Estimating Strategic Complementarities in Credit Unions Outsourcing Decisions. Ron Borzekowski and Andrew Cohen. Federal Reserve Board of Governors

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1 Estiating Strategic Copleentarities in Creit Unions Outsourcing Decisions Ron Borzekowski an Anrew Cohen Feeral Reserve Boar of Governors March 2005 (Preliinary Version) Abstract: We exaine the choice between internal or external provision of inforation technology ( IT ) services for US creit unions. Epirically, the likelihoo that a creit union outsources its IT is increasing in the nuber of other creit unions in the sae geographic arket who outsource. We estiate a gae theoretic oel in which the siultaneous outsourcing ecisions of the other creit unions in the arket are containe as an arguent in the relative costs associate with outsourcing. We aapt ethos for estiating entry oels with ultiple equilibria to a setting with strategic copleentarities, where ultiple equilibria are eneic. Since our approach oes not require one to select a particular equilibriu when there is ore than one, we can assess how firs coorinate to particular equilibria in ifferent types of arkets. Our ain epirical fining is that the likelihoo of coorination failure is U-shape in the nuber of firs in the arket, reaching its iniu at five firs. JEL Coes: L1, C5, C7 The opinions expresse are those of the authors an o not necessarily represent those of the Boar of Governors or its staff. We are grateful to Brian McManus an Elie Taer for useful iscussions.

2 I. Introuction We exaine the choice between internal or external provision of inforation technology ( IT ) services for US creit unions. Creit unions ay aintain their own systes for tracking loans an eposit accounts or they ay choose to outsource these systes fro external proviers. Epirically, the likelihoo that a creit union outsources its syste is increasing in the nuber of other creit unions in the sae geographic arket that outsource. This regularity ay be ue to characteristics coon to the creit unions in close proxiity or characteristics of the arket itself that ake it ore or less likely that creit unions ake the sae technology choice. It ay also be ue to copleentarities whereby one creit union s ecision to outsource affects the relative costs associate with outsourcing for the other creit unions in the arket. Anecotal evience suggests that copleentarities ay result fro counication aong creit unions through local associations an state level leagues an which is further facilitate by creit unions non-profit status an lack of significant copetitive overlap. Even if copleentarities exist, it is nonetheless possible that a lack of coorination aong participants in the arket can lea to suboptial outcoes. In this paper, we attept to easure the size of strategic copleentarities in the cost function for IT an to assess the egree of coorination such copleentarities engener. To accoplish this, we estiate a gae theoretic oel in which creit unions outsource if oing so achieves a cost savings relative to aintaining their own systes. The ecision to outsource is oele as a gae in which the siultaneous outsourcing ecisions of the other creit unions in the arket are containe as an arguent in the relative costs associate with outsourcing. The gae gives rise to an econoetric oel

3 that is incoplete. In particular, the existence of ultiple pure strategy Nash equilibria ( PSNE ) iplies that the likelihoo associate with those outcoes is not well-efine, precluing stanar axiu likelihoo estiation. Researchers estiating payoff functions in these types of gaes have confronte coplications cause by existence of ultiple equilibriu outcoes, ating back at least to the work of Bresnahan an Reiss (1990). In recent papers, Taer (2002), Ciliberto an Taer (2004) ( CT ), an Anrews, Berry an Jia (2004) ( ABJ ) have ae significant progress on this proble. In particular, these papers have suggeste ethos for estiating payoff paraeters assuing only that observe outcoes represent a PSNE, without specifying any selection rule in cases where the oel is consistent with ultiple PSNE. This literature has been inspire by the stuy of entry gaes an other gaes in which players actions are strategic substitutes in the sense that a given agent s payoff is ecreasing in the nuber of other players taking the sae action. In contrast, our stuy of creit union outsourcing is an exaple of a gae with strategic copleentarities where a given agent s payoff is increasing in the nuber of other players who take the sae action. In this case, ultiple PSNE are also eneic; though the probles arising fro ultiplicity in these gaes have receive significantly less attention, in large part because the structural oeling of social interactions an network externalities (two leaing exaples involving copleentarities aong agents actions) has been a relatively recent phenoenon. To estiate the extent of copleentarities in creit unions outsourcing ecisions, we aapt the intuition in CT an propose a etho for estiating payoff paraeters that places no restrictions on which outcoe obtains when the oel is consistent with 2

4 ultiple PSNE. We provie an algorith to solve for the set of PSNE that is easy to ipleent an whose coputational buren increases linearly, rather than exponentially, in the nuber of players. 1 Because we are able to overcoe the iensionality proble, we o not nee to restrict the outcoe space (as in CT an ABJ) nor o we nee to observe the choices of the sae firs in every (or inee any) arket. We also provie conitions uner which the paraeters are point (rather than set) ientifie. Using these estiates, we are then able to assess the how the egree of coorination aong creit unions varies with the size of the arket (i.e., the nuber of creit unions). We fin that the probability that the observe outcoe is pareto oinate by another outcoe, when the observe outcoe is consistent with ultiple PSNE, is U-shape reaching its iniu at five creit unions. In aition to the ethoological contributions, unerstaning how other firs ecisions affect the payoffs associate with outsourcing is of broaer interest as the practice of procuring intereiate services outsie of the fir is gaining in prevalence in the financial services inustry an the US econoy as a whole. The reainer of the paper is organize as follows. Section 2 presents backgroun on creit unions, presents the ata an iscusses potential copleentarities in creit unions outsourcing ecisions. Section 3 presents our theoretical oel an characterizes the set of PSNE for this gae. Section 4 presents the econoetric fraework an estiation strategy. Section 5 presents the results an section 6 conclues. 1 The ethos propose to eal with ultiplicity in this context suffer fro a version of the curse of iensionality where aing players iposes a substantial coputational buren. Other researchers have confronte this issue by aking ancillary assuptions to restrict the iensionality of the outcoe space, though this approach essentially ignores useful inforation. 3

5 II. Backgroun In this section we iscuss a creit union s choice between proviing IT in-house or outsourcing the IT function. We begin by consiering the unique nature of creit unions an then contrast the two technologies. We present soe suary statistics along with a probit regression that ocuents the relationship between a creit union s outsourcing ecisions an the outsourcing ecisions of other creit unions in the local arket. Creit Unions an Outsourcing Creit unions are non-profit, cooperative financial institutions owne an run by their ebers. In 2003, over 80 illion people were ebers of the approxiately 9,700 creit unions in the Unite States. The institutions hel roughly $533 billion in eposits which accounts for nearly 10% of total consuer savings. Critically for our analysis, each creit union ust efine a fiel of ebership, efining what custoers the creit union can serve. This grouping can be base on resience in a given counity, eployent with a given eployer or ebership in a given organization or association. While a particular consuer ay possibly be eligible for ebership in ore than one creit union, the ebership requireent liits irect copetition between these organizations. As a result, creit unions are ore likely to share inforation than irect copetitors woul be. This is evient in the existence of creit union leagues in every state where eber institutions share news an inforation as well as in the foration of creit union service organizations (CUSOs). 4

6 Inforation technology is crucial in the operation of creit unions, as it is throughout the financial services inustry. The central role of inforation technology in the operation of financial services firs is often creite with fostering the increasing oinance of larger financial institutions ost able to capture the increasing returns to scale inherent in the technology. An alternative organization of econoic activity, however, has large outsie suppliers of these services capturing the sae returns to scale by aggregating the nees of saller iniviual firs. The sae inforation an counication technologies that ay ai consoliation ay also foster the existence of saller financial services firs an to the evolution of an inustry that specializes in proviing IT services to financial institutions. As of 2003, alost all creit unions have ove fro anaging their accounts on paper to soe for of coputerization. However, creit unions vary substantially in how they structure their IT systes, utilizing both fors of organization just iscusse. While any creit unions choose to anage an store inforation in-house using software an harware that is typically purchase, a substantial fraction hire an external fir to anage an store inforation outsie of the creit union. For creit unions that choose to anage their IT in-house, the priary costs relate to the harware an software that nee to be purchase (or evelope) an aintaine, in aition to ongoing labor costs for the personnel that anage the systes. Harware costs involve a fixe upfront investent while software usually requires soe upfront costs an then soe ongoing licensing fees. The size an quality of the local labor arket will affect the labor cost associate with keeping IT in-house. 5

7 In contrast, outsourcing IT requires saller outlays for harware an for an internal IT staff. Originally, coputers at a creit union that outsource their IT were ub terinals connecte point to point by eicate telephone lines to the service provier s coputers. More recently, the creit union nee only purchase stanar PCs an the ability to connect to the Internet. In either case, the harware costs are far less than running an internal syste. The labor costs are also substantially lower uner this option since, by efinition, the ain harware an ata infrastructure resie at the service provier. For exaple, security, harware an software aintenance, an upgraes at the ata center are not irect expenses of the creit union, but rather of the outsourcer. For these services, the creit union usually pays the service provier an initial fee an then an ongoing fee, relate to the nuber of custoers that they have. The initial fee is often less than an initial licensing fee for software; however, the ongoing fee ay be higher. Conversations with inustry participants inicate that on average the total fees pai to a service provier are about 30% higher than the fees for licensing coparable software. Presuably for those firs that outsource, the savings on harware an labor as well as any net benefits outweigh this cost ifference. Discussions with soe of the larger proviers of creit union IT services inicate that the largest non-pecuniary ifference between these two options rests in flexibility. In ters of anaging IT, this inclues losing the ability to control the tiing of upgraes, or to use soe thir party software. For operations, this ay inclue the inability to custoize reports or to query ata in particular ways. Outsourcing ay even liit the proucts a creit union coul offer to those supporte by the service provier. However, given the econoies of scale inherent in IT, it is also possible that procuring a new 6

8 service (such as payent car processing) fro an outsie fir is cost effective while a given creit union using their own IT woul not fin it worthwhile. In total, the relative pecuniary an non-pecuniary costs an benefits of each option eterine what choice the fir will ultiately ake. A conversation with Leslie M. Mua, Presient an CEO of Fiserv provie soe interesting insight into why strategic copleentarities ay an ay not exist in the outsourcing ecision. Mr. Mua suggeste that (1) the nuber of firs in a given geographic arket who choose to outsource their IT oes not affect the costs associate with proviing outsourcing services to any fir in that arket; an (2) creit unions are uch ore likely to exhibit geographic clustering in their outsourcing ecisions than are coercial banks. 2 Thus, potential spillovers that woul reuce the costs associate with a given technology appear to take the for of creit unions sharing inforation about how to use each technology ore efficiently. Inee, further anecotal evience suggests that it woul not be uncoon for one creit union to contact another creit union using the sae IT to iscuss probles as they arise. Furtherore, there is a sufficient egree of flexibility in the outsourcing option, while less than that associate with internal IT provision, so that consultation aong creit unions that outsource their IT oes take place regaring various ay to ay IT-relate operations. Data 2 This conversation took place at the conference on Bank Structure an Copetition in Chicago, IL on Thursay, May 6,

9 The priary ata source for this stuy is the Secon Quarter 2003 Call Report fro the National Creit Union Ainistration (NCUA). Besies reporting financial ata, creit unions are aske about the inforation technology that they eploy. In particular, the following question is aske: Inicate in the box at the right the nuber of the stateent below which best escribes the syste the creit union uses to aintain its loan an share recors: 1=Manual Syste (No Autoation) 3=Venor On-Line Service Bureau 5=Other 2=Venor Supplie In-House Syste 4=CU Develope In-House Syste We classify all creit unions using a service bureau (choice 3) as outsourcing while those using any other systes as using in-house software. Of the 9615 creit unions that file in June 2003, the 232 still using anual systes are roppe fro the saple. Our oel easures strategic copleentarities aong firs using in-house software as well as aong firs that outsource. As escribe above, the presence of these effects for those using in-house software results priarily fro inforation sharing aong firs. While only 1.3% of the universe uses self-evelope software an another 1.7% is liste as other, grouping these with those using in-house software ay bias our results away fro fining evience of network effects aong the creit unions using inhouse systes. This effect ay be sall to begin with, given that even those using venor supplie in-house software ay be using systes fro ifferent venors. In orer to estiate the oel, we restrict the ata to creit unions in arkets with 16 or fewer creit unions. This liits the saple to 3570 creit unions in 1101 arkets. On average, 24.8% of these outsource their IT. Table 1 shows this breakown by arket, isplaying the actual istribution of arket outcoes given the nuber of firs in the arket. For exaple, both firs outsource in 8% of the two-fir arkets 8

10 that we observe, while neither fir outsources in 61% of the two-fir arkets that we observe. The inepenent variables, suarize in Table 2, inclue two creit union specific an two arket level variables that are assue to influence the relative costs of outsourcing. The creit union variables inclue the log of assets an the nuber of branches for the given fir. Many IT ecisions vary with the size of the fir an using assets as a proxy for size is the logical choice in this case. To hanle the large skew in this variable, the iniu is only $83,000 while the axiu in the saple is over $2.5 billion, the log of assets is use in the estiation. The branch variable captures the variation in costs of ipleenting an aintaining either an in-house or outsource syste over a larger nuber of locations. A priori, it is not clear however whether the relative costs of outsourcing will be larger or saller for creit unions with larger branch networks. Following conventional practice, a creit union s arket is efine as the MSA or non-msa county in which it is locate. Given this efinition, the arket level variables that we use inclue a uy for non-msa arkets an the overall population of the arket. Rural arkets are likely to be ore istant for outsourcers an therefore have a higher relative cost (an lower tenency) to eploy that strategy. These arkets ake up 82% of the arkets in the saple but contain only 55% of the creit unions. Population is a proxy for the availability of local IT services an we expect creit unions in larger arkets to be less likely to outsource. 9

11 Probit Results In orer to escribe the ata, we first estiate siple probits an linear probability oels. The results of these estiations are in Table 3. In each case, the probability of outsourcing is the epenent variable. The regressors inclue the size of the fir, the size of the branch network, the population an classification of its arket as well as the nuber of other firs who o an o not outsource their IT systes. The first colun shows the coefficient estiates fro the probit results while the secon translates those into arginal effects. As expecte, larger firs are ore likely to outsource while rural firs, those with large branch networks an those in ore populate arkets are less likely. With the exception of the rural uy, each of these coefficients is statistically significant. The variables esigne to easure strategic copleentarities are also significant an show strong evience that these effects exist. Conitional on the nuber of creit unions in the arket, it is iportant to note that an aitional fir outsourcing iplies one fewer using in-house software. Thus, the effect of having an aitional outsourcer is the ifference between the first two coefficients in the table. In the basic probit results, this iplies that having an aitional fir outsourcing increases the probability that a given fir will outsource by over 5.5%, evaluate at the ean of the explanatory variables. Bias ay arise in the probit oel for two relate reasons. First, there ay be oitte variables affecting the choices of firs that are correlate within a given arket. If there is any interepenency aongst firs ecisions, these oitte variables will cause correlation between the unobservable an the nuber of other firs who 10

12 outsource, resulting in biase estiates. A secon source of bias erives fro oel isspecification. If the oel aits ultiple equilibria then the likelihoo can not be escribe using a probit oel. We attept to aress the first of these concerns by estiating an instruental variables linear probability oel which is presente in the last colun of Table 3. (Both concerns are aresse in the structural oel.) In these estiates, the nuber of firs who outsource (insource) is instruente by the sus an eans of the two creit union specific variables for other firs in the arket. This approach is suggeste by Manski (1993), an has been use by Gowrisankaran an Stavins (2003) to estiate network effects. Colun 2 presents the non-instruente version of the linear probability oel for coparison. Both linear probability oels show results siilar to the probit. The effects for assests an branches are saller in these oels as is the effect of population. Interestingly, the copleentarities in the linear probability oel closely irror those in the probit. Accounting for the enogeneity of these variables with the instruents reuces these effects soewhat, particularly for the effect of other firs in-sourcing ecisions. III. Theoretical Moel In this section, we present our basic theoretical oel that captures the interepenence of creit unions outsourcing ecisions in equilibriu. We explore characteristics coon to all PSNE an are able to place iportant restrictions on the set of PSNE. These restrictions infor our estiation strategy which is iscusse in Section 4. While we borrow fro the entry literature in our estiation approach, iportant 11

13 ifferences between entry gaes an our gae allow us to exten the existing econoetric ethos that have been applie to entry oels. Let y j = 1 if fir j outsources its IT, y j = 0 if fir j provies IT internally, an let Y be the vector that captures all firs IT ecisions. The nuber of creit in each arket, N, is assue to be exogenously given an, in our saple, it varies across arkets. where We assue that each creit union s IT costs take the following for: i i 3.1 φ ( Y) = X β + g( K ) + ε y + X β + h( N K 1) + ε ( 1 y ) i i 1 1, i i i 0 0, i i X i is a vector of exogenous covariates that shift IT costs for each fir epening on whether the fir outsources its IT or provies it internally. K i = y, is the nuber j i j of creit unions in the arket, other than i, that outsource their IT. The functions g( K i ) i an h( N K 1) capture the effect on i s costs of the nuber of other creit unions in the arket who choose to outsource an provie IT internally, respectively. Finally, ε1,i an ε 0,i are fir-specific cost shifters that are unobserve to the econoetrician. We can now efine the increental cost of outsourcing as: i i 3.2 ci( Y) = Xi( β1 β0) + g( K ) h( N K 1) + ( ε1, i ε0, i) Cobining ters, we can rewrite 3.2 as follows: = + + ε i 3.3 c ( Y) X β f ( K ) where β β1 β0 i i i i i i =, f ( K ) g( K ) h( N K 1) =, an εi = ε1, i ε0, i. 12

14 A fir will choose to outsource, i.e., choose y i = 1, if c i < 0 i 3.4 εi < Xiβ f ( K ) A PSNE of this gae is a profile Y = ( y1,... y N ) such that 3.5 X β f ( K i ) ε 0 y < = 3 i i i i.e., all firs that outsource achieve a cost savings, an that those who o not outsource woul not achieve a cost savings if they chose to o so. At this point, it is useful to iscuss the ore iportant assuptions ebee in our specification of the increental costs of outsourcing in 3.3 as well as the aitional assuption of strategic copleentarity. These assuptions are:, or Assuption 1: X i an N orthogonal to ε i Assuption 2a: g( ) an h( ) epen on only K -i (as oppose to Y ) Assuption 2b: g( ) an h( ) o not vary across firs 4 Assuption 3: g( ) an h( ) are weakly ecreasing. XX FILL IN DISCUSSION HERE XX Properties of the Set of Pure Strategy Nash Equilibria In general, the gae we stuy aits ultiple PSNE. As iscusse below, this poses significant ifficulties for estiating the oel as we have specifie it. We are able, however, to place soe restrictions on the set of PSNE outcoes that will allow us 3 This is not precisely correct since a fir ay either provie IT internally or outsource if c i =0. We ignore this in our presentation here since the istributional assuptions (naely that the error ter has continuous support) we use to estiate the oel iply that c i =0 happens with zero probability. 4 Note that g an f coul be peritte to vary across firs with no ifference in our theoretical results. 13

15 to ake use of recent techniques evelope in Taer, ABJ an CT for estiating oels with ultiple PSNE. In particular, we show that there are at ost N +1 PSNE in our 2 gae an that it is sufficient to search over only N +1 outsourcing configurations in orer to fully characterize the set of PSNE. This is in contrast to the 2 N possible PSNE in the entry gae stuie by CT, whose estiator we aapt. This result stes fro our assuptions that the increental costs of outsourcing fro equation 3.3 are weakly ecreasing in the nuber of firs who outsource, an epen only on the nuber, not the ientities, of the other firs who outsource. 2 N to N +1. We can now establish Proposition 1 which reuces the set of possible PSNE fro Proposition 1: Any PSNE in which N firs choose to outsource, will always involve the sae N firs choosing to outsource. Proofs of all propositions are presente in an appenix. Proposition 1 iplies that there are at ost N + 1 PSNE, where N is the nuber of firs in arket. 5 This observation significantly siplifies the search for possible PSNE. Proposition 2 shows that outsourcing ecisions ust follow the orering of firs increental outsourcing costs, an that (siilar to Berry 1992) this orering oes not epen on the outsourcing ecisions of other firs in the arket. Proposition 2: The N firs with the lowest increental costs of outsourcing will all choose to outsource in any PSNE in which N firs outsource. 5 The aitional PSNE is the outcoe in which no fir outsources. 14

16 The ability to orer the firs outsourcing costs for any strategic interaction, N allows us to establish Proposition 3, which iplies that there are at ost + 1 PSNE. 2 Proposition 3: If there is a PSNE in which N firs choose to outsource, a PSNE in which 1 N + firs choose to outsource oes not exist. Propositions 1-3 place significant restrictions on the set of PSNE of our gae. We will use these restrictions below to oify the estiators propose by Taer, ABJ an CT so that they can be applie in a parsionious way to the estiation of strategic copleentarities. IV. Econoetric Moel The purpose of this section of the paper is to outline our estiation strategy. Before oing so, however, we iscuss how the previous literature has approache the estiation of iscrete gaes with ultiple PSNE in the context of entry gaes, network effects, an social interactions. After introucing the estiator, we iscuss our algorith for ientifying the entire set of PSNE. Finally, we iscuss ientification of the oel s paraeters. Previous stuies of gaes with ultiple PSNE The nature of the gae we consier, an inee any gae with strategic copleentarities, preclues straightforwar axiu likelihoo estiation ue to the fact that the likelihoo will not be well-efine if the gae aits ultiple equilibria. Previous work on network effects an social interactions has confronte the issue of ultiplicity either by aking assuptions that guarantee a unique equilibriu or by 15

17 assuing a particular selection rule. We eploy a ifferent tact. In particular, we ake the inial assuption that each arket is in a PSNE outsourcing configuration, without specifying any selection rule. In general, our inial assuptions ay not be enough to ientify the paraeters of interest, but they are capable of ientifying the set of paraeters that are consistent with the gae as we specify it below. We o, however, erive sufficient conitions for point ientification of the oel s paraeters. We approach the estiation of our oel in a anner that is conceptually siilar to entry gaes stuie by Bresnahan an Reiss (1990), Berry (1992), Taer (2002), an CT. Bresnahan an Reiss (1991) show that for rich enough error structures, their entry gae will always ait ultiple PSNE (as inee is the case in our setup). In the entry context, actions by other firs (naely whether to enter or not) are strategic substitutes. In an entry gae with two firs, for exaple, it will be possible to fin values of the unobservables such that either fir, but not both, ay enter in equilibriu. To the extent that the ecision to outsource is associate with network effects, firs actions are strategic copleents. In the exaple above, for soe values of the exogenous variables, it will be possible to fin values of the unobservables such that either both firs outsource or neither fir outsources. In the presence of ultiple PSNE, therefore, the likelihoo of a given profile of strategic choices is not well efine. While a nuber of solutions to the proble of ultiple equilibria have been suggeste for the case of strategic substitutes, ost are not appropriate in our context. Bresnahan an Reiss (1990) an Berry (1992) suggest focusing on the total nuber of entrants (which, for any entry oels is constant across PSNE), rather than their 16

18 ientity. 6 The nuber of firs who choose to outsource, however, is not constant across PSNE since the outsourcing ecisions of ifferent firs are strategic copleents. The stuy of ultiplicity in gaes with copleentarities has been otivate by increase interest in social interactions an network externalities. Brock an Durlauf (2004) consier the ientification of iscrete choice oels with social interactions. They avoi the issue of ultiplicity by assuing that iniviual preferences are a function of one s beliefs, which are self-confiring in equilibriu, about how other ebers of the group will behave. They recognize that the assuption will be probleatic for saller size groups which we observe if one consiers the creit unions in a arket to be a group. Furtherore, since the copleentarities in our context relate to the sharing of inforation about a particular IT option, it is the actual IT ecisions of the other creit unions that affect a given fir s costs, not beliefs about what these ecisions will be. The literature on network effects has also confronte the issue of ultiplicity. Sweeting (2004) stuies a gae with iperfect inforation that aits only two possible equilibria an shows that the ultiplicity facilitates ientification of firs incentive to coorinate. Augerau, Greenstein an Rysan (2004) assue a unique equilibriu uring estiation an then check to see if there are other equilibria ex-post. Ackerberg an Gowrisankaran (2003) structurally estiate network effects in the ACH Banking Inustry by expressing the likelihoo as a weighte average of the likelihoos associate with the Pareto worst an best equilibria in each arket. The probability that either equilibriu is the one that is actually playe is estiate assuing a constant selection rule. 6 Davis (2003) uses a siilar approach, focusing on the total level of output (for exaple the nuber of retail outlets in a arket) as oppose to the output of each fir. 17

19 We aapt an exten ethos that have been use to accooate the existence of ultiple PSNE in entry oels. In particular, papers by Taer (2003), CT, an ABJ have propose ethos for estiating oels that have ultiple PSNE without iposing any structure on which equilibriu is selecte. Taer (2003) shows that, while the likelihoo ay not be well efine, one can erive inforative bouns on the likelihoo that can be use as the basis of a oifie axiu likelihoo (MML) estiator. CT an ABJ use siilar approaches to for a oifie iniu istance (MMD) estiators that can be use in settings with ore than two potential entrants. An alternative approach that we o not choose to aopt is that suggeste in Bajari, Hong an Ryan (2004) who consier ynaic entry gaes in an iperfect inforational setting. They ientify equilibriu selection rules fro exclusion restrictions on variables that affect equilibriu selection but not firs objective functions. Our strategy is otivate by our interest in characterizing the egree of coorination aong firs. We opt for an estiation approach that requires no assuptions about equilibriu selection, but afterwar allows us to analyze certain efficiency aspects of the selecte PSNE when ultiple PSNE exist. Aaptation of the MMD estiator Our general approach, which follows fro CT, is to boun the probability of observing certain outcoes, an to fin the set of paraeters for which the true probabilities of those outcoes lie within the bouns given by our oel. In the previous section, we showe that uner our assuptions the iensionality of the space of potential equilibriu outcoes is significantly reuce. In this section, we show that the 18

20 structure of the set of PSNE for this gae can be copletely characterize using a siple algorith. The large nuber of potential outcoes associate with gaes where actions are strategic substitutes akes ipleentation of the MMD estiator alost ipossible. 7 CT, for exaple, collapse the nuber of potential entrants to five by cobining siilar types of firs into a single fir for this very reason. Our algorith is easy to ipleent for any nuber of firs. Furtherore, we o not nee to observe the sae firs in every (or even any) arket. Taer (2003) consiers a oel with two potential entrants, where, for certain values of the unobservables, either fir ay enter as a onopolist, but not both. Intuitively, bouns on the likelihoo are obtaine by assuing ifferent selection rules for the region of the errors which yiels ultiple PSNE. The likelihoo of (1,0), where (, ) y y entry by fir i correspons to y i =1, can not be larger than the likelihoo that 1 2 (1,0) is a PSNE. On the other han, the likelihoo of (1,0) ust be at least as large as the likelihoo that (1,0) is the unique PSNE. That is, the upper boun, P (1, 0) is obtaine by assuing that (1,0) always obtains in the region in which both (1,0) an (0,1) are Nash equilibriu outcoes. The lower boun, P (1,0), is obtaine by assuing that (1,0) never obtains in the region in which both (1,0) an (0,1) are Nash equilibriu outcoes. 8 Taer (2002) an CT exten this intuition to the case of several potential, N, entrants. They propose a oifie iniu istance (MMD) estiator that requires obtaining a consistent non-paraetric estiate, p( Y ), of the epirical probability of 7 Cohen (2004), however, presents a anageable algorith for coputing the set of PSNE for the special case where the effect of entry on profits epens on the nuber but not the ientity of the entrants. 8 For a graphical representation, see Figure 1 in Taer (2003). 19

21 observing each profile of entrants. The loss function for the MMD estiator is then efine in the following way: 4.1 LY (, X, θ ) = 0 if PY (, X; θ) py ( ) PY (, X; θ) LY (, X, θ) = PY (, X; θ) py ( ) if PY (, X; θ) > py ( ) LY (, X, θ) = py ( ) PY (, X; θ) if PY (, X; θ) < py ( ) where is a istance easure. The upper an lower bouns are fore in the exact sae way as the case with two potential entrants, however, the calculation of the region in which a particular outcoe is a unique equilibriu is uch ore coplicate because one ust check each N of the possible 2 configurations at each raw of the unobservables. As iscusse above, CT reuce the nuber of possible outcoes by luping siilar types of firs together (for exaple, all sall airlines are treate as one fir). Recall that each creit union akes its outsourcing ecision base on the increental costs relative to running its own syste. We assue that each fir s observe ecision represents an eleent of a PSNE profile. This iposes the inial conition that each fir that chooses to outsource gains a cost avantage, an no fir who provies its own syste coul outsource at a lower cost. These conitions are capture in equation 3.5 above. We efine a PSNE as a vector Y such that 3.5 hols for all firs in the arket. We can efine the set of PSNE, Λ( X ; ) Θ, as follows: 4.2 Λ( X ; Θ ) = { Y : ci( Y, Xi; Θ ) < 0 yi = 1 i = 1... N} 20

22 Upper an lower bouns on the probability of a particular strategy profile, Y k are erive by calculating the probability that Y k is a PSNE an the probability that Y k is the only PSNE, respectively. In particular: 4.3 P ( X; Θ ) = Pr Y Λ( X; Θ) k 4.4 P ( X ) Y ( X ) ( X ) k k ; Θ = Pr k Λ ; Θ Λ ; Θ = 1 where Λ( X ; Θ ) is the carinality of Λ ( X ; ) these bouns. That is, 4.5 P ( X; Θ) Pr [ Y ] P ( X; Θ ) k k k Θ. The true probability ust lie between These bouns for the basis for our estiator. Following the suggestion of Manski an Taer (2002) an CT, we for a oifie iniu istance estiator by choosing the paraeters that iniize the following loss function: ( P ( X ; ) Pr [ Y ]) * I( P ( X ; ) Pr[ Y ]) 2 ( Pk( X; Θ) Pr [ Yk] ) * I( Pk( X; Θ ) < Pr[ Yk] ) M N 2 k k k k = 1 k= 1 Θ Θ > + The loss function punishes values of the paraeters for which the true probabilities fall outsie of the probability bouns that are generate by the oel. Since the true probabilities are not observe, we replace Pr[ ] Pr [, ] k Y with a consistent estiate, Y X N. The first stage estiator preicts outsourcing configurations as a function of the observable characteristics of each fir, observable arket characteristics, observable characteristics of the other firs in the arket, an the nuber of other firs in the arket. k 21

23 V. Results We paraeterize each fir s relative costs of outsourcing, fro equations 3.3, as follows: 5.1 c ( Y) i i ( ) i ( ) i α + Xiβ + δ1k + δ2log K + 1 = i λ 1 N K 1 λ2log N K + εi where K i = yj an i j i X inclues two arket specific variables (the log of population an a uy for rural arkets) an two fir specific variables (the log of assets an the nuber of branches). The specification allows for the copleentarities to enter both linearly an logarithically into the costs of both outsourcing an internal IT provision. We follow Berry s (1992) suggestion an oel each fir s unobservable as a coposite of fir an arket level unobservables. In particular, we let ε = 1 ρη + ρµ 2 i i where η i is an ii stanar noral fir-level unobservable, an µ is an ii stanar noral arket-level unobservable. The weighte average of the two, by construction, is stanar noral. The correlation between the unobservables for firs in the sae arket 2 is ρ. Our MMD estiator is given by the iniu of the following criterion function: 5.2 = 1 2 ( PY ( X ; ) Pr [ Y X ]) * I( PY ( X ; ) Pr [ Y X ]) ( 2 PY ( ; ) Pr [ ]) * ( ( ; ) Pr X Θ Y X I PY X Θ < [ Y X] ) Θ Θ > + M 22

24 where Y is the observe outsourcing configuration for arket. Pr [ ] Y X is a consistent non-structural estiate of the probability of observing Y given the set of exogenous variables X. Finally, PY ( X ; Θ ) is the probability, erive fro the structural oel, that Y is the unique PSNE while PY ( X; Θ ) is the probability that Y is a PSNE. Specification of the first stage estiates The ost obvious way to obtain non-structural estiates of the probability of observing Y is to use a bin approach or a paraetric ultinoial qualitative response oel. We o not opt for either approach for several reasons. First, we o not observe the sae firs across arkets. Therefore, it is not possible to efine the outcoes in a way that will be coparable across arkets. In aition, our ecision to inclue arkets with a large nuber of participants in the analysis preclues the use of a ultinoial 16 qualitative response oel ue to the large nuber of possible outcoes, 2, in the largest arkets. Our solution is to begin with a qualitative response oel, an then to ecopose it into two anageable pieces. We begin by consiering the estiation of Pr ( Y X ) where X is the atrix of exogenous variables coprise of eleents that iffer across firs as well as arketspecific variables. We assue that Pr ( ) RKk X is well efine, where k particular set of rankings of firs relative outsourcing costs fro the set RK of all RK RK is a possible ( N!) set of rankings. This assuption allows us to ecopose the probability of observing a particular outsourcing profile, Y, as follows: 23

25 * * 5.3 Pr ( Y X) = Pr ( Y X, yi = N )* Pr ( yi = N X) We ecopose the ultinoial probability on the left han sie as the ultinoial probability conitional on a particular nuber of outsourcers ultiplie by the probability of observing that nuber of outsourcers. Given our assuption that the probability of any given ranking of firs outsourcing costs is well efine, then: where * 5.4 ( ) ( ) ( ) Pr Y X, yi = N = Pr rki X < rkj X, yi = 1, yj = 0 Y rk i is the cost ranking for fir i. We estiate 5.4 using a rank orer logit whose inex is a polynoial function of with a Poisson * X. We estiate Pr ( y i = N X ) regression whose ean is a polynoial function of oents of istribution is right-truncate at N. X, an whose Coputation of set of PSNE We eploy siulation ethos to calculate the bouns iplie by our oel on the probability of observing a given outsourcing configuration, Y k, for a given set of paraeters. For a each set of raws, r, fro the ultivariate istribution of ε, we copute the set of PSNE, Λ ( ) r X ; Θ, using the algorith escribe in Appenix 2. This algorith is feasible as the nuber of firs increases, an oes not require one to liit the space of possible outcoes. This is ue to the fact that one nee not check whether each possible outsourcing configuration is a PSNE. Instea, one nee only check for PSNE in the nuber of firs who outsource, which can be one generically (as escribe in the Appenix). Proposition 1 above iplies that if there is a PSNE in which a given nuber of firs outsource, there ust be only one such PSNE. Once the existence of a 24

26 PSNE involving a particular nuber of firs is verifie, fining the exact equilibriu configuration siply entails ranking the firs by X i β + ε ir. Siulation Estiators for PY ( X ; Θ ) an PY ( X ; Θ ) The MMD estiator in 5.2 requires one to copute the probability that the outcoe Y is (1) the unique PSNE, PY ( X; Θ ), an (2) a PSNE, PY ( X; Θ ). We use an iportance sapling technique that takes quasi-rano raws fro the istribution of µ, the arket-specific unobservable. For each raw, we fin the set of fir-specific unobservables, 5.5 A = { η : Y Λ µ } A, that is consistent with the observe outcoe. That is, A is an N iensional rectangle with enpoints: ( ) ( ) i i i i α + Xiβ + δ1k + δ2log K + 1 λ1 N K 1 λ2log N K + ρµ, if y 1 2 i = 1 ρ an, ( ) ( ) i i i i α + Xiβ + δ1k + δ2log K + 1 λ1 N K 1 λ2log N K + ρµ, if y 0. 2 i = 1 ρ Since the fir-specific unobservables are ii an inepenent of the arket specific unobservable, one can, for each siulate raw µ, easily (1) copute analytically the probability that a given outcoe is a PSNE an (2) take rano raws fro the set A. 25

27 The siulation estiator for the upper boun on the likelihoo of observing Y is: D D PY ( X; Θ ) = Pr( η A) = 1 The siulation estiator for the lower boun on the likelihoo of observing Y is: D ( ; ) PY X Θ = I ηi A Λ = 1 *Pr( η A ) D = 1 The inepenence of η i an µ allows us to take a single raw fro A for each µ. Paraeter estiates Table 4 presents the values of the paraeters that iniize the MMD criterion function in 5.2. For purposes of coparison, the right han panel presents the estiates we woul have obtaine ha we estiate each fir s cost iniization proble as a probit ignoring the existence of ultiple PSNE. We also ignore correlation aong the errors of firs in the sae arket, though this correlation is estiate to be quite sall in the structural oel. Copleentarities appear to be present in both outsourcing an internal IT provision. That is, firs costs appear to ecrease to the extent that aitional firs are aopting the sae technology. The copleentarities for internal IT provision are increasing at a ecreasing rate (with a non-trivial effect on the log ter), while 26

28 copleentarities for outsourcing appear to be linear (with the log ter alost equal to zero). In general, the MMD will ientify the set of paraeters consistent with the behavioral oel, but the paraeters ay not be point ientifie. Chernozhukov, Hong an Taer (2003) provie a etho for calculating the bouns on the paraeters that cover the ientifie set with the esire probability. Appenix 3 iscusses ientification of the oel. We provie ifferent sets of sufficient conitions for point ientification, epening on the paraetric specification. Assuing that these conitions are satisfie allows us to use bootstrap techniques to calculate stanar errors which will generally be tighter than the bouns iscusse in Chernozhukov, Hong an Taer. STANDARD ERRORS TO BE COMPUTED XXX. Measures of Coorination Failure The power of our oeling approach can be seen as we attept to easure the extent of coorination aong creit unions in each arket. Because our oel is structural, we can siulate equilibriu outcoes. Because we o not ipose a selection rule in the presence of ultiple PSNE, we can copare observe outcoes to other PSNE outcoes consistent with the observe outcoe to aress which PSNE are selecte. Table 5 characterizes the observe outcoes in ters of the likelihoo of ultiplicity an the cost characteristics of alternative PSNE when they are present. All of our exercises conition on the observe outcoe being a PSNE. That is, we integrate over the region of the error ters, η i an µ, that are consistent with Y being an 27

29 eleent of Λ, the set of PSNE. The first colun of Table 5 presents the probability that there are other PSNE, given that the observe outcoe is a PSNE, i.e., 5.8 Pr ( Λ> 1 Y Λ ) As one woul expect, the prevalence of ultiple PSNE increases with the nuber of firs in the arket. The secon colun of table 5 presents the probability of a lower total cost PSNE, given the existence of ultiple PSNE, i.e., ' ( i Y Λ i Y < Y Y Y ) Λ Λ > ' 5.9 α ( ) α ( ) where ( Y ) Pr 0, 1 α i captures each fir s realize IT cost, in outsourcing configuration ' α ( Y ) is not ientifie in our oel, however, α ( Y Λ) α ( Y ) i i i Y. Λ is ientifie. Once again, as the nuber of firs in the arket increases so o PSNE that woul reuce the total costs. This shoul not necessarily be construe as a coorination failure, since soe firs ay be better off in an alternative PSNE while others ight be worse off. The thir colun presents the nuber of firs who woul be better off (i.e., incur lower costs) in the least total cost equilibriu, assuing that it is not the observe outcoe, i.e., 5.10 ' ( αi ( Λ) α ( ) 0 ) i < ' ' α ( ) α ( ) I Y Y E Y, Y Λ i Y Λ i Y < 0 The iea here is that arkets with lots of firs that woul be better off in the lower total cost PSNE are, on average, less coorinate than those arkets with fewer firs that woul be better off in the least cost equilibriu. To obtain a easure that will allow us to copare arkets with ifferent nubers of firs, we copute the expecte proportion of 28

30 firs that woul be better off in the least cost PSNE, when one exists. This easure, which is siply 5.10 ivie by the total nuber of firs, is presente in the fourth colun. Here we observe the U-shape pattern relating coorination to the nuber of firs in the arket. Coorination gets progressively stronger, with a lower proportion of firs better off in the least total cost PSNE, as one goes up to five firs, at which point it gets progressively weaker looking at arkets with two firs up to five firs, an then weaker oving above five firs. A ore efinitive easure of coorination failure is to exaine the likelihoo that the observe outcoe is pareto oinate by another PSNE, given that ultiple PSNE exist. This easure is presente in the far right colun of table 5. Once again, a U- shape pattern of coorination failure is observe, with the iniu being at five firs. Figure 1 plots our two easures of coorination failure against the nuber of firs in our observe arkets. VI. Conclusion This paper extens the literature on the estiation of gae theoretic oels of fir behavior in several iensions. Our exaple of creit unions outsourcing ecisions is a gae with strategic copleentarities an ultiple PSNE. It represents an ieal laboratory for estiating copleentarities since creit unions are known to counicate uch ore than firs that copete irectly with one another. We show how to apply techniques evelope for the estiation of entry oels with ultiple PSNE to our proble. Our algorith for fining the fixe points of firs best reply corresponences uses prior inforation about the properties of the set of PSNE 29

31 to substantially alleviate our coputational buren. In aition, we show that the ethos evelope by CT an ABJ can be generalize to gaes with any players where one oes not observe the sae set of agents across arkets. The fact that our oel oes not ipose any equilibriu selection rule when ultiple PSNE are present allows us to characterize the egree of coorination in the observe outcoes. Our ain epirical fining is the U-shape relationship between easures of coorination failure an the nuber of firs. 30

32 Appenix 1: Proof of Propositions Proposition 1: Any PSNE in which N firs choose to outsource, will always involve the sae N firs choosing to outsource. Proof: Suppose not. Then there ust exist at least one fir such that: i ( ) c ( Y ') >0 where y = y' j j, an both Y an ' c Y <0, i Y are NE. Since fir i outsources in profiley, its increental costs are ci( Y) = Xiβ + f yj < 0 by the fact that Y is a NE. j i In configuration Y ', i oes not outsource. It s increental costs, if it were to outsource, ust be positive if Y ' is a NE. Those costs are given by: ci( Y' ) = Xiβ + f y' j = Xiβ + f yj + 1 = ci( Y) + f yj + 1 f yj j i j i j i j i But c ( Y' ) ust be negative because ( ) 0 i ci Y < an f is weakly ecreasing. This contraicts the supposition that Y ' is a NE. QED. Proposition 2: The N firs with the lowest increental costs of outsourcing will all choose to outsource in any pure strategy Nash equilibriu in which N firs outsource. Proof: Since the outsourcing ecision of a fir affects all other firs costs equally, the ranking of increental costs oes not epen on how any other firs choose to outsource. That is, c Y < c Y c Y' < c Y'. ( ) ( ) ( ) ( ) i N j N i N j N This also iplies that c Y < c Y X β + ε < X β + ε ( ) ( ) i N j N i i j j Therefore, it is possible, without loss of generality, to rank firs increental costs in the following way: c1 < c2 <... < c N. 9 9 The error ters insure that ties occur with zero probability. 31

33 In an equilibriu in which N firs outsource, there ust be N firs with ci 0 an N firs with c > 0. N j By proposition 1, there ust be only one set of N firs such that ci 0. Therefore, in a NE with N firs outsourcing, it ust be the N firs with the lowest increental cost that choose to outsource. That is, if Y is a NE, then c ( Y )... c ( Y ) 0 c ( Y )... c ( Y ) < < < < < QED. 1 N N N N+ 1 N N N Proposition 3: If there is a PSNE in which N firs choose to outsource, a PSNE in which 1 N + firs choose to outsource oes not exist. Proof: Suppose Y N is a NE profile in which N firs outsource. By Proposition 2, it ust be possible to arrange firs by their increental costs in the following way. c Y <... < c Y < 0 < c Y <... c Y ( ) ( ) ( ) ( ) 1 N N N N+ 1 N N N The fact that the copleentarity in the cost of outsourcing epens only on the c Y = c Y ecisions of the other firs in the arket, iplies that ( ) ( ) Therefore, c ( Y ) QED. + + > an Y + 1 cannot be a NE. N 1 N 1 0 N N+ 1 N N+ 1 N+ 1 32

34 Appenix 2: Algorith to copute set of PSNE outsourcing configurations (1) For a given arket, calculate the atrix of increental outsourcing costs, C ( Θ ), for a particular raw,, for all firs uner each possible nuber of outsourcers. The eleent corresponing to the i th row an t th colun, c (suppressing the epenence on the paraeters), represents the increental outsourcing costs for fir i assuing that t 1 other firs outsource. That is, ( 1) log( ) ( ) log ( 1) cit = Xiβ + δ1 t + δ2 t λ1 N t λ2 N t+ + εi for t = 1,..., N (2) Calculate the atrix of firs best replies, at each raw, to t-1 firs outsourcing, R Θ as follows ( ) r r it it = 1 if cit < 0 = 0 if c 0 it (3) Calculate the nuber of firs in every PSNE. Let R() be the th t t colun of ( ) R Θ. it R, i.e., the profile in which no firs outsource, is a PSNE if r1 1 () N i i= 1 = 0. That is, each fir s outsourcing costs are copute assuing no other firs choose to outsource. If, R Λ X Θ. given this assuption, it is not optial for any fir to outsource, then () ( ) 1 ; For configurations in which at least one fir chooses to outsource, the following two conitions are necessary an sufficient to conclue that R () is a PSNE, i.e., t R () ( X; t ) Λ Θ. N C1: r, it i= 1 N = t C2: r, + 1 = t it i= 1 The first conition says that there are exactly t firs who fin it optial to outsource given that t 1other firs choose to outsource. The secon conition says that the N t firs who (if C1 is satisfie) woul not outsource given that t 1 other firs outsource, also woul not outsource given that t other firs choose to outsource. 33