A Geometric Approach to Inference in Set Identi ed Entry Games"

Transcription

1 TSE 943 Jul 28 A Geometric Approach to Inference in Set Identi ed Entr Games" Christian Bontemps and Rohit Kumar

2 A Geometric Approach to Inference in Set-Identified Entr Games Christian Bontemps Rohit Kumar Ma 5 28 Abstract In this paper we consider inference procedures for entr games with complete information. Due to the presence of multiple equilibria we know that such a model ma be set identified without imposing further restrictions. We complete the model with the unknown selection mechanism and characterize geometricall the set of predicted choice probabilities in our case a convex poltope with man facets. Testing whether a parameter belongs to the identified set is equivalent to testing whether the true choice probabilit vector belongs to this convex set. Using tools from the convex analsis we calculate the support function and the extreme points. The calculation ields a finite number of inequalities when the explanator variables are discrete and we characterized them once for all. We also propose a procedure that selects the moment inequalities without having to evaluate all of them. This procedure is computationall feasible for an number of plaers and is based on the geometr of the set. Furthermore we exploit the specific structure of the test statistic used to test whether a point belongs to a convex set to propose the calculation of critical values that are computed once and independent of the value of the parameter tested which drasticall improves the calculation time. Simulations in a separate section suggest that our procedure performs well compared with existing methods. Kewords: set identification entr games convex set support function ENAC & Toulouse School of Economics Université de Toulouse 2 allee de Brienne 3 Toulouse France christian.bontemps@tse-fr.eu. We gratefull acknowledge financial support from the Agence Nationale de Recherche ANR-5-BLAN-24-. This research has received financial support from the European Research Council under the European Communit Seventh Framework Program FP7/27-23 grant agreement N Toulouse School of Economics Université de Toulouse Capitole 2 allee de Brienne 3 Toulouse France sirohi.rohit@gmail.com

3 Introduction This paper provides an estimation procedure for empirical models of entr and market structure also called entr games which ma be set identified. Entr games are ver popular in the empirical Industrial Organization (IO) literature because the allow researchers to stud the nature of firms profits and the nature of competition between firms from data that are generall eas to collect. The were popularized b the seminal works of Bresnahan and Reiss (99a) Bresnahan and Reiss (99b) and Berr (992). However the econometric analsis of entr games is complicated b the presence of multiple equilibria a problem that affects the standard estimation strateg. Without additional assumptions the model is indeed incomplete. proposed in the literature. Various solutions have been First assumptions can be added regarding the (unknown) selection mechanism in regions of multiple equilibria. Reiss (996) considers a specific order of entr and Born and Vuong (984) randoml draw the equilibrium selection. Baari et al. introduce a parametric specification of the selection mechanism and Grieco (24) extends it to a nonparametric function of observables and non-observables. Alternativel it is sometimes possible to estimate an entr model from the observation of some outcome that is independent of the true selection mechanism. In their seminal work with heterogeneous firms Bresnahan and Reiss (99a) report that the model uniquel predicts the number of active firms. Berr (992) generalizes the estimation of the profit function from the observed number of active firms for more than two plaers. A tremendous number of empirical applications have followed this path (see the surve b de Paula (22)) notabl including Mazzeo (22) Reiss (996) Cleeren et al. and Sampaio (27) among others. However adding assumptions leads to a misspecified model and working on certain invariant outcomes ma lead to a suboptimal procedure (see Tamer (23)). The recent and blossoming literature on partial/set identification following earlier work b Manski (995) makes it possible to estimate a model that does not uniquel predict actions b using bounds. Following Tamer (23) the presence of multiple equilibria does not impl partial identification 2 but the literature provides inference methods that are eligible in both cases. Ciliberto and Tamer (29) See Berr and Reiss (27) for a surve. 2 However identification at infinit as in Tamer (23) ma lead to inference procedures that are non standard; see Khan and Tamer. 2

4 were the first to use bounds to estimate an entr game with complete information. In this paper we complete the model with the selection mechanism η( ) and characterize the set of predicted choice probabilities generated b the variation of η( ) in the space of admissible selection mechanisms. Our first contribution is to characterize more deepl the geometric structure of this set. The set is a convex poltope with man facets (because we focus on pure strateg Nash equilibria) and the number of facets increases exponentiall with the number of plaers. Alternative equilibrium concepts have been proposed in the literature (as in Aradillas-Lopez and Tamer (28) or Beresteanu et al. ). Changing the equilibrium concept affects some of the calculations provided in this paper and sometimes increases the complexit but does not alter the general philosoph. Moreover Nash equilibrium is the most commonl used solution concept. In this paper we derive a closed-form expression for the support function of this poltope the extreme points (or vertices) of which can also be calculated as a function of the primitives of the model. These vertices are indeed characterized b an order of outcome selection in the regions of multiple equilibria. Each vertex is also geometricall defined b the intersection of some supporting hperplanes. We are able to define the cone of outer normal vectors of these hperplanes and thereb the inequalities that are binding in this point. Testing whether a parameter belongs to the identified set is equivalent to testing whether the true choice probabilit vector belongs to this convex set. Following Rockafellar (97) the support function defines a continuum of inequalities that have to be satisfied for an point in the set. This characterization has alread been used in the set identification literature b in particular Beresteanu and Molinari (28) Beresteanu et al. and Bontemps et al. (22). Geometricall the support function in a given direction leads to an inequalit that detects whether the point of interest belongs to the same halfspace than the convex set itself. This continuum of inequalities simplifies here toward a finite number because of the specific structure of the entr game considered in this paper. However when the number of plaers increases the number of faces of the poltope increases exponentiall and therefore the smallest number of inequalities necessar to have a sharp characterization - from 6 in a game with three plaers to more than one million in a game with six plaers. The standard approach for moment inequalit models is to first evaluate all the moments 3

5 then eliminate the ones that are far from being binding (see Andrews and Soares 2 for the unconditional case). Our second contribution is to provide a geometric selection procedure that does not require us to evaluate all the moments and for which the computational cost is polnomial in the dimension of the outcome space (2 N ). The idea is to recursivel identif the extreme point of the set that is the most relevant for demonstrating that our point the true choice probabilit vector belongs to the poltope. We then test onl the inequalities related to the facets of this extreme point and do not evaluate the other inequalities. The number of relevant moments grows at the rate 2 N as in Ciliberto and Tamer (29) but with a sharper characterization of the set. This is a considerable improvement in terms of computational burden with respect to alternative sharp characterization methods (such as the two step approaches of Andrews and Soares and Romano et al. (24)). Furthermore and more important we develop a test procedure for the hpothesis of whether the true choice probabilit vector belongs to this poltope that exploits the specific structure of the test statistic. When we test whether a parameter θ belongs to the identified set the feasible test statistic depends on the support function of the poltope which is fixed and the choice probabilit vector which is estimated. The asmptotic distribution depends on θ in a specific manner. It allows us to derive critical values for our test procedures that are calculated once and valid for an θ. This is a tremendous simplification with respect to a general moment inequalit model. Some of the recommended test procedures can be conservative but one which exploits the maximum number of facets at an extreme point is not. This maximum number can be determined b brute force using the geometric structure of the set; we also provide an upper bound. Simulations highlight that our method works well with the sample sizes that are usuall considered in the empirical IO literature (from 5 to 2 observations) and outperforms existing inference procedures. This paper belongs to the growing literature on set identification and lies at the intersection between the moment inequalit literature and the literature on set identification that exploits the geometric structure. A model is generall set identified when the data are incomplete because of some missing information (a censorship mechanism two-sample combination or multiple equilibria). In entr games the unknown selection mechanism in regions of multiple equilibria is the missing one but it is naturall bounded between and. These bounds lead to moment inequalities (see 4

6 for example Ciliberto and Tamer (29)). Moment inequalit procedures are now well developed in the partial identification literature which includes contributions b among others Chernozhukov et al. (27) Rosen (28) Andrews and Soares Bugni Cana Romano and Shaikh (28) Romano and Shaikh Chernozhukov et al. (25) Andrews and Shi (23) or Aradillas-Lopez and Rosen (27). However in most cases the structure of the missing information is not exploited. Other contributions exploit this missing information b characterizing the identified set directl or indirectl through a convex set and then through the support function. The support function provides a collection of moment inequalities but these moment inequalities have a particular structure and are exploited differentl than in the general moment inequalit literature (see in particular the two surves b Molchanov and Molinari (25) and Bontemps and Magnac (27)). Papers that have exploited the structure of the missing information either use random set theor (introduced in econometrics b Beresteanu and Molinari 28) optimal transport (Galichon and Henr ) or complete the model as in Bontemps et al. (22). Fundamentall all methods are intended to provide a sharp characterization of the identified set and lead to a collection of moment inequalities that are specificall determined using the tools of the approach considered. The all lead to the same set of inequalities. In this spirit the can be seen as equivalent. Redundant ones are eliminated through the characterization of what Galichon and Henr call the coredetermining class. In our case we also characterize the core-determining class i.e. the collection of facets of the convex set b a necessar and sufficient condition. However in addition to these equivalences our geometric approach is able to select the subsets of moments that are close to binding without having to evaluate all of them and to determine the maximum number of moments that are binding. Section 2 considers the general entr game with N plaers and no explanator variable. Explanator variables do not pla a specific role in the procedure. We characterize the regions with multiple equilibria and compute the support function of the poltope i.e. the set of predicted choice probabilit vectors for a given parameter value. The sequence of inequalities that need to be verified is characterized. The aim of Section 3 is to more deepl exploit the geometr of the poltope to propose a more efficient estimation procedure. We first provide a necessar and sufficient condition for 5

7 eliminating an redundant inequalities. We then propose our geometric selection procedure which consists of determining the local extreme point of the poltope and onl testing the inequalities relative to the facets defining this extreme point. Section 4 calculates the asmptotic distribution of the test statistic used and proposes different calculations of the critical value that are computed once for all. Section 5 proposes a few Monte Carlo simulations to assess the performance of our inference procedure and compare it with existing procedures. Section 6 considers the case with discrete explanator variables and Section 7 concludes the paper. All the proofs and algorithm details are provided in a supplemental appendix. 2 Entr game with N plaers We formalize the entr game with N plaers/firms. For exposition purposes we first consider a model without explanator variables and then extend it in section 6. We first define some notations before characterizing the identified set. 2. Setup and notations 2.. The model Let N denote the total number of firms that can enter an market. Following Berr (992) we introduce a model of market structure where the profit function π im of firm i in a market m is assumed to be independent of the identit of the firm s competitors. All firms decide simultaneousl whether to enter the market (the action is a im = ) if in a pure strateg Nash equilibrium their profit is positive (π im > ). Otherwise π im <= and the action is a im =. The profit function is assumed without loss of generalit to be linear in the explanator variables π im = β i + α i ( i a im = π im >. a m ) + ε im () Following the literature on entr games we assume that α i < i.e. the presence of more competitors decreases a firm s profit. 3 The unobserved components ε im i =... N are drawn from 3 The case in which α i > could be handled equivalentl. Gualdini (28) considers in a network formation game these two cases. 6

8 a known distribution (up to some parameter vector γ). The econometrician does not observe their values whereas all firms within a market observe them before deciding whether to enter (complete information case). An (separable) parametric form π im ( ) = f i i a m; θ + ε im could have been considered 4 without loss of generalit as long as the function f i ( ; θ) is strictl decreasing in its first argument. For identification we first need a scale normalization and thus we assume that the variance of each shock ε im is equal to unit. of ε m We denote b F ( ; γ) the cumulative distribution function = (ε m... ε Nm ) and we assume that the distribution is continuous with full support. Henceforth we use the notation θ for all the parameters in the model (θ R l ) 5 and we omit the subscript m for notational convenience The multiplicit of pure strateg Nash equilibria For a given market an outcome is the vector of actions (in N ) taken b the firms. There are obviousl 2 N possible outcomes from (... ) to (... ). We denote b Y this set of possible outcomes. Y K denotes the subset of outcomes with K active firms in equilibrium i.e. plaing action. There is outcome with active firms N outcomes with active firm and d K = ( N K) outcomes with K active firms for K N. For each K we label the outcomes as (K) =... d K according to a predefined order. 6 Globall we order the outcomes in Y first b their number of active firms then according to the predefined order within each Y K : Y = () ()... () d Y Y... (K) (N) d K.... Y K Y N (K)... It is well known that the model has multiple equilibria i.e. there are regions of realizations of ε in which we cannot uniquel predict each firm s action. Consequentl there is no one-to-one mapping between the collection of possible outcomes and the regions of ε given an parameter value θ. There are three was to address this issue. The first is to add additional restrictions for example a rule that determines the order of entr in these regions of multiple equilibria. This approach is purel ( ) 4 or more generall f i i a m X; θ when there are explanator variables. 5 θ = (β... β N α... α N γ) 6 The order for K = is ((... ) (... )... (... ) ) and so forth. 7

9 ad hoc and ma lead to misspecification. The second approach consists of studing an outcome that is invariant in the regions of multiple equilibria. Berr (992) shows that the number of active firms is constant in the specification considered in this paper. We exploit this specific structure below. However basing the inference on the number of active firms onl can be ver inefficient (see for example Tamer (23); our simulation results in Section 5 quantif the loss in some particular examples). The third approach follows the recent literature on set identification. What is missing from the model is the selection of a given equilibrium in the regions of multiple equilibria. Not observing this selection mechanism 7 generates some uncertaint that generall transmits into the parameters of interest. It leads to set/partial identification in some cases. One can therefore define moment inequalities that are satisfied b the points of this identified set (Ciliberto and Tamer 29 Galichon and Henr 2 Beresteanu et al. 2). Our approach belongs to this literature and we now present it. The difference between our work and the existing literature is emphasized later. 2.2 From the set of choice probabilities to the identified set Let P (K) (θ) =... d K denote the probabilit of the outcome (K). We stack all the P (K) first for a given K into the vector P (K) (θ) and then all the P (K) (θ) with K varing from to N into a vector P (θ) of size 2 N. The aim of this section is to characterize the set A(θ) to which P (θ) belongs. For a given parameter θ a given K in... N and a given in... d K we define b U (K) (θ) the region of ε R N where the outcome (K) is the unique pure strateg Nash equilibrium and P(U (K) (θ)) is its probabilit. We now characterize the regions of multiple equilibria which are necessar to define A(θ) 2.2. The regions of multiple equilibria Henceforth we sa that a subset S Y K of cardinalit d (K) i that (K) i... (K) i d... (K) is in multiplicit (or are in multiplicit) if there exists a non-empt region of ε R (K) (θ) in which i d S the prediction of the game is an of these outcomes. We denote its area b (K) S (θ).8 We therefore 7 It will be introduced more formall below; see Definition. 8 (K) S (θ) = R (K) S (θ) df (ε; γ). 8

10 define b S (K) the collection of subsets S Y K in multiplicit. Note that not all subsets of cardinalit greater than two are elements of S (K) and we now present a necessar and sufficient condition for S to be in multiplicit. Define N (resp.n ) the set of indices of firms that alwas pla action (resp. ) across S. n and n are their cardinalities. N and N being fixed there are ( N n n ) K n possible outcomes in YK corresponding to the remaining choice of the K n firms which pla action among the N n n remaining ones. S should contain all these possibilities to be in multiple equilibria and it is formalized in the next proposition. Proposition A set S Y K of cardinalit d is in multiplicit if and onl if d = ( N n n K n ). When K = (or N ) an collection of more than two outcomes is in multiplicit. However this is not the case for other values of K. For example when N = 4 and K = 2 we consider the following subsets: S = S 2 = S is in multiplicit because N and N being fixed the collect all possible outcomes leading to K = 2. However S 2 is not in multiplicit. N and N are empt and consequentl the subset should contain d = ( 4 2) i.e. the 6 outcomes with K = 2 to be in multiplicit. The following proposition counts the number of multiple equilibria regions. Proposition 2 The cardinalit of S (K) i.e. the number of regions of multiple equilibria for. K N is equal to K S (K) = n = N K n = ( )( N N n n n ). When K = the number of regions of multiple equilibria is N n=2 ( N n) i.e. all possible combinations of more than two outcomes. Table I displas the number of regions for various values of N and K. It appears that for K 2... N 2 the number of multiple equilibria regions is far less important than what it would be if all the combinations were possible. [Include Table I] 9

11 2.2.2 The set of predicted choice probabilities We also define the subset of S (K) that contains one specific outcome (K) S (K) = S S (K) : (K) S as and define the selection mechanism η( ) as in Definition 2 of Galichon and Henr. Definition (Equilibrium selection mechanism) An equilibrium selection mechanism is a conditional probabilit η( ε; θ) of given ε such that the selected value of the outcome variable is actuall an equilibrium predicted b the game. We denote b E the set of selection mechanisms. Following Berr and Tamer (27) and Galichon and Henr we can define the probabilit of observing outcome (K) according to the model. This probabilit depends on the parameter vector θ and on the unknown selection mechanism η that selects equilibrium (K) P (K) in the regions of multiple equilibria that predicts this outcome. (θ η) = P ( U (K) (θ) ) + S S (K) A(θ) is the set generated b the variation of η in E: A(θ) = η ( (K) R (K) S (θ) ε; θ ) df (ε; γ). P R 2N : η E P = P (θ η). Equation is a parametrization of the set A(θ). This parametrization is indexed b the regions R (K) (K) S (θ). The set is contained in the cube where each coordinate P varies between P ( U (K) (θ) ) and P ( U (K) (θ) ) + df (ε; γ) because η( ) lies between and. However the true set S S (K) R (K) S (θ) is strictl included in the cube A characterization of the identified set Let P = P (θ η ) be the true choice probabilities generated b the true (unknown) parameter θ and the true (unknown) selection mechanism η. The identified set Θ I is defined as the collection of points θ such that P can be rationalized with a selection mechanism Θ I = θ R l such that η E P = P (θ η). (3) 9 See Appendix D Figure 9 for a visual illustration of the case with three plaers and K =.

12 Without further restrictions η represents an conditional probabilities in the space R 2N R N Θ. The following is easil verified: θ Θ I if and onl if P A(θ). (4) We therefore need to be more precise about the structure of A(θ) to be able to verif the second part. Let B K (θ) be the set of choice probabilities described b the vector P (K) (θ η) when η varies in E. The following result holds: Proposition 3 A(θ) is a convex set of R 2N and where B K (θ) is a convex set in R d K. A(θ) = B (θ) B (θ) B 2 (θ)... B N (θ) The convexit of the sets is due here to the linearit of the points P (θ η) in η( ) in Equation. For the cartesian product consider two different ε and ε 2 in R (K ) S (θ) and R (K 2) S (θ) for K K 2 ; the equilibrium selection mechanism is equal to zero when Y K and ε = ε 2 or when Y K2 and ε = ε. Observe that B K (θ) shrinks to a singleton when the number of active firms in equilibrium is or N. Θ I the identified set is not convex but it can be characterized b verifing that a point P belongs to a convex set A(θ). following sub-conditions: Using Proposition 3 we can decompose this condition into the P A(θ) iff P (K) B K (θ) K... N. At this stage we make the following comment. As the sum of probabilities is equal to we do not have to check one of the probabilities and therefore we can choose to eliminate either point B (θ) or point B N (θ) from our testing procedure as it is redundant. However it would be easier to account for the fact that we are on the simplex i.e. the sum of the probabilities of each outcome is equal to one. Let n (K) be the true probabilit of having K entering firms induced b the true parameter θ. The model predicts n K (θ). Obviousl an θ in the identified set satisfies the equalities: n () = n (θ) for all... N. We thank one of the referees for highlighting this issue.

13 Instead of dropping one equalit this is equivalent to checking the N + inequalities: n () n (θ) for all... N. The last equivalence result holds because the sums of each side of both the inequalities across are equal to and all the terms involved are positive or null. In the Monte Carlo section we compare on our specific examples the two solutions. 2.3 The support function and a first selection of moment inequalities A convex set can be uniquel characterized b man functions but the literature on set identification mainl uses the support function. One of its advantages is that the Hausdorff distance between two convex sets the most commonl used distance is defined as the supremum of the difference between their support functions. Following Beresteanu and Molinari (28) or Bontemps et al. (22) we use the support function to generate the inequalities satisfied b P. We first recall what the support function of a convex set is and how it generates the inequalities that are the basis of our inference procedure. The support function of a convex set A R d is defined as: δ (q; A) = sup x A for all directions q R d. Following Rockafellar (97) it uniquel characterizes the set A: q x. P A q R d q P δ (q; A). (5) This construction is illustrated in Figure. It is sufficient to restrict our attention to the unit sphere due to the homogeneit propert of the support function but we do not impose it here. The support function of a convex set in a given direction locates the supporting hperplane in this direction. It defines an inequalit that is satisfied b an point of the identified set. The support function implicitl gathers all the inequalities that define the convex set into a single function. If the set is smooth there is a continuum of such inequalities; if it is a poltope onl a finite number of inequalities is necessar to characterize the set. Kaido and Santos (24) show that when the set is convex using the support function leads to an efficient estimator of the convex identified set. We now turn to the calculation of the support function of B K (θ) for an direction. Let q R d K be a given direction. We assume the following order among the coordinates of q: q i q i2... q idk. 2

14 We also partition S (K) as follows: we denote O (K) i = S (K) i and b O (K) i 2 the subset S (K) i 2 of elements that are not in O (K) i i.e. S (K) i 2 \S (K) i and more generall O (K) i = S (K) i \ k< S (K) i k for an d K. We now provide a closed-form expression for the support function in this direction. Proposition 4 Let q R d K and assume q i q i2... q idk. The support function in the direction q δ (q; B K (θ)) is equal to: δ (q; B K (θ)) = It is reached at the point β K q d K = q i P ( U (K) i (θ) ) + d K q i = ( = vec P ( U (K) (θ) ) + S O (K) S S O (K) i (K) S (θ)... P( U (K) d K (θ). (6) (θ) ) + S O dk (K) S (θ) ). Consequentl B K (θ) is a poltope and its vertices are included in the set of points β K q q R d K. B K (θ) has at most d K! vertices. Each extreme point of B K (θ) and therefore its support function can be calculated from the knowledge of the non-zero values of (K) S (θ) S S(K). This number of non-zero values is the number of multiple equilibria regions and we saw in Proposition 2 that this number is much smaller when K is not or N than the number of subsets of cardinalit greater than two of a set with cardinalit d K i.e. 2 d K d K. Consequentl the parametrization of B K (θ) is numericall tractable for moderate values of N. Furthermore each non-zero value (K) (θ) can easil be calculated or simulated from the knowledge of the distribution of ε. S We can now extend this result to the calculation of the support function of the full set for an direction of R 2N. We adopt the standard notation: q = vec ( q q... q N ) where qk is the direction related to the set B K (θ) (i.e. q K R d K ) and vec( ) denotes the vertical concatenation. Corollar 5 The support function of A(θ) in direction q is equal to N δ (q; A(θ)) = δ ( q K ; B K (θ)). (7) K= The last proposition combined with Equation (5) is the basis of our inference procedure. It generates a continuum of inequalities that have to be satisfied for an parameter of the identified 3

15 set. However since A(θ) is a poltope it is necessar and sufficient to test the inequalities in a finite set of directions. The particular geometric structure of A(θ) permits us to define these directions. We denote the completion of a direction q in R d K b c K (q) its corresponding direction in R 2N : where d is the vector with d zeros. ( ) c K (q) = vec K i= d q N i i=k+ d i Let Q K be the set of non-null directions of R d K with coordinates that are either one or zero. There are 2 d K directions in Q K. Let Q be the union of directions in Q K completed. The next proposition shows that it is sufficient to check the inequalities in Q to characterize A(θ). Proposition 6 θ Θ I iff P A(θ) iff q Q q P δ (q; A(θ)). Following the proposition there are at maximum N K= (2d K ) inequalities. However this number can become ver large when N 5 (22 for 5 plaers and more than million for 6 plaers). The next section introduces strategies to reduce the number of inequalities to test. However we first compare our approach to existing ones. Optimal transport random sets or completion of the model Our approach consists in characterizing the set A(θ) through its support function and extreme points. This can be done after having completed the model with the unknown selection mechanism η( ) and finding which selection mechanisms generate the extreme points. The geometric structure induced b the multiplicities allows us to exhibit the inequalities that are satisfied b an parameter of the identified set. Galichon and Henr use the optimal transport theor applied to games and in particular the notion of core-determining class to generate the relevant inequalities that characterize sharpl the identified set; in parallel Beresteanu et al. emphasize that an entr game is a model with convex predictions. The use random set theor and in particular the support function to generate the moment inequalities. The re is no closed-form for this support function. Both methods are numericall challenging for a game with 6 plaers even when considering onl pure 4

16 strateg equilibria. Ciliberto and Tamer (29) bounds the set A(θ) b a cube which is easier to characterize. This approach can handle games with a moderate number of plaers above 6 like our method but bounding component b component makes the estimated set larger and sharpness is not attained. Fundamentall whether one uses random set theor and the capacit functional the optimal transport approach of Galichon and Henr and their core-determining class (see also Chesher and Rosen 27 for a generalization) or the approach presented in this paper all these methods are intended to derive a sufficient set of inequalities satisfied b the parameters in a specific manner. Each method has its specificities. Some analog could be made with Bontemps et al. (22) and Beresteanu and Molinari (28). Both use the support function introduced b Beresteanu and Molinari to estimate the identified set (which in this case is convex). Both are considering for the interval censoring case the same identified set with their specific approach. Completing the model allows Bontemps et al. (22) to consider cases where the identified set has kinks and faces. The Aumann expectation considered in Beresteanu et al. is our set A(θ). We both use the support function to characterize the inequalities necessar to estimate the model. However our approach allows us to go deeper into the geometric analsis of the set A(θ) and this is the obective of the next section. 3 Using the geometr of A(θ) to select inequalities In this section we present two strategies for reducing the number of inequalities. The first consists of calculating the core-determining class introduced b Galichon and Henr and generalized b Chesher and Rosen (27). The second consists of exploiting further the geometr of the set A(θ) to propose a geometric section procedure of the inequalities without having to evaluate them. 3. The core-determining class of an entr game The convex set B K (θ) can be characterized b at most 2 d K inequalities. The number of facets of the set corresponds to the minimum number of inequalities sufficient to characterize this convex set. Galichon and Henr refer to this as the core-determining class. We provide a characterization 5

17 of the core-determining class in an entr game from the geometric stud of the multiplicit structure of the model. For the text to be self contained we borrow some definitions from Galichon and Henr. Definition 2 (Choquet capacit) A Choquet capacit L on the set Y is a set function L : C Y [ ] that is normalized i.e. L( ) = and L(Y) = and monotone i.e. L(C) L(B) for an C B Y. Definition 3 The smallest class Ω of subsets of Y is called core-determining for the Choquet capacit L on Y if P(C) L(C) holds for all C Ω; then P(C) L(C) holds for all C Y. A(θ) is characterized b its support function. Thus we define the Choquet capacit for a subset (K) i... (K) i m YK as L( (K) i... (K) i m ) = δ ( c K (e (K) i...i m ); A(θ) ) (8) where e (K) i...i m d K with ( e (K) i...i m ) = if = i k for some k... m otherwise and c K ( ) is the completion operator defined above that completes a vector in R d K into a vector in R 2N. For a collection of subsets C = S i Y K : i p the Choquet capacit is defined as L(C) = L ( i S i ). Clearl L(Y) = as probabilities sum to. L is also monotone as it is the sum of quantities that are positive. We define the concept of connectedness which is useful for the exposition. For a subset C Y K we define the (undirected) graph generated b C as Γ C = (C E) where there is an edge between two vertices if the are in multiplicit with eventuall some additional outcomes that are onl in C (no outcome from Y K \ C). For an graph Γ = (V E) we sa that C V is connected in the graph Γ if there is a path of elements of E connecting an pair of nodes of C. Definition 4 (Well connectedness) A subset C Y K is called well connected in Y K if Y K \ C is connected in the graph Γ YK \C. Recall that an undirected graph Γ = (V E) is a collection of vertices/nodes and edges/links that link these vertices. A graph Γ is connected if an pair of vertices are connected in Γ. 6

18 Note that there is a multiple equilibria involving all outcomes in Y K. Therefore the graph Γ YK is connected and ever C Y K is connected in graph Γ YK. The notion of well connectedness extends the notion of connectedness b imposing restrictions on the complementar of C. Note that the graph Γ Y is not connected as there is no multiplicit between Y K and Y K for K K. Thus Γ YK is a component of Γ Y. 2 We collect all well-connected subsets of Y K as Ω K = C Y K : C well connected in Y K Galichon and Henr present some models in which the core-determining class can be of much lower cardinalit than 2 Y b exploiting the monotonicit propert in certain supermodular games. However their approach does not provide a wa to find the core-determining class for a general entr game. Chesher and Rosen (27) provide a sufficient condition to characterize the core-determining class of set-identified models that can be written into what the call a generalized instrument variable model. To the best of our knowledge our entr game has a different structure. Our next proposition provides a complete characterization of the core-determining class for our entr model through a necessar and sufficient condition. Proposition 7 A collection Ω of subsets of Y is core-determining class for our entr model if and onl if Ω = Ω K : K =... N. Subsection C. in the Appendix provides a simple algorithm to calculate the core-determining class in our model using this proposition. It is applied for the particular examples of N = 4 and K = 2 in Subsection C..2. The core-determining class is a useful concept because we eliminate redundant inequalities. However it does not significantl reduce the number of inequalities in our entr game. For example when N = 6 and K = 3 it eliminates fewer than 3 inequalities from a total of 2 2 = The number of facets at an extreme point We now need to determine the number of facets of the convex set at a given extreme point. This number is the number of inequalities that are binding at this point. Then this number is used later 2 Recall that for an undirected graph Γ = (V E) components of Γ are subgraphs H i k i= such that each H i is connected and H i is not connected to H for i. 7

19 in the inference procedure. We know from Proposition 4 that B K (θ) is included in an hperplane of R d K. It is due to the fact that the sum of the choice probabilities of all outcomes in Y K is constant. Following the convex literature an exposed face is the intersection between a supporting hperplane and the convex set. Henceforth we call facets of B K (θ) the (d K ) and (d K 2)- exposed faces of B K (θ) with an abuse of language. Each facet is related to one supporting hperplane which defines one inequalit. The number of facets of all the subsets B K (θ) is the number of inequalities necessar to sharpl characterize A(θ). From Proposition 4 the maximum number of extreme points of B K (θ) is d K!. We first consider the case in which K = as a benchmark (b smmetr it also characterizes the case in which K = N ) before considering the general case The case of K = Observe first that the geometr of a set B K (θ) is the same as that of the set B N K (θ). We characterize the geometr of the sets B (θ) that will serve as a benchmark for considering the other sets. Observe that d = N. Proposition 8 The convex set B (θ) has d! extreme points. Each of them is the intersection of d supporting hperplanes. First following Proposition 2 we know that an subset of Y of cardinalit greater than 2 corresponds to a multiple equilibria region. Consequentl () S (θ) for an subset S Y is nonzero and following Proposition 4 an change in the order gives a different point. Let i i 2... i d be an order of the coordinates that defines an extreme point B () i i 2...i d (θ) and C () i...i d be the cone of directions q such that δ (q; B (θ)) = q B () i...i d (θ). Each direction defines a supporting hperplane (or facet) of B (θ) at B () i...i d (θ). An direction q in the cone can be written as q = (q i q i2 )e i + (q i2 q i3 )e i i (q id q i d )e i i 2...i d + q i d e i i 2...i d. All the coefficients except the last one are positive. The cone is therefore generated b e i e i i 2... e i i 2...i d e i i 2...i d or e i i 2...i d. 3 3 Remark that e ii 2...i d = (... ). 8

20 In other words there are onl d supporting hperplanes of B (θ) at this point and it is sufficient to check the inequalities related to these hperplanes/facets for a point locall around B () i...i d (θ) The general case For a general value of K man collections of outcomes cannot be a prediction of a multiple equilibria region. Consequentl there are fewer than d K! extreme points but more important there are more than d K facets at some of these extreme points. When an extreme point is determined we can ultimatel check onl the d K hperplanes defined as for the case of K =. However ignoring some of the facets would certainl lead to a non-sharp characterization of the set. We now provide an algorithm to define these facets and an upper bound of the number. Let i i 2... i dk be an order of the coordinates that defines an extreme point B (K) i i 2...i dk (θ). We now want to determine the facets of B K (θ) at this extreme point. The algorithm detailed below is based on the idea that when some multiple equilibria region does not exist between a series of outcomes the corresponding indices can be swapped if the are in consecutive ranks without changing the point B (K) i i 2...i dk (θ) in the space. It increases the number of inequalities that are binding at this extreme point (intuitivel it gathers all the hperplanes defined with one specific order). Step : Start with L K = (K) i and set k = 2. Step k: Find the largest m such that (K) i k (K) i k+... (K) i m are not in multiplicit even eventuall with elements in (K) i k. We add to LK all the parts of (K) (K) i k+... (K) i m oined with (K) i class D K : (K)... (K) i k and purged from the element that are not in the core-determining i 2 L K = L K (K) i (K) i 2... (K) i k P (K) (K) i k+... (K) DK DK defined in Section C. collects the redundant directions. Then update to k +. Iteration: Repeat the previous step for k = 2... d K steps. L K can be converted into equivalent support directions. An element (K) i ields to the direction c K (e (K) i...i m ) following Equation (8). 9 i k i k i m... (K) i m of L K

21 We provide a simple illustration of this algorithm in section C.2. The local geometr of set A(θ) in the extreme point considered is L = L K : K =... N. (9) Note that the composition of L is specific to each extreme point. We now provide an upper bound of its cardinalit that is valid for an extreme point. This number allows us to provide an invariant inference method valid for each extreme point. The next proposition provides an upper bound on the maximum number of local moments for an extreme point. Proposition 9 An extreme point of B K (θ) for < K < N is the intersection of at most 2 lmax + (d K l max )2 lmax supporting hperplanes where l max is the maximum cardinalit of a subset of (K)... (K) d K that cannot be in multiplicit. The upper bound on the number of facets passing through an extreme point of B K (θ) can be improved further on a case-b-case basis. However note that the upper bound given in Proposition 9 is much smaller than the cardinalit of the core-determining class. Table II gives the maximum number of facets for N = 3 to A geometric selection procedure We now present a selection procedure that full exploits the geometr of the set A(θ). The procedure first selects the extreme point of the set that seems the closest to the vector P and then evaluates onl the inequalities associated with this extreme point i.e. tests the directions that are the outer normal vector of the supporting hperplanes of A(θ) at this point. Following Proposition 4 an extreme point is determined b an order in the coordinates (note that a priori two different orders could lead to the same phsical point). The algorithm is intended to determine this order in a recursive manner b exploiting the order of the coordinates of P after an appropriate and recursive normalization. We explain the steps in non-technical detail below and formalize them in Appendix C.2:. Do the following for ever K... N. Select P (K) the sub-vector of P that corresponds to the outcome with K active firms. 2

22 2. Determine the cube that contains B K (θ) b calculating the minimum and maximum of each coordinate. Then determine which coordinate of P (K) is the furthest from the center of the cube. 3. Assume this is the th coordinate. (a) If it is on the maximum side the extreme point is of tpe B (K)?...? (θ) and we now have to determine the next component. To do so we proect P (K) on the face and we repeat the previous calculation b taking into account that we are on the face that maximizes the th coordinate. (b) If it is on the minimum side we know that our extreme point will be of the form B (K)?...?(θ) and we now have to determine the next component. To do so we proect P (K) on this face and we repeat the previous calculation b taking into account that we are on the face that minimizes the th coordinate. 4. Repeat the following steps until having found one order of coordinates. Once the local extreme point B (K) i i 2...i dk is determined we can focus on the directions of the local supporting hperplanes derived in section 3.2 to do our test in (6) instead of considering the whole set Q K the cardinalit of which is 2 d K. Proposition shows that the procedure is sharp for N = 3. Proposition The following procedure provides a sharp characterization of the identified set for N = 3. Find a local extreme point B (K) i i 2...i dk. Find the outer normal of the supporting hperplanes of B K (θ) in B (K) i i 2...i dk. Do the previous steps for K in... N. Evaluate the inequalities in all the directions collected onl adding the ones related to B (θ) and B N (θ). 2

23 Repeat that for an θ. However it is difficult to prove sharpness in a more general case due to the difficult of globall characterizing all the facets. We evaluate this procedure for N = 4 in the Monte Carlo section and results highlight that we are sharp too. 4 Estimation and inference Following the results derived above we now adopt the approach developed in Beresteanu and Molinari (28) and Bontemps et al. (22) for testing a point in a convex set: θ Θ I (P) P A(θ) q G T (q; θ) = δ (q; A(θ)) q P min q G T (q; θ). The set of minimizers G is either the set of directions Q characterized in Proposition 6 (normalized or not) or the core-determining class Ω characterized in Proposition 7 (normalized or not); P is the true choice probabilit. The test based on T ( ) is infeasible because we do not observe P. We now characterize the feasible test statistic and its asmptotic distribution under the null and derive strategies to calculate the critical values. Throughout this section we assume that we observe a sample of n i.i.d. markets in which the same N firms (known to the econometrician) compete. 4. The asmptotic distribution of the test statistic Let T n (q; θ) be the empirical counterpart of T (q; θ): T n (q; θ) = δ (q; A(θ)) q ˆPn where ˆP n = n [(Y i = )... (Y i = n 2 N )] is the empirical frequenc vector. i= Note that the onl random part in T n comes from the estimation of the choice probabilities. In particular the support function with θ being given is known and not random. The variance of T n (q; θ) is therefore ver specific because it reduces to q Σ q where Σ denotes the variance of ˆP n 22

24 Σ = diag(p ) P (P ). Moreover an empirical estimator ˆΣ can be used b plugging in ˆP n in place of P in its expression. Under the assumption that the markets are i.i.d. we have: n ( ˆPn P ) d n N ( Σ ). In this section we want our asmptotic result to be valid not onl for the true probabilit but also uniforml in the neighborhood of the true probabilit following the set identification literature. We impose the following uniform integrabilit condition: Assumption (Uniform integrabilit) The class P satisfies ( 2 lim sup sup E P (Y = ) µ (P ) ( ) µ (P ) λ P P...2 N µ (P )( µ (P ))) µ (P )( µ (P )) > λ = (UI) where µ (P ) = E P (Y = ). Assumption UI ensures the uniform convergence of our test statistics over the class of probabilit distributions P. This condition is clearl satisfied over the class of probabilit distributions such that µ (P ) ε for each and some ε >. The literature on inference in partiall identified models has largel focused on the construction of confidence regions C n. There is an open debate about whether one should cover an point of the identified set (and in particular the true value) or the identified set entirel. Unless stated otherwise we are mainl interested in covering an point in the identified set with some pre-specified probabilit α i.e. lim inf n inf P P inf P (θ C n) α. θ Θ I (P ) Following Bontemps et al. (22) our inference method is based on T n (q; θ) rescaled b n: and its normalized version ξ n (θ) = n min q G T n(q; θ) ξ n(θ) = n min q G T n (q; θ) q ˆΣq. A point θ belongs to the confidence region C n if the test based on ξ n (θ) or ξ n(θ) is not reected. We now calculate the asmptotic distribution of the test statistics ξ n (θ) and ξ n(θ). 23

25 Proposition Let Q θ be the set of minimizers of T (q; θ) in G. Let Z be a random vector of R 2N distributed according to the normal distribution with variance Σ. We have d ξ n (θ) q Q n θ q Z if P A(θ) ξ n (θ) a.s if P / A(θ) n and similarl ξ n(θ) ξ n(θ) d min q Q n θ a.s n q Z q Σq if P A(θ) if P / A(θ) Under assumption UI these results are uniforml valid over P P. The asmptotic distribution of the test is standard onl when P belongs to a facet but does not belong to an other facet. In fact the set Q θ reduces to a singleton q and following the argument of Bontemps et al. (22) the argmin of T n q n tends asmptoticall to q and can therefore be plugged in to replace the true unknown q. In this case: ntn (q n ; θ) d n q Z. The right-hand side is a normal distribution the variance of which can be estimated b q n ˆΣq n. However when P does belong to more than one facet the asmptotic distribution is not standard as Q θ is the cone of outer normal vectors of the facets of A(θ) in P. The following subsections show how to propose some calculations of critical values. 4.2 A global bound approach We first propose conservative bounds. The have the advantage of being calculated simpl and once for all for an θ thereb considerabl simplifing the inference procedure. The can be used in a first step to locate the confidence region. This approach is based on the idea that we can triviall define two outer sets of Q θ and using these bound the asmptotic distribution of ξ n (θ) and ξ n(θ). We make the assumption that the directions in G are normalized and therefore are on the unit sphere. Simulations not presented in the paper highlight the interest to have the same norm across all directions tested. We thus have: Q θ G S. 24

26 where S = c K (q) : q S d K for K =... N. 4 Consequentl min q Z min q Q θ q G q Z min q Z. Let c (G α) and c (S α) be the α quantile of the bounding distributions and let the confidence region be defined as the collection of points for which the test statistic ξ n (θ) is above a given threshold: q S C(c) = θ Θ : ξ n (θ) c () The next proposition shows that the confidence regions built using our procedures have asmptoticall and uniforml over P the correct coverage rate for an point of the identified set. Proposition 2 Under Assumption UI the following holds: lim inf n inf inf P ( θ C(c (S α)) ) lim inf P P θ Θ I n inf inf P (θ C(c (G α))) α. P P θ Θ I Additionall the two critical values can be simulated easil. Observe first that c (S α) is the α quantile of min K=...N Z K. 5 Second an algorithm is provided in Section C.4 in the Appendix to simulate the critical value c (G α) with a number of calculations polnomial in N. Similar bounds and a similar algorithm can be proposed for the standardized test statistic ξ n(θ) which we summarize in the next proposition. Proposition 3 Let Z be a 2 N random vector distributed according to a standard normal variable N( I 2 N ) let Σ /2 be a square-root of Σ let c 2 (S α) be the α quantile of min K=...N Z K q and let c 2 (G α) be the α quantile of inf ˆΣ /2 Z q G q ˆΣq Assumption UI lim inf n inf inf P ( θ C (c 2 (S α)) ) lim inf P P θ Θ I n and C (c) = θ Θ : ξ n(θ) c. Under inf inf P (θ C (c 2 (G α))) α. P P θ Θ I Observe that because we bound the asmptotic distribution of both ξ n (θ) and ξ n(θ) uniforml for θ Θ I the confidence regions built from our global bounding strateg entirel cover the identified set. We therefore have the following stronger result: 4 S d K is ust the unit sphere of dimension d K and c K (.) is the completion operator defined before. 5 Vector Z can be subdivided according to K as Z = (Z Z Z 2... Z N ). 25