Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
147,419 characters · 34 sections · 116 citation commands
Stable Outcomes and Information in Games: An Empirical Framework
In dynamic strategic settings where firms can react after observing their opponents' choices, our intuitions suggest that firms' actions would change over time. Interestingly, we often see firms reach a certain steady state in which no firm changes its decision even when it can. For example, major exporters' decisions to export products to specific markets remain unchanged for a long period ciliberto2021superstar. Airline firms' decisions to operate between cities tend to be persistent ciliberto2009marketstructure. Food-service retailers operate in a local market over a long horizon, knowing precisely the identities of the competitors operating nearby. In all these examples, each firm's action constitutes a best response to the observed actions of the opponents.
The prevalence of incomplete information in the real world makes the phenomenon particularly interesting. If opponents' actions are observable at the steady state, rational firms will use the observation to update their beliefs. For example, while a coffee chain's own research might find a given neighborhood unattractive, observing that Starbucks\textemdash a chain known to have leading market research technology\textemdash enter the neighborhood may make it think twice.\footnote{According to Tom O'Keefe, the founder of Tully's Coffee, Tully's early business expansion strategy was to “open across the street from every Starbucks” because “they do a great job at finding good locations.” goll2000seattles.} If there is no further revision of actions, it must be that each firm holds beliefs refined by their observations of opponents' actions, but no further updating is possible.
Although stable outcomes in the presence of information asymmetries are common in the real world, it is not straightforward to model the data generating process. The main difficulty arises from requiring that the firms' beliefs and actions be consistent with each other. On the one hand, firms' beliefs must support the realized actions as optimal. On the other hand, each firm's beliefs must incorporate its private information as well as the information extracted from observing its opponents' decisions. Static Bayes Nash equilibrium, although a popular modeling choice, does not account for the possible revision of actions after opponents' actions are observed. Modeling convergence to stable outcomes via a dynamic games framework may be feasible but is likely non-trivial and reliant on ad hoc assumptions. In this paper, we develop a tractable equilibrium notion that satisfies the consistency requirement and facilitates econometric analysis when the analyst observes a cross-section of stable outcomes at some point in time.
We propose a solution concept dubbed Bayes stable equilibrium as a basis for analyzing stable outcomes in the presence of incomplete information. Bayes stable equilibrium is described as follows. A decision rule specifies a distribution over action profiles for each realization of the state of the world and players' private signals. Suppose that, after the state of the world and private signals are realized, an action profile is drawn from the decision rule, and the action profile is publicly recommended to the players. The decision rule is a Bayes stable equilibrium if the players always find no incentives to deviate from the publicly recommended action profile after observing their private signals and the action profile.
We justify Bayes stable equilibrium using a version of rational expectations equilibrium � la radner1979rational. First, we argue that rational expectations equilibrium, appropriately defined for our setting, provides a simple approach to rationalizing stable outcomes under incomplete information. We define rational expectations equilibrium by adopting the “outcome function” approach of liu2020stability, who uses a similar approach to define the notion of stability in two-sided markets with incomplete information. Next, we show that Bayes stable equilibrium characterizes the implications of rational expectations equilibria when the analyst can only specify the minimal information available to the players. Thus, Bayes stable equilibrium is useful as it allows the analyst to be “informationally robust” in the same sense as the Bayes correlated equilibrium of bergemann2016bayescorrelated. The informational robustness property is attractive since it is often difficult for the analyst to know the true information structure governing the data generating process.
Assuming that the analyst observes a cross-section of stable outcomes, we characterize the identified set of parameters using Bayes stable equilibrium as a solution concept. The corresponding identified set has a number of attractive properties. First, the identified set is valid for arbitrary equilibrium selection rules and robust to the possibility that the players actually observed more information than specified by the analyst. We let the model be “incomplete” in the sense of tamer2003incomplete, and the parameters are typically partially identified. Second, when strong assumptions on information are made, the Bayes stable equilibrium identified set collapses to the pure strategy Nash equilibrium identified set studied in \citet*{beresteanu2011sharpidentification} and galichon2011setidentification. Third, everything else equal, the Bayes stable equilibrium identified set is (weakly) tighter than the Bayes correlated equilibrium identified set studied in magnolfi2021estimation. While Bayes stable equilibrium and Bayes correlated equilibrium both facilitate estimation of games with weak assumptions on players' information, the former is stronger as it leverages the assumption that players' actions are observable to each other at equilibrium situations.
We propose a computationally tractable approach to estimation and inference. We show that checking whether a candidate parameter enters the identified set (asking whether we can find an equilibrium consistent with data) solves a linear program. Furthermore, we propose a simple approach to constructing confidence sets for the identified set by leveraging the insights from horowitz2021inference. The key idea is to construct convex confidence sets for the conditional choice probabilities, which are the only source of sampling uncertainty. Checking whether a candidate parameter belongs to the confidence set amounts to solving a convex program.
As an empirical application, we use our framework to analyze the strategic entry decisions of McDonald's and Burger King in the US. We estimate the model parameters using Bayes stable equilibrium and explore the role of informational assumptions on identification. We also use the model to simulate the impact of increasing access to healthy food in Mississippi food deserts. We find that popular assumptions on players' information may be too strong, as the corresponding identified set can be empty. On the other hand, making no assumption on players' information produces an identified set that is too large, indicating that some assumptions on information are necessary to produce informative results. We show that an informative identified set can be obtained under an intermediate assumption, which is also credible; this specification assumes that McDonald's has accurate information about its payoff shocks while Burger King may minimally observe nothing. We also compute the identified sets under the Bayes correlated equilibrium assumption and find that the Bayes stable equilibrium identified sets are substantially tighter under the same assumptions on players' information: the volume\textemdash measured as the product of the projection intervals\textemdash under Bayes stable equilibrium is at most 5% of that under Bayes correlated equilibrium.
Our work adds to the literature on the econometric analysis of game-theoretic models by designing a framework that applies to a class of situations characterized by stable outcomes (see depaula2013econometric and aradillas-lopez2020theeconometrics for recent surveys).\footnote{In his survey on the econometrics of static games, aradillas-lopez2020theeconometrics classifies existing papers around five criteria: (i) Nash equilibrium versus weaker solution concepts; (ii) the presence of multiple solutions; (iii) complete- versus incomplete-information games; (iv) correct versus incorrect beliefs; (v) parametric versus nonparametric models. To place our work in these categories, this paper (i) develops a new solution concept that is weaker than complete information pure strategy Nash equilibrium but stronger than Bayes correlated equilibrium; (ii) admits a set of equilibria; (iii) allows a general form of incomplete information which accommodate standard assumptions as special cases; (iv) assumes that players have correct beliefs; (v) imposes parametric assumptions on the payoff functions and the distribution of unobservables.} Our framework would be well-suited when (i) it is reasonable to assume that the realized actions represent best responses to the observed decisions of the opponents, (ii) the stability of outcomes is not driven by high costs of revising actions, and (iii) the analyst observes cross-sectional data of firms' stable decisions at some point in time.\footnote{This idea behind cross-sectional analysis of games is accentuated in ciliberto2009marketstructure: “The idea behind cross-section studies is that in each market, firms are in a long-run equilibrium. The objective of our econometric analysis is to infer long-run relationships between the exogenous variables in the data and the market structure that we observe at some point in time, without trying to explain how firms reached the observed equilibrium.” (pp.1792-1793).}
Our framework differs from the usual Nash framework. To account for stable outcomes, we assume players can observe opponents' actions and react. In contrast, in static Nash frameworks, players are not allowed to change their “one-shot” actions and therefore may be subject to regrets after observing the realized actions of their opponents.\footnote{The empirical literature has been aware that the Nash framework is subject to ex-post regret when information is incomplete or players are using mixed strategies. See, for example, the discussions in draganska2008discrete, einav_not_2010, and ellickson2011structural. } Furthermore, we are not aware of dynamic models (e.g., frameworks based on Markov perfect equilibrium) that can straightforwardly handle stable outcomes in incomplete information environment.
Bayes stable equilibrium allows the researcher to work with weak assumptions on players' information. An early work in this spirit is grieco2014discrete, which considers a parametric class of information structures that nests standard assumptions. Our work is most closely related to recent papers that use Bayes correlated equilibrium as a basis for informationally robust econometric analysis: magnolfi2021estimation applies Bayes correlated equilibrium to static entry games (which are also considered in this paper), \citet*{syrgkanis2021inference} to auctions, and gualdani2020identification to static, single-agent models.\footnote{There is also a strand of literature that studies the possibility that firms might have biased beliefs (see aguirregabiria2020identification and aguirregabiria2020firmstextquoteright for a review). The works in this literature assume that the econometrician knows the true information structure of the game but firms may not have correct beliefs. In contrast, we assume that firms have correct beliefs but the econometrician does not know the true information structure.}
We contribute to the literature on the econometrics of moment inequality models by proposing a simple approach to constructing confidence sets based on the idea of horowitz2021inference.\footnote{Recent development in inference with moment inequality models has introduced many alternative approaches for constructing confidence sets (see ho2017partial, canay2017practical, and molinari2020microeconometrics2 for recent surveys). However, to the best of our knowledge, most are not directly applicable to our setup, primarily due to the presence of a high-dimensional nuisance parameter and a large number of inequalities. A feasible strategy for inference is the subsampling approach of \citet*{chernozhukov2007estimation}, which is also used in magnolfi2021estimation and \citet*{syrgkanis2021inference}.} Our approach is new in the context of econometric analysis of game-theoretic models and applicable under alternative solution concepts such as pure strategy Nash equilibrium or Bayes correlated equilibrium.
Our work also relates to the game theory literature in two dimensions. First, our solution concept adopts the idea of rational expectations equilibrium pioneered by radner1979rational to capture how players refine their information based on market observables in equilibrium situations. Our approach closely follows the logic in liu2020stability, which uses the same idea to define the notion of stability in two-sided markets with incomplete information. Compared to other works that study solution concepts based on rational expectations equilibrium (e.g., green1987posterior, minehart1999expost, minelli2003information, and kalai2004largerobust), we do not assume that actions are generated by a product of individual strategy mappings nor that players' types are fully revealed after actions are realized. Second, our solution concept also adds to the recent literature that studies solution concepts with informational robustness properties (e.g., Bergemann and Morris bergemann2013robustpredictions,bergemann2016bayescorrelated,bergemann2017belieffree and doval2020sequential).
Finally, our empirical application contributes to the literature on entry competition in the fast-food industry. Existing empirical works that study strategic entries by the top burger chains include toivanen2005marketstructure, thomadsen2007product, yang2012burgerking, gayle2015choosing, igami2016unobserved, yang2020learning, and aguirregabiria2020identification. In particular, yang2020learning, which studies strategic entries in the Canadian hamburger industry, shares a similar motivation that players extract information from the opponents' actions, but uses a dynamic games framework to explicitly model the learning process. Our empirical work is distinguished by the use of novel datasets and its focus on exploring the role of informational assumptions. To the best of our knowledge, we are the first to study the impact of local food environment on burger chains' strategic entry decisions.\footnote{For a list of works in economics that study issues related to food deserts, see allcott2019fooddeserts and the references cited therein.}
The rest of the paper is organized as follows. Section (ref) introduces the notion of Bayes stable equilibrium in a general finite game of incomplete information and studies its property. Section (ref) sets up the econometric model and provides identification results. Section (ref) provides econometric strategies for computationally tractable estimation and inference. Section (ref) applies our framework to the entry game played by McDonald's and Burger King in the US. Section (ref) concludes. All proofs are in Appendix (ref).
We consider empirical settings characterized by two properties. First, the setting is dynamic in the sense that players can revise their actions after observing the opponents' actions.\footnote{We use “dynamic” to mean that each player can react to the realized actions of the opponents. However, we do not introduce standard dynamic games assumptions (e.g., finite number of periods, timing of moves, etc.) to model players' interactions. } Second, players' actions are readily and publicly observed by others. In other words, we focus on certain “steady-state” situations in which all players publicly observe each other's realized actions, yet no deviation occurs even when they have the opportunity to do so. Our objective is to describe such situations as a static equilibrium. When conducting econometric analysis, we will assume that the analyst observes a cross-section of stable outcomes.
In this section, we introduce Bayes stable equilibrium as a solution concept that solves the consistency problem and facilitates econometric analysis while allowing for weak assumptions on players' information. Throughout the paper, we assume that the state of the world remains persistent enough to abstract away from transitions over time, and that the costs of revising actions are sufficiently low so that we can ignore them.\footnote{The zero adjustment cost assumption is not essential for the key ideas of this paper. In the real world, the costs of revising actions are not zero. However, the relevant question is whether high adjustment costs drive stable outcomes. We treat adjustment costs as negligible compared to the long-run profits obtained at stable outcomes. This is in the same spirit as the empirical matching models surveyed in chiappori2016theeconometrics; the stable matching condition abstracts away from the costs of entering into or exiting a marriage. Similar assumptions assumptions are commonly used for econometric models of network formation although forming networks can be costly in reality (see, e.g., de2020econometric). The assumption is useful for motivating an alternative to the Nash framework and simplifying exposition. See Appendix (ref) for further discussion.} We formalize the idea in a general class of discrete games of incomplete information, following the notation of bergemann2016bayescorrelated.
We proceed as follows. In Section (ref), we lay out the game environment. In Section (ref), we formalize the notion of stable outcomes and motivate our solution concept. In Section (ref), we argue that rational expectations equilibrium � la radner1979rational can be used as a baseline solution concept for rationalizing stable outcomes in the presence of incomplete information. In Section (ref), we introduce Bayes stable equilibrium. Then, in Section (ref), we show that Bayes stable equilibrium characterizes the implications of rational expectations equilibria when the players might observe more information than assumed by the analyst. In Section (ref), we compare the proposed solution concepts to pure strategy Nash equilibrium and Bayes correlated equilibrium. Finally, in Section (ref), we discuss issues around the existence and uniqueness of the proposed solution concepts.
Let $\mathcal{I}=\left\{ 1,2...,I\right\} $ be the set of players. The players interact in a finite game of incomplete information $\left(G,S\right)$.\footnote{Throughout this paper, we assume that the state space is finite. The assumption simplifies the notation. In addition, even though continuous state space can be used, we will eventually need to discretize the space for feasible estimation. magnolfi2021estimation and \citet*{syrgkanis2021inference} take similar discretization approaches for estimation with Bayes correlated equilibria. } A basic game $G=\langle\mathcal{E},\psi,\left(\mathcal{A}_{i},u_{i}\right)_{i=1}^{I}\rangle$ specifies the payoff-relevant primitives: $\mathcal{E}$ is a finite set of unobserved states; $\psi\in\Delta\left(\mathcal{E}\right)$ is a common prior distribution with full support; $\mathcal{A}_{i}$ is a finite set of actions available to player $i$, and $\mathcal{A}\equiv\times_{i=1}^{I}\mathcal{A}_{i}$ is the set of action profiles; $u_{i}:\mathcal{A}\times\mathcal{E}\to\mathbb{R}$ is player $i$'s von Neumann\textendash Morgenstern utility function. An information structure $S=\langle\left(\mathcal{T}_{i}\right)_{i=1}^{I},\pi\rangle$ specifies the information-related primitives: $\mathcal{T}_{i}$ is a finite set of signals (or types), and $\mathcal{T}\equiv\times_{i=1}^{I}\mathcal{T}_{i}$ is the set of signal profiles; $\pi:\mathcal{E}\to\Delta\left(\mathcal{T}\right)$ is a signal distribution, which allows players' signals to be arbitrarily correlated. The interpretation is that the state of the world $\varepsilon\in\mathcal{E}$, which is drawn from the prior $\psi$, is not directly observed by the players, but each player $i$ receives a private signal $t_{i}\in\mathcal{T}_{i}$ whose informativeness about $\varepsilon$ depends on the signal distribution $\pi$. The game is common knowledge to the players. As highlighted by bergemann2016bayescorrelated, the separation between the basic game and the information structure facilitates the analysis on the role of information structures.
In empirical applications, there is a finite set of exogenous observable covariates $\mathcal{X}$. We can augment the notation and let $\left(G^{x},S^{x}\right)$ describe the game in markets with characteristics $x\in\mathcal{X}$. Indexing each game by $x$ is justified by assuming that $x$ is common-knowledge to the players and that the game primitives are functions of $x$. We suppress the dependence on $x$ for now.
The following two-player entry game serves as a running example as well as a baseline model for our empirical application.
Let us formalize the notion of stable outcomes and motivate our solution concept.\footnote{The term “stability” has been used in different ways in the theory literature depending on the context. Our notion of stability is the closest to “stable matching” defined in liu2020stability under incomplete information matching games (the canonical complete information stable matching is a special case). There is also “hindsight (or ex-post) stability” of kalai2004largerobust, whose motivation is very similar to ours but differs in that it also requires players' types to be revealed after the play. To the best of our knowledge, the term “Bayes stable equilibrium” has not been used in the literature.} Suppose that, at some point in time, the state of the world is $\varepsilon$, the private signals are $t=\left(t_{1},...,t_{I}\right)$, and the players' decisions are $a=\left(a_{1},...,a_{I}\right)$. Assume that each player $i$ observes her private signal $t_{i}$ as well as the outcome $a$. What are the conditions for having no deviation at this situation? A necessary condition is that each player $i$ holds a belief $\mu^{i}\in\Delta\left(\mathcal{E}\right)$ that gives no incentive to deviate from the status quo outcome $a$ unilaterally.
In addition to actions being optimal with respect to the beliefs, a sensible equilibrium would require that the beliefs reflect each player's private information as well as the information revealed from observing opponents' decisions.
But how do these beliefs arise? In general, static Bayes Nash equilibrium will not generate stable outcomes and stable beliefs; players may have incentives to revise their actions after directly observing opponents' decisions and updating beliefs by inverting the equilibrium strategies.\footnote{There may be special classes of games where ex post regret does not arise or can be limited in a Bayes Nash equilibrium. kalai2004largerobust studies hindsight stability in a special class of games with many players. Mathevet2022 study a class of information structures called “single meeting schemes” in which a subset of players participant in a meeting and get equally informed while the non-participants stay uninformed. In this case, the informed players are not subject to regret in a pure Bayes Nash equilibrium because they have (weakly) more information than others and thus can predict all players' actions (although the uninformed players may regret their actions after observing the actions of the informed players).} While it is natural to ask whether we can use a noncooperative dynamic game to model convergence to a pair of stable decisions and stable beliefs, such route is likely to be non-trivial and dependent on ad hoc assumptions. In the following sections, we propose a simple and pragmatic approach to the problem.
Before introducing Bayes stable equilibrium, which will be the solution concept we take to econometric analysis, we define a version of rational expectations equilibrium � la radner1979rational that offers a simple conceptual framework for rationalizing stable outcomes in the presence of incomplete information. To define rational expectations equilibrium appropriately in our setting, we follow liu2020stability and use the “outcome function” approach described as follows.\footnote{An analog of an outcome function in liu2020stability is the matching function that maps players' types to an observable match. In noncooperative games settings, minehart1999expost and minelli2003information have made similar attempts to connect rational expectations equilibrium to games without price. While their definition of rational expectations equilibrium refers to strategy profiles, we take a “cooperative” approach and use outcomes functions, which are not necessarily the product of individual strategy mappings.}
Let a game $\left(G,S\right)$ be given. Let $\delta:\mathcal{T}\to\Delta\left(\mathcal{A}\right)$ be an outcome function in $\left(G,S\right)$; an outcome function specifies a probability distribution over action profiles at each realization of players' signals.
Assume that $\delta$ is common knowledge to the players. Suppose that, after the state of the world $\varepsilon\in\mathcal{E}$ and the signal profile $t\in\mathcal{T}$ are realized according to the prior distribution $\psi\left(\cdot\right)$ and the signal distribution $\pi\left(\cdot\vert\varepsilon\right)$, an action profile $a\in\mathcal{A}$ is drawn from the outcome function $\delta\left(\cdot\vert t\right)$, and the players publicly observe $a$. Each player $i$, having observed his private signal and the realized action profile $\left(t_{i},a_{i},a_{-i}\right)$, updates his beliefs about the state of the world $\varepsilon$ using Bayes' rule, and decides whether to adhere to the observed outcome (play $a_{i}$) or not (deviate to $a_{i}'\neq a_{i}$). If $\delta$ is such that the players always find the realized action profiles optimal, we call it a rational expectations equilibrium of $\left(G,S\right)$. Let $\mathbb{E}_{\varepsilon}^{\delta}\left[u_{i}\left(a_{i}',a_{-i},\varepsilon\right)\vert t_{i},a_{i},a_{-i}\right]$ denote the expected payoff to player $i$ from choosing $a_{i}'$ conditional on observing private signal $t_{i}$ and action profile $\left(a_{i},a_{-i}\right)$.
The outcome function $\delta:\mathcal{T}\to\Delta\left(\mathcal{A}\right)$ represents a reduced-form relationship between players' information and the outcome of the game. We are agnostic about the details on how $\delta$ came about. However, it is assumed that the players agree on $\delta$, and use it to infer opponents' information after observing the realized decisions. Thus, $\delta$ serves as the players' “model” for connecting the uncertainties to the observables.
There is nothing conceptually new; we simply apply the idea of rational expectations equilibrium to our setting. Rational expectations equilibrium refers to a mapping from agents' information to observable market outcomes such that the agents do not have incentives to deviate after observing the realized market outcomes. The key idea is that if the final market outcome is observable and depends on agents' signals about the state of the economy, then the agents must be able to learn others' information based on their observation of the market outcome. The agents are said to have “rational expectations” because they refine their information based on the information available at the equilibrium situation. In radner1979rational, there is a price function (or a forecast function) that maps agents' signals to market price. The agents use their observation of their price to not only calculate their budget but also to infer others' information via the price function. In liu2020stability, there is a matching function that maps agents' signals to a (two-sided) match. The agents use their observation of a match to infer others' information before assessing whether they have (unilateral and pairwise) incentives to deviate from a given match. Although the exact definition varies by economic environment\textemdash depending on endogenous outcomes of the model and agents' optimality conditions\textemdash the logic is parallel.
In a rational expectations equilibrium, outcomes and beliefs are determined simultaneously such that the stability condition ((ref)) is satisfied. If the environment\textemdash the state of the world and the players' signals\textemdash stays unchanged and the outcomes are generated by a rational expectations equilibrium, the realized decisions persist over time. In the econometric analysis, we assume that the analyst observes these decisions at some point in time.
Let us introduce Bayes stable equilibrium. Let $\left(G,S\right)$ be given. A decision rule in $\left(G,S\right)$ is a mapping $\sigma:\mathcal{E}\times\mathcal{T}\to\Delta\left(\mathcal{A}\right)$ that specifies a probability distribution over action profiles at each realization of state and signals. Assume that $\sigma$ is common knowledge to the players. Suppose the data generating process is described as follows. First, the state of the world $\varepsilon\in\mathcal{E}$ is drawn from $\psi\left(\cdot\right)$, and the profile of private signals $t\in\mathcal{T}$ is drawn from $\pi\left(\cdot\vert\varepsilon\right)$. Next, an action profile $a\in\mathcal{A}$ is drawn from $\sigma\left(\cdot\vert\varepsilon,t\right)$ and publicly observed by the players. Then, each player $i$, having observed her private signal and the realized action profile $\left(t_{i},a_{i},a_{-i}\right)$, updates her belief about the state of the world $\varepsilon$ using Bayes' rule and decides whether to adhere to the observed outcome (play $a_{i}$) or not (deviate to $a_{i}'\neq a_{i}$). If the players always have no incentives to deviate from the realized action profiles, we call $\sigma$ a Bayes stable equilibrium.
It is helpful to interpret $\sigma$ as the recommendation strategy of an omniscient mediator. The mediator commits to $\sigma$ and announces it to the players at the beginning of the game. Then, after observing the realized $\left(\varepsilon,t\right)$, the mediator draws an action profile $a$ from $\sigma\left(\cdot\vert\varepsilon,t\right)$ and publicly recommends it to the players. The Bayes stable equilibrium condition requires that the publicly recommended action profiles are always incentive compatible to the players.
Note that an outcome function $\delta$ does not depend on the state of the world $\varepsilon$ whereas a decision rule $\sigma$ can. The measurability of an outcome function with respect to players' information reflects the requirement that if any outcome is to be achieved, it cannot depend on what they do not know. On the other hand, a decision rule allows the realized action profiles to be correlated with the unobserved state. In the next section, we show that the correlation arises because Bayes stable equilibrium captures the implications of rational expectations equilibria when the players might observe extra signals about the state of the world that are unknown to the analyst.
We can simplify the obedience condition ((ref)) so that the equilibrium conditions are linear in the decision rule. Given that player $i$ observes signal $t_{i}$ and recommendation $\left(a_{i},a_{-i}\right)$, the expected payoff from choosing $a_{i}'$ is
Then, after cancelling out the denominator, which is constant across all possible realizations of $\varepsilon\in\mathcal{E},t_{-i}\in\mathcal{T}_{-i}$, the obedience condition ((ref)) can be rewritten as\footnote{Using a similar argument, we can express the rational expectation equilibrium conditions for an outcome function $\delta$ in $\left(G,S\right)$ as: $\sum_{\varepsilon,t_{-i}}\psi_{\varepsilon}\pi_{t\vert\varepsilon}\delta_{a\vert t}u_{i}\left(a,\varepsilon\right)\geq\sum_{\varepsilon,t_{-i}}\psi_{\varepsilon}\pi_{t\vert\varepsilon}\delta_{a\vert t}u_{i}\left(a_{i}',a_{-i},\varepsilon\right),\ \forall i,t_{i},a,a_{i}'.$}
Since $\sigma$ enters the expression linearly, finding a Bayes stable equilibrium solves a linear feasibility program, a feature that renders estimation computationally tractable.
In Section (ref), we have argued that an analyst can use rational expectations equilibrium as a description of stable outcomes under incomplete information situations. More often than not, however, it is difficult for the analyst to know the true information structure governing the data generating process. Attempts to characterize all feasible predictions (joint distribution on states, signals, and actions) of a model by a direct enumeration over all possible information structures are likely to be futile since the set of information structures is large. How might the analyst proceed without making strong assumptions on players' information?
We show that Bayes stable equilibrium provides a tractable characterization of all rational expectations equilibrium predictions that can arise when the players might observe more information than assumed by the analyst. Thus, Bayes stable equilibrium serves as a tool for analyzing stable outcomes with weak assumptions on players' information. The informational robustness property closely resembles that of Bayes correlated equilibrium (established in Theorem 1 of bergemann2016bayescorrelated), namely that Bayes correlated equilibrium provides a shortcut to charactering all Bayes Nash equilibrium predictions that can arise when the players might observe more information than specified by the analyst.
We formalize the idea as follows. First, to capture the idea that players observe more information under one information structure than under another, we introduce the notion of expansion defined in bergemann2016bayescorrelated.
Intuitively, $S^{*}$ is an expansion of $S$ if each player is allowed to observe more signals under $S^{*}$ than under $S$. In other words, in $S$, each player $i$ observes a private signal $t_{i}$, whereas in $S^{*}$, each $i$ gets to observe an additional signal $\tilde{t}_{i}$ generated by an augmenting signal distribution $\lambda$. The notion of expansion defines a partial order $\succsim_{E}$ on the set of information structures.
Let $\mathcal{P}_{\varepsilon,t,a}^{BSE}\left(G,S\right)$ be the set of joint distributions on $\mathcal{E}\times\mathcal{T}\times\mathcal{A}$ that can arise in a Bayes stable equilibrium of $\left(G,S\right)$. Let $\mathcal{P}_{\varepsilon,t,a}^{REE}\left(G,S\right)$ be defined similarly. Note that if $S^{*}\succsim_{E}S$, a joint distribution on $\mathcal{E}\times\mathcal{T}^{*}\times\mathcal{A}$ induce a marginal on $\mathcal{E}\times\mathcal{T}\times\mathcal{A}$. The following theorem states that by considering Bayes stable equilibrium of $\left(G,S\right)$, we can capture all joint distributions on $\mathcal{E}\times\mathcal{T}\times\mathcal{A}$ that can arise in a rational expectations equilibrium under some information structure that is more informative than $S$.
The proof of the theorem closely follows that of bergemann2016bayescorrelated Theorem 1. The “$\subseteq$” direction is established by taking the equilibrium decision rule $\sigma:\mathcal{E}\times\mathcal{T}\to\Delta\left(\mathcal{A}\right)$ as an augmenting signal function that generates a “public signal” $a$ that is commonly observed by the agents. We then construct a trivial outcome function $\delta$ that places unit mass on the recommended outcome, i.e., $\delta\left(\tilde{a}\vert a\right)=1$ if and only if $\tilde{a}=a$. Then the rational expectations equilibrium condition for $\delta$ in the game with augmented information structure is implied by the obedience condition for $\sigma$. Conversely, the “$\supseteq$” direction is established by integrating out the “extra signals” $\tilde{t}_{i}$ from the rational expectations equilibrium condition, which directly implies the obedience condition for the induced decision rule $\sigma\left(a\vert\varepsilon,t\right)\equiv\sum_{\tilde{t}}\lambda\left(\tilde{t}\vert\varepsilon,t\right)\delta\left(a\vert t,\tilde{t}\right)$.
Theorem (ref) can be framed in terms of marginal distributions on the action profiles. This characterization is more relevant for econometric analysis since typical data contain information on players' decisions but not the signals nor the state of the world. Let $\mathcal{P}_{a}^{BSE}\left(G,S\right)$ be the set of marginal distributions on $\mathcal{A}$ that can arise in a Bayes stable equilibrium of $\left(G,S\right)$. Let $\mathcal{P}_{a}^{REE}\left(G,S\right)$ be defined similarly.
Intuitively, allowing more information to the players should shrink the set of equilibria because it tightens the obedience constraints. The following corollary formalizes the idea.
In this section, we compare our solution concepts to other existing solution concepts that have been frequently employed for empirical analysis. First, we compare rational expectations equilibrium and pure strategy Nash equilibrium, which are decentralized solution concepts. We show that our framework attains pure strategy Nash equilibrium as a special case. Second, we compare Bayes stable equilibrium and Bayes correlated equilibrium, which are centralized solution concepts that rely on a mediator analogy. We show that Bayes stable equilibrium refines Bayes correlated equilibrium as the former imposes stronger restrictions than the latter. We also provide an overview of how our framework relates to the literature.
The following theorem says that pure strategy Nash equilibrium arises as a special case of rational expectations equilibrium (or Bayes stable equilibrium) when strong assumptions on players' information are made.
Theorem (ref).(ref) states that when information is complete, rational expectations equilibrium is observationally equivalent to pure strategy Nash equilibrium. A rational expectations equilibrium outcome function $\delta$ is just a selection device over pure strategy Nash outcomes. It also implies that, when players have complete information, a rational expectations equilibrium exists if and only if there is at least one pure strategy Nash equilibrium action profile at each $\varepsilon\in\mathcal{E}$ (on the support of $\psi$).
Theorem (ref).(ref) states that when $\varepsilon$ is simply a vector of player-specific payoff shocks\textemdash a common assumption for empirical models of discrete games\textemdash we can use weaker informational assumptions to rationalize pure strategy Nash outcomes. Intuitively, when each player $i$ observes his type $\varepsilon_{i}$ and an outcome $a$ in an equilibrium situation, opponents' types $\varepsilon_{-i}$ are payoff-irrelevant. In a pure strategy Nash equilibrium, $i$ uses its knowledge of $\varepsilon_{-i}$ to predict $a_{-i}$. However, in a rational expectations equilibrium, $i$ observes $a_{-i}$, so $\varepsilon_{-i}$ plays no role for $i$. Therefore, under the rational expectations equilibrium assumption, it is sufficient that player $i$ observes $\varepsilon_{i}$ in order to support pure strategy Nash outcomes.
Note that under the assumptions in the theorem, there is no material difference between an outcome function and a decision rule because players' signals exhaust information about the state of the world, so Bayes stable equilibrium and rational expectations equilibrium are equivalent.
Bayes stable equilibrium refines Bayes correlated equilibrium because equilibrium conditions for the former are stronger. To describe Bayes correlated equilibrium, suppose that an omniscient mediator commits to a decision rule $\sigma:\mathcal{E}\times\mathcal{T}\to\Delta\left(\mathcal{A}\right)$ in $\left(G,S\right)$ and announces it to the players so that $\sigma$ is common knowledge to the players. After the state $\varepsilon$ and signal profile $t$ are drawn from $\psi$ and $\pi$ respectively, the mediator observes $\left(\varepsilon,t\right)$ and draws an action profile $a$ from the decision rule $\sigma\left(\cdot\vert\varepsilon,t\right)$. Then, the mediator privately recommends $a_{i}$ to each player $i$. Each player $i$, having observed his private signal $t_{i}$ and the privately recommended action $a_{i}$, decides whether to follow the recommendation (play $a_{i}$) or not (deviate to $a_{i}'\neq a_{i}$). If the players are always obedient, then the decision rule is a Bayes correlated equilibrium of $\left(G,S\right)$.
Formally, a decision rule $\sigma:\mathcal{E}\times\mathcal{T}\to\Delta\left(\mathcal{A}\right)$ in $\left(G,S\right)$ is a Bayes correlated equilibrium if for each $i\in\mathcal{I}$, $t_{i}\in\mathcal{T}_{i}$, and $a_{i}\in\mathcal{A}_{i}$, we have \[ \mathbb{E}_{\left(\varepsilon,a_{-i}\right)}^{\sigma}\left[u_{i}\left(a_{i},a_{-i},\varepsilon\right)\vert t_{i},a_{i}\right]\geq\mathbb{E}_{\left(\varepsilon,a_{-i}\right)}^{\sigma}\left[u_{i}\left(a_{i}',a_{-i},\varepsilon\right)\vert t_{i},a_{i}\right] \] for all $a_{i}'\in\mathcal{A}_{i}$ whenever $\text{Pr}^{\sigma}\left(t_{i},a_{i}\right)>0$, or more compactly,
The only difference between Bayes stable equilibrium and Bayes correlated equilibrium is that the former assumes each player $i$ observes $\left(a_{i},a_{-i}\right)$ whereas the latter assumes each $i$ observes only $a_{i}$, but not $a_{-i}$. While the Bayes correlated equilibrium conditions ((ref)) integrate out opponents' actions $a_{-i}$ since each player $i$ needs to form expectation over $a_{-i}$, Bayes stable equilibrium conditions ((ref)) condition on $a_{-i}$ because $a_{-i}$ is observed to $i$ at the equilibrium situation. The following is immediate.
Outcomes generated by a Bayes correlated equilibrium may be subject to regret; a player who observes the realized decisions of the opponents might want to revise her action. In contrast, Bayes stable equilibrium explicitly requires that such regret is absent. When information is complete, Bayes correlated equilibrium reduces to the canonical correlated equilibrium, whereas Bayes stable equilibrium reduces to pure strategy Nash equilibrium in the sense described in Theorem (ref). When there is a single player, the two solution concepts are identical because there is no informational feedback from observing opponents' actions.
Although the relationship between our solution concepts and static Nash equilibrium can be gleaned from the above theorems, we provide a compact review and discuss connections to the related literature for the readers.
Let $\mathcal{P}_{\varepsilon,a}^{SC}\left(G,S\right)$ denote the set of predictions (joint distributions on $\mathcal{E}\times\mathcal{A}$) that can arise in game $\left(G,S\right)$ under solution concept $SC$. We use $\mathcal{P}_{\varepsilon,a}^{NE}\left(G,S\right)$ and $\mathcal{P}_{\varepsilon,a}^{PSNE}\left(G,S\right)$ to represent the set of (mixed-strategy) Bayes Nash equilibrium predictions and pure strategy Bayes Nash equilibrium predictions respectively. Note that $\mathcal{P}_{\varepsilon,a}^{PSNE}\left(G,S\right)\subseteq\mathcal{P}_{\varepsilon,a}^{NE}\left(G,S\right)$ since pure strategies are special cases of mixed strategies. The predictions of various solution concepts are related to each other in the following way.
In the literature on econometric models of games, it has been common to assume that the unobserved state variable $\varepsilon$ is a vector of $\varepsilon_{i}$'s that only enter firm $i$'s payoff (see our running example and Theorem (ref).(ref)). Under this structure on payoffs and states, most papers have assumed that the data generating process can be described by a Nash equilibrium with information structure set to either $S^{complete}$ or $S^{incomplete}$ (an important exception is grieco2014discrete who considers a flexible, but parametric, information structure that nests both). Examples of works that use pure strategy Nash equilibrium under $S^{complete}$, which is a special case of our framework, include Bresnahan and Reiss (bresnahan_entry_1990,bresnahan1991empirical,bresnahan_entry_1991), tamer2003incomplete, ciliberto2009marketstructure, bajari2010identification, kline_identification_2015, and aradillas2022inference. Examples of works that use pure strategy (Bayes) Nash equilibrium under $S^{private}$ include seim_empirical_2006, pesendorfer_asymptotic_2008, sweeting_strategic_2009, aradillas-lopez_semiparametric_2010, and bajari2010estimating. Since generally $\mathcal{P}_{\varepsilon,a}^{PSNE}\left(G,S^{complete}\right)\neq\mathcal{P}_{\varepsilon,a}^{PSNE}\left(G,S^{private}\right)$, the two sets of papers rely on different model predictions.
magnolfi2021estimation motivate their analysis by arguing that researchers often do not know whether the true information structure is $S^{complete}$ or $S^{private}$ and propose using Bayes correlated equilibrium under $S^{private}$. Bayes correlated equilibrium summarizes the implications of Nash equilibrium with unknown information structure in the sense that $\mathcal{P}_{\varepsilon,a}^{BCE}\left(G,S^{private}\right)=\bigcup_{\tilde{S}\succsim S^{priavte}}\mathcal{P}_{\varepsilon,a}^{NE}\left(G,\tilde{S}\right)$, as established by bergemann2016bayescorrelated. \citet*{syrgkanis2021inference} apply Bayes correlated equilibrium to common-value and private-value auctions.
There has been no work that develops an empirical framework to tackle the regret problem associated with Nash equilibrium.\footnote{yang2020learning allows for information updating after observing opponents' actions in the context of fast-food industry, but models the interaction as a dynamic game. In contrast to his framework that requires panel data, our framework allows the researcher to work with cross-sectional data. } Rational expectations equilibrium provides a simple framework for capturing steady state situations in which players observe opponents' actions but do not deviate. Similarly to Bayes correlated equilibrium, Bayes stable equilibrium provides robustness to informational assumptions because $\mathcal{P}_{\varepsilon,a}^{BSE}\left(G,S\right)=\bigcup_{\tilde{S}\succsim_{E}S}\mathcal{P}_{\varepsilon,a}^{REE}\left(G,\tilde{S}\right)$. Complete information pure strategy Nash equilibrium arises as a special case of our framework because $\mathcal{P}_{\varepsilon,a}^{PSNE}\left(G,S^{complete}\right)=\mathcal{P}_{\varepsilon,a}^{BSE}\left(G,S^{complete}\right)\subseteq\mathcal{P}_{\varepsilon,a}^{BSE}\left(G,S\right)$ for any information structure $S$. However, the predictions under rational expectations equilibrium or Bayes stable equilibrium are generally unrelated to incomplete information Nash equilibrium predictions, e.g., $\mathcal{P}_{\varepsilon,a}^{PSNE}\left(G,S^{private}\right)\not\in\mathcal{P}_{\varepsilon,a}^{BSE}\left(G,S^{private}\right)$. Bayes stable equilibrium predictions are tighter than Bayes correlated equilibrium predictions ($\mathcal{P}_{\varepsilon,a}^{BSE}\left(G,S\right)\subseteq\mathcal{P}_{\varepsilon,a}^{BCE}\left(G,S\right)$ for any $\left(G,S\right)$). In the empirical application, we show that Bayes stable equilibrium can lead to a substantially tighter identified set compared to Bayes correlated equilibrium; leveraging the assumption that market outcomes are readily observed by the players can add substantial identifying power.
The reader may wonder about high-level conditions for the existence and uniqueness of Bayes stable equilibrium. Unfortunately, we do not have results applicable to a large class of games relevant to empirical work. The existence and uniqueness of Bayes stable equilibrium are generally not guaranteed. For instance, in the matching pennies game, there is no Bayes stable equilibrium because there is always one player who wants to deviate. In the battle of the sexes game, there is a continuum of Bayes stable equilibria because any decision rule that represents a mixture over the two pure strategy Nash equilibrium action profiles corresponds to a Bayes stable equilibrium.
The task is actually non-trivial even for complete information pure strategy Nash equilibrium (which Bayes stable equilibrium boils down to when information is complete) in discrete games environment because we cannot use the standard fixed point theorems that depend on continuity, convexity, and compactness. Existence and uniqueness of pure strategy Nash equilibria in discrete games are typically checked numerically (e.g., ciliberto2009marketstructure's algorithm enumerates over all action profiles at each state to find all pure strategy Nash equilibria). Similarly, the existence and uniqueness of Bayes stable equilibrium should be checked on a case-by-case basis.
Fortunately, this paper provides positive results. First, knowledge about the existence and uniqueness of complete information pure strategy Nash equilibrium can be used to infer the existence and uniqueness of Bayes stable equilibrium (see Theorem (ref)). The researcher can apply this result when dealing with a class of games for which the existence and uniqueness of pure strategy Nash equilibrium are well-understood (e.g., two-player entry games). Second, numerically checking for existence of a Bayes stable equilibrium (or a rational expectations equilibrium) can be done quickly by solving a linear program. To the best of our knowledge, a linear programming approach to checking the existence of complete information pure strategy Nash equilibrium is new.
In this section, we describe the econometric model. We characterize the identified set under the assumption that data are generated by a Bayes stable equilibrium and discuss its properties.
Let us denote observable market covariates as $x\in\mathcal{X}$ where $\mathcal{X}$ is a finite set; $x$ is common knowledge to the players and observed by the econometrician. At each $x\in\mathcal{X}$, the player interact in a game $\left(G^{x,\theta},S^{x}\right)$ where $G^{x,\theta}=\langle\mathcal{E},\psi^{x,\theta},\left(\mathcal{A}_{i},u_{i}^{x,\theta}\right)_{i=1}^{I}\rangle$ is the basic game, $S^{x}=\langle\left(\mathcal{T}_{i}\right)_{i=1}^{I},\pi^{x}\rangle$ is the information structure, and $\theta\in\Theta$ is a finite-dimensional parameter the analyst wish to identify.\footnote{It is without loss to assume that $\mathcal{E}$ and $\mathcal{T}$ do not depend on $x$ because we can use $\mathcal{E}\equiv\cup_{x}\mathcal{E}^{x}$ and $\mathcal{T}\equiv\cup_{x}\mathcal{T}^{x}$. In principle, we can also let $\theta$ enter the information structures, which would make the information structures be part of the objects the econometrician wants to identify. In this paper, however, we focus on the case where $\theta$ only enters the payoff functions and the distribution of the payoff shocks.} We maintain the assumption that the set $\mathcal{E}$ is finite in order to make estimation feasible.\footnote{If the benchmark distribution of unobservables is continuous, it will be discretized. Increasing the number of points in $\mathcal{E}$ can make the discrete approximation more accurate at the expense of increased computational burden. See Appendix (ref) for the details on how we make discrete approximations to continuous distributions.} The parameter $\theta$ enters the prior distributions $\psi^{x,\theta}\in\Delta\left(\mathcal{E}\right)$ and the payoff functions $u_{i}^{x,\theta}:\mathcal{A}\times\mathcal{E}\to\mathbb{R}$. As standard in the empirical literature, we assume that the state of the world is a vector of player-specific payoff shocks, i.e., $\varepsilon=\left(\varepsilon_{1},\varepsilon_{2},...,\varepsilon_{I}\right)$ and $u_{i}^{x,\theta}\left(a,\varepsilon\right)=u_{i}^{x,\theta}\left(a,\varepsilon_{i}\right)$.
The data $\left\{ \left(a_{m},x_{m}\right)\right\} _{m=1}^{n}$ represent a cross-section of action profiles and covariates in markets $m=1,...,n$ that are independent from each other. Let $\phi^{x}\in\Delta\left(\mathcal{A}\right)$ denote the conditional choice probabilities that represent the probability of observing each action profile conditional on covariate value $x$. We assume that the econometrician can identify $\phi^{x}$ at each $x\in\mathcal{X}$ as $n\to\infty$. The set of baseline assumptions for identification analysis is summarized below.
Given Assumption (ref), the identified set of parameters can be defined when the solution concept and the information structure are specified. For any game $\left(G^{x,\theta},S^{x}\right)$, let $\mathcal{P}_{a}^{SC}\left(G^{x,\theta},S^{x}\right)$ be the set of feasible probability distributions over action profiles under solution concept $SC$.
In words, a candidate parameter $\theta$ enters the identified set $\Theta_{I}^{SC}\left(\tilde{S}\right)$ if at each $x\in\mathcal{X}$, the observed conditional choice probabilities $\phi^{x}$ can arise under some equilibrium.
Let us translate the observational equivalence between rational expectations equilibrium and Bayes stable equilibrium (Corollary (ref)) in terms of identified sets. Consider the following assumption.
Assumption (ref) says that there is a true parameter $\theta_{0}$ underlying the data generating process, and that at each $x\in\mathcal{X}$, the true information structure is some $\tilde{S}^{x,0}$ that is an expansion of $S^{x}$. In practice, we will consider a scenario where the econometrician knows the baseline information structure $S^{x}$, which describes the minimal information available to the players but not the true information structure $\tilde{S}^{x,0}$. Then, under Assumptions (ref) and (ref), the econometrician has to admit all information structures that are expansions of $S^{x}$. This approach contrasts with the traditional approach that assumes the econometrician knows the true information structure exactly.
However, directly working with Assumption (ref) is computationally infeasible because it requires searching over the set of information structures, which is large. We show that Assumption (ref) can be replaced with the following assumption, which does not rely on unknown information structures.
The following theorem is the consequence of Corollary (ref); Assumption (ref) and Assumption (ref) are observationally equivalent.
Theorem (ref) says that in order to compute the identified set when the data are generated by some rational expectations equilibrium but with an unknown information structure, we can proceed as if the data are generated by a Bayes stable equilibrium with known information structure.
magnolfi2021estimation and \citet*{syrgkanis2021inference} develop a similar approach for informationally robust estimation of games, but use Bayes correlated equilibrium as the solution concept. They assume that the underlying data generating process is described by Bayes Nash equilibria, whereas we rely on rational expectations equilibria. Also see gualdani2020identification for the single-agent case.
Our identification results make no assumptions on the equilibrium selection rule. The Bayes stable equilibrium identified set under Assumptions (ref) and (ref) is valid even when the data are generated from a mixture of multiple equilibria. The convexity of the set of Bayes stable equilibria (readily verified from the equilibrium conditions ((ref)) since $\sigma$ enters the expression linearly) makes the single equilibrium assumption innocuous. For example, if the data are generated by two equilibria $\sigma^{1}$ and $\sigma^{2}$ with mixture probability $\lambda$ and $\left(1-\lambda\right)$, then since $\sigma^{\lambda}\equiv\lambda\sigma^{1}+\left(1-\lambda\right)\sigma^{2}$ is another equilibrium that generates the same joint distributions, it is as if the data were generated by a single equilibrium $\sigma^{\lambda}$.\footnote{\citet*{syrgkanis2021inference} Lemma 2 presents a general argument on why it is without loss to assume that the data are generated by a single equilibrium if the set of predictions is convex.}
Recall from Example (ref) that in $S^{complete}$ each player $i$ observes the realization of $\varepsilon$, and in $S^{private}$ each player $i$ observes the realization of $\varepsilon_{i}$. We let $\Theta_{I}^{SC}\left(S^{complete}\right)$ denote the identified set when $S^{x}=S^{complete}$ at every $x\in\mathcal{X}$; $\Theta_{I}^{SC}\left(S^{private}\right)$ is defined similarly. Finally, we write $S^{1}\succsim_{E}S^{2}$ if and only if $S^{1,x}\succsim_{E}S^{2,x}$ at every $x\in\mathcal{X}$. The following theorem shows the relationship between identified sets.
First, Theorem (ref).(ref) says that a stronger assumption on information leads to a tighter identified set. The result directly follows from Corollary (ref), which says that the feasible set of equilibria shrinks when more information is available to the players. A consequence of Theorem (ref).(ref) is that we will have $\Theta_{I}^{BSE}\left(S^{complete}\right)\subseteq\Theta_{I}^{BSE}\left(\tilde{S}\right)\subseteq\Theta_{I}^{BSE}\left(S^{null}\right)$ for any $\tilde{S}$, i.e., the tightest identified set is obtained when $S^{complete}$ is assumed and the loosest identified set is obtained when $S^{null}$ is assumed. Note that $\Theta_{I}^{BSE}\left(S^{null}\right)$ corresponds to the identified set that makes no assumption on players' information.
Second, Theorem (ref).(ref), which is a consequence of Theorem (ref), says that Bayes stable equilibrium and pure strategy Nash equilibrium are observationally equivalent when $S^{complete}$ is assumed.\footnote{When Assumption (ref).(ref) is imposed, rational expectations equilibrium and Bayes stable equilibrium are identical under $S^{private}$ and $S^{complete}$. This is because a profile of players' signals is equal to the state of the world, so conditioning on players' information is equivalent to conditioning on the state of the world.} Furthermore, due to Assumption (ref).(ref), Bayes stable equilibrium can deliver the same identified set under $S^{private}$ which is weaker than $S^{complete}$. Thus, if the researcher takes Bayes stable equilibrium (or rational expectations equilibrium) to be a reasonable notion for the given empirical setting, pure strategy Nash equilibrium outcomes can be rationalized with informational assumptions that are weaker than the complete information assumption.
Finally, Theorem (ref).(ref), which follows from Theorem (ref), says that for any baseline assumption on players' information, the Bayes stable equilibrium identified set is a subset of the Bayes correlated equilibrium identified set.
We use a two-player entry game (our running example) to numerically illustrate the identifying power of various informational assumptions in the spirit of aradillas-lopez_identification_2008. We also compare the identifying power to that of Bayes correlated equilibrium studied in magnolfi2021estimation.
Each player's payoff function is $u_{i}^{\theta}\left(a_{i},a_{j},\varepsilon_{i}\right)=a_{i}\left(\kappa_{i}a_{j}+\varepsilon_{i}\right)$. We assume $\left(\varepsilon_{1},\varepsilon_{2}\right)$ follows a bivariate normal distribution with zero mean, unit variance, and zero correlation. As a discrete approximation to the prior distribution, we use a grid of 30 points for each $\mathcal{E}_{i}$ and a Gaussian copula to assign appropriate probability mass on each grid point $\left(\varepsilon_{1},\varepsilon_{2}\right)$.\footnote{Computational details can be found in Appendix (ref).} We set $\theta=\left(\kappa_{1},\kappa_{2}\right)=\left(-1.0,-1.0\right)$ and generate choice probabilities using the pure strategy Nash equilibrium assumption with arbitrary selection rule.\footnote{Specifically, we generate population choice probability by finding a feasible $\sigma:\mathcal{E}\to\Delta\left(\mathcal{A}\right)$ which satisfies the inequalities in ((ref)) as described in Section (ref).}
To construct the identified sets, we take the distribution of unobservables as known, and collect all points $\left(\kappa_{1},\kappa_{2}\right)$ compatible with the given solution concept and informational assumptions. We plot the convex hulls of the identified sets in Figure (ref).
Figure (ref)-(a) shows the BSE identified sets obtained under different baseline information structures. The identified sets shrink as the informational assumptions get stronger. We omit the complete information case since $\Theta_{I}^{BSE}\left(S^{private}\right)=\Theta_{I}^{BSE}\left(S^{complete}\right)$. Setting the baseline information structure as $S^{null}$ generates an identified set that is quite permissive while using $S^{private}$ generates a tight identified set. Note that $\Theta_{I}^{BSE}\left(S^{null}\right)$ amounts to making no assumption on what the players minimally observe, and $\Theta_{I}^{BSE}\left(S^{private}\right)$ is equal to the PSNE identified set. Similarly, Figure (ref)-(b) plots the BCE identified sets obtained under different baseline information structures. It shows that stronger assumptions on information lead to tighter identified sets. Assumptions on players' information play a crucial role in determining the size of the identified set. In this sense, imposing strong assumption on players' information may be far from innocuous because it places strong restrictions for identification.
As stated in Theorem (ref).(ref), comparing Figure (ref)-(a) and (ref)-(b) shows that, for any given baseline information structure, the corresponding BSE identified set is a subset of the corresponding BCE identified set. In our example, under the same informational assumption, the BSE identified set can be substantially tighter than the BCE identified set, illustrating the identifying power of leveraging observability of opponents' actions in the equilibrium conditions.
We propose a computationally attractive approach for estimation and inference. In Section (ref), we show that whether a candidate parameter enters the identified set can be determined by solving a single linear feasibility program. In Section (ref), we show that this property can be combined with the insights from horowitz2021inference to make construction of confidence sets simple and computationally tractable: determining whether a candidate parameter enters the confidence set amounts to solving a convex feasibility program. Finally, in Section (ref), we provide some practical suggestions for computational implementations.
We provide a computationally attractive characterization of the identified set. \citet*{syrgkanis2021inference} uses a similar characterization, but with Bayes correlated equilibrium. Bayes stable equilibrium and Bayes correlated equilibrium share similar computational property since decision rules enter the equilibrium conditions linearly in both cases.
Let $\Theta_{I}\equiv\Theta_{I}^{BSE}\left(S\right)$ denote the sharp identified set. Let $\partial u_{i}^{x,\theta}\left(a_{i}',a,\varepsilon_{i}\right)\equiv u_{i}^{x,\theta}\left(a_{i}',a_{-i},\varepsilon_{i}\right)-u_{i}^{x,\theta}\left(a_{i},a_{-i},\varepsilon_{i}\right)$ denote the gains from unilaterally deviating to $a_{i}'$ from outcome $\left(a_{i},a_{-i}\right)$ given $\varepsilon_{i}$. Recall our notation that $\sigma^{x}\in\Delta_{a\vert\varepsilon,t}$ if and only if $\sigma_{a\vert\varepsilon,t}^{x}\geq0$ for all $a,\varepsilon,t$ and $\sum_{a\in\mathcal{A}}\sigma_{a\vert\varepsilon,t}^{x}=1$.
Theorem (ref) says that for any candidate $\theta\in\Theta$, whether $\theta\in\Theta_{I}$ can be determined by solving a single linear feasibility program. The first condition ((ref)) states that the nuisance parameter $\sigma^{x}$ should be a decision rule that satisfies the Bayes stable equilibrium conditions. The second condition ((ref)) states that the observed conditional choice probabilities must be consistent with those induced by the equilibrium decision rule. Given a candidate $\theta$ as fixed, $\psi_{\varepsilon}^{x,\theta}$, $\pi_{t\vert\varepsilon}^{x}$, $\partial u_{i}^{x,\theta}$, and $\phi_{a}^{x}$ are known objects. Also note that $\sigma^{x}\in\Delta_{a\vert\varepsilon,t}$ represent constraints that are linear in $\sigma^{x}$. Then, since the variables of optimization $\sigma^{x}$ enter the constraints linearly, the program is linear.
Since our empirical framework obtains pure strategy Nash equilibrium as a special case, the complete information pure strategy Nash equilibrium identified set can be computed using linear programs as well. Let $\Theta_{I}^{PSNE}$ be the sharp identified set obtained under the pure strategy Nash equilibrium assumption and no assumption on the equilibrium selection rule. As a corollary to Theorem (ref) and Theorem (ref), whether $\theta\in\Theta_{I}^{PSNE}$ can also be determined via a single linear feasibility program. Thus, Bayes stable equilibrium identified sets embed the pure strategy Nash equilibrium identified set studied in \citet*{beresteanu2011sharpidentification} and galichon2011setidentification as a special case.
We leverage the insights from horowitz2021inference and propose a simple approach to inference on the structural parameters.\footnote{horowitz2021inference describe methods for carrying out non-asymptotic inference when the partially identified parameters are solutions to a class of optimization problem. While we leverage the insights from their work, we focus on asymptotic inference with multinomial proportion parameters.} The key idea behind our approach is summarized as follows. In discrete games, all information in the data is summarized by the conditional choice probabilities, as apparent in Theorem (ref). The statistical sampling uncertainty arises only from the estimation of the unknown population conditional choice probabilities, which are multinomial proportion parameters. Then, if we control for the sampling uncertainty associated with the estimation of the conditional choice probabilities, we can conduct inference on the structural parameters of interest. This strategy is feasible given that the number of multinomial proportion parameters to estimate is small relative to the sample size. Thus, we construct a confidence set for the conditional choice probabilities, and translate inference on the conditional choice probabilities to inference on the structural parameters using the characterizations in Theorem (ref).\footnote{A similar idea has been used by kline2016bayesian who propose a Bayesian method for inference. They leverage the idea that a posterior on the reduced-form parameters (the conditional choice probabilities) can be translated to posterior statements on $\theta$ using a known mapping between them.}
Let $\phi\equiv\left(\phi^{x}\right)_{x\in\mathcal{X}}$ be the population choice probabilities. Let us make the dependence of the identified set on $\phi$ explicit by writing \[ \Theta_{I}\equiv\Theta_{I}\left(\phi\right). \] In other words, the identified set is constructed by inverting the mapping from the structural parameters to the conditional choice probabilities; if we know $\phi$ accurately, then we can obtain the population identified set.
When there is a finite number of observations, $\phi$ is unknown. However, we are able to construct a confidence set for $\phi$ that accounts for the sampling uncertainty. Let $\alpha\in\left(0,1\right)$. We assume that the econometrician can construct a convex confidence set $\Phi_{n}^{\alpha}$ that covers $\phi$ with high probability asymptotically.
Leading examples of $\Phi_{n}^{\alpha}$ are box constraints or ellipsoid constraints; the former will be characterized by constraints that are linear in $\phi^{x}$ and the latter will be characterized by those quadratic in $\phi^{x}$. For example, we can construct simultaneous confidence intervals for each $\phi_{a}^{x}\in\mathbb{R}$ such that the probability of covering all $\left\{ \phi_{a}^{x}\right\} _{a\in\mathcal{A},x\in\mathcal{X}}$ simultaneously is asymptotically no smaller than $1-\alpha$.
Define the confidence set for the identified set as
By construction, if $\Phi_{n}^{\alpha}$ covers $\phi$ with high probability, then $\widehat{\Theta}_{I}^{\alpha}$ covers $\Theta_{I}$ with high probability.
Theorem (ref).(ref) follows directly from ((ref)) and the assumption on $\Phi_{n}^{\alpha}$. To understand Theorem (ref).(ref), note that $\theta\in\widehat{\Theta}_{I}^{\alpha}$ if and only if, for all $x\in\mathcal{X}$, there exist $\sigma^{x}:\mathcal{E}\times\mathcal{T}\to\Delta\left(\mathcal{A}\right)$ and $\phi^{x}\in\Delta\left(\mathcal{A}\right)$ such that ((ref)), ((ref)), and $\phi\in\Phi_{n}^{\alpha}$ are satisfied. Compared to the population program described in Theorem (ref), which treated $\phi$ as known constants, we make $\phi$ part of the optimization variables and impose convex constraints $\phi\in\Phi_{n}^{\alpha}$. Since all equality constraints are linear in $\left(\sigma,\phi\right)$ and inequality constraints are convex in $\left(\sigma,\phi\right)$, the feasibility program is convex (see boyd2004convexoptimization). Note that the computational tractability comes from the fact that $\phi$ enters the restrictions in Theorem (ref) in an additively separable manner; letting $\phi$ be part of the optimization variable does not disrupt the linearity of the constraints with respect to the variables of optimization.
Finally, we note that computation can be made faster by constructing $\Phi_{n}^{\alpha}$ as linear constraints since then $\theta\in\widehat{\Theta}_{I}^{\alpha}$ can be determined via a linear program. In our empirical application, we construct $\Phi_{n}^{\alpha}$ as simultaneous confidence intervals for the multinomial proportion parameters $\phi$ using the results in fitzpatrick1987quicksimultaneous.\footnote{See Appendix (ref) for details. We also provide Monte Carlo evidence that the proposed method has desirable coverage probabilities even when $\mathcal{X}$ has many elements.}
We propose a practical routine for obtaining the confidence set $\widehat{\Theta}_{I}^{\alpha}$. Theorem (ref) says that for any candidate $\theta$, we can determine whether $\theta\in\widehat{\Theta}_{I}^{\alpha}$ by solving a convex (feasibility) program. This feature is attractive, but it only provides us a binary answer (“yes” or “no”).
As commonly done in existing works on partially identified game-theoretic models (e.g., ciliberto2009marketstructure, \citet*{syrgkanis2021inference}, magnolfi2021estimation), we define a non-negative criterion function $\widehat{Q}_{n}^{\alpha}\left(\theta\right)\geq0$ with the property that $\widehat{Q}_{n}^{\alpha}\left(\theta\right)=0$ if and only if $\theta\in\widehat{\Theta}_{I}^{\alpha}$. The value of $\widehat{Q}_{n}^{\alpha}\left(\theta\right)$ for each $\theta$ can be obtained by solving a convex program. The advantage of using a criterion function is that the value of $\widehat{Q}_{n}^{\alpha}\left(\theta\right)$ gives us information on the distance between $\theta$ and the identified set. Moreover, the gradients of the criterion functions provide information on which directions to descend in order to spot a local minimum.
Let $\left\{ w^{x}\right\} _{x\in\mathcal{X}}$ be the set of strictly positive weights for each bin $x\in\mathcal{X}$. The choice of weights can be arbitrary although we will choose values proportional to the number of observations at each bin $x$. Let $q^{x}\in\mathbb{R}$ and $q\equiv\left(q^{x}\right)_{x\in\mathcal{X}}$. Let $\widehat{Q}_{n}^{\alpha}\left(\theta\right)$ be the value of the following convex program.
Intuitively, $q^{x}\geq0$ measures the minimal violation of the inequalities necessary at bin $x$; when all equilibrium conditions can be satisfied, the solver will drive the value of $q^{x}$ to zero.\footnote{This formulation uses the fact that $\max\left\{ z_{1},...,z_{K}\right\} $ can be obtained by solving $\min t$ subject to $z_{k}\leq t$ for $k=1,...,K$.} The solution to ((ref)) measures the weighted average of the minimal violations of the equilibrium conditions required to make $\theta$ compatible with data. Also note that the choice of weights do not affect the results if the researcher is only interested in the set of $\theta$'s whose criterion function values are exactly zero.
The following summarizes the properties of the criterion function approach.
In particular, Theorem (ref).(ref) says that, due to the envelope theorem, we can obtain the gradients for free when we evaluate the criterion function at each point (assuming the analytic derivatives of $\psi^{x,\theta}$ and $u_{i}^{x,\theta}$ are available). In practice, we need to identify the minimizers of $\widehat{Q}_{n}^{\alpha}\left(\theta\right)$ in order to numerically approximate $\widehat{\Theta}_{I}^{\alpha}$. However, doing so by conducting an extensive grid search over the whole parameter space can be computationally costly especially when the dimension of $\theta$ is high. Due to Theorem (ref).(ref), one can use gradient-based optimization algorithms to identify a minimizer of the criterion function.\footnote{When program ((ref)) has a manageable number of variables, then the nested minimization problem $\min_{\theta}\widehat{Q}_{n}^{\alpha}\left(\theta\right)$ can be solved more efficiently as a single joint minimization problem using a large-scale nonlinear solver su2012constrained. We use this approach for our empirical application in the next section.} The ability to quickly identify $\arg\min_{\theta}\widehat{Q}_{n}^{\alpha}\left(\theta\right)$ is advantageous since we can quickly test whether the identified set is empty, or restrict the search to points near the minimizer.
For our empirical application, we use a heuristic approach to approximate $\widehat{\Theta}_{I}^{\alpha}$. The idea is to identify a minimizer of the criterion function and run a random walk process starting from the minimizer in order to collect nearby points that have zero criterion function values. This way we avoid the need to evaluate points that are far from the identified set. See Appendix (ref) for details.
We apply our framework to study the entry game by McDonald's and Burger King in the US using rich datasets. Entry competition in the fast food industry fits our framework well due to two stylized facts. First, the decisions on whether or not to operate outlets are highly persistent, indicating that the firms' decisions are publicly observed. Tables (ref) and (ref) report the three-year transition probability of the firms' decisions and the market outcomes $\left(a_{MD},a_{BK}\right)$ (where $a_{i}=1$ if firm $i$ is present in the market and $a_{i}=0$ otherwise), measured for all urban census tracts (which correspond to our definition of markets) in the contiguous US over 1997-2019. For instance, the probability that McDonald's has an outlet in operation in a local market three years later conditional on it having an outlet in operation today is 0.95. Together with the assumption that the costs of revising decisions are sufficiently low, the evidence supports the claim that firms' decisions are best-responses to opponents' decisions that are readily observed.\footnote{In the model, we assume that the costs of revising actions are zero. We discuss the validity of the assumption in this setting in Appendix (ref). }
Second, information asymmetries and information spillover from observing others' decisions are common features in the industry. It is well-documented that competitors take extra scrutiny over the locations where McDonald's opens new outlets in order to take advantage of McDonald's leading market research technology.\footnote{See ridley2008herding and yang2020learning who provide anecdotal evidence on how competing firms learn about the profitability of a location from entries of leading firms such as McDonald's and Starbucks. For example, according to The Wall Street Journal, “In the past, many restaurants... plopped themselves next to a McDonald's to piggyback on the No. 1 burger chain's market research.” leung2003aglutted} Our notion of equilibrium accounts for this phenomenon.
Using the proposed framework, we estimate the entry game under different baseline information structures in order to explore the role of informational assumptions on identification. We also compare our results to those obtained under Bayes correlated equilibrium, which also allows estimation with weak assumptions on players' information. We then perform a counterfactual policy exercise that studies how the market structures in Mississippi food deserts respond after increasing access to healthy food.
We combine multiple datasets to construct the final dataset for structural estimation of the entry game. In the final dataset, the unit of observation is a market (urban tract). Each observation contains information on the firms' market entry decisions and the observable characteristics of the firms and the market.
Our primary dataset comes from Data Axle Historical Business Database, which contains a (approximately) complete list of fast-food chain outlets operating in the US between 1997 and 2019 at an annual level.\footnote{This database contains location information for a detailed list of business establishments in the US from 1997 to 2019. The provider attempts to increase accuracy by using an internal verification procedure after collecting data from multiple sources. The dataset is approximately complete in the sense that the list is not free of error. However, we compare the number of burger outlets in the data and the number reported in external sources and confirm that the information is highly accurate for the case of burger chains. See Appendix (ref) for details.} The advantage of this dataset is that it provides the address information of the burger outlets across all regions of the US. The use of this dataset to study strategic entry decisions is new.\footnote{We are not the first to study the entry game between McDonald's and Burger King in the US. gayle2015choosing uses 2011 cross-sectional data hand-collected using the online restaurant locator on the brands' websites. However, they define a local market as an “isolated city” that is more than 10 miles away from the closest neighboring city, which is larger than our definition that uses a census tract. Moreover, they focus on examining assumptions on the order of entries.}
Although we use panel data to investigate the persistence of decisions over time, we use cross-section data to estimate the structural model. The idea is to illustrate that the econometrician can use cross-sectional data as a snapshot of the stable outcomes of the markets at some point in time.\footnote{If we wanted to exploit the information available in panel data, we would need to model the dependence of observations across time. However, given that market environments usually seem to stay very stable over time, it is not clear how to leverage the information for structural estimation. For simplicity, we focus on analyzing a single cross-section (which also represents a typical dataset available to researchers).} We use the 2010 cross-section since it was the last year for which decennial census data were available. We describe the main features of our dataset below. Further details on data construction are provided in Appendix (ref).
Markets are defined as 2010 urban census tracts in the contiguous US. A census tract is classified as urban if its geographic centroid is in an urbanized area defined by the Census. The final data contain 54,944 markets. We code $a_{i}=1$ if firm $i$ had an outlet operating in the market.\footnote{McDonald's (resp. Burger King) has more than one outlets in 1.5% (resp. 0.3%) of the markets.} The unconditional probabilities of market outcomes are $\left(\hat{\phi}_{00},\hat{\phi}_{01},\hat{\phi}_{10},\hat{\phi}_{11}\right)=\left(0.74,0.06,0.15,0.05\right)$ where $\hat{\phi}_{a}$ is the sample frequency of outcome $a=\left(a_{MD},a_{BK}\right)$.
We use two firm-specific variables that have been used in existing works: distance to headquarters (zhu2009marketstructure, zhu2009spatial, yang2012burgerking) and own outlets in nearby markets (toivanen2005marketstructure, igami2016unobserved, yang2020learning). Variable distance to headquarter measures the distance between the center of each market to the firms' respective headquarters. The associated exclusion restriction is valid if the cost of operating an outlet increases with its distance to own headquarter, but is unrelated to the distance to opponents' headquarters. Variable own outlets in neighboring markets is constructed by finding all outlets in tracts that are adjacent to a given tract. The underlying assumption is that an outlet's profit can be affected by an own-brand outlet in a neighboring market, but not by a competing brand's outlet in a neighboring market; competition with opponents occur only within each market.
Summary statistics are provided in Table (ref). Continuous variables are discretized to binary variables by using cutoffs around their medians. Clearly, the entry probability of McDonald's is higher. McDonald's is more likely to have an outlet present in adjacent markets. The distance to headquarter is higher for Burger King on average because Burger King has its headquarter in Florida while McDonald's has its headquarter in Chicago.
Market environment variables control for the determinants of profitability that are common across firms. We obtain the following variables to describe market environments. First, we have an indicator for whether a tract has many eating or drinking places; the variable is obtained from the National Neighborhood Data Archive (NaNDA) which provides business activity information at the tract-level. Second, we have an indicator for whether a tract has high income per capita; the variable is from the census. Finally, from the Food Access Research Atlas, we obtain indicators for whether a tract has low access to healthy food and whether a tract is classified as a food desert. A tract is classified as having low access to healthy food if at least 500 or 33 percent of the population lives more than 1/2 mile from the nearest supermarket, supercenter, or large grocery store.\footnote{ USDA use supermarkets, supercenters, and large grocery stores that offer a full range of food products\textemdash including fresh meat and poultry, produce, dairy, dry and packaged foods, and frozen foods\textemdash to calculate access to healthy food. To construct the list of stores, USDA combines a list of stores authorized to accept Supplemental Nutrition Assistance Program (SNAP) benefits and a list of stores from Trade Dimensions TDLinx (see ver2012access). The list of stores flagged as healthy food providers by the USDA serves as a proxy for access to healthy and affordable food but does not count other retailers that might offer healthy options (e.g., convenience stores, drugstores, dollar stores, military commissaries, and warehouse club stores).} A tract is classified as a food desert if it has low income and low access to healthy food, where the criteria for low-income are from the U.S. Department of Treasury's New Markets Tax Credit program.
The last rows of Table (ref) shows that 85% of all urban census tracts are classified as having low access to healthy food and 33% are classified as food deserts. In the counterfactual analysis, we select food deserts in Mississippi and investigate the impact of increasing access to healthy food on the strategic entry decisions of the firms.
Before estimating the structural model, we examine the data patterns using simple probit regressions. Each market $m$ contains binary decisions of each firm $a_{im}\in\left\{ 0,1\right\} $ where $a_{im}=0$ if firm $i$ stays out in market $m$ and $a_{im}=1$ if $i$ stays in. We pool the decisions of the firms in each market (so that the unit of observation is $\left(i,m\right)$) and regress the binary decisions on market characteristics. Table (ref) reports the average marginal effects computed from the regression results.
Table (ref) conveys three messages. First, the presence of own outlets in neighboring markets and distance to headquarter are negatively correlated with entry decisions. This appears to be consistent with our prior that these variables have a negative impact on potential profits. Second, the number of eating and drinking places strongly affects the burger chains' entries. This is presumably because districts with high concentration of food services are also places with high traffic of people who eat out. Finally, low access to healthy food is positively correlated with entry decisions. That is, the burger chains are more likely to enter a market when there are fewer healthy substitutes for food.
While Table (ref) provides a helpful snapshot for what drives the chains' entry decisions, the estimates are likely to be biased since they ignore the fact that firms' decisions affect each other. Such consideration is crucial not only for estimating the parameters of the model but also for studying a policy experiment. In the next section, we estimate the entry game using Bayes stable equilibrium as a solution concept.
We posit a canonical entry game that extends the running example to incorporate covariates in the payoff functions. Let us recall the notation. We use $i=1,2$ to denote McDonald's and Burger King respectively. In each market $m$, firm $i$ can choose a binary action $a_{im}\in\left\{ 0,1\right\} $ where $a_{im}=1$ if $i$ stays in and $a_{im}=0$ if $i$ stays out. The payoff function is specified as \[ u_{i}^{x_{m},\theta}\left(a_{im},a_{jm},\varepsilon_{im}\right)=a_{im}\left(\beta_{i}^{T}x_{im}+\kappa_{i}a_{jm}+\varepsilon_{im}\right). \] That is, the payoff from operating in the market is $\beta_{i}^{T}x_{im}+\kappa_{i}a_{jm}+\varepsilon_{im}$ where $x_{im}$ represents market covariates, $a_{jm}$ represents whether the opponent is present, and $\varepsilon_{im}$ is firm $i$'s payoff shock that is not observed by the econometrician. Each $\varepsilon_{im}$ can include firm-specific payoff determinants (e.g., customers' loyalty to brand and managerial ability) and common payoff determinants (e.g., local food preference, local price level, the degree of competition from other restaurants). We model $\left(\varepsilon_{1m},\varepsilon_{2m}\right)\in\mathbb{R}^{2}$ as being normally distributed with zero mean, unit variance, and correlation coefficient $\rho\in[0,1)$.\footnote{For example, suppose that $\varepsilon_{im}\equiv\nu_{im}+\xi_{m}$ where $\nu_{im}\overset{iid}{\sim}N\left(0,\sigma_{\nu}^{2}\right)$ for $i=1,2$, and $\xi_{m}\overset{iid}{\sim}N\left(0,\sigma_{\xi}^{2}\right)$. Then $\text{Var}\left(\varepsilon_{im}\right)=\sigma_{\nu}^{2}+\sigma_{\xi}^{2}$, $\text{Cov}\left(\varepsilon_{1m},\varepsilon_{2m}\right)=\sigma_{\xi}^{2}$, and $\text{Corr}\left(\varepsilon_{1m},\varepsilon_{2m}\right)=\sigma_{\xi}^{2}/\left(\sigma_{\nu}^{2}+\sigma_{\xi}^{2}\right)$. Normalizing the variance of $\varepsilon_{im}$ to one scales the coefficients $\beta_{i}$ and $\kappa_{i}$ to units equal to the standard deviation of $\varepsilon_{im}$. Our approach of modeling $\varepsilon_{im}$'s as jointly normally distributed with arbitrary correlation follows magnolfi2021estimation. ciliberto2009marketstructure models each $\varepsilon_{im}$ as a sum of independent firm-specific and market-specific random shocks and estimates the associated covariance matrix.} The payoff from staying out is normalized to zero. Our specification of the payoff functions is quite standard in the literature.\footnote{A more flexible specification might add a richer set of covariates or let the spillover effects $\kappa_{i}$ be a function of the observable covariates as done in ciliberto2009marketstructure. We keep the specification parsimonious.}
We estimate the parameters under the baseline information assumptions specified previously in Example (ref): $S^{null}$, $S^{1P}$, and $S^{private}$. To recap, $S^{null}$ is the information structure in which each player observes nothing; in $S^{1P}$, Player 1 observes (only) $\varepsilon_{1}$ whereas Player 2 observes nothing; in $S^{private}$, Player 1 observes $\varepsilon_{1}$ and Player 2 observes $\varepsilon_{2}$.
Under the Bayes stable equilibrium assumption, the baseline information structures should be interpreted as specifying what the players minimally observe. Then estimating the model with $S^{null}$ as the baseline information structure amounts to making no assumption on players' information. On the other hand, if the baseline information structure is set to $S^{private}$, then the identified set is robust to all cases in which the players observe at least their payoff shocks. Finally, setting the baseline information structure to $S^{1P}$ amounts to assuming that McDonald's has good information about its payoff shocks whereas Burger King might minimally have no information about its payoff shock. This assumption relaxes the standard assumption on information (namely the information structure is fixed at either $S^{private}$ or $S^{complete}$) and is consistent with the anecdotal evidence that McDonald's is a leader in the market research technology.
In order to keep the model parsimonious and reduce the computational burden, we take some steps before estimation, which are described as follows (see Appendix (ref) for further details). First, we assume that the coefficients for common market-level variables (eating places, income per capita, and low access to healthy food) are identical across the two players.\footnote{This assumption is not without loss and can be refuted on the basis that each chain might react differently to market environment. However, we believe it is reasonable given that McDonald's and Burger King are close substitutes to each other.} We also assume that the coefficients of the firm-specific variables (distance to headquarter and the presence of own-brand outlets in nearby markets) are non-positive. Second, while the benchmark distribution of the latent variables $\left(\varepsilon_{1m},\varepsilon_{2m}\right)$ is continuous, we use discretized normal distribution for feasible estimation. Third, we discretize each variable to binary bins; since there are 7 variables in the covariates, this gives $2^{7}=128$ discrete covariate bins. Conditional choice probabilities are non-parametrically estimated using the observations within each bin. Fourth, to construct confidence sets for the conditional choice probabilities, we used simultaneous confidence bands based on the method described in fitzpatrick1987quicksimultaneous; using simultaneous confidence bands makes the evaluation of the criterion function a linear program.
Table (ref) reports projections of the 95% confidence sets obtained under the Bayes stable equilibrium assumption with different baseline information structures. There are three main findings related to the role of informational assumption. First, making no assumption on players' information leads to an uninformative identified set. The confidence set under $S^{null}$ is quite large, and we cannot determine the signs of the parameters. Therefore, being utterly agnostic about players' information does not give us enough identifying power to draw meaningful conclusions.
Second, standard assumptions on information may be too strong. It is quite standard to assume that each player $i$ observes (exactly) $\varepsilon_{i}$ or $\left(\varepsilon_{i},\varepsilon_{-i}\right)$. Setting baseline information structure as $S^{private}$ nests all these cases. However, we find that the identified set under $S^{private}$ is empty, suggesting the possibility of misspecification.\footnote{Specifically, we consistently find that the minimum of the criterion function under $S^{private}$ is strictly greater than zero. This is also true even if we do not use sign constraints or reduce the nominal level to a very low level (e.g., $\alpha=0.0001$).} Thus, assuming that each player observes at least their $\varepsilon_{i}$ may be too strong. Since the Bayes stable equilibrium identified set under $S^{private}$ is equivalent to the pure strategy Nash equilibrium identified set (see Theorem (ref).(ref)), the pure strategy Nash equilibrium assumption would also be rejected.\footnote{The emptiness of the identified set is not driven by the possibility of non-existence of Bayes stable equilibrium. When the competition effects parameters have the same signs, there exists at least one pure strategy Nash equilibrium at each state, implying the existence of a Bayes stable equilibrium. Of course, the emptiness of the identified set might be due to misspecification in payoff functions, distribution of errors, etc. Our statements are conditional on having these specifications correct.}
Third, we find that setting the baseline information structure to $S^{1P}$ can produce an informative identified set. Recall that the identified set under $S^{1P}$ makes the assumption that McDonald's has accurate information about its payoff shock, but Burger King's information can be arbitrary. This assumption is consistent with the anecdotal evidence that McDonald's has superior information on the potential profitability of each market, and Burger King tries to free-ride on McDonald's information by observing what McDonald's does.\footnote{ Note that Burger King's extraction of McDonald's information is feasible when the errors are correlated. For instance, suppose that local food taste variable $\xi$ enters both $\varepsilon_{1}$ and $\varepsilon_{2}$ and that McDonald's observes $\varepsilon_{1}$ via its research technology. In a rational expectations equilibrium, McDonald's decision reveals partial information about $\xi$, which in turn Burger King can use to infer its $\varepsilon_{2}$. Such refinement of information is not allowed in the static Bayes Nash equilibrium framework.} Table (ref) shows that, even if we substantially relax the assumption on Burger King's information, we can determine the signs of the most parameters. For example, we can see that burger chains are more likely to enter in markets that have low access to healthy food. We can also learn that the firms' payoff shocks are highly correlated to each other.
In conclusion, we find that the informativeness of the identified set crucially depends on the underlying assumption on players' information. At least in our empirical application, it is difficult to draw a meaningful economic conclusion without making assumptions on players' information. On the other hand, under the maintained solution concept, the model rejects the popular assumptions made in the literature, namely that each firm $i$ observes at least its $\varepsilon_{i}$. A credible intermediate case $S^{1P}$, which is consistent with our knowledge of the market research technology in the fast food industry, delivers strong identifying power.\footnote{Following a reviewer's comment, we have also tried an alternative specification that assumes $\varepsilon_{im}=\nu_{im}+\xi_{m}$ where $\nu_{im}\overset{iid}{\sim}N\left(0,\sigma_{\nu}^{2}\right)$ and $\xi_{m}\overset{iid}{\sim}N\left(0,\sigma_{\xi}^{2}\right)$. In the estimation stage, we normalized to variance of $\varepsilon_{im}$ to one. We estimated the model under the assumption that each player $i$ minimally observes $\nu_{im}$ (but remained agnostic to whether firm $i$ observes $\xi_{m}$ or $\nu_{-im}$). This specification is weaker than $S^{private}$, but it is neither stronger nor weaker than $S^{1P}$ as McDonald's may not observe $\xi_{m}$ and Burger King observes at least $\nu_{2m}$. We found that the identified set is non-empty. Thus, an alternative specification can be used to relax strong assumption on players' information. However, this approach requires imposing an additional structure on the unobservables. Moreover, it increases computational costs by increasing the dimensionality of optimization problems that use discretized distributions. }
We compare the Bayes stable equilibrium identified sets to the Bayes correlated equilibrium identified sets studied in magnolfi2021estimation. The Bayes correlated equilibrium identified sets are reported in Table (ref). We can readily see that the Bayes correlated equilibrium assumption produces a much larger set for each baseline information structure. Even when we set $S^{private}$ as the baseline information structure, it is not easy to learn the signs of many parameters. For example, we cannot determine whether low access to healthy food promotes or deters entries by the burger chains.
Comparing Tables (ref) and (ref) suggests that if the researcher is willing to accept the Bayes stable equilibrium assumption, it can add significant identifying power while providing the same kind of informational robustness as Bayes correlated equilibria. At least in the context of our empirical application, we believe it is reasonable to assume that McDonald's decisions that we observe in the data represent best-responses to the observed decisions of Burger King and vice versa.
Consumption of fast food is driven not only by consumers' taste for fast food but also the availability of food substitutes in the neighborhood. Following the recent surge of interest on the relationship between food deserts and food consumption patterns (see, e.g., allcott2019fooddeserts and kolb2021retail), we study the impact of accessibility to healthy food on the entry decisions of fast-food chains. Specifically, we consider a policy experiment to predict changes in market structure in Mississippi food deserts after increasing access to healthy food measured by supermarket entries.\footnote{Our policy experiment relies on consumers' substitution patterns between fast-food restaurants and supermarkets (which include supercenters and large grocery stores). Although shorter distance to providers of healthy food does not necessarily translate to healthier diet (allcott2019fooddeserts and kolb2021retail), easier access to supermarkets leads consumers to lower their visits to fast-food chains due to reduced travel costs and increased availability of alternative (healthy or equally unhealthy but cheaper) food substitutes. Note that our indicator for access to supermarket corresponds to having a supermarket within 1/2-mile distance which amounts to less than 10-minutes walking distance while allcott2019fooddeserts consider supermarket entries within 10-15 minutes driving distance. } Mississippi is often called one of the “hungriest” states in the US.\footnote{For example, Mississippi has been identified as the most food insecure state in the country since 2010 according to Feeding America. See \url{https://mississippitoday.org/2018/05/04/mississippi-still-the-hungriest-state/}.} Mississippi had 664 census tracts in 2010, and 329 of them are classified as urban tracts, which correspond to our definition of markets. Out of 329 urban tracts, 185 tracts (approximately 56%) are classified as food deserts, according to the U.S. Department of Agriculture. According to the definition of food deserts, all of these tracts are classified as having low access to healthy food.
We conduct a policy experiment as follows. We select the 185 tracts classified as food deserts in Mississippi and then increase access to healthy food. This amounts to changing the low access indicator from one (low access) to zero (high access) in all these markets. In reality, such policy would correspond to increasing healthy food providers (large grocery stores, supermarkets, or supercenters) by providing subsidies or tax breaks. We then recompute the equilibria in these markets and report the weighted average of the bounds associated with each measure of market structure.\footnote{Our counterfactual analysis corresponds to a partial equilibrium analysis. We abstract away from considering how entry or exit in each market can affect the burger chains' decisions in neighboring markets and the responses of healthy food providers.} See Appendix (ref) for computational details.
We report the results of the counterfactual analysis in Table (ref). The first column reports the estimates obtained from the data of the 185 markets corresponding to Mississippi food deserts. For example, the probability of observing McDonald's enter the market in Mississippi food deserts is 0.30, much larger than the unconditional probability obtained using all markets, which was around 0.20.
The second and third columns report the bounds obtained before (“Pre” has low access indicators set to one) and after the counterfactual policy (“Post” has low access indicators set to zero) using the $S^{1P}$-Bayes stable equilibrium identified set. The bounds are pretty wide because we have considered all parameters in the identified set and made no assumption on the equilibrium selection. However, they shift in the expected directions. For example, the bounds on the expected number of entrants shift from $[0.28,1.01]$ to $[0.15,0.79]$. Since the mean number of entrants in the data was $0.47$ and the post-counterfactual bounds are $\left[0.15,0.79\right]$, the maximal change we can expect is $0.15-0.47=-0.32$. In some cases, we can make a stronger statement: while the unconditional probability of observing McDonald's enter in data was 0.30, the upper bound in the Post-regime decreases to 0.23, so we can expect that the probability of McDonald's enter to decrease by at least 0.07.
Our results suggest that meaningful counterfactual statements may be made even with weak assumptions on players' information. The bounds do not depend on specific assumptions on equilibrium selection and admit all information structures that are expansions of the baseline information structure.\footnote{Our predictions are conservative because we do not make any assumptions on how the information structure or the equilibrium selection rule might change after the counterfactual policy.} Hence our approach can also serve as a useful tool to conduct sensitivity analysis for researchers who want to see whether their predictions are driven by assumptions on equilibrium selection or what the players know.
For comparison, in the last two columns, we report the counterfactual results obtained using the $S^{1P}$-Bayes correlated equilibrium identified set. One can readily see that the bounds are quite large compared to the Bayes stable equilibrium counterpart. For example, we cannot make any statement about the probability of Burger King's entry after the counterfactual policy is implemented. Table (ref) shows that Bayes correlated equilibrium predictions can be too permissive, especially when no assumption is imposed on what equilibrium might be selected in the counterfactual world.
This paper presents an empirical framework for analyzing stable outcomes with weak assumptions on players' information. We propose Bayes stable equilibrium as a framework for analyzing stable outcomes, which appear in various empirical settings. Our framework can be an attractive alternative to existing methods for practitioners who want to work with an empirical game-theoretic model and be robust to informational assumptions. Furthermore, we believe the proposed computational algorithms can also be helpful in similar settings, especially since reducing computational burden remains a fundamental challenge in the literature.
We believe there are many exciting avenues for future research. First, providing a non-cooperative foundation to our solution concepts remains an open question. While we can imagine a dynamic adjustment process that converges to stable outcomes, how to formalize this idea is yet unclear. Second, it will be interesting to find reasonable ways of imposing equilibrium selection. While Bayes stable equilibrium (or Bayes correlated equilibrium) has the informational robustness property, the set of predictions may be too large, limiting our ability to make sharp predictions for counterfactual analysis. Finding ways to sharpen predictions without sacrificing robustness to information will be helpful. Third, our counterfactual analysis is limited to a partial equilibrium analysis. It will be interesting to think about ways to model the strategic interactions of healthy food providers and unhealthy food providers together. Finally, there are other forms of informational robustness that our model cannot handle but is empirically interesting. For example, it might be natural to assume that McDonald's has superior information relative to Burger King while being agnostic about their specific information structure. Our model only specifies what the players minimally observe and is agnostic about the relative information across players. Studying alternative forms of informational robustness and corresponding econometric frameworks should be interesting.