenseb and one for the rivate synergy arameter. Ulku (2013) tackles multi-dimensional valuations but he essentially reduces the multi-dimensionality into single-dimension, and his techniques are not alicable in our framework. The comuter science literature made some rogress to find the almost otimal mechanism with algorithms (Cai et. al 2011). We can generalize our main model where there are more A local bidders than B local bidders. For examle, if there are more A local bidders, we can draw two valuations for each one of the additional A local bidders from the distribution function F and assign the maximum of the two values as their valuation. This formulation guarantees that A > B and our results hold. If there are more B local bidders, with a model where one draws two values and assigns the minimum to the additional B local bidders, A < B is ossible, 26 then the global bidder has to deal with cases in which it wins license A first, and our results may not hold. We do not believe that assuming more B local bidders will bring new insights, and we have already analyzed a model where any auction might end with a ositive robability. Our main contribution was our simulations that calculates the robability of ine cient allocations in the SAA auction by using ex-ost valuations. We showed that the robability of ine cient allocation can be u to 9 er cent for some distribution functions. To our best knowledge, our aer is the first one calculating these robabilities. We believe that this is a relevant information for Industry Canada, and other olicy makers. 27 Our other contribution is comaring the revenue of the simultaneous ascending auction (SAA) with those of the VCG auction. We find the following relation between the ine cient allocations and the revenue of the auctions. If the valuations are such that the global bidder wins the licenses with an ex-ost loss (which is an ine cient outcome), then the revenue of 26 The minimum of two values can be aroximately 1 or even exactly 1, the highest ossible value so A < B is ossible with a ositive robability. 27 Needless to say, more research has to be done to determine the ercentage of ine cient allocation since we do not know the exact distribution of valuations. Bajari and Fox (2013) estimate the synergy value to be around 73 er cent of the valuations for one of the US sectrum auctions. 19

22 the SAA auction for these valuations would be higher than the revenue of the VCG auction. When local bidders win the license(s) ine ciently (in the SAA auction), the VCG mechanism gives a higher revenue. In our simulations, there are many cases where the local bidders win ine ciently. We interret this as a suorting evidence of the exosure roblem. The global bidder does not bid high enough from an e ciency oint of view, and hence, the local bidders win the licenses ine ciently. Our simulation code is a comlex one so increasing the number of local bidders even to 2 would be a roblem for calculating ine cient allocations. However, calculating revenue for the SAA and VCG auction with simulations for more than one local bidder seems more feasible. We are not sure whether such a code will bring new insights. We use only one global bidder. Albano et. al (2006) writes In fact, for intermediate values of 2 (0, 1) and if v 1 and v 2 are di erent...; showing the existence of a PBE is already roblematic in this case. In their aer, is used to denote synergy and their v 1 and v 2 are and v 0B in our aer, resectively. We use low/moderate synergy at the exense of giving u using more than one global bidder. We also note that Goeree and Lien (2014) uses one global bidder when they assume substitutability of items for local bidders in subsection 4.1 of their aer. Kagel and Levin (2005) is another examle where one global bidder is used in the literature. When there are two global bidders, one must write each global bidder s first order conditions searately and solve it simultaneously. The di culty arises from the fact that while solving these equations, one needs the distribution of the other global bidder s otimal dro out rice since E 1 0 (and E 2 0)willbefunctionsofthe other global bidder s otimal dro out rice. 6 Aendix Proof of Proosition 1. We will rove that there is a unique otimal dro-out rice with lemma 6 and lemma 7 below. First we will show that there exists a unique 0 2 (v 0B, min{ +, 1}) thatsolves 20

23 J( 0)=0. Then,wewillrovethatonly0, 0,and1satisfyFOC = J()h() =0. Then, we will show that 0 satisfies the second order condition so it is a local maximum. Then, we will show that 0 and 1 cannot be the maximizer, and, 0 is the unique maximizer. Lemma 6 There exists a unique 0 such that J( 0)=0given the conditions a) to d) in Proosition 3. Proof of Lemma 6 a). In this case, we have the assumtions of + < 1and R v0a + G( A,m A )d A +(v 0B ) < 0. We will show that this imlies 0 < (which in turn imlies E 1 0 > 0). R v0a + We have already shown in the text that J(, m A )=E 2 0 E 1 0,whereE 2 0(, m A )= (v 0A +v 0B + A )g( A, m A )d A + R 1 (v + 0B )g( A, m A )d A and E 1 0(, m A )= R v0a (v 0A A )g( A, m A )d A. To rove uniqueness, we will show that J(, m A )ismonotonicallydecreasingfrom0to,anditisositivewhen = v 0B and is negative when =. Hence, there must be a unique root in the interval v 0B <<. J(, m A )= R + ( + v 0B + A )g( A, m A )d A + R 1 + (v 0B )g( A, m A )d A R v0a ( A )g( A, m A )d A By using R 1 g( A, m A )d A = 1 since g( A, m A )isarobabilitydensityfunctiononthe suort [, 1], we have (v 0B ) R 1 g( A, m A )d A = v 0B,wecanre-writeitas R v0a J(, m A )= (v 0A A )g( A, m A )d A + R + (v 0A + A )g( A, m A )d A + (v 0B ) We will use integration by arts, R udv = uv R vdu, twiceinwhatfollowstore-writej(.). First we assume that u = A and v = G( A, m A ); then assume that u = + A and v = G( A, m A ), we have J(, m A )= ( A )G( A, m A ) + R G( A, m A )d( + A ) R v0a + G( A, m A )d( + A )+(v 0B ) +( A )G( A, m A ) + R v0a = G( A, m A )d A + R + G( A, m A )d A +(v 0B ) = R + G( A, m A )d A +(v 0B ) 21

24 We take artial derivative of J(, m A A ) G( A, m A )d A ] 1. We will rove that this is negative since the [R G( A, m A )d A ] is negative. The term inside is non-negative since it is a cumulative distribution function. We must show A,m A ) ale 0 to rove this. one can easily see that this is correct (as increases the cumulative distribution conditional on decreases), we will give a formal roof by using Leibniz s A,m A R A f(v)dv R 1 f(v)dv )m = (m A )f() (R A f(v)dv) m A 1 ( R 1 f(v)dv)m A +(m A )f() ( = (m A)f()( R A f(v)dv) m A 1 ( R 1 f(v)dv)m A [ 1+ R A f(v)dv R ] 1 f(v)dv R A f(v)dv) m A ( R 1 f(v)dv)m A +1 = (m A)f()( R A f(v)dv) m A 1 ( R 1 [ 1+F ( f(v)dv)m A A, m A )] < 0 (ale 0onlyif A =1). Thus, J(, m A )isamonotonicallydecreasingfunctionof2(0, ). If = v 0B,thenJ( = v 0B,m A )= R v 0B G( A v 0B,m A )d A + R + v 0B G( A v 0B,m A )d A = R + G( A v 0B,m A )d A > 0. If =, J( =,m A )=0+ R + G( A,m A )d A +(v 0B ) < 0byour assumtion. Hence, there is a unique root, 0,suchthatv 0B < 0 <. b) In this case, we have the assumtions of + <1and R + G( A,m A )d A + (v 0B ) 0whichwewillshowthatthisimlies 0. And this condition in turn imlies that E 1 0 =0,sincetheglobalbidderdrosfrombothlicenses. E 2 0(, m A )= J(, m A )= R + (v 0A + v 0B + A )g( A, m A )d A + R 1 (v + 0B )g( A, m A )d A = R v0a + G( A, m A )d A +(v 0B ). When,wetakeartialderivativeofJ(, m A )withresectto, we A ) [R A, m A )d A ] 1= G( A, m A )+ R A,m A ) A 1 < 0, A,m A < 0. Thus, J(, m A )isamonotonicallydecreasingfunctionof in (, 1). Our assumtion R + G( A,m A )d A +(v 0B ) 0imliesthatJ( =,m A ) 0. If = +, thenj( = +, m A )=(v 0B ) < 0. Thus, there is a unique solution, 0,intheinterval[, + ). 22

25 c) In this case, we have the assumtions of + 1and R 1 G( A,m A )d A +(v 0B + 1) < 0 which imlies < 0. And this condition in turn imlies that E 1 0 > 0. That is, E 2 0(, m A )= R 1 (v 0A + v 0B + A )g( A, m A )d A and E 1 0(, m A )= R A )g( A, m A )d A. J(, m A )=E 2 0(, m A ) E 1 0(, m A )= R 1 ( + v 0B + A )g( A, m A )d A R v0a ( A )g( A, m A )d A Similar to art a, we use integration by arts twice. First, we assume that u = and v = G( A, m A )forthefirstintegral;thenassumethatu = + v 0B + A and v = G( A, m A ) for the second integral. As a result, J(, m A + >1) = ( A )G( A, m A ) v 0B + A )G( A, m A ) 1 + R 1 G( A, m A )d A = R G( A, m A )d A +( + v 0B + 1) + R 1 G( A, m A )d A =( + v 0B + 1) + R 1 G( A, m A )d A We take artial derivative of J(, m A )withresectto, A = [R 1 G( A, m A )] 1 < 0 ( A R v0a G( A, m A )d A +( + It is negative since [R 1 G( A, m A )] is negative. And we have already shown A,m A ale 0. Thus, J(, m A )ismonotonicallydecreasingfunctionof in (0, ). If = v 0B,thenJ( = v 0B,m A )= R 1 G( A v 0B,m A )d A +( + + >1. If =, J( =,m A )=0+ R 1 G( A,m A )d A +(v 0B + assumtion. Hence, there is a unique root in the interval v 0B < 0 <. 1) > 0since 1) < 0byour d) In this case, we have the assumtions of + 1and R 1 G( A,m A )d A + (v 0B + 1) 0 which imlies 0. And this condition in turn imlies that E 1 0 =0, and E 2 0(, m A )=J(, m A )= R 1 ( + v 0B + A )g( A, m A )d A = R 1 G( A, m A )d A +( + v 0B + 1). When >,wetakeartialderivativeofj(, m A )withresectto, we A = [R 1 G( A, m A )d A ] 1 < 0, A,m A < 0. 23

26 Thus, J(, m A ) is monotonically decreasing function of in (, 1). Our assumtion R 1 G( A,m A )d A +(v 0B + 1) 0imliesthatJ( =,m A ) 0. If =1,thenJ( =1,m A )=(v 0B + + 2) < 0. Since v 0B + + <2by our average valuation is less than 1 assumtion in the roosition. Thus, there is a unique solution, 0,intheinterval[, 1). Lemma 7 a) Only 0,1, and 0 solves the FOC. b) 0 satisfies the second order condition. c) 0 and 1 cannot maximize the exected ayo. d) d 0 dm A < 0. That is, when the number of active firms in license A auction decreases, the otimal dro-out rice will increase. Proof a) From lemma 6, we can write the FOC = J(, m A )h() withtheabuseof notation since J(, m A )takesdi erentformsinartatod. ItiseasytoseethatFOCis equal to zero only if J(, m A )=0orh() =0. Inlemma6,weshowedthatthereisaunique 0 2 (v 0B, min{ +, 1}) thatmakesj(, m A ) = 0. Hence, FOC( 0)=0. Thisroves 0 solves FOC. Now, we will show that only =0or =1solvesh() =0. SinceH() =[1 (1 F ()) 2 ] m B,wehaveh() =2m B (1 F ())f()[1 (1 F ()) 2 ] m B 1. Then h() =0only if 1 F () =0whichimlies =1,or1 (1 F ()) 2 =0whichimlies =0. There is no 2 (0, 1) that makes h() =0sincebyourassumtionf() > 0when2 (0, 1] and m B 1. b) FOC = J(, m A )h(), hence SOC = J 0 (, m A )h() +J(, m A )h 0 (). When = 0, J( 0)=0asshowninlemma6,hence,SOC = J 0 ( 0,m A )h( 0). This is negative since we have showed in lemma 6 that J(, m A )isadecreasingfunctionandh() > 0forany 2 (0, 1). c) While it is straightforward to see that droing out at =0cannotbeotimal,we will rovide the roof. When we consider = 0, the exected ayo function is written as in 24

27 art a and c of Proosition 1 since the otimal dro out rice is assumed to be =0<. We have shown that J() isadecreasingfunctionin(0, )andj( 0)=0. Thesetwo facts imly that that J() > 0forany 2 (0, 0). Since h() > 0for 2 (0, 1), we have FOC = J()h() > 0forany 2 (0, 0). Since J() andh() arecontinuous,foc > 0 imlies that the global bidder can increase its ayo by bidding higher than 0. Hence, 0 cannot be the maximizer. One can similarly rove that =1cannotbethemaximizerby using art b and d of Proosition 1. Therefore 0 is the only rice that satisfies the FOC and SOC and it is unique. d) Next, we show that as the number of active firms in license A auction decreases, the otimal dro-out rice will increase. We will use the imlicit function theorem for this: 28, d 0 dm A 0,m A 0,m A 0 < 0. We 0,m 0 since J( 0) = 0, and the inequality holds since we have roved in lemma 6 that J 0 ( 0) < 0. = J 0 ( 0)h( 0)+J( 0)h 0 ( 0)=J 0 ( 0)h( 0) < 0. The equality holds By using Leibniz s rule for di erentiation under the integral sign, we take artial derivative of FOC( 0,m A )= R + G( A 0,m A )d A +(v 0B 0) h( 0)withresecttom A, since h( 0) > 0,m A 0,m A = R A 0,m A d A h( 0 )= R + ln(f ( A, m A ))G( A 0,m A )d A h( 0 ) < 0, A 0,m A =ln(f ( A 0,m A ))G( A, m A ) < 0. Hence, we show 0,m A < 0holds. 0,m 0 < 0,m A < 0, we have, d 0 dm A 0,m A 0,m A 0 By the imlicit function theorem, we show that the otimal dro-out rice increases as the number of local firms, m A,decreases. < 0 Proof of Lemma 3. If the global bidder wins only one license and has an ex-ost loss, then clearly 0 > B >v 0B must hold since the global bidder will wait until the rice reaches To rove the ine ciency for this case, we will use the inequalities v 0B < B 28 We treat m A as a continuous variable here although it is discrete. We do not see any harm in doing this since the function is continuous in m A and the result will be valid even when m A is discrete. 29 Note that the global bidder cannot win license A first since local bidders minimum valuations are 25

28 and + < A.ThelatteroneholdssincetheglobalbidderdoesnotwinlicenseA. This immediately shows that we have + v 0B + < A + B ;thee cientoutcomeisthatthe local bidders must have won both licenses. Hence, the outcome is ine cient. If the global bidder wins two licenses, winning license B imlies 0 > B and winning license A imlies A < +. The global bidder makes a loss by our assumtion, that is + v 0B + A B < 0whichimliesv 0B B < A < 0. Hence, we have v 0B < B. Hence, we showed that 0 <v 0B < B. The loss condition imlies + v 0B + < A + B which roves that the outcome of global bidder winning two licenses is ine cient since local bidders must have won the licenses. Proof of Proosition 4. When the global bidder wins the license(s) with an ex-ost loss, then the revenue of VCG auction, R VCG, is lower than the revenue of SAA, R SAA.We know that B >v 0B by lemma 3. There are two cases that we must consider, (i) global bidder wins both licenses with an ex-ost loss and (ii) global bidder wins only license B with an ex-ost loss. We have already exlained that global bidder cannot win license A only and makes a loss since it will bid like a local bidder after losing license B. i) The revenue of SAA auction is R SAA = A + B,andsincetheglobalbidderwinsboth licenses and makes a loss, we have A + B > + v 0B +. We know that + > A since global bidder wins license A and B in the SAA auction. In a VCG auction, the local bidders will win the licenses if < A.TheaymentoflocalbidderB can be calculated as follows. If local bidder B does not articiate in the auction, the welfare of others (in the e cient allocation) is A + v 0B since local bidder A will win license A and global bidder will win license B. If local bidder B articiates in the auction, the welfare of others (in the e cient allocation) will be A.Thedi erenceistheaymentoflocalbidderbwhichisv 0B. A symmetric calculation will show that the local bidder A s ayment will be. Hence, R VCG = + v 0B <R SAA = A + B by the global bidder s loss condition and that >0. If > A,thenthee cientallocationisthatglobalbidderwinslicenseaandlocal bidder B wins license B. The global bidder s ayment will be A since if the global bidder assigned to B and maximum valuations are assigned to A. 26