An ordinal approach to the empirical analysis of games with monotone best responses

Lazzati, Natalia; Quah, John K.-H.; Shirai, Koji Article An ordinal approach to the empirical analysis of games with monotone best responses Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Lazzati, Natalia; Quah, John K.-H.; Shirai, Koji (2025) : An ordinal approach to the empirical analysis of games with monotone best responses, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 16, Iss. 1, pp. 235-266, https://doi.org/10.3982/QE2192 This Version is available at: https://hdl.handle.net/10419/320326 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Quantitative Economics 16 (2025), 235–266 1759-7331/20250235 An ordinal approach to the empirical analysis of games with monotone best responses Natalia Lazzati Department of Economics, University of California at Santa Cruz John K.-H. Quah Department of Economics, National University of Singapore Koji Shirai School of Economics, Kwansei Gakuin University We develop a nonparametric and ordinal approach for testing pure strategy Nash equilibrium play in games with monotone best responses, such as those with strategic complements/substitutes. The approach makes minimal assumptions on unobserved heterogeneity, requires no parametric assumptions on payoff functions, and no restriction on equilibrium selection from multiple equilibria. The approach can also be extended in order to make inferences and predictions. Both model-testing and inference can be implemented by a tractable computation procedure based on column generation. To illustrate how our approach works, we include an application to an IO entry game. Keywords. Revealed preference, monotone comparative statics, single-crossing differences, supermodular games, revealed monotonicity axiom. JEL classification. C1, C6, C7, D4, L1. Natalia Lazzati: [email protected] John K.-H. Quah: [email protected] Koji Shirai: [email protected] For helpful discussions and comments, the authors are grateful to S. Berry, J. Fox, K. Hirano, T. Hoshino, A.Kajii,Y.Kitamura,B.Kline,E.Krasnokutskaya,C.Manski,W.Newey,T.Sekiguchi,J.Stoye,B.Strulovici, S. Takahashi, Y. Takahashi, and especially X. Tang. Various versions of this project have been presented to audiences at the following events and we are grateful for their comments: seminars at University of Arizona, Johns Hopkins, Kyoto, Louvain (CORE), New York University, Rice University, Simon Fraser University, the National University of Singapore, Northwestern University, University of Paris (Dauphine), Queensland, Shanghai University of Finance and Economics, University of Southern California, Singapore Management University, Stanford, UC Davis, UC San Diego, the Canadian Economic Theory Conference (Vancouver, 2017), the Conference on Econometrics for Incomplete Models (CeMMAP and Northwestern, 2018), 13th Greater New York Metropolitan Area Econometrics Colloquium (Princeton University, 2018), and the Econometric Society North American Summer Meeting (UC Davis, 2018). Koji Shirai gratefully acknowledges financial support from the Japan Society for Promotion of Science (KAKENHI 19K00155) and the hospitality of Johns Hopkins University during his visit in the 2019–2020 academic year. ©2025 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE2192 236 Lazzati, Quah, and Shirai Quantitative Economics 16 (2025) 1. Introduction Economic analysis is often concerned with the effect of an exogenous or strategic variable on an agent’s decision: Would a consumer buy more of good A if the price of good B falls? Would a firm follow its rival when the latter raises its price? Is someone more likely to join a demonstration if more people are participating? The theory of monotone comparative statics identifies the single-crossing property (see Milgrom and Shannon (1994)) as a sufficient (and, in a specific sense, necessary) condition for optimal choices to be monotone with respect to opponents’ strategies and exogenous variables. The empirically relevant follow-up question is the following: What kind of observed choice behavior are necessary and sufficient for the recovery of payoff functions obeying the single-crossing property? The contribution of this paper is to answer this revealed preference question and to show that it forms the basis of an econometric analysis of games with strategic complements. One obvious and important area of application of our results is to the study of entry games (as in Bresnahan and Reiss (1990), Berry (1992), or Ciliberto and Tamer (2009)) and other games that arise in the empirical IO literature. In these papers, firms’ entry decisions are modeled as games of complete information, where each firm’s decision on whether or not to enter a given market is a best response to the entry decisions taken by other firms in that market. The payoff functions are assumed to depend on observable variables in a specific parametric form while the unobserved component is additively separable. The unobserved component is heterogenous across markets and belongs to a known class of distributions. Entry decisions by firms across many markets are observed, from which one could then estimate firms’ payoff functions. A major issue in this work concerns the effects of strategic interaction and market characteristics in terms of its direction and size: How often does the entry of another firm encourage or deter entry? To what extent does an exogenous variable (such as market size) encourage or deter the entry of other firms? Our approach has as its starting point a data set of the same type as the papers cited above. With this data set, we can test whether firms are playing pure strategy Nash equilibria (PSNE), subject to single-crossing restrictions on its payoff functions. For example, we can test the hypothesis that a firm’s entry into a market is encouraged when the market is large and discouraged when another firm is also entering. Our method works without imposing any parametric assumptions on payoff functions, without assuming that unobserved heterogeneity is additive or that its distribution belongs to a particular family, and without assumptions on equilibrium selection. By specifying a joint distribution on the payoff functions, we allow for correlation or other forms of dependence among firms’ payoff functions, which is important in many settings (see Chen, Christensen, and Tamer (2018)). To pass our test means that the hypothesis that the data are explained as PSNE by firms with payoff functions satisfying single-crossing restrictions cannot be refuted. At its most basic, our approach provides a way for researchers to test the general (nonparametric) features of a model, before the implementation of a more restrictive parametric model that could be used for inference and prediction. In some cases, the Quantitative Economics 16 (2025) An ordinal approach to the empirical analysis 237 confirmation of monotone features, which are part of our test could also facilitate estimation procedures.1Beyond this, since our test recovers the distributions on firms’ payoff functions that satisfy single-crossing restrictions and agree with the observations, the procedure can also be extended for the purposes of inference and prediction (when the data set passes the test). While we write of recovering “payoff functions,” what we are really recovering are a player’s preference over different actions, conditional on covariates and the actions of other players; this is as it should be, because in an environment where only PSNE are played, the information recovered from the data has to be just ordinal. The specific preference property we test (or when making inferences, assume)—the single-crossing property—is also an ordinal property. Our econometric approach is similar to that in Kitamura and Stoye (2018)(henceforth KS).2This paper tests a random utility model of consumer demand. In the first step, it is assumed that the population distribution of consumer demand at a linear budget set B,whichwedenotebyP (·|B), is known for a finite collection of budget sets B. Then one could formulate necessary and sufficient conditions under which the stochastic demand system PKS ={P(·|B)}B∈Bis generated by a population of utility-maximizing consumers, under the conditional independence assumption; this assumption requires the distribution of utility functions (which generates the distribution of demand) to be the same at each budget set B∈B. The characterization of PKS in KS is facilitated by the well-known characterization of utility-maximizing demand behavior for a single consumer, known as the strong axiom of revealed preference (SARP). The second step in the KS approach is to show how the characterizing conditions on PKS could be statistically tested for an actual data set, with empirical frequencies estimated at each budget set B∈B. The key observation in our paper is that a two-step procedure similar to that implemented in KS could also be used for analyzing specific classes of games. Suppose that there is a large population of groups, with each group playing the same game. We assume that the population distribution over joint action profiles at a given vector of covariate values x,whichwedenotebyP (·|x), is known for a finite set of covariate val1This information could be used to build a mapping from specific moments of the data to the identified set of relevant parameters. For instance, in two-player games the sign of the strategic interaction parameters allows us to identify outcomes that could occur only as a unique equilibrium; it follows that the probabilities of these outcomes (conditional on various observable variables) do not depend on any equilibrium selection mechanism and can be nicely related to payoff relevant parameters (see Tamer (2003) and Kline and Tamer (2016)). Shape restrictions can also reduce the size of the identified set of relevant parameters (see, e.g., Matzkin (2007)) and allow for the more efficient use of small sample data sets (see, e.g., Beresteanu (2005,2007)) . 2Our approach is also close in spirit, though not in specifics, with the nonparametric random utility models in Tebaldi, Torgovitsky, and Yang (2023), Deb, Kitamura, Quah, and Stoye (2023), Apesteguia, Ballester, and Lu (2017), Hoderlein and Stoye (2014), Manski (2007), McFadden (2005), McFadden and Richter (1991), and Marschak (1960). As far as we know, our paper is the first to exploit this nonparametric approach to study games. Note that Kitamura and Stoye’s empirical approach (and hence ours) is based on linear programming, which can be also found in earlier works such as Honoré and Tamer (2006) and Chernozhukov, Fernández-Val, Hahn, and Newey (2013). 238 Lazzati, Quah, and Shirai Quantitative Economics 16 (2025) ues  X.3(The set  Xtakes the place of Bin the KS model.) We then formulate necessary and sufficient conditions under which the set of choice distributions P={P(·|x)}x∈ Xis consistent with a population of groups made up of agents having payoff functions that satisfy single-crossing conditions and playing PSNE, under the assumption of conditional independence (which, in this case, means that the distribution of payoff function profilesacrossgroupsisthesameatdifferentx∈ X). The second step in our approach shows how these conditions on Pcould be statistically tested on an actual data set, with empirical frequencies over action profiles at different covariate values; for this second step, we simply follow the statistical procedure in KS. As in KS, the sampling framework requires that for each x∈ X,thereareNxobservations of action profiles such that Nx/N →ρx∈(0, 1),whereN=x∈ XNx→∞. Similar to KS, the characterization of P={P(·|x)}x∈ Xrequires that we find necessary and sufficient conditions under which the joint actions from a single group at different covariate values are consistent with our hypothesis of PSNE play and payoff functions satisfying single-crossing conditions (with respect to opponents’ actions and covariates). Since, unlike KS, there is no ready-made characterization for this class of games, we need to develop it ourselves. We show that this hypothesis can be characterized by a property we call the revealed monotonicity (RM) axiom. This axiom plays the role of SARP in the KS model. When the data set passes the test, our approach is in turn useful for making inference and prediction in the spirit of Deb et al. (2023), which deals with a version of the consumer model. For example, we can estimate the fraction of players who are effectively nonstrategic, in the sense that their actions depend only on covariate values and are independent of what other players do. We can also bound the proportion of groups which (at a given covariate vector) has a particular equilibrium profile as a PSNE (along the lines of the analysis in Aradillas-Lopez (2011)); note that this potentially differs from the observed fraction of groups playing that action profile, not just because of sampling variation, but also because a given action profile could be a nonchosen PSNE when there aremultiplePSNE. The procedure in KS is hard to implement when there is a large number of budget sets and Smeulders, Cherchye, and De Rock (2021) propose a column generation method to deal with this difficulty. This method is also applicable in our setting and is useful in easing the computational burden of our test when (e.g.)  Xis a big set. In our paper, we develop a new result on column generation that allows for this method to be used, not just for testing but also inference. The rest of the paper is organized as follows. In Section 2, we provide an outline of how our procedure works in the context of an entry game and contrast it with a parametric approach. Section 3presents our main results at the population level. We introduce the revealed monotonicity axiom and use it to characterize those distributions over joint actions that are consistent with our hypothesis; properties of the underlying 3Variation of feasible sets (as in KS) can be included in our analysis of games (see Lazzati, Quah, and Shirai (2018)), but we have avoided it, in order not to burden the reader with too many model features and also because our empirical application does not have such variation. (See also Carvajal (2004) for a related result.) Quantitative Economics 16 (2025) An ordinal approach to the empirical analysis 239 Table 1. P={P(·|x2=(0, 0)),P (·|x2=(0, 1)),P (·|x2=(1, 0))}. x2=(0, 0) Firm 2 NE Firm 1 N3/12 3/12 E4/12 2/12 x2=(0, 1) Firm 2 NE Firm 1 N1/12 5/12 E3/12 3/12 x2=(1, 0) Firm 2 NE Firm 1 N2/12 4/12 E2/12 4/12 distribution over payoff function profiles can also be recovered. Section 4explains how the population-level analysis in Section 3can be implemented on finite sample data. In this section, we also introduce and extend the column generation method of Smeulders, Cherchye, and De Rock (2021). To illustrate our approach, we carry out an empirical analysis of entry decisions made by airlines; this is found in Section 5. The Supplemental Material (Lazzati, Quah, and Shirai (2024)) contains some additional theoretical/empirical results as well as the omitted details of the statistical procedure. 2. Motivating example There is a large empirical literature modeling oligopoly entry decisions. We shall use this model to illustrate the basic question we are interested in and the approach we propose to address this question. For simplicity, we treat the case of two firms. Let yi∈{N,E} be the action set of firm i,whereEmeans that the firm enters the market and Nthat it stays out and let xibe a real-valued, finite-dimensional vector of exogenous profit shifters (covariates) that affect firm i’s profit and are observed by the other firm and the researcher. We assume that there is a large population of markets, with each market consisting of a Firm 1 and a Firm 2 that make their entry decisions simultaneously. The designation of a player as Firm 1 or Firm 2 is made by the researcher and based on observable characteristics; for example, in Kline and Tamer (2016), one firm is the “Low-Cost Carrier” and the other firm is “Other Airlines” (see Section 5). There is a finite set of realized profit shifters, which we denote by  X. For each (x1,x2)∈ X, we suppose that the population distribution of joint action profiles P(·|x1,x2)is known to the researcher. We denote this set of distributions by P={P(·|(x1,x2))}(x1,x2)∈ X.Table1gives an example of Pwhere there is only variation in x2and it takes three possible vector values; for example, the box on the left tells us that P((E,N)|x2=(0, 0)) =4/12. We are interested in developing a procedure, which allows us to identify those Pthat are compatible with our model of firm entry. Of course, in any empirical analysis these characterizing conditions on P would have to be statistically tested on an actual data set with sampling variation (as we explain in detail in Section 4). Confining our discussion to Pat this stage allows us to focus on the more distinctive aspects of our analysis. We now describe the model, which (potentially) generates P.Wedenotethepayoff/profit of Firms 1 and 2 by 1(y1,y2,x1)and 2(y1,y2,x2), respectively. We postulate that entry decisions are generated as pure strategy Nash equilibria (PSNE) of an entry game between Firms 1 and 2. We allow for multiple PSNE and impose no restriction on 240 Lazzati, Quah, and Shirai Quantitative Economics 16 (2025) how firms select among these equilibria. There remains unobserved market heterogeneity even after conditioning on (x1,x2); this heterogeneity is captured by a joint distribution on (1,2), which in turn leads to a distribution over joint actions P(·|x1,x2).We assume that there is conditional independence, in the sense that the distribution over (1,2)does not vary with the realized value of (x1,x2). Lastly, we postulate that the firms’ profit functions satisfy single-crossing restrictions (see Milgrom and Shannon (1994)). In this context, it means that Firm 1’s entry into the market is encouraged when the profit shifter x1takes higher values and is discouraged when Firm 2 chooses to enter. Formally, we require 1E,y 2,x 1> 1N,y 2,x 1=⇒ 1E,y 2,x 1> 1N,y 2,x 1(1) whenever x 1≥x 1and either y 2=y 2or y 2=Eand y 2=N. (A similar requirement is imposed on 2.) For example, in Ciliberto and Tamer (2009), 1(y1,y2,x1)=α 1x1+δ11y2+ε1if y1=E, 0ify1=N,(2) where 1E=1and1N=0. In this specification, the entry of Firm 2 alters the profit of Firm 1byδ1and unobserved heterogeneity in payoff functions is captured by ε1,whichenters the profit function additively. It is straightforward to check that our single-crossing restrictions are satisfied if δ1<0andα1>0. Note, however, that the converse is not true, that is, there are distributions over payoff functions satisfying (1)thatcannot be represented in the additive form given by (2), for any distribution on ε1. We say that Pis consistent with the single-crossing model, or SC-rationalizable,if there is a joint distribution of payoff functions (1,2)that satisfy our single-crossing conditions (1) such that the resulting distribution of PSNE (given some equilibrium selection rule) coincides with P(·|x1,x2)for each x∈ X. We would like to answer the following question: What conditions on Pcharacterize SC-rationalizability? In other words, when presented with P, how could we check if it is SC-rationalizable? We first observe that our model does have structural implications for P. Suppose the observable profit shifters weakly increase entry-by-entry from (x 1,x 2)to (x 1,x 2);4then, at any particular realization 1of Firm 1’s payoff function, if it prefers to enter when the other firm enters at (x 1,x 2), then the single-crossing condition guarantees that it will continue to prefer entry at (x 1,x 2). The same argument applies to Firm 2, and so we conclude that if (E,E)is the Nash equilibrium at (x 1,x 2)for a given realized profit function profile (1,2), then it will be the unique Nash equilibrium at (x 1,x 2)for this realized profile. Aggregating across all profiles, we establish that P(E,E)|x 1,x 2≥P(E,E)|x 1,x 2, provided conditional independence holds. This inequality constitutes a restriction on P but it is not the only restriction imposed by our model. We now sketch out the procedure 4Formally, (x 1,x 2)is weakly higher than (x 1,x 2)in the product order (see footnote 8for its formal definition). Quantitative Economics 16 (2025) An ordinal approach to the empirical analysis 241 Table 2. Distribution of types rationalizing the choice distributions in table 1. x2=(0, 0)x2=(0, 1)x2=(1, 0) Action Profiles Action Profiles Action Profiles Type Weight N,NN,EE,NE,EN,NN,EE,NE,EN,NN,EE,NE,E 1 1/12 1/12 1/12 1/12 2 2/12 2/12 2/12 2/12 3 2/12 2/12 2/12 2/12 4 1/12 1/12 1/12 1/12 5 1/12 1/12 1/12 1/12 6 2/12 2/12 2/12 2/12 7 3/12 3/12 3/12 3/12 Sum 1 3/12 3/12 4/12 2/12 1/12 5/12 3/12 3/12 2/12 4/12 2/12 4/12 for systematically checking whether Pis SC-rationalizable, using Ppresented in Table 1 as an example. Given a particular realization (1,2), the firms will choose an action profile (either (E,E),(E,N),(N,E),or(N,N)) at each realization of x2,andasx2takes different values the action profile of the two firms may change. We shall refer to the map from x2to the action profile as a group type. Notice that even though firms’ profit functions may be heterogenous in infinitely many ways, its manifestation in behavior must be finite, since there are only finitely many possible actions and the realized covariates (x1,x2) take values in the finite set  X. To be precise, there are in total 43=64 group types, but not all are consistent with PSNE play and single-crossing payoff functions. For example, as we have already explained, a group type where (E,E)is played at x2=(0, 0)and (N,N)at x2=(0, 1)is not compatible with single-crossing. On the other hand, it is quite clear a group type where (N,E)is played at all three values of x2can be justified with single-crossing profit functions. Ascertaining if Pcan be rationalized involves a two-step procedure. First, we must identify all single-crossing group types, in the sense that the action profile (y1,y2)at each value of (x1,x2)could be generated as PSNE from payoff functions satisfying (1). This is do-able because we show in Section 3that these group types are characterized by an easy-to-check condition called the revealed monotonicity axiom.Second,wehave to check whether there are weights on these group types that could account for the observed distribution of action profiles; this involves solving a system of linear inequalities. We claim that Pdepicted in Table 1can be rationalized. To understand why, we list in Table 2seven possible group types. One could check that each of these group types is consistent with the single-crossing property. When these types are represented in the 242 Lazzati, Quah, and Shirai Quantitative Economics 16 (2025) population with the weights indicated in Table 2, they generate the distribution of entry decisions observed in Table 1. (Compare the entries in Table 1with the last row of Table 2.) Lastly, we point out that while Pis SC-rationalizable, it is not compatible with a model where profit functions have the form (2), so the latter specification does involve a loss of generality. Indeed, with this specification, Firm 2’s profit upon entry is π2(E,y1,x21,x22,ε2)=α21x21 +α22x22 +δ211y1+ε2,(3) where (α21,α22)>0andδ21 <0.5Whether the boost to profits of an increase in x21 is greater or smaller than that obtained from the same increase in x22 depends on whether α21 is bigger or smaller than α22 and is independent of the realization of ε2. So, it excludes the case where the realization of ε2influences the relative benefit of higher x21 versus higher x22. To see why this parametric model cannot explain the choice distributions in Table 1, suppose instead that it does. Then P(E,E)|x1,(1, 0)−P(E,E)|x1,(0, 0) =με1:π1(E,E,x1,ε1)≥0×{ε2:−δ21 ≥ε2≥−α21 −δ21}, where μis the probability measure on the space of (ε1,ε2); similarly, P(E,E)|x1,(0, 1)−P(E,E)|x1,(0, 0) =με1:π1(E,E,x1,ε1)≥0×{ε2:−δ21 ≥ε2≥−α22 −δ21}. Since the former equals 2/12 while the latter equals 1/12, we conclude that α22 <α 21. However, 1 12 =P(N,N)|x1,(0, 0)−P(N,N)|x1,(1, 0) =με1:π1(E,N,x1,ε1)≤0×{ε2:0≥ε2≥−α21} and 2 12 =P(N,N)|x1,(0, 0)−P(N,N)|x1,(0, 1) =με1:π1(E,N,x1,ε1)≤0×{ε2:0≥ε2≥−α22}, which tells us that α22 >α 21. So, we obtain a contradiction. In Supplementary Appendix A1, we provide a more elaborate discussion of the contrast between the observable restrictions imposed by a linear parametric model and our (more general) nonparametric model. In particular, using simulations based on an extended version of the above example, we show that the difference between the two models is also picked up at the sample level: the method of Kline and Tamer (2016)(correctly) finds that the data are inconsistent with the linear model, whereas our method (also correctly) finds that the data are consistent with the more general model. 5We are grateful to Aureo De Paula for suggesting that we construct an example with this specific feature. Quantitative Economics 16 (2025) An ordinal approach to the empirical analysis 249 SC-rationalizable boils down to finding a positive solution to a set of equations linear in the unknowns τBfor all B ∈B.15 Remark 4. It is part of the definition of SC-rationalizability that the distribution of = (i)i∈Nis independent of x. Suppose we drop this condition but still require all payoff functions to consist of single-crossing functions; then it is easy to see that the payoff functions and equilibrium selection rules will induce a distribution over group types in Bat each x,whichwemaydenoteby(τB x)B∈B, such that the following counterpart of (7) holds: P(y|x)= {B∈B:B(x)=y} τB xfor all y∈Yand x∈ X.(8) This condition is trivially true in the sense that one could always find (τB x)B∈Bsuch that it holds. Conditional independence imposes the additional requirement that τB x=τB x for any x,x ∈ X, and this condition in combination with (8)isobviouslyequivalent to (7). One could imagine situations where the modeler has different views of how the distribution of (and hence the distribution of the associated group types) varies with x, which may be more permissive than or different from conditional independence; these could be incorporated as further conditions on τB xthat could be tested in combination with (8). Obviously, such a test will remain a linear test if the added conditions are linear in τB x. 3.4 Recovering properties of a rationalizing distribution P When Pis SC-rationalizable, we are also able to extract information about this rationalization through the properties of (τB)B∈Bthat solve (7). In particular, let SC∗beasubset of single-crossing payoff functions (including all of its strictly increasing transformations) and let B∗=B∈B:thereis∈SC∗that rationalizes B.(9) By a straightforward adaptation of the proof of Theorem 2(see the Supplemental Appendix), we can show that max B∈B∗ τB:τBB∈Bsolves (7)=max∈SC∗dP:P rationalizes P(10) Notice that the left-hand side of this equation is straightforward to compute when B∗ and Bare known, since it simply involves solving a linear program. Thus we can find the greatest possible weight on a given set of payoff profiles, for any distribution that 15In some problems, it may not be computationally feasible to find all the elements of B,butinthose cases, one could still test for SC-rationalizability by progressively enlarging the set of single-crossing types (see Section 4.1). 250 Lazzati, Quah, and Shirai Quantitative Economics 16 (2025) rationalizes P.16 We give two cases where this exercise is useful, both of which are empirically implemented in Section 5. Other examples can be found in the Supplementary Appendix A3. Application 1. Bounds on the role of strategic interaction While our model allows for the possibility that each player reacts strategically to other players in the game, it is conceivable that the conditional choice distributions could be explained more simply, without appealing to strategic effects for one or more players in the game. To be specific, suppose we wish to check whether it is possible to regard a subgroup Nof the players as nonstrategic. Let SC∗be the payoff profiles in SC such that idoes not depend on y−ifor every i∈Nand let B∗be its corresponding set of group types (as defined by (9)). The types in B∗can be characterized by a stricter version of the RM axiom: a group type is in B∗if and only if it obeys the RM axiom and, for each i∈N,we require that y ∈B(x),y∈B(x),andx i≥x i=⇒ y i≥y i. With this characterization, we can construct B∗. If we find that max B∈B∗ τB:τBB∈Bsolves (7)=1, we conclude (by (10)) that Pcan be SC-rationalized without requiring the players in N to be strategic; on the other hand, if the upper bound is strictly below 1, then we must incorporate strategic interactions among these players to SC-rationalize P. Application 2. Probability bounds for Nash equilibrium profiles Given a strategy profile yand covariate x, we pose the following question: Among all the possible SC-rationalizations of P, what is the greatest fraction of groups, which could have yas a pure strategy Nash equilibrium at x?Here,x∈Xmay or may not be an element of  X,andwhenx/∈ X, the answer to this question provides information on how the game would be played at an hitherto unobserved covariate value. However, the question is interesting even when x∈ X. To see why, notice that there is a distinction between P(y|x), the observed fraction of groups in the population that play yat x, and the fraction of groups for which yis a Nash equilibrium. The former is typically smaller than the latter because groups might have multiple Nash equilibria. Thus some groups who play strategy profiles other than y may also have yas a Nash equilibrium.17 The distinction between P(y|x)and the greatest possible weight on those groups, which have yas a Nash equilibrium at x=xis relevant, 16To obtain min{∈SC∗dP:P rationalizes P}, we use the similarly easy-to-prove identity min B∈B0 τB:τBB∈Bsolves (7)=min∈SC∗dP:P rationalizes P, where B0={B∈B: B can only be rationalized by ∈SC∗}. 17In our empirical application of an entry game with two firms, if (E,E)or (N,N)is played by a pair of firms, then it has to be their unique equilibrium, but any pair that plays (E,N)may also have (N,E)as another (albeit unselected) equilibrium. Thus if P(E,N|x)and P(N,E|x)are the observed probabilities of action profiles (E,N)and (N,E), respectively, then the probability that (E,N)(similarly, (N,E))isaNash equilibrium profile at x=xis no greater than P(E,N|x)+P(N,E|x). Quantitative Economics 16 (2025) An ordinal approach to the empirical analysis 251 because if the gap is small, then we are sure that changing the equilibrium selection scheme cannot significantly increase the frequency with which yis played. This means (e.g.) that a policymaker who wants yto be played more often must alter payoffs in some way and it is not possible to simply convince players to coordinate on a different equilibrium. An earlier analysis of questions of this type can be found in Aradillas-Lopez (2011), which focuses on a different class of games. To answer our question, let SC∗={∈SC :y∈NE(,x)}and let B∗be its corresponding set of group types. We can check whether B belongs to B∗by using the RM axiom. Indeed B ∈B∗if and only if the (possibly) multivalued group type Bdefined as follows obeys the RM-axiom: B(x)={B(x),y}and B(x)=B(x)for every x∈ X\{x}. The proportion of the population which has yas a PSNE cannot exceed max{B∈B∗τB: (τB)B∈Bsolves (7)}and can equal this number.18 4. The statistical procedure This section outlines the statistical procedure that implements the results in the previous section, which are based on population distributions. The test of SC-rationalizability is explained in Section 4.1 and relies on the statistical hypothesis testing proposed by Kitamura and Stoye (2018). The efficient implementation of this test when Bis large (and cannot be fully listed) uses the column generation approach proposed in Smeulders, Cherchye, and De Rock (2021). Section 4.2 outlines the procedure (in essence provided by Deb et al. (2023)) to obtain confidence intervals for the weights on certain group types; the efficient implementation of this procedure requires a nontrivial extension of the column generation method in Smeulders, Cherchye, and De Rock (2021)andwe provide this in Proposition 3. 4.1 Statistical hypothesis testing We begin with a matrix reformulation of the characterization given in Theorem 2.Each generalized group type B :  X⇒Ycan be represented as a vector b=(by,x)Y× Xsuch that by,x=1ify∈B(x)and by,x=0 otherwise. Conversely, for any b∈{0, 1}|Y× X|corresponds to a generalized group type, with a vector b∈{0, 1}|Y× X|representing a single-valued group type if and only if y∈Yby,x=1ateveryx∈ X. Similarly, since Pconsists of | X| distributions on Y, it can be captured by the column vector p∈[0, 1]|Y× X|,wherethe (y,x)-th entry of pis P(y|x)(and hence, y∈Ypy,x=1 for each x∈ X). In what follows, we shall abuse notation and use Bto denote both the set of group types obeying the RM axiom and also the vectors corresponding to those types. We denote by Bthe matrix where each column represents a group type in B.Theorem2states that Pis SC-rationalizable if and only if there is τ∈B, the set of distributions on B,that 18Our analysis here gives the most optimistic estimate on the possibility of switching the equilibrium action to y, in the sense that it assumes that every group type, which can be rationalized by an element in SC∗, actually does have a payoff function profile in SC∗. We could also find the most conservative estimate of the proportion of the population that could switch to yby changing equilibrium selection rules; this is explained in Supplementary Appendix A6.2. 252 Lazzati, Quah, and Shirai Quantitative Economics 16 (2025) solves Bτ=p.(Bcould be thought of as elements of the standard (|B|−1)-simplex.) We would like to test if the data is consistent with the SC-rationalizability of P.Equivalently, letting PSC ={Bτ:τ∈B}(i.e., the set of SC-rationalizable distributions in vector form), our null hypothesis is min η∈PSC(p−η)·(p−η)=0. (11) The data set consists of Nxobservations of the action profiles at each realization of x∈ X. We assume that Nx/N →ρx∈(0, 1)at each x∈ X,asN=x∈ XNx→∞.We denote the empirical distribution over action profiles by Q=Q(·|x):x∈ X, and we estimate Pby this sample analog. As in the case with P, we can represent Qby a column vector q∈[0,1]|Y×X|where the (y,x)-th entry is equal to Q(y|x). The testing procedure by Kitamura and Stoye (2018) depends on the simple, but important observation that Bτ=pholds for some τ∈B, if and only if Bτ=pholds for some τ≥0 (Theorem 3.1 in their paper). Thus, by letting A={Bτ:τ≥0}, the null hypothesis is equivalent to whether plives in this convex cone, that is, min η∈A(p−η)·(p−η)=0. (12) Given this, following Kitamura and Stoye (2018), we adopt the test statistic JN:=min η∈AN(q−η)·(q−η)=min τ∈R|B| + N(q−Bτ)·(q−Bτ). (13) Calculating the critical value. Note that we cannot simply adopt a solution to the problem (13) as the bootstrap estimator for the empirical choice distribution, due to the possible discontinuity of the limiting distribution of JN. Addressing this issue involves introducing a tuning parameter and considering the corresponding tightened problem. We follow the procedure by Smeulders, Cherchye, and De Rock (2021), which is a modification of the one in Kitamura and Stoye (2018). Choose B⊂Bso that it contains a basis of the space spanned by B, and define TκN= {τ∈R|B| +:τb≥κN/|B|for all b∈B},withκNbeing selected so that κN↓0and√NκN↑ ∞as N→∞. (See Supplementary Appendix A5, for the procedure for constructing B and a detailed justification of our procedure.19) Letting AκN={Bτ:τ∈TB κN},weadopt η∗=argmin η∈AκN N(q−η)·(q−η)=argmin τ∈TB κN N(q−Bτ)·(q−Bτ). (14) 19In the original formulation by Kitamura and Stoye (2018), positive weights are required on all elements in B, which is inconvenient when applying the column generation procedure described later in this subsection. The modification of that approach by Smeulders, Cherchye, and De Rock (2021) (which we are using here) requires positive weights only for the types in B. Quantitative Economics 16 (2025) An ordinal approach to the empirical analysis 253 as the bootstrap estimator of the empirical choice frequency. Compared to the problem (13), the feasible set in the minimization problem is tightened by the tuning parameter, with positive weights required for the elements in B. We then generate a bootstrap sample q(r)(for r=1, 2, ,R) using standard nonparametric bootstrap resampling from η∗ and recenter this sample by setting q(r):=(q(r)−q)+η∗.With q(r), we can calculate the bootstrap test statistic J(r) N:=min η∈AκN N q(r)−η· q(r)−η=min τ∈TB κN N q(r)−Bτ· q(r)−Bτ, (15) and the empirical distribution of J(r) Nallows us to obtain the p-value p=#{J(r) N>J N}/R. The null hypothesis that qis a sample from some p∈A(equivalently, p∈PSC)isnot rejected if the p-value is greater than the critical value. Column Generation. A major hurdle in implementing the above test is that the computation of JNand J(r) Ninvolves B, which is often too large to be listed in its entirety. We cope with this problem by applying the column generation procedure in Smeulders, Cherchye, and De Rock (2021). This procedure involves first testing a more stringent version of the model corresponding to a strict subset B0of B, which is completely known. For instance, we may choose the “starter” set B0to be the set of constant types,inwhich every player takes the same action regardless of opponents’ actions and covariates; these group types obviously obey the RM axiom. Then the set B0is progressively enlarged by including more group types from B, up to the point where further additions will not improve the model’s ability to explain the data. To be precise, let B0bethematrixwherethecolumnsareelementsofB0.Wecan calculate JN,0 :=min τ∈R|B0| + N(q−B0τ)·(q−B0τ). (16) Obviously, JN,0 ≥JNand we could check if it is possible to decrease JN,0 by including some b∈B. We say that b∈Bimproves B0if, when bis included in B0,thenewvalue of JN,0 is strictly lower. The following result, which follows from the convex projection theorem, provides a necessary and sufficient condition for B0to be improvable. Proposition 1. A set of group types B0is improved by some b∈B,if and only if max b∈B(q−η0)·(b−η0)>0, (17) where η0=B0τ0and τ0=argminτ∈R|B0| + (q−B0τ)·(q−B0τ). To solve problem (17) without fully enumerating B, we must find a computationally efficient way to characterize B. Conveniently for us, the RM axiom—and hence the set B—can be characterized as solutions to an integer linear programming problem. Proposition 2. We can construct a matrix Cand a column vector θ,both with nonnegative integer entries,such that for any b∈{0, 1}|Y× X|,we have b∈Bif and only if Cb≤θ. 254 Lazzati, Quah, and Shirai Quantitative Economics 16 (2025) The formulae for Cand θare found in our proof of this proposition in the Supplemetnal Appendix. Combining this result with Proposition 1,B0is improved by some b∈B, if and only if max(q−η0)·(b−η0),subjecttob∈{0, 1}|Y× X|and Cb≤θ, (18) is strictly positive. If it is, we add this bto B0and then repeat the process. In other words, we recalculate JN,0 and η0based on the new B0, and try to find another element in B that improves B0by checking if (18) has a strictly positive solution. Since Bis finite, this algorithm must terminate, and at the end we can be sure we have found B0such that JN,0 =JN. The column generation procedure described above can be also applied to the computation of J(r) Ndefined by (15). Since the constraint in problem (15) requires positive weights on B, this set needs to be contained in the initial choice of B0.20 Summary. Below is a step-by-step summary of the test procedure.21 I Obtain the test statistic JNdefined in (13) as follows: (i) Based on B0, solve the minimization problem (16)togetJN,0 and η0. (ii) Check the value of (18). Ifit is strictly positive, then update B0by adding a solution of (18) and go to (i). Else,setJN=JN,0 and Stop. II Obtain the tightened estimator η∗in (14) as follows: (i) Obtain B⊂Busing the procedure explained in Supplementary Appendix A5.2. (ii) Set Bas B0, and run the procedure in Step I replacing R|B0| +in problem (16) with TB0 κN={τ∈R|B0| +:τb≥κN/|B|for b∈B}. When it stops, the resulting η0 is η∗. III Obtain the bootstrap test statistics J(r) Ndefined in (15)forr=1, 2, ,R: (i) Obtain the recentered bootstrap sample  q(r)=(q(r)−q)+η∗. (ii) Set Bas B0, and run the procedure in Step I replacing qand R|B0| +in problem (16)with q(r)and TB0 κN, respectively. When it stops, adopt the resulting JN,0 as J(r) N. IV Lastly, calculate the p-value p=#{J(r) N>J N}/R. 20Note that, even if B0contains B, and Bcontains a linear basis of B,theconical hull of B0need not coincide with the conical hull of B(though the linear hull of B0of course coincides with the linear hull of B). So, it is still possible for B0to be improvable. 21The Supplementary Appendix (Section A5.2) provides a couple of shortcuts that improves the computation time. Quantitative Economics 16 (2025) An ordinal approach to the empirical analysis 255 4.2 Inference on types Suppose we have a data set that is consistent with SC-rationalizability (in the sense that the null hypothesis (12) is not rejected) and would now like to form a confidence interval on b∈B∗τb, the total weight of a subset of single-crossing group types B∗(see Section 3.4). To do this, we follow the procedure in Deb et al. (2023). The problem of determining whether a given weight of B∗falls within the confidence interval can be determined by testing a suitably modified version of the null hypothesis (12), with PSC replaced by a different set of distributions. To be specific, suppose we would like to find the upper bound of the confidence interval. For each β∈(0, 1),welet PSCβ;B∗=Bτ:τ∈Band  b∈B∗ τb≥β, and test the null hypothesis min η∈PSC(β;B∗) (p−η)·(p−η)=0 (19) at some significance level ¯ p. We then use binary search to obtain the maximal value of βunder which the null hypothesis is not rejected; the resulting maximal value of β corresponds to the supremum of the 100(1−¯ p)% confidence interval of β. For a given β, the test statistic is22 JN(β):=min η∈PSC(β;B∗) N(q−η)·(q−η) =min τ∈BN(q−Bτ)·(q−Bτ)subject to  b∈B∗ τb≥β. (20) As in the preceding subsection, it may not be possible to fully enumerate Bor B∗,and so a version of the column generation procedure outlined there is needed. This in turn requires an extension of Proposition 1, which we now explain. Let B0⊂Bbe such that B0∩B∗=∅, and let us calculate JN,0(β)=min τ∈B0 N(q−B0τ)·(q−B0τ)s.t.  b∈(B0∩B∗) τb≥β, (21) where B0is the standard (|B0|−1)-simplex. We say that B0is improvable given problem (20),ifJN,0(β)>J N(β). The following proposition is the counterpart of Proposition 1 and provides a necessary and sufficient condition for a given B0to be improvable. Proposition 3. If the set B0⊂Bis improvable given problem (20), then there is a pair of types {b∗,b},with b∗∈B∗and b∈Bsuch that (q−η0)·βb∗+(1−β)b−η0>0, (22) 22As pointed out in Deb et al. (2023), unlike the case of the preceding subsection (see (13)), τmust be chosen from the simplex B, rather than the nonnegative orthant. 256 Lazzati, Quah, and Shirai Quantitative Economics 16 (2025) where η0=B0τ0with τ0being the distribution that achieves JN,0(β).Conversely,suppose there is b∗∈B∗and b∈Bsuch that (22)holds;then {b∗,b}improves B0given problem (20). We already know (from Proposition 2) that we can construct a matrix Cand a column vector θso that b∈Bif and only if Cb≤θ.Supposethat,inaddition,wecanconstruct amatrixC∗and a column vector θ∗with integer entries so that, for any b∈{0, 1}|Y× X|, we have b∗∈B∗⇐⇒ C∗b∗≤θ∗.Thenapair{b∗,b}obeying (22) exists, if and only if the problem max(q−η0)·βb∗+(1−β)b−η0(23) s.t. b,b∗∈{0, 1}|Y× X|and C∗O OC b∗ b≤θ∗ θ has a positive optimal value. Note that every B∗in our empirical application has a matrix characterization like the one described above (see Supplementary Appendix A4 for the specific construction). If there is a pair {b,b∗}that improves B0,thenweupdateB0by including the pair in B0and recalculate JN,0(β). Since Bis finite, this process terminates and we obtain JN,0(β)=JN(β). To find the valid critical value, we need a suitable tightening that imposes strictly positive weights on a certain subset of group types. The tightening here must depend on βand its formulation is rather involved, so we postpone this discussion to Supplementary Appendix A5.3. That said, once we have constructed a suitably tightened subset of Bby some tuning parameter κN, the rest of the procedure is similar to the one outlined in the preceding subsection. Denoting this subset by B κN(β;B∗)and letting PSC κNβ;B∗=Bτ:τ∈B κNβ;B∗and  b∈B∗ τb≥β, (24) the bootstrap estimator and the recentered bootstrap samples can be obtained as in (14)–(15), after replacing the sets AκNand TκNby PSC κN(β;B∗)and B κN(β;B∗), respectively. We can also implement here the column generation procedure we used before. The details of the procedure, including a step-by-step summary, are provided in Supplementary Appendix A5.3. 5. Empirical illustration We apply our results in the preceding sections to an entry game using a data set taken from Kline and Tamer (2016). The data set contains the entry decisions of airlines in 7882 markets, where a market is defined as a trip between two airports irrespective of intermediate stops. Airline firms are divided into two categories: LCC (low cost carriers) and OA (other airlines).23 In Kline and Tamer’s analysis (and in ours), the two categories 23The data were collected from the second quarter of the 2010 Airline Origin and Destination Survey (DB1B). The low cost carriers are AirTran, Allegiant Air, Frontier, JetBlue, Midwest Air, Southwest, Spirit, Sun Country, USA3000, and Virgin America. A firm that is not a low cost carrier is, by definition, an “other airline”. Quantitative Economics 16 (2025) An ordinal approach to the empirical analysis 257 are treated as two firms. Thus, in each market, the two firms, LCC and OA, can either both enter a market, both stay out, or one could enter with the other staying out. This data set also contains information on two covariates: market presence (MP) and market size (MS). Market presence is a marketand airline-specific variable. For each airline and for each airport, one counts the number of markets that the airline serves from that airport and divide it by the total number of markets served from that airport by any airline; the market presence variable for a given market and airline is the average of these ratios at endpoints of that market/trip. The construction and inclusion of this covariate is not novel and follows Berry (1992). Since the airlines are aggregated into two firms, the market presence variable is also aggregated: the market presence for LCC (resp., OA) is the maximum among the actual airlines in the LCC category (resp., OA category). The second covariate, market size, is a market-specific variable (shared by all airlines in that market) and is defined as the population at endpoints of the corresponding trip. Furthermore, Kline and Tamer (2016) discretize these variables, where each of them takes value 1 if the variable is higher than its median value and 0 otherwise. Thus, in our data set, there are three binary covariates, MPLCC ,MP OA, and MS, and markets are partitioned into eight groups according to realizations of them. Formally, X={0, 1}3,and in this case, it also holds that  X=X. Note that MS simultaneously influences the payoffs of both LCC and OA, and hence the covariates affecting LCC’s payoff can be written as xLCC =(MPLCC ,MS ),andsimilarly,xOA =(MPOA,MS ). Observations in the data set can be used to calculate the empirical choice distributions that we include in Table 3. It consists of eight blocks, with the markets in each block sharing the same covariates. For example, there are 1271 markets with (MPLCC ,MP OA,MS )=(0, 0, 0), of which around 30% are not served by either airline and about 68% are served only by airlines in the OA category (an action profile is written as (yLCC ,yOA)∈{E,N}×{E,N}). The entries in Table 3seem “reasonable,” in the sense that it appears as though a firm’s entry is encouraged whenever its market presence is large or the market size is large, and it is deterred by the entry of the other firm. For example, going from (0, 0, 0)to (1, 0, 0)(so the market presence of LCC has increased), both Q(N,N)and Q(N,E)fall, while Q(E,N)and Q(E,E)both increase. Testing SC-rationalizability. Our hypothesis is that, in each market, two firms (LCC and OA) are playing a pure strategy Nash equilibrium in a game of strategic substitutes with monotone effects from covariates. The payoff function of LCC, say, LCC(yLCC ,yOA,MP LCC ,MS ), is required to obey single-crossing differences in (yLCC;(−yOA,MP LCC ,MS )), and similarly, the payoff function of OA, OA(yOA,yLCC , MPOA,MS ), is required to obey single-crossing differences in (yOA;(−yLCC ,MP OA, MS)). This ensures that a firm’s entry is discouraged by the opponent’s entry and enhanced by an increase in own covariates. The data set is supposed to arise from a population of those firms, with unobserved heterogeneity generating a distribution of realizations of payoff functions =(LCC ,OA),whichwedenotebyP , and an equilibrium selection rule. Employing the statistical test in Section 4.1, we find a p-value of 0.138, and hence, the hypothesis that the empirical choice frequencies are explained by our modeling restric- 258 Lazzati, Quah, and Shirai Quantitative Economics 16 (2025) Table 3. Empirical distribution across each realization of covariates. (MPLCC ,MP OA,MS )=(0, 0, 0)1271 markets Q(N,N)Q(N,E)Q(E,N)Q(E,E) 0.304 0.682 0.006 0.009 (MPLCC ,MP OA,MS )=(0, 1, 0)763 markets Q(N,N)Q(N,E)Q(E,N)Q(E,E) 0.190 0.785 0.003 0.022 (MPLCC ,MP OA,MS )=(1, 0, 0)1125 markets Q(N,N)Q(N,E)Q(E,N)Q(E,E) 0.194 0.367 0.253 0.186 (MPLCC ,MP OA,MS )=(1, 1, 0)782 markets Q(N,N)Q(N,E)Q(E,N)Q(E,E) 0.122 0.542 0.050 0.286 (MPLCC ,MP OA,MS )=(0, 0, 1)869 markets Q(N,N)Q(N,E)Q(E,N)Q(E,E) 0.159 0.823 0.001 0.017 (MPLCC ,MP OA,MS )=(0, 1, 1)1039 markets Q(N,N)Q(N,E)Q(E,N)Q(E,E) 0.078 0.889 0.000 0.033 (MPLCC ,MP OA,MS )=(1, 0, 1)677 markets Q(N,N)Q(N,E)Q(E,N)Q(E,E) 0.106 0.326 0.306 0.261 (MPLCC ,MP OA,MS )=(1, 1, 1)1356 markets Q(N,N)Q(N,E)Q(E,N)Q(E,E) 0.055 0.501 0.021 0.423 tions cannot be refuted at 5% (or 10%) significance level. We choose the tuning parameter κN=10−3logNx/Nx,whereNx=minx∈ XNx, and the number of bootstrap samples as R=2000.24 Note that having a p-value strictly less than 1 means that JNdefined in (13) is strictly positive, that is, there is a strictly positive distance between our empirical distribution and PSC, the set of (exactly) SC-rationalizable distributions. Using our R code on a desktop computer with Apple M1 processor and 16 GB RAM, the p-value was calculated in less than 3 minutes. In this setting, the set X= Xhas exactly eight elements, and hence, the number of possible group types is 48≈65,000. In this small environment, it is in fact not difficult to check the RM axiom for each of these group types. Doing that, we find that only 482 types satisfy single-crossing (equivalently, satisfy the RM axiom). This gives a sense of the“empiricalbite”ofourtest:thedatasethastobeexplainedbyusingaverysmall fraction (less than 1%) of all possible group types. Significance of strategic interactions. Having established that the data set is (statistically) SC-rationalizable, we can now go on to explore its properties. In particular, we can assess the extent to which strategic interactions play a role in explaining the data, in the sense discussed in Section 3.4, by considering the subclasses of single-crossing group types that correspond to: (i) the LCC firm having a payoff function that is independent of the actions of OA; (ii) the OA firm having a payoff function that is independent of the actions of LCC; and (iii) both firms having payoff functions that are independent of the other firm’s action. Applying the procedure explained in Section 4.2, we find that the greatest possible weights on these three subclasses of single-crossing group types are (i) 0.923, (ii) 0.790, and (iii) 0.789 (within 5% significance level). 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