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Estimating Discrete Games of Complete Information: Bringing Logit Back in the Game
Econometric models of strategic interactions have become standard empirical tools in various fields of economics following recent theoretical and computational developments \citep*{ellickson_structural_2011, de2013econometric, ho2017partial, aradillas-lopez_econometrics_2020, kline2021moment}. However, researchers face challenges in estimating discrete games of complete information due to model incompleteness that arises with multiple equilibria tamer_incomplete_2003. Existing econometric methods remain computationally difficult because they require repeated equilibrium computation, Monte Carlo simulation, and/or a grid search over the parameter space.\footnote{For instance, \citet*{ciliberto2021superstar}, in studying the entry decisions of superstar exporters using the popular ciliberto2009marketstructure algorithm, reports that their computation took a week to run.} Researchers in the partial identification literature have identified computational barriers as the main force preventing wider adoption of moment inequality models \citep*{kline2021moment, canay2023user}.\footnote{Computational concerns have led researchers to use non-sharp outer sets in various applications with partially identified models; see, e.g., \citet*{blundell2007changes}, dickstein2018exporters, honore2020selection, sheng2020structural, \citet*{ciliberto2021marketstructure}, gualdani2021econometric, and berry2023instrumental, among many others. More broadly, computational tractability and numerical stability have been viewed as a key challenge in a wide range of topics in the applied structural econometrics literature. For example, recent contributions on reducing the computational burden for demand estimation include \citet*{dube2012improving}, \citet*{lee2015computationally}, \citet*{salanie2019fast}, \citet*{conlon2020best}, \citet*{conlon2023incorporating}, and \citet*{canay2023user}.}
In this paper, I develop simple, flexible, and scalable approaches for estimating a large class of finite static discrete games of complete information that covers most empirical models considered in the literature. I separately consider both unordered and ordered actions.\footnote{I also show my approach applies to models with vector-valued actions (e.g., fan2025estimating).} I assume pure strategy Nash equilibrium but make no assumption on the equilibrium selection rule.\footnote{I do not consider mixed strategy equilibria. While my approach does not extend to them, it can still help narrow the identified set when mixed strategies are allowed in the data-generating process.} I also allow for market-level unobserved heterogeneity. In summary, I construct a set of generalized likelihood-based conditional moment inequalities that are convex in (a subvector of) structural model parameters under standard econometric assumptions and use convex programming tools to estimate the identified set. My approach removes the need for extensive equilibrium enumeration, Monte Carlo simulation, and grid search.
My strategy relies on finding an identified set characterized by easy-to-handle moment inequalities. I build on galichon2011setidentification, which characterizes the sharp identified set using a finite number of conditional moment inequalities that require observed conditional probabilities of events not to contradict the generalized likelihood of the events.\footnote{The generalized likelihood of an event represents the maximal probability of the event the model can admit. In other words, the generalized likelihood computes the probability of an event when the underlying equilibrium selection rule is set to maximize its probability. } From there, I choose a subset of the sharp identifying inequalities and define an outer set characterized by conditional moment inequalities of the form
where $\theta$ is the structural model parameter vector of interest. I show, separately for unordered and ordered-actions cases, there is an appropriate choice of the index set $\mathcal{J}$ that renders each $f_j$ in (ref) convex in a subvector $\gamma$ of $\theta = (\gamma, \sigma)$ under the standard logit assumption on unobservables. The convexity in a subvector of parameters allows me to leverage the numerical stability and scalability of convex programs in estimating the parameters.
Construction of moment inequalities that are convex in $\gamma$ relies on finding conditions under which the corresponding generalized likelihood functions are log-concave in $\gamma$. Thus, my strategy is to find conditions that restore the log-concavity of likelihood functions present in single-agent point-identification analogs. When actions are unordered, I assume Type-1 extreme value distribution for players' idiosyncratic payoff shocks and use the same set of conditional moment inequalities used in \citet*{andrews_confidence_2004}.\footnote{Although I use the same set of constraints, my derivation of computationally tractable closed-form conditional moment inequalities based on the multinomial logit assumption is new.} When actions are ordered, I assume that players' payoff shocks follow the standard logistic distribution. I then select a subset of identifying inequalities such that the associated generalized likelihood functions are easy-to-compute probabilities over hyperrectangles and log-concave under the standard logistic distribution assumption. In both cases, I leverage the linear payoff assumption and the log-concavity arising from the logit distributional assumptions. The idea extends to games with vector-valued decisions such as those in fan2025estimating.
My framework accommodates market-level unobserved heterogeneity that affects all players' payoffs. In a typical application, $\gamma$ is the parameter vector that controls the deterministic part of players' payoff, and $\sigma$ is a low-dimensional parameter that controls the variance of the unobserved heterogeneity. When there are no common unobservables, each $f_j$ is convex in the entire vector $\theta$, so (ref) characterizes a convex identified set. When there are common unobservables, I use Pr\'ekopa's theorem to establish the convexity of each $f_j$ with respect to a subvector $\gamma$. I then use convex programs to estimate the identified set quickly at each candidate $\sigma$ and take their union over $\sigma$.
I also propose a simple strategy for constructing confidence sets for the identified set. I follow horowitz2022inference and koh2023stable and control for sampling uncertainty by constructing simultaneous confidence intervals for the conditional choice probabilities. I can nest the simultaneous confidence intervals in the optimization problems without disturbing the convexity to allow for quick computation of confidence sets.
My identified sets are non-sharp because I lose some identifying information by dropping non-convex moment inequalities.\footnote{Yet, selecting a subset of moment inequalities need not lead to a loss in identifying power since some identifying inequalities may be redundant galichon2011setidentification.} However, even if the researcher is interested in obtaining the tightest set possible, narrowing the search space to an easy-to-compute outer set can lead to significant savings in computational costs. Furthermore, it is straightforward to sharpen the identified set further using existing methods such as \citet*{beresteanu_sharp_2011, galichon2011setidentification, henry2015combinatorial, chesher2020econometric, koh2023stable, luo2024selecting}.\footnote{\citet*{li2024discordant} caution against using non-sharp identified sets, as outer sets can be misleading when the model is misspecified. Since obtaining sharp identified sets for discrete games is well understood, the outer sets developed in this paper can be further refined into sharp sets. My methodology streamlines the estimation of sharp-identified sets by substantially reducing the overall computational burden, thereby complementing their perspective.} Focusing on a subset of moment inequalities to increase tractability has been a popular approach for empirical works \citep*{ciliberto2009marketstructure, pakes_alternative_2010, nosko2010competition, eizenberg2014upstream, pakes_moment_2015, wollmann2018trucks, ciliberto2021marketstructure, aradillas2022inference, canay2023user, fan2025estimating}. Using numerical and empirical examples, I show that my identified sets are not only easy to compute but also tight in practice.
I consider two empirical examples to illustrate the usefulness of my framework. In the first empirical application, I consider the binary entry game between Walmart and Kmart. I follow ellickson_structural_2011 and use a simplified version of jia_what_2008's model. I apply my methodology for unordered actions. I compare my outer set to ellickson_structural_2011's estimates (which rely on equilibrium selection assumptions) and the sharp set. My identified set is close to those obtained under alternative approaches and delivers informative bounds. Moreover, my approach takes only 9.31 seconds to obtain the projection intervals of the 9-dimensional identified set without parallelization, which is approximately 5,000 times faster than galichon2011setidentification's approach.\footnote{The gap in computational time is very conservative. It can be several orders of magnitude larger if the researcher increases the number of simulation draws and the number of candidate parameter points to explore in the grid search algorithm.}
In the second empirical application, I study McDonald's, Burger King, and Wendy's strategic entry decisions under an ordered-response game framework, assuming the chains can open up to two outlets in each market. My identified set produces informative bounds and takes less than an hour to compute the projection intervals of the 14-dimensional identified set without parallelization. My results suggest that McDonald's has a higher baseline profit, is less influenced by network economies, and has a higher degree of concavity in its payoff function relative to other chains. Overall, my simulation and empirical examples suggest that my outer sets are easy to compute and informative.
This paper contributes to three strands of literature. First, it adds to a body of research that develops econometric methodologies for estimating discrete games of complete information with weak assumptions on equilibrium selection rules \citep*{ciliberto2009marketstructure, bajari2010identification, galichon2011setidentification, beresteanu_sharp_2011, henry2015combinatorial, pakes_moment_2015, kline_bayesian_2016, aradillas2022inference, koh2023stable, fan2025estimating}.\footnote{My econometric model follows the “generalized discrete choice” framework, which makes distributional assumptions on unobservables ciliberto2009marketstructure. An alternative is the “profit inequality” framework of pakes_alternative_2010, which avoids distributional assumptions on structural errors but typically produces outer sets as the framework does not focus on the sharpness question. The two frameworks take fundamentally different approaches. Still, the profit inequality approach is also generally computationally burdensome. See pakes_alternative_2010, pakes_moment_2015, kline2021moment, canay2023user.} For most existing algorithms, the computational burden can be prohibitive with large games. A notable exception is fan2025estimating, which introduces a scalable estimation algorithm estimating an outer set when firms simultaneously make a vector of binary actions.\footnote{fan2025estimating's algorithm removes the need for repeated equilibrium enumeration and Monte Carlo simulation. Yet, their algorithm still requires a grid search over the parameter space and is limited to a certain class of games. My framework also removes the need for such grid search and extends to fan2025estimating's setting.} This paper also relates to gowrisankaran2024computable, which develops a computationally tractable estimation algorithm for dynamic games with many discrete ordered investment choices by leveraging the discrete concavity structure.
My paper complements the above works by introducing simple and scalable algorithms for estimating a large class of discrete games. The common understanding in the literature has been that the standard linear payoff and logit assumptions do not necessarily lower the estimation cost because existing estimation methods do not optimize smooth functions. My framework provides a novel way to leverage the assumptions to restore the log-concavity of generalized likelihood functions for scalable estimation. In particular, the logit distributional assumption has been common under incomplete information game settings seim2006empirical, aguirregabiria2007sequential, pesendorfer2008asymptotic, bajari2010estimating, vitorino2012empirical, xiao2018identification but less so under complete information game settings because the simultaneity feature of Nash equilibrium disrupts the researcher from obtaining closed-form logit expressions.\footnote{bajari2010identification use the multinomial logit formula to model the equilibrium selection probabilities, but their method still requires Monte Carlo integration over high-dimensional latent variables and repeated enumeration over all Nash equilibria.} Therefore, this paper challenges the conventional wisdom that logit has no bite in complete information discrete game settings.
This paper also relates to the literature on conditional moment inequalities andrews2013inference, chernozhukov2013intersection, lee2013testing, lee2018testing, armstrong2015asymptotically and works developing econometric methods that can leverage the linearity or convexity of the moment inequalities to facilitate computation beresteanu2008asymptotic, kaido2014asymptotically, gafarov2019simple, andrews2023inference, cho2024simple. Instead of relying on generic algorithms for moment inequality models, I exploit the structure of the pure Nash equilibrium to construct convex moment inequalities. To the best of my knowledge, leveraging the computational tractability of the logit assumption for partially identified models is new.
Finally, this paper relates to empirical works on retail chains' strategic entry decisions. My empirical results complement the existing works that use alternative game-theoretic assumptions by revealing how competitive pressure compares with cannibalization concerns toivanen2005market, yang2012burger, gayle2015choosing, igami2016unobserved, aguirregabiria2020identification, yang2020learning, koh2023stable.\footnote{toivanen2005market uses bresnahan_entry_1991's model. aguirregabiria2020identification uses toivanen2005market's data to study the dynamic entry game of McDonald's and Burger King, where each firm can have multiple stores in the market. However, they use a dynamic discrete game model with incomplete information, and the entry decision is binary in each stage.} Furthermore, this paper also relates to recent papers on retail firms' strategic entry decisions when network economies are present jia_what_2008, holmes2011diffusion, ellickson2013estimating, nishida2015estimating, aradillas2022inference.
The rest of the paper is organized as follows. Section (ref) introduces the econometric problem. Section (ref) discusses the representation of generalized likelihood functions. Section (ref) considers games with unordered actions. Section (ref) considers ordered actions. Section (ref) applies my approach to an entry game between Walmart and Kmart. Section (ref) studies an empirical application to the chain entry game by McDonald's, Burger King, and Wendy's. Section (ref) concludes. All proofs are in Appendix (ref).
Before delving into the details of discrete games assumptions, I review galichon2011setidentification's characterization of the sharp identified set with a finite number of inequalities. I also overview how my framework gains computational advantages relative to existing approaches.
\paragraph{Incomplete Econometric Model} An incomplete econometric model is a tuple
where $\mathcal{Y}$ is the finite set of feasible endogenous outcomes; $\Omega$ is the set of exogenous states; $F$ is the distribution over $\Omega$; $\Theta$ is the parameter space; the correspondence $G_\theta: \Omega \rightrightarrows \mathcal{Y}$ describes what equilibrium outcomes are feasible at each state. Model (ref) is “incomplete” when $G_\theta$ predicts multiple outcomes without spelling out which will be realized. The model is “complete” when there is an equilibrium selection rule $\pi_\theta : \Omega \to \Delta (\mathcal{Y})$ that places positive weights only on equilibrium outcomes (i.e., $\pi_\theta (y \vert \omega) > 0$ only if $y \in G_\theta(\omega)$).\footnote{In the canonical entry game (e.g., tamer_incomplete_2003), $y \in \mathcal{Y}$ is a profile of firms' entry decisions, $\omega \in \Omega$ captures market factors like size and unobserved tastes, $\theta \in \Theta$ contains parameters for payoffs and payoff shock covariance. Firms' payoff shocks follow $F$, assumed to be multivariate normal. Each each $y \in G_\theta(\omega)$ is a pure strategy Nash equilibrium action profile that can arise at state $\omega$.}
\paragraph{Data Generating Process} A true parameter $\theta_0 \in \Theta$ and an equilibrium selection rule generate the data. Let $\omega=\left(x,\xi\right)$, where $x\in\mathcal{X}$ and $\xi \in \Xi$ denote variables that are observable and unobservable to the econometrician, respectively. I assume $\mathcal{X}$ is discrete and has full support.\footnote{Assuming $\mathcal{X}$ is discrete with full support simplifies the exposition by avoiding measure-theoretic details that are typically irrelevant in empirical applications. In practice, continuous covariates are often discretized for computational tractability (see, e.g., ciliberto2009marketstructure).} The econometrician observes a cross-sectional data $\left\{ y_{m},x_{m}\right\} _{m=1}^{n}$, where $m$ indexes units of independent observations (e.g., market), and $n$ represents the total number of observations. Each realized outcome $y_m$ is selected from $G_{\theta_0}(\omega_m)$ according to an equilibrium selection rule.
\paragraph{Identification Problem} The econometrician uses the data to identify the true parameter based on (ref) but without knowing the true equilibrium selection rule. I assume that with large sample ($n \to \infty$), the econometrician can identify the vector of conditional choice probabilities (CCP) $\phi:\mathcal{X} \to \Delta(\mathcal{Y})$; each $\phi\left(y\vert x\right) \in [0,1]$ denotes the probability of observing outcome $y$ at observable state $x$. Thus, I treat $\phi$ as a constant known to the econometrician.
\paragraph{A Generalized Likelihood-Based Characterization of Sharp Identified Set}
The sharp identified set is the collection of parameters at which the model (ref) is compatible with the observed data; in the current setting, the econometrician cannot reject $\theta$ if some (unknown) selection mechanism can generate $\phi$. I build on galichon2011setidentification's Theorem 1, which characterizes the sharp identified set using a finite set of conditional moment inequalities that do not depend on the unknown equilibrium selection rule.
A conditional choice probability function (or vector) $q_\theta : \mathcal{X} \to \Delta (\mathcal{Y})$ is feasible if there exists an equilibrium selection mechanism $\pi_\theta : \Omega \to \Delta (\mathcal{Y})$ such that $q_{\theta} (y \vert x) = \mathbb{E}\left[ \pi_\theta (y \vert \omega) \vert x \right]$ for all $y \in \mathcal{Y}$ and $x \in \mathcal{X}$. Function
is called the generalized likelihood function; each $\mathcal{L}_\theta(A \vert x)$ represents the maximal probability that event $A$ can occur given covariate $x$. Thus, Theorem (ref) says that a conditional choice probability vector is feasible if and only if it does not contradict the maximal probabilities admissible by the model.
To understand Theorem (ref), it is useful to note that a probability distribution $\pi_\theta(\cdot \vert \omega) \in \Delta (\mathcal{Y})$ is an equilibrium selection mechanism if and only if\footnote{Characterization (ref) does not appear in galichon2011setidentification.}
Characterization (ref) simply says that a proper equilibrium selection mechanism does not place positive weights on any outcome that is not an equilibrium; otherwise, there exists a non-equilibrium outcome $y$ at which $\pi_\theta(y \vert \omega) > 0$ but $G_\theta(\omega) \cap \{y \} = \emptyset$, contradicting (ref). Taking conditional expectation on (ref) produces the conditional moment inequalities (ref). Galichon and Henry prove that (ref) is not only necessary, as implied from (ref), but also sufficient for an arbitrary conditional choice probability vector to be feasible.
Theorem (ref) implies that a candidate parameter $\theta$ enters the sharp identified set if and only if the observed conditional choice probability vector is feasible at $\theta$:
Galichon and Henry's characterization (ref) is significant as it eliminates the need to deal directly with an equilibrium selection rule that may be infinite-dimensional. However, I argue below that their estimation algorithm remains computationally burdensome as it still hinges on a combination of equilibria enumeration, simulation, and grid search.
Theorem (ref) generates conditional moment inequalities using all subsets of $\mathcal{Y}$. However, restricting attention to a smaller subset does not necessarily yield a non-sharp identified set, as some inequalities may be redundant. The minimal subset that fully characterizes the sharp identified set, known as the core-determining set, is studied in galichon2011setidentification and luo2024selecting. My approach builds on this idea, seeking to balance computational tractability with identifying power by selectively dropping moment inequalities.
Existing methods for estimating discrete games of complete information are often subject to a high computational burden for three reasons. First, they require repeated enumeration of all possible equilibria, which amounts to computing the correspondence $G_\theta(\omega)$ at many values of $\omega$ and $\theta$; this cost rises exponentially with the number of players and actions. Second, they require simulation of latent variables to approximate probabilities (e.g., using $\mathcal{L}_\theta(A \vert x) \approx \frac{1}{S} \sum_{s=1}^S \mathbb{I} \{ G_\theta(\omega^s) \cap A \neq \emptyset \}$); this cost rises exponentially with the dimension of the unobservables. Third, they require repeating the test at many points in the parameter space; the cost of grid search rises exponentially with the dimension of the parameter space. Thus, the overall computational cost of finding the identified set is roughly $C_\text{total} \approx C_\text{solving a game} \times C_\text{simulation} \times C_\text{grid search}$, with each component on the right-hand side being subject to the curse of dimensionality.
To the best of my knowledge, virtually all existing algorithms are subject to at least one of the three problems, with generic algorithms being subject to all \citep*{ciliberto2009marketstructure, bajari2010estimating, beresteanu_sharp_2011, galichon2011setidentification, henry2015combinatorial, pakes_moment_2015, aradillas2022inference, koh2023stable, fan2025estimating}. My econometric approach reduces the computational burden by several orders of magnitude by addressing these challenges directly. Specifically, I characterize a subset of generalized likelihood functions as log-concave in the parameters, which enables the use of convex optimization tools. This reformulation eliminates the need for exhaustive equilibrium enumeration, high-dimensional simulation, and costly grid search, dramatically improving computational efficiency.
I argue that while working with the sharp identifying inequalities (ref) is computationally burdensome, it is possible to speed up the estimation dramatically. The trick is to find (i) a subset of identifying inequalities and (ii) distributional assumptions on unobservables that render the conditional moment inequalities convex in (a subvector of) $\theta$. Let $\mathcal{A} \subseteq 2^\mathcal{Y}$ and define $\Theta_I^\mathcal{A}$ as the identified set characterized by a subset of sharp-identifying conditional moment inequalities
Since $\phi$ is a known constant vector, if the generalized likelihood functions in (ref) are log-concave in $\theta$, constraints (ref) become convex in $\theta$, and estimating $\Theta_I^\mathcal{A}$ becomes computationally tractable.\footnote{For example, to find the projection intervals of $\Theta_I^\mathcal{A}$ along the first dimension, one can minimize $q^\top \theta$ with respect to $\theta$ subject to constraint (ref), using $q=(1,0,...,0)$ and $q=(-1,0,...,0)$. Both optimization problems are convex.} Thus, my approach aims at finding conditions that restore the log-concavity of the generalized likelihood function that would be present in the single-agent analogs.
In general, it is difficult to ensure that (ref) is log-concave in the full vector $\theta$. However, in the following sections, I show that for both unordered and ordered action cases, I can find conditions that allow the generalized likelihood functions to be log-concave in a subvector of $\theta$.
Assumption (ref) allows me to express the identified set $\Theta_I^\mathcal{A}$ as a union of convex sets as follows. Define $\Theta_I^\mathcal{A}(\sigma) \equiv (\Gamma_I^\mathcal{A}(\sigma),\sigma)$, where $\Gamma_I^\mathcal{A}(\sigma) \equiv \{\gamma \in \Gamma: \; (\gamma, \sigma) \in \Theta_I^\mathcal{A}\}$ is the set of $\gamma$'s satisfying (ref) for a given $\sigma$; each $\Theta_I^\mathcal{A}(\sigma)$ represents the “slice” of $\Theta_I$ at $\sigma$.
Figure (ref) visualizes the idea of Theorem (ref). In a typical discrete game application, $\sigma$ is a low-dimensional parameter that controls the variance of the unobserved heterogeneity that affects all players' payoffs, so the overall computational cost from applying Theorem (ref) remains low. When there are only player-specific unobservables but no unobserved heterogeneity (common unobservables), Assumption (ref) holds with $\gamma = \theta$ (i.e., log-concavity holds with respect to the entire parameter vector), so $\Theta_I^\mathcal{A}$ becomes a convex set (without the need to take the union).
I consider static discrete games of complete information with finite players and actions. I specify the game primitives with a tuple $\langle \mathcal{I}, (\mathcal{Y}_i)_{i \in \mathcal{I}}, ( u_i^\theta )_{i \in \mathcal{I}} \rangle$, where $\mathcal{I}=\left\{ 1,...,I\right\}$ is the set of players, $\mathcal{Y}_i$ is the finite set of actions available to player $i$, and $u_i^\theta: \mathcal{Y} \times \Omega \to \mathbb{R}$ is player $i$'s payoff function. I assume that the payoff functions are differentiable. To represent an action profile $y=(y_1, ..., y_I) \in \mathcal{Y} \equiv \times_{i=1}^I \mathcal{Y}_i$, I also use $y=\left(y_{i},y_{-i}\right)$ where $y_{-i}=\left(y_{1},...,y_{i-1},y_{i+1},...,y_{I}\right)$ denotes the actions of $i$'s opponents. I assume the players observe the realized state $\omega$ and the game is common knowledge to the players. Parameter $\theta$ governs the players' payoffs and the distribution of states. The solution concept is pure strategy Nash equilibrium.\footnote{My econometric strategy does not directly apply to cases involving mixed-strategy Nash equilibria. In general, accounting for mixed strategy Nash equilibria significantly increases the computational complexity, as finding all such equilibria in a game is PPAD-complete daskalakis2009complexity. koh2023stable argues that considering mixed strategy Nash equilibria can be inappropriate for certain empirical problems because firms may opt to change their actions if they perceive deviation as profitable ex-post. Nevertheless, my approach can help narrow the search area and save computation time significantly, even when mixed-strategy Nash equilibria are allowed.} Thus, $G_\theta(\omega)$ describes the set of all complete information pure strategy Nash equilibrium action profiles at state $\omega$. The model is incomplete \`{a} la tamer_incomplete_2003---it is silent on which equilibrium is selected.
Recall that $\omega=\left(x,\xi\right)$ where $x$ represents observable covariates and $\xi$ represents latent variables. Covariates $x\in\mathcal{X}$ can include common and player-specific observable characteristics. I assume that $\xi=\left(\lambda,\varepsilon\right)$, where $\lambda\in\Lambda$ is the payoff shock common to all players with distribution $H_{\theta}$, and $\varepsilon=\left(\varepsilon_{1},...,\varepsilon_{I}\right)$ collects player-specific idiosyncratic payoff shocks where each $\varepsilon_{i}$ follows distribution $F_{\theta}$. I assume the econometrician observes the conditional choice probability vector $\phi$. I make the following standard econometric assumptions.
Assumptions (ref) and (ref) model players' payoff shocks. Assumptions (ref) and (ref) model players' payoff functions. Assumption (ref) is necessary to establish the log-concavity of the generalized likelihood functions in $\gamma$ while integrating out the unobserved heterogeneity term $\lambda$. While Assumption (ref) appears non-standard, it is typically satisfied in empirical applications, as I show in the example below.\footnote{Each $v_i^\theta$ may be linear in $\theta = (\gamma, \sigma)$ as well as in $(\gamma, \lambda)$, but the latter is the important assumption for my results. Note that this assumption also applies to settings where $v_i^\theta$ includes “second-stage” profits estimated separately. For example, in entry game settings, one may specify the payoff function as $v_i^\theta(y,x,\lambda) = \pi_i(y,x) + c_i^\theta(y_i,x,\lambda)$, where $\pi_i(y,x)$ is a known variable profit function that captures operating profits conditional on entry (estimated from demand and supply data) and $c_i^\theta(y_i,x,\lambda)$ is the entry cost ciliberto2021marketstructure, fan2025estimating. The entry model identifies $\theta$, which governs the fixed cost of entry.} I defer making specific assumptions on the distribution of $\varepsilon_i$ since they depend on whether actions are unordered or ordered. I use a two-player entry game as a running example throughout the paper to fix ideas.
Recall that the computational tractability of my approach hinges on Assumption (ref). To meet Assumption (ref), I seek to find conditions that ensure the following assumption holds. Define the generalized likelihood function conditional on $x$ and $\lambda$ as
In the absence of unobserved heterogeneity (i.e., $\lambda = 0$ almost surely), Assumption (ref) boils down to Assumption (ref) with $\gamma = \theta$, and establishing the log-concavity of $\mathcal{L}_\theta(A \vert x)$ is relatively easy (assuming moment inequalities and distributional assumptions are appropriately chosen). However, the presence of the unobserved heterogeneity complicates the problem; even if $\mathcal{L}_\theta(A \vert x, \lambda)$ is log-concave in $\theta$, its expectation over $\lambda$ may not be log-concave in $\theta$ because log-concavity is not preserved under summation boyd2004convexoptimization. Yet, I can overcome the problem by leveraging Pr\'ekopa's theorem, which says that the log-concavity of a function is preserved under marginalization.\footnote{Let $H:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}$ be a log-concave function, and let $M\left(y\right)=\int_{\mathbb{R}^{m}}H\left(x,y\right)dx$. Pr\'ekopa's theorem says, if $H$ is non-negative, real-valued, and measurable, then $M\left(y\right)$ is log-concave in $y$. See boyd2004convexoptimization Chapter 3 and the references therein.} The following lemmas serve as key stepping stones for establishing the log-concavity of the generalized likelihood function with respect to a subvector of $\theta$.
Lemma (ref) is due to the law of iterated expectations. Lemma (ref) is due to Pr\'ekopa's theorem; since the integrand in (ref) is log-concave in $(\gamma,\lambda)$, marginalizing out $\lambda$ preserves the log-concavity with respect to $\gamma$.
In sum, the key step to finding a computationally tractable identified set is to find a subset of sharp-identifying inequalities and distributional assumptions on unobservables that ensure Assumption (ref) holds, which in turn ensures Assumption (ref) under standard econometric assumptions. In the following sections, I focus on characterizing conditions that yield Assumption (ref). I show how to choose the index set $\mathcal{A}$ and leverage logit-based distributional assumptions to find closed-form log-concave generalized likelihood functions, which I numerically integrate over $\lambda$.\footnote{This approach is reminiscent of the random coefficient approach, which facilitates computation by first finding the closed-form logit choice probabilities conditional on realized coefficients and integrating them over the random coefficients.}
In this section, I consider games with unordered (multinomial) actions. I construct a computationally tractable outer set by choosing $\mathcal{A}$ as a singleton class and assuming players' payoff shocks follow the Type-1 extreme value distribution. I also show that the outer set is tight in practice.
I consider an identified set $\Theta_I^\mathit{ABJ}$ defined by
I obtain the conditional moment inequalities (ref) by selecting $\mathcal{A}$ as the singleton class---a collection of all action profiles in $\mathcal{Y}$.\footnote{Singleton class consists of elements of $2^\mathcal{Y}$ that are singleton. For example, if $\mathcal{Y}=\{(0,0),(0,1),(1,0),(1,1)\}$, then its singleton class is $\{\{(0,0)\},\{(0,1)\},\{(1,0)\},\{(1,1)\}\} \subsetneq 2^\mathcal{Y}$.} Intuitively, (ref) requires that the observed probability of each action profile not be larger than the model-implied probability that the action profile is a possible equilibrium. I refer to the outer set as the ABJ set since \citet*{andrews_confidence_2004} uses analogous identifying restrictions.\footnote{\citet*{andrews_confidence_2004} (superseded by andrews2012inference) generates moment inequalities using simulation and different distributional assumptions (multivariate normal) on players' payoff shocks. In contrast, I show that combining the same set of restrictions with the logit assumption can dramatically speed up the estimation, which is new.} I combine my choice of $\mathcal{A}$ with the following distributional assumption to establish the computational tractability of the ABJ set.
Lemma (ref) derives the log-concavity of $\mathcal{L}_\theta(y \vert x, \lambda)$ to meet Assumption (ref). I obtain Lemma (ref).(ref) from the Nash equilibrium assumption that each agent faces a single-agent choice problem given opponents' actions. Each $\mathcal{M}_i^\theta(y_i \vert y_{-i},x, \lambda)$ represents the probability that action $y_i$ is a best response for agent $i$ who takes $y_{-i}$, $x$, and $\lambda$ as given. The generalized likelihood $\mathcal{L}_\theta(y \vert x, \lambda)$ represents the maximal probability that $y$ can be observed, but this is just the product of probabilities that each $y_i$ is the best response to the agent who takes $y_{-i}$ as fixed. Lemma (ref).(ref) uses the fact that each (ref) represents a single-agent choice probability. Lemma (ref).(ref) uses the fact that each (ref) is a composition of a log-concave function with a linear function, so it is log-concave in $(\gamma, \lambda)$. The log-concavity of $\mathcal{L}_\theta (y \vert x, \lambda)$ with respect to $(\gamma, \lambda)$ follows from it being a product of log-concave functions.\footnote{It is interesting that Lemma (ref) also alludes to the possibility of incorporating a nested logit assumption, which has not been used in multi-agent settings. Although a variant of Lemma (ref) should provide closed-form expressions for the generalized likelihood functions (which still provide significant computational savings), the convexity property will be lost.} In sum, I obtain the following characterization.
Thus, Theorem (ref) meets Assumption (ref), so I can trigger Theorem (ref): the ABJ-identified set is a union of easy-to-compute convex sets. In Appendix (ref), I show that extending Theorem (ref) to settings where players make vector-valued decisions such as in fan2025estimating is straightforward.
I illustrate the ABJ set using the running example. To fix the idea, suppose that the common shock is absent. It is straightforward to verify that $\theta=\left(\beta_{1},\beta_{2},\Delta_{1},\Delta_{2}\right)\in\Theta_{I}^\mathit{ABJ}$ if and only if, for all $x\in\mathcal{X}$,
where $\underline{\epsilon}_i = -x_i^\top \beta_i$, $\overline{\epsilon}_i = \underline{\epsilon}_i - \Delta_i$, and $F\left(z\right)\equiv1/\left(1+e^{-z}\right)$ is the standard logistic cumulative distribution function.
To visualize (ref), recall $\mathcal{L}_\theta (A \vert x) = \mathbb{P} (G_{\theta}^{-1}(A \vert x) )$, where $G_{\theta}^{-1}(A \vert x) \equiv \{\xi: \; G_\theta(x,\xi) \cap A \neq \emptyset\}$ is the set of unobservables that support some action profile in $A$ as an equilibrium outcome. Figure (ref) shows an example of $G^{-1}_\theta\left(y\vert x \right)$ with $y = (1,0)$ in the $\left(\epsilon_{1},\epsilon_{2}\right)$-space. A common feature of $G^{-1}_\theta\left(y\vert x \right)$'s are that they are rectangles, whose probabilities are log-concave under the standard logistic distribution assumption. So, Figure (ref) shows that I can restore the log-concavity of the generalized likelihood function if I choose an appropriate event $A$.
Note that the identifying power of the ABJ set can be gleaned from (ref); the identifying inequalities prevent $\theta$ from being too arbitrary. For example, if $\phi((1,1)\vert x)$ is bounded away from zero, $\Delta_i$ cannot be too negative since driving $\Delta_i \to -\infty$ implies $\overline{\epsilon}_i \to \infty$, which is incompatible with the second inequality of (ref).
Given its non-sharpness, is the ABJ set sufficiently tight? The following numerical example suggests it is. Consider a two-player entry game where the players' payoffs are given by $u_i^\theta(y_i, y_{-i}) = y_i (\beta_i + \Delta_i y_{-i} + \epsilon_i)$. For $i=1,2$, I set $\beta_i = 0$ and $\Delta_i = -0.5$ and assume i.i.d. standard logistic distribution for each $\epsilon_i$. I assume a symmetric equilibrium selection rule. The model yields a choice probability vector $\left(\phi_{00},\phi_{10},\phi_{01},\phi_{11}\right)=\left(0.250,0.304,0.304,0.142\right)$.
As a first experiment, I assume that the econometrician knows the value of $(\beta_1, \beta_2) \in \mathbb{R}^2$ but wants to estimate $(\Delta_1, \Delta_2) \in \mathbb{R}^2$. Figure (ref) plots the ABJ set in blue and the sharp set in red in the $(\Delta_1, \Delta_2)$-space. Both sets contain the true parameter represented as a star node. The ABJ set is convex, as expected.\footnote{In this example, the sharp identified set also appears to be convex. However, the sharp identified set is non-convex when the normal distribution assumption is used (see, e.g., beresteanu_sharp_2011).} Although the ABJ set is wider than the sharp set, their projection intervals are essentially identical.
As a second experiment, I also consider the case where the econometrician estimates $(\beta_1, \beta_2, \Delta_1, \Delta_2)$; I jointly estimate the four parameters (without assuming, e.g., $\beta_{1}=\beta_{2}$ and $\Delta_{1}=\Delta_{2}$). Table (ref) reports the projection intervals of the sharp set and the ABJ set. The table shows that the two sets deliver essentially identical projection intervals, hinting that the ABJ set preserves considerable identifying power despite not being sharp.
Econometric models of discrete games with multinomial actions have most commonly been applied to settings with binary actions. My approach is well-suited to these cases, and Section (ref) provides an empirical illustration. While I focus on an outer set, the sharp identified set for this class of models is well understood. By first narrowing attention to my outer set and then applying existing methods, one can obtain the sharp identified set with substantial computational savings.
My framework also naturally extends to games in which each player chooses a vector of actions. For example, fan2025estimating study a setting where firms make multiple product-level entry decisions simultaneously. In Appendix (ref), I demonstrate how my method can be readily applied to their context.
Games with three or more unordered actions have received comparatively little attention in the literature. The main challenge in these settings is that pure strategy Nash equilibria may fail to exist. In Appendix (ref), I show that my approach remains straightforward to apply by examining a supermarket pricing game from ellickson2008supermarket under the assumption of incomplete information. Using simulated data, I illustrate that even under complete information, the model remains tractable, though the parameters are only partially identified. Finally, I discuss how my approach can still be valuable even when there are concerns about misspecification due to the potential non-existence of pure strategy Nash equilibria.
In this section, I switch attention to games with ordered actions games. I construct a computationally tractable outer set by choosing $\mathcal{A}$ to include a sequence of actions from below or above and assuming players' payoff shocks follow the standard logistic distribution.
I follow aradillas2022inference's econometric framework.\footnote{Although I follow aradillas2022inference's framework, I take a different approach to estimation and inference. aradillas2022inference use chesher2017generalized's characterization of the sharp identified set, but I apply galichon2011setidentification's characterization. I also provide additional results for characterizing pure strategy Nash equilibria.} Each player can choose from an ordered set $\mathcal{Y}_i = \left\{ s_i^0, s_i^1,...,s_i^{L_i} \right\} \subseteq\mathbb{R}$ where $s_i^l < s_i^{l+1}$ for $l=0,...,L_i-1$, and $s_i^{L_i} < \infty$.\footnote{It is common to have $s_i^l=l$ so that $\mathcal{Y}_i = \{0,1,2,...,L_i\}$. My econometric strategies also apply when $L_i$ is unbounded above.} I extend the action space and introduce $s_i^{-1}$ and $s_i^{L_i+1}$ such that $u_{i}^{\theta}\left(\tilde{y}_{i},\cdot,\cdot,\cdot\right)\equiv-\infty$ if $\tilde{y}_{i}\in\{s_i^{-1}, s_i^{L_i+1} \}$. I also use $y_{i}^{-}$ and $y_{i}^{+}$ as lower and upper adjacent actions to $y_{i}$ respectively, i.e., if $y_i = s_i^l$, then $y_i^- = s_i^{l-1}$ and $y_i^+ = s_i^{l+1}$.
Let $z\equiv\left(x,\lambda\right)\in\mathcal{Z}\equiv\mathcal{X}\times\Lambda$. I consider the following assumptions.
The above assumptions are standard in empirical works. Assumption (ref) ensures that the best-response action is unique and follows a threshold-crossing strategy. Assumption (ref) ensures that the thresholds for characterizing best responses are non-overlapping. Assumption (ref) ensures that the thresholds are increasing in opponents' actions. Finally, Assumption (ref) ensures that the idiosyncratic payoff shocks have log-concave and closed-form cumulative probabilities.
Lemmas (ref) characterizes Nash equilibria using threshold-crossing rules. Lemma (ref) shows that the thresholds are non-overlapping, have closed forms, and are increasing in players' actions. Given $y_{-i}$, threshold $e_i^\theta(s_i^l,y_{-i},z)$ represents the point at which $y_i=s_i^l$ starts to be the best-response action as $\varepsilon_i$ increases from below. For example, $y_i=s_i^0$ is the best-response action starting from $\varepsilon_i = -\infty$, and $y_i=s_i^1$ becomes the best-response as $\varepsilon_i$ exceeds point $e_i^\theta(s_i^1,y_{-i},z)$. The following example illustrates the lemmas in the case players' actions are binary.
I propose a computationally tractable outer set by considering events of the form “each player's action is higher (or lower) than a certain action.” Let $\iota = (\iota_1, ..., \iota_I)$ with $\iota_i \in \{l,h\}$. I consider $\mathcal{A} \subseteq 2^\mathcal{Y}$ of form
where, for a given $y \in \mathcal{Y}$,
Set (ref) includes all actions that are either lower or higher than a given action for each player; it is defined with the intent of leveraging the threshold crossing structure delineated in Lemma (ref). Also note $\min A^\iota(y)$ and $\max A^\iota(y)$ are well-defined as\footnote{For example, consider a two-player binary action game. The set of action profiles is $\mathcal{Y} = \{(0,0), (0,1), (1,0), (1,1)\}$. Let $y = (1,0)$ be given. Then $A^{(l,l)}(y) = \{(0,0), (1,0)\}$, $A^{(h,h)}(y) = \{(1,0),(1,1)\}$, $A^{(h,l)}(y) = \{(1,0)\}$, and $A^{(l,h)}(y) = \{(0,0), (1,0), (0,1), (1,1) \}$. Note that $\min A^{(l,h)}(y) = (0,0)$ and $\max A^{(l,h)}(y) = (1,1)$.}
The following lemma shows that each generalized likelihood function $\mathcal{L}_\theta (A^\iota(y) \vert z)$ is bounded above by a closed-form log-concave function.
Intuitively, Lemma (ref) obtains a tractable upper bound on the generalized likelihood functions by finding the smallest hyperrectangles in the $\varepsilon$-space that cover the non-rectangular region of $\varepsilon$'s that correspond to $\mathcal{L}_\theta(A^\iota(y) \vert z)$. To ensure the log-concavity, I restrict $\mathcal{A}$ to contain a sequences of actions from below or above for each player. Lemma (ref) implies Assumption (ref), so I obtain the following computationally tractable outer set.
Theorem (ref) is similar to Theorem (ref) but with the generalized likelihood function indexed by $\iota$ that determines whether actions below or above $y$ are being considered.\footnote{I interpret each $\overline{\mathcal{L}}_\theta^\iota(y \vert x)$ as a form of generalized likelihood although, strictly speaking, it is obtained from upper bounds of generalized likelihood functions as seen in Lemma (ref).}
To visualize the intuition behind Lemma (ref), I return to the running example. Lemma (ref) uses the fact that the smallest $I$-dimensional rectangle that covers $G_\theta^{-1}(A^\iota(y) \vert z)$ (i.e., all realization of $\varepsilon$ that support some outcome in $A^\iota(y)$ as an equilibrium) is easy to characterize due to the monotonicity of the thresholds with respect to players' actions. Take, for example, $y=(1,0)$ and $\iota = (h,h)$ so that $A^{(h,h)}((1,0)) = \{(1,0),(1,1)\}$. Figure (ref) shows $G_\theta^{-1}(\{(1,0),(1,1)\} \vert z)$, the set of $\varepsilon$'s that support either $(1,0)$ or $(1,1)$ as an equilibrium outcome. Clearly, computing $\mathcal{L}_\theta(\{(1,0),(1,1)\} \vert z) \equiv \mathbb{P} (G_\theta^{-1}(\{(1,0),(1,1)\} \vert z))$ is complicated by the non-rectangular shape of $G_\theta^{-1}(\{(1,0),(1,1)\} \vert z)$.
The idea of Lemma (ref) is to “fix” this computational problem by finding the smallest rectangle that covers $G_\theta^{-1}(\{(1,0),(1,1)\} \vert z)$. It is easy to see that $\mathcal{L}_\theta(\{(1,0),(1,1)\} \vert z) \leq 1 - F(\underline{\varepsilon}_1)$: the right-hand side is the probability over the rectangle $(\underline{\epsilon}_1,+\infty) \times (-\infty, +\infty)$. In other words, the generalized likelihood of the event $A^{(h,h)}((1,0))$ is bounded above by an easy-to-compute log-concave function. A full iteration over all $y$ and $\iota$ gives
The above restrictions generate identifying power by ruling out extreme values of competitive effects and intercepts.
As an illustrative example, I consider a version of the multi-store entry game considered in aradillas2022inference. Let $\mathcal{Y}_i = \{0,1,...,L_i\}$ be the number of stores for firm $i=1,...,I$. I assume $I=2$. The payoff functions are modeled as
where I assume $\Delta_i > 0$ and $\eta > 0$ to ensure actions are strategic substitutes and the payoff function is strictly concave in own actions. Assumptions (ref), (ref), (ref), and (ref) are straightforward to verify. Lemma (ref) applies and the thresholds in (ref) are given as\footnote{For every $y$, $\underline{e}_i^\theta (y,z) = -z_i^\top \gamma_i + \Delta_iy_{-i} + \eta(2y_i-1)$ and $\overline{e}_i^\theta (y,z) = -z_i^\top\beta +\Delta_iy_{-i} + \eta (2y_i+1)$.}
Figure (ref) shows the structure of equilibria when $L_i = 2$, $\beta_0=\beta_1 = \sigma = 0$, and $\Delta = \eta$. Intuition of Theorem (ref) can be gleaned again from the figure. Take, for example, $y=(0,0)$ and $\iota = (h,l)$ so that $A^\iota(y) = \{(0,0), (1,0), (2,0) \}$. Although $\mathcal{L}_\theta(A^\iota(y))$, which is the probability over $\varepsilon$'s that support at least one outcome in $A^\iota(y)$ as an equilibrium, covers non-rectangular area, $\overline{\mathcal{L}}_\theta^\iota(y)$ covers a rectangular area defined by $(\varepsilon_1, \varepsilon_2) \in (-\infty, +\infty) \times (-\infty, 3\Delta)$. Clearly, $\mathcal{L}_\theta(A^\iota(y)) \leq \overline{\mathcal{L}}_\theta^\iota(y)$ but the latter is significantly easier to compute and log-concave in $\Delta$.
I simulate data assuming parameter values in Table (ref) and $\sigma = 0.5$; I assume the econometrician knows $\sigma = 0.5$ to simplify the simulation exercise. I assume each $\varepsilon_i$ independently follows the standard logistic distribution, and $\lambda$ follows the standard normal distribution. I take 100,000 draws of $\varepsilon_1$, $\varepsilon_2$, and $\lambda$. I assume a symmetric equilibrium selection rule whenever I find multiple equilibria. I report the projection intervals of the identified set and the execution time in Table (ref). The projection bounds are tight. The total runtime of the algorithm took only 2.85 seconds.
To test the proposed methodology, I apply my approach to ellickson_structural_2011's entry game between Walmart and Kmart discount stores.\footnote{ellickson_structural_2011 simplifies jia_what_2008's model to illustrate traditional discrete games estimation methodologies that rely on assumptions on the equilibrium selection rules to render the traditional likelihood approach applicable. Specifically, while the original model in jia_what_2008 assumes that the players are playing a single network game across a large number of markets, ellickson_structural_2011 assumes that the entry games were played independently across markets. I refer the reader to jia_what_2008 and ellickson_structural_2011 for full details of the model and dataset.} I consider this setting because binary-action entry games serve as the canonical example in the literature. I can compare my results, which are robust to equilibrium selection assumptions, to those reported in ellickson_structural_2011. Moreover, the underlying dataset is simple, transparent, and publicly available on the authors' and publisher's websites.
Walmart and Kmart decide whether to operate a store in a well-defined local market. The firms' strategic entry choices are modeled as a static discrete game. Each firm can enter or stay out. The firms' payoff functions are specified as (ref); the competitive effects parameter is identical across players and non-positive (i.e., entry decisions are strategic substitutes). Each local market is defined as a county. There are 2,065 markets. Firms have complete information. The solution concept is pure strategy Nash equilibrium.
Table (ref) reports the summary statistics of the original dataset. To facilitate the estimation, I discretize continuous variables (population, retail sales per capita, urban, and distance to Benton) to binary variables.\footnote{Using auxiliary regressions that regress firms' entry decisions on exogenous covariates with and without discretization, I confirm that the discretization does not change the regression coefficients significantly.} Specifically, I find the median of each variable, classify each observation as above or below the median and replace each value with the within-group mean. After dropping covariate bins with no observations, I have $\vert \mathcal{X} \vert = 40$ covariate bins.
I estimate the ABJ-identified set. To account for sampling uncertainty, I compute the confidence set for the identified set following the methods described in horowitz2022inference and koh2023stable. Let $\Theta_I(\phi)$ be the identified set, where I make the dependence on the conditional choice probability vector $\phi$ explicit. I first construct the confidence set for the conditional choice probabilities $\Phi_n^\alpha$ such that
I then define the confidence set as
The confidence set defined as (ref) covers the identified set $\Theta_I$ with probability at least $1-\alpha$ asymptotically because $\phi$ is the only source of sampling uncertainty, which has been controlled by $\Phi_n^\alpha$.\footnote{See shi2015simple and hsieh2022inference for related approaches to inference.} Following koh2023stable, I construct $\Phi_n^\alpha$ to be simultaneous confidence intervals to maintain computational tractability. The projection intervals of the confidence set are easy to compute. I also apply the same method to compute the confidence set for the sharp set. I provide computational details in Appendix (ref).
When estimating the model, I find evidence of potential model misspecification: Even after accounting for the sampling uncertainty associated with the first-stage conditional choice probability estimation, it is challenging to find parameters that satisfy the conditional moment inequalities exactly.\footnote{Misspecification-robust estimation and inference for partially identified models is an active area of research. See, for example, andrews2024misspecified.} This may be due to multiple reasons, including discretization error, finite sample error, numerical error, and restrictive parametric assumptions, which are difficult to disentangle. To facilitate comparison between different estimation strategies, I adhere to ellickson_structural_2011's specification but slightly relax the moment inequality conditions.\footnote{For example, when computing the sharp set, I allow for an extra margin of error and relax each inequality (ref) by 0.0025. See Appendix (ref) for computational details.}
Table (ref) reports the estimation results. Column “Ellickson and Misra (2011)” reports the estimates from the original paper. Column “BR” reports the estimates using bresnahan_entry_1991's methodology, which assumes symmetric players. Column “Berry” uses berry1992estimation's approach, which relies on assumptions on the order of moves: specification “Profit” assumes that a more profitable firm moves first; “Walmart” assumes that Walmart is the first mover; “Kmart” assumes that Kmart is the first mover.
The last two columns report the ABJ and sharp set projection intervals, which do not rely on arbitrary assumptions on the equilibrium selection rule. I assume the common shock $\lambda$ is drawn from the standard normal distribution. I find the parameter $\sigma$ difficult to identify because I cannot reject a wide range of values. So, I estimate the ABJ set and the sharp set at $\sigma = 0$ and $\sigma =3$ (which imply a correlation coefficient of 0 and 0.85 for players' unobservables, respectively) and take the union of the projection intervals.\footnote{Correlation parameter can be difficult to identify in two-player entry games. kline_bayesian_2016 also finds it difficult to estimate the correlation parameter between players' observables in their empirical example studying the strategic entry decisions of airlines.} To compute the sharp set, I conduct a grid search on 100,000 candidate parameters in the parameter space, constructed using a Halton sequence. For comparison to ellickson_structural_2011's estimates, I report the coefficients after normalizing them to be relative to the standard deviation of the players' unobserved shocks.
Table (ref) shows that my approach works well. The ABJ set is quite tight, qualitatively similar to ellickson_structural_2011's estimates, and takes little time to compute. A comparison to the sharp set also confirms this conclusion. Computing the ABJ set is at least three orders of magnitude faster than computing the sharp set. Attempts to increase the accuracy of the sharp set can make the gap significantly larger.\footnote{For example, I increasing the number of draws from the parameter space to 10 million increases the computational time by 100 folds.} The ABJ set is also qualitatively similar to the sharp set. Note that the sharp set is not a strict subset of the ABJ set because I use different estimation algorithms for the two identified sets.\footnote{I also relax the moment inequality conditions to handle potential numerical and misspecification errors. A grid search on a coarse grid usually exaggerates the tightness of the identified set. For example, suppose the true identified set is $\Theta_I = [0.1,2.9]$, but the candidate parameters are $\{0,1,2,3\}$. Since $\theta = 0$ and $\theta=3$ are rejected, the implied identified set would be $\widehat{\Theta}_I = [1,2]$, which is only $(2-1)/(2.9-0.1) \approx 35\%$ of the true interval. I slightly relax the threshold for my criterion function to avoid this problem. Refining the sharp set by taking a larger number of draws would substantially increase the computational burden.}
As a second and more involved empirical application, I consider the chain entry game by the top 3 burger chains in the US: McDonald's, Burger King, and Wendy's. In 2019, these three firms were responsible for over 70% of the sales among the top 20 burger chains in the US technomic2019. I assume that a three-player entry game provides a good approximation to the chains' strategic entry decisions despite the presence of fringe players.
I estimate a three-player chain entry game where each firm can choose the number of outlets from $\mathcal{Y}_i = \{0,1,...,L_i\}$. I model each firm's payoff function as (ref). My primary dataset is the 2019 cross-section of the Data Axle Historical Business Database, which contains the firms' outlet locations. Following koh2023stable, I define markets as 2010 urban census tracts in the US. Table (ref) reports the distribution of outlet numbers by firms in urban tracts. I set the maximal action as $L_i = 2$ and encode all actions higher than $L_i$ as $y_i=L_i$.
To control for market characteristics, I obtain the number of eating and drinking places at each tract in 2017 from the National Neighborhood Data Archive (NaNDA) database esposito2020nanda. I also obtain indicators for whether each tract is classified as having low income and low access to food (also referred to as “food deserts”) in 2010 by the Food Access Research Atlas, constructed by the US Department of Agriculture Economic Research Service ERS_USDA_FARA. Finally, I use the number of own-firm outlets per 100,000 people in the tract's county as a firm-specific demand shifter that captures network effects.
Table (ref) reports the summary statistics. To facilitate the estimation, I discretize variables to binary variables in the same manner as explained in Section (ref), which gives $\vert \mathcal{X} \vert = 32$ covariate bins. To obtain the first-stage estimates of the conditional choice probability vector, I regress the $\vert \mathcal{A} \vert = 27$ possible outcomes on the exogenous covariates using a multinomial logistic model and estimate the predicted probabilities by evaluating the estimated regression model at the discretized covariates. I follow the same steps described in Appendix (ref) to construct the confidence set, only replacing nonparametric frequency estimates of conditional choice probabilities with those obtained from parametric multinomial logit estimates.
Table (ref) reports the estimation results. The projection intervals represent the union of those estimated at $\sigma \in \{1.2,2\}$ and take the union.\footnote{I first check the non-emptiness of confidence set $\widehat{\Theta}_I(\sigma)$ at $\sigma$'s ranging from $0$ to $3$ with step size $0.2$. I find that the moment inequality violations are minimized at $\sigma \in [1.2,2]$, with violations taking values very close to zero in this range. The model fit deteriorates when moving away from $\sigma$'s in this range. When finding the projection intervals, I relax the moment inequality constraints slightly to be conservative and accommodate numerical errors.} All parameter estimates are normalized to be relative to the standard deviation of the players' unobservables. To avoid the dimension of the structural parameter from being too high, I make two assumptions. First, I assume firms have the same coefficients with respect to the common market characteristics. Second, I assume their presence exerts the same level of externality on competitors' profits (for instance, McDonald's entry decreases Burger King and Wendy's profits by the same amount).
I find that the confidence set is quite tight and informative. Burger chains' profit per outlet is higher in markets with low income and low access to food. Having a larger number of eating places increases store profitability; for example, adding 10 more eating places boosts profit per store by 0.2-0.3 units. The network effects are positive; store profitability increases with a denser network within a county, likely due to lower distribution costs and higher brand equity. The concavity parameters can be quite high, explaining why firms rarely open multiple outlets. In particular, McDonald's concavity parameter is the highest, meaning McDonald's is most conscious about cannibalization between stores. McDonald's baseline profit, captured by the intercept, is also higher than the other chains. Finally, the competitive effects parameters can be significant based on the upper bounds of the projection intervals. Note that I cannot rule out the null effects for Burger King and Wendy's concavity and competitive effects parameters since the projection intervals contain zero.
Computing the projection intervals of the 14-dimensional confidence set took 2811 seconds ($\approx 0.78$ hour) without parallelization. The total computational time for estimating a 14-dimensional identified set via grid search will likely be several orders of magnitude larger. In conclusion, the proposed approach can quickly generate an informative identified set of parameters.
In this paper, I introduced a novel approach to estimating discrete games of complete information, providing separate methods for settings with unordered and ordered actions. Under standard logit assumptions on unobservables, I developed conditional moment inequalities that are convex in a subset of parameters. Through numerical and empirical examples, I demonstrated that the proposed approach is straightforward to implement and scales well to games with many players and actions.
There are several promising directions for future research. First, it would be valuable to apply similar strategies for constructing convex conditional moment inequalities in other contexts. Second, improving the scalability of the algorithm is an important challenge. For instance, developing numerical methods that avoid coarse discretization or applying the approach to large-scale games, such as the network entry game in jia_what_2008, would be interesting. Finally, exploring alternative approaches to inference is a natural next step. I conjecture that the computational framework developed here is compatible with many existing inferential methods canay2023user.
\paragraph{Disclosure Statement} During the preparation of this work the author used Open AI's ChatGPT 5.0 in order to check grammar and improve clarity of the writing. After using this tool/service, the author reviewed and edited the content as needed and takes full responsibility for the content of the published article. Wharton Research Data Services (WRDS) was used in preparing part of the data set used in the research reported in this manuscript. This service and the data available thereon constitute valuable intellectual property and trade secrets of WRDS and/or its third-party suppliers. The author declares that no financial or personal conflicts of interest influenced the research presented in this paper.