A Decomposition Approach to Counterfactual Analysis in Game-Theoretic Models

bibunit[econometrica] \begin{abstract} Decomposition methods are often used for producing counterfactual predictions in non-strategic settings. When the outcome of interest arises from a game-theoretic setting where agents are better off by deviating from their strategies after a new policy, such predictions, despite their practical simplicity, are hard to justify. We present conditions in Bayesian games under which the decomposition-based predictions coincide with the equilibrium-based ones. In many games, such coincidence follows from an invariance condition for equilibrium selection rules. To illustrate our message, we revisit an empirical analysis in Ciliberto/Tamer:09:Eca on firms' entry decisions in the airline industry. { Key words: Counterfactual Analysis, Game-Theoretic Models, Bayes Correlated Equilibria, Decomposition Method} { JEL Classification: C30, C57} \end{abstract} \@startsection{section}{1} \z@{0.6\linespacing\@plus\linespacing}{.6\linespacing}{Introduction} One of the central goals of empirical research in economics is to quantify the effects of new policies that are yet to be implemented. Examples include analyzing the effects of increasing minimum wages on labor outcomes, the effects of different government legislation in healthcare and the effects of different market characteristics on firm entry, to name only a few. To obtain appropriate counterfactual predictions in strategic environments, researchers typically specify a game-theoretic model and estimate structural parameters of the model, the results of which are used for generating post-policy predictions. The main virtue of this approach is that its predictions are incentive compatible, that is, agents have no incentive to deviate from the strategies yielding the post-policy predictions. However, it requires specification of the game to fine details, often made to address computational challenges in implementation. These challenges frequently arise from the presence of multiple equilibria in the game, which may result in identified sets that are too large for meaningful policy analysis. In fact, it is not uncommon that they produce conflicting conclusions (e.g., see Aguirregabiria/Nevo:12:WP, p.111). An alternative approach for counterfactual analysis is to use decomposition methods. Such methods extrapolate the observed relationship between an outcome and an explanatory variable to a counterfactual environment. They are widely used in labor economics. For instance, a researcher studying the effects of minimum wages on wage inequality (e.g., DiNardo/Fortin/Lemieux:96:Eca) would first estimate this relationship in the observed data, then use those estimates with different (counterfactual) minimum wages to evaluate the distribution of wages under the new policy. The decomposition-based approach has several practical merits: it is computationally simple, yields point-identified predictions, and does not rely on detailed specifications (e.g., of payoffs or unobserved heterogeneity). Furthermore, statistical inference is often straightforward: we can just use bootstrap. However, to the best of our knowledge, decomposition-methods have been used mostly in non-strategic settings. What prevents us from using decomposition methods for counterfactual analysis in a strategic setting? Decomposition methods assume that the causal relationship between the outcome variable and the policy variable remains the same in the counterfactual environment. This assumption can be violated if an agent has an incentive to deviate from the equilibrium strategies in the original game after a new policy. In such cases, we cannot put forth a decomposition-based prediction as a sound counterfactual prediction. In this paper, we present a set of sufficient conditions under which the predictions obtained from the decomposition method coincide with the equilibrium-based predictions (i.e., the predictions defined in terms of the equilibria in the game in combination with the unknown equilibrium selection rule in the data generating process). These results are derived in a general strategic environment, where the policy of interest affects a component of the model with sampling variation (e.g., observed variables in the payoff function).\footnote{Throughout the paper, we focus on counterfactual analysis where a policy of interest changes an observed random vector constituting the payoff state and the target of prediction is the action of the agents (or a known function of such actions). While this restriction covers a large set of policy analysis settings, it does exclude many important situations of counterfactual analysis such as those that focus on a change in the welfare. We clarify this restriction on the scope later in the paper.} For a formal analysis, we consider a generic game, which includes games with various information structures, with the solution concept of Bayes Correlated Equilibria of Bergemann/Morris:16:TE (which includes Nash equilibria and other concepts as special cases). It turns out that the coincidence between equilibrium-based predictions and those from decomposition methods applies to both complete information and incomplete information games. It does not depend on whether we use Nash equilibria as a solution concept or not. Rather, the coincidence depends on the class of counterfactual policies that are considered. The sufficient conditions for the validity of decomposition methods can be summarized as follows: (a) the policy alters only a publicly observed component of the payoff state (observed by players and the researcher), (b) the policy keeps the payoff state within its support in the pre-policy game, and (c) the equilibrium selection rule conditional on a payoff state remains the same if the set of equilibrium actions conditional on the payoff state remains the same after the policy. When these conditions are met, researchers can use a decomposition-based prediction as a counterfactual prediction because this prediction is the same as the equilibrium-based one. Condition (a) is satisfied by many policies, including those that affect taxes, tariffs or laws, which are often observed by all agents. Most of all, complete information games satisfy this condition immediately. Condition (b) is also satisfied in many empirical applications. Even when the condition is not met, we show below that we can obtain bounds for the equilibrium-based prediction using decomposition-based predictions. These bounds are generally informative and easy to estimate using data. The invariance condition (c) in this paper requires that, for each value of the payoff state, if the pre-policy set of equilibrium actions and the post-policy set of equilibrium actions are the same, so are their selection probabilities. While the invariance condition is not innocuous, it seems intuitive because, if the equilibrium action profiles remain the same at a payoff state after the policy, this means that the policy has not altered the prediction of the game at the payoff state. If the equilibrium selection probability at the payoff state nevertheless changed after the policy, our counterfactual predictions would depend on variations which have nothing to do with the prediction of the game-theoretic model. Furthermore, the invariance is already widely used in empirical work in various disguises. For example, this condition is implicitly used when we focus on a specific equilibrium played (e.g., a Pareto superior equilibrium, or the most profitable for a specific firm as in Jia:08:Eca), assumes that the same equilibrium is played after the policy (as discussed in Aguirregabiria/Mira:10:JOE), or parametrizes the equilibrium selection rule and uses the estimated selection function for counterfactual analysis (e.g., Bajari/Hong/Ryan:10:Eca). In their analysis of counterfactual predictions on games, Aguirregabiria/Mira:13:WP explicitly considered an invariance condition for the equilibrium selection rule though differently from ours. On the other hand, there are alternative approaches that are fully agnostic about the equilibrium selection rule or even about part of the structure of the game, such as the partial identification approach of Ciliberto/Tamer:09:Eca and Haile/Tamer:03:JPE. Our results can be applied to many empirical settings, including counterfactual exercises in entry games where a policy changes part of the market characteristics (e.g., Jia:08:Eca) or regulatory policy (e.g., Ciliberto/Tamer:09:Eca), in auction markets where the focus is on the impact of a change in reserve prices on auction outcomes, in various policy settings in labor economics such as increases in taxes or minimum wages on employment, among others. (We provide details on the validity and its implementation below.) In all of these examples, under the invariance condition, it suffices to run a decomposition-based prediction to recover the equilibrium-based prediction without further assumptions on payoffs or distributions of unobserved heterogeneity. Overall, our results show the tradeoffs that researchers face when they perform counterfactual analysis using a game-theoretic model. They can use the decomposition approach which is more computationally tractable and does not require specifying the fine details of the model, at the expense of focusing on a rather narrower class of counterfactual policies under the invariance of the equilibrium selection rules. Hence, although the decomposition approach requires weaker assumptions (e.g., on parametrizations and information structure), it cannot generally be used for other counterfactuals, including those that involve changes in parts of the model that do not have sample variation (e.g., to structural parameter values) or those that are players' private information. It further requires the invariance condition on the equilibrium selection rules described above. The right balance in the tradeoff will depend on the details of the empirical settings and the policy questions of interest. The primary contribution of our paper is to provide formal results that clarify these tradeoffs for researchers. Finally, we emphasize that even if one were to use decomposition-based predictions to point- or interval-identify the equilibrium-based predictions, it does not eliminate entirely the need to use a game-theoretic model for counterfactual analysis. On the contrary, to check the validity of decomposition-based predictions for counterfactual analysis, we need to clarify the strategic environment, the agents' information structure, and then consider what components of the strategic environment are invariant to the policy of interest. We believe that our results are useful for this step, as they show which specifications are needed (and which ones are not) for the use of decomposition-based predictions in such settings. Once the validity of decomposition-based predictions is confirmed, we do not need to specify further details of the game for counterfactual analyses. As an illustration of our results, we revisit the empirical application of Ciliberto/Tamer:09:Eca. Using a model of a complete information entry game, they studied the U.S. airline market and assessed the effects on entry from a repeal of the Wright amendment, a law that restricted entry in routes using Dallas Love Field Airport. Due to the multiplicity of equilibria, they pursued a set identification approach and reported maximum entry probabilities as counterfactual predictions. In our application, we produce a decomposition-based prediction and compare it with the prediction from Ciliberto/Tamer:09:Eca. We also compare these predictions with the actual results following the repeal of the Wright amendment in 2014. We find that the decomposition-based predictions using the pre-policy data perform well out-of-sample. As a second empirical application, we follow Goolsbee/Syverson:08:QJE and revisit the effects of a shock which decreases Southwest's costs of operating a given market on its competitors' decisions. Our focus, though, is on its effects on competitors' entry, rather than prices. We find that other airlines' entry behavior is consistent, on average, with deterrence. However, this effect is heterogeneous across markets, and driven primarily by small markets. In large markets, they appear to “accommodate” Southwest (i.e., the competitors do not change their own entry behavior when Southwest has a lower cost of operating that same market). This is consistent with evidence in other industries (e.g., Tenn/Wendling:14:ReStat). We then extend this exercise to settings where multiple competitors potentially have lower costs to operate that same route. While airlines are generally more likely to enter when the number of potential competitors decreases from low benchmarks (i.e., when they are likely to become monopolists), this is not the case when very competitive markets become slightly less competitive. Related Literature The idea that we may not need to identify a full structural model to identify policy effects of interest goes back at least to Marschak:53:Cowles, as pointed out by Heckman:10:JEL. See Wolpin:13 for further examples and discussion. Recent approaches exploring similar insights include the sufficient statistics approach (most notably in public finance, e.g., Chetty:09:ARE; see Kleven:20:WP for a recent overview) and policy relevant treatment effects proposed by Ichimura/Taber:00:NBER and Heckman/Vytlacil:01:AERPP. See Heckman:10:JEL for a review of this literature. We convey a similar message in this paper by studying conditions for the validity of decomposition-based predictions in strategic settings. The decomposition approach is widely used in economics, most notably in labor economics. In the decomposition approach, the researcher estimates their model and then decomposes the variation in the outcome into the effects from different covariates. One example is the study of the effects of minimum wages on wage inequality (e.g., DiNardo/Fortin/Lemieux:96:Eca): while there might be other mechanisms that affect wage inequality (e.g., unionization), once the model has been estimated, we can check how the distribution of wages would have differed if minimum wages had increased. Salient examples of the decomposition approach include the early work of Oaxaca:73:IER and Blinder:73:JHR, Juhn/Murphy/Pierce:93:JPE, the nonparametric/semiparametric approach of DiNardo/Fortin/Lemieux:96:Eca, and other extensions which has been very widely applied, see Fortin/Lemieux/Firpo:11:Handbook for a survey. The connection between the decomposition methods and methods of program evaluations have been pointed out by Fortin/Lemieux/Firpo:11:Handbook. Kline:11:AERPP shows that the estimated decomposition-based prediction can be interpreted as a reweighting estimator in the program evaluation setting. More relevant to our paper is the decomposition approach used for counterfactual policy predictions. Rothe:10:JoE and Chernozhukov/Fernandez/Melly:13:Eca provided inference on the full counterfactual distributions. Hsu/Lai/Lieli:22:JBES introduced the quantile counterfactual treatment effects on a different population, and showed how we can identify and perform inference on the treatment effects using an invariance condition. While the main emphasis of this literature is on the general method of inference on various counterfactual distributions, our focus is on presenting a generic set of conditions in game-theoretic models under which such decomposition-based predictions can be accepted as valid predictions. A common way to generate counterfactual predictions in a strategic environment is to specify and estimate a game-theoretic model, and then use the predictions from its equilibria. In many cases, point-identification of these models is not possible due to multiple equilibria. Researchers often either choose one equilibrium from the game (e.g Jia:08:Eca) or conduct inference on the identified set allowing for all equilibria (e.g., Ciliberto/Tamer:09:Eca). In light of these challenges, a recent literature studies point-identification of counterfactual predictions without identifying the full details of the model. These developments have been centered around dynamic discrete choice models – e.g., Aguirregabiria/Suzuki:14:QME, Norets/Tang:14:ReStud, Arcidiacono/Miller:20:JOE, Kalouptsidi/Scott/Souza:20:QE. While many structural models within this class are unidentified - see the discussion in Aguirregabiria/Suzuki:14:QME, some counterfactuals are point-identified such as those characterized by linear changes in payoffs (the so called “additive transfers counterfactuals” in Kalouptsidi/Scott/Souza:17:IJIO). See also Kalouptsidi:Kitamura:Lima:Souza:20:NBER for partial identification of counterfactuals in a similar context. Jun/Pinkse:20:JOE explored various approaches to produce a point-decision on partially identified counterfactual predictions from a game. Recently, Gu/Russell/Stringham:21:WP found a useful characterization of an identified set for counterfactual predictions in a general class of discrete-outcome models. The message in this paper is related to that of Kocherlakota:19:JME. Using a model of a dynamic game between the private sector and the government, he showed that to obtain an optimal policy, we can just use predictions by regressing policy payoffs on policies using historical data, without relying on a structural macroeconomic model. There are major differences between his framework and ours. First, his model is a dynamic model where the policy-maker is a player of the game, whereas ours is a static one in which the policy-maker is outside the model. Second, his model is a macroeconomic model where the analysis is based on one equilibrium which generated the data. In our setting, many independent copies of a static game are observed so we need to deal with the problem of multiple equilibria. We use the solution concept of BCE due to its generality, which makes it easy to demonstrate that the decomposition method is not limited to a specific equilibrium concept. Furthermore, through the use of BCE, we show that our result holds even if the counterfactual policy changes players' information structures in specific ways -- see Section (ref). BCE has recently gained interest as a solution concept in an empirical setting due to its robustness properties and computational tractability. Magnolfi/Roncoroni:22:ReStud adopt it in their study of entry decisions in the supermarket sector. They focus on characterizing the identified set obtained under a weak assumption on the information structure of the game, and use the set to study a policy that changes a covariate (presence of large malls). Other empirical examples using BCE and obtaining partial identification include Gualdani/Sinha:20:WP on discrete choice models and Syrgkanis/Tamer/Ziani:18:WP on auctions. Finally, Bergemann/Brooks/Morris:22:AER present a general framework to produce a set of counterfactual predictions in games exploiting the requirement that observations be consistent with some unknown information structure of the games. Their counterfactual policies are concerned with a change of payoff functions or the set of action profiles, whereas in our setting, the policies are mainly a change in the distribution of the payoff state vector. More importantly, our emphasis is on the use of decomposition-based predictions without requiring the researcher to specify details on the payoff functions and the distribution of the payoff states. This paper is organized as follows. In Section 2, we begin with a brief overview of our main results using a simple example. Then, we present the formal results of the validity of decomposition-based predictions for generic Bayesian games and provide discussions, extensions and examples. In Section 3, we provide empirical applications. Section 4 concludes. The mathematical proofs, details on generalizations of our results, details on the data and the implementation of decomposition-based predictions are found in the Online Appendix to this paper. \@startsection{section}{1} \z@{0.6\linespacing\@plus\linespacing}{.6\linespacing}{Counterfactual Analysis Using Game-Theoretic Models} \@startsection{subsection}{2} \z@{.5\linespacing\@plus.7\linespacing}{.5\linespacing}{An Overview with a Simple Example} \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Environment} To fix ideas, we revisit a complete information entry game that has been widely used in the empirical literature. There are two firms, $i=1,2$, that choose a binary action $Y_i \in \{0,1\}$, $Y_i = 1$ representing entry in the market, and $Y_i=0$ staying out of the market. As in many empirical models used in the literature, we may consider specifying the payoff function parametrically. While our main results do not require such specifications, we consider one possible parametrization of the payoff function (e.g., Bresnahan/Reiss:91:JOE, Jia:08:Eca, Ciliberto/Tamer:09:Eca): \begin{align} u_i(Y,W_i) = Y_i(-\delta Y_{-i} + X_i'\beta_i + \varepsilon_i), \end{align} where $W_i = (X_i, \varepsilon_i)$ is the payoff state with $X_i$ and $\varepsilon_i$ denoting the characteristics of firm $i$ observable and unobservable by the researcher, $\delta>0$ is a parameter measuring the effect of competition and $\beta_i$ coefficient vector for the observed covariates. We denote the set of equilibria as $\mathcal{E}$. For illustration purposes, we assume that $\mathcal{E}$ is a finite set. \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Counterfactual Predictions:} Our focus is on predicting the entry probability for the firms when a policy changes the payoff state $(X,\varepsilon)$ into $(f(X),\varepsilon)$, for some map $f$. In this setting, for example, Ciliberto/Tamer:09:Eca studied a change in an observable policy (modeled as a dummy variable in the payoff state), Jia:08:Eca a change in the market size (i.e., increasing the population size by 10%), Grieco:14:RAND a change in the presence of a supercenter, and Magnolfi/Roncoroni:22:ReStud a decrease in the presence of large malls. All such policies change an observable payoff state $X$ to $f(X)$, without affecting the unobserved $\varepsilon$. Given policy $f$, our target parameter is the average entry probabilities of the two firms after the (observable) payoff state is changed from $X$ into $f(X)$, as predicted by the entry game model. We call this \bi{Average Equilibrium-based Prediction (AEP)}. It can be written as: \begin{align} \mathsf{AEP} \equiv \mathbf{E}\left[\sum_{g \in \mathcal{E}_f} g(f(X),\varepsilon) e_f(g \mid f(X),\varepsilon) \right], \end{align} where the expectation $\mathbf{E}$ is over the joint distribution of $(X,\varepsilon)$, $\mathcal{E}_f$ is the set of post-policy equilibria, and $e_f$ denotes the post-policy equilibrium selection rule, i.e., the probability of choosing equilibrium $g \in \mathcal{E}_f$ after the policy $f$, given state $W = (f(X), \varepsilon)$. It is usually difficult to estimate this prediction because we need to recover both the post-policy equilibria, $\mathcal{E}_f$ and the equilibrium selection rule, $e_f$, from pre-policy data. \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{The Decomposition-Based Prediction} An alternative approach to prediction in this environment is to extrapolate the pre-policy relationship between $X$ and $Y$. One example, which we call the \bi{Average Decomposition-based Prediction (ADP)} due to its link to decomposition methods in labour economics, is: \begin{align} \mathsf{ADP} = \mathbf{E}\left[ m(f(X))1\{f(X) \in \mathbb{S}_X \}\right] , \end{align} where $m(x) = \mathbf{E}[Y \mid X = x]$ and $\mathbb{S}_X$ denotes the support of $X$. By extrapolating pre-policy relationships, the ADP implicitly uses the \textit{pre-policy} equilibrium selection mechanism ($e(\cdot)$) and set of equilibria ($\mathcal{E}$) for predictions, even if both may change following the policy. The practical advantage of using the ADP is clear: we do not need to recover the set of equilibria or the equilibrium selection rule from data for the counterfactual prediction. The quantity is point-identified, and statistical inference on it is simple. However, its predictions may fail to be incentive compatible in the post-policy game. \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Main Results} Our main contribution is to provide sufficient conditions for $\mathsf{ADP}$ to serve as useful bounds for the target parameter $\mathsf{AEP}$. The sufficient conditions are that the policy $f$ affects only the publicly observable component of $X$ and the pre and post-policy equilibrium selection rules satisfy a type of invariance condition (described below), which is applied in many empirical settings. Under these conditions, we show that \begin{align} \mathsf{ADP} \le \mathsf{AEP} \le \mathsf{ADP} + \Delta, \end{align} where $\Delta$ is the $2$-dim vector with the same entry $P\left\{f(X) \notin \mathbb{S}_X \right\}$, $\mathsf{ADP}$ and $\mathsf{AEP}$ are $2$-dim vectors and the inequalities are point-wise. The entry of $\Delta$ represents the probability that the policy sends the payoff state outside of the support of the payoff state in the pre-policy game. It is identified and can be estimated because $X$ is observable. It tends to increase as the policy $f$ transforms the payoff state further away from the support of the payoff state in the pre-policy game $G$. Hence, we can identify bounds on the counterfactual predictions of the game relying only on the nonparametric causal structure of the game above without invoking more detailed specifications of the game, such as the parametric specification of payoffs as in ((ref)). Furthermore, there is no need to compute the set of equilibria under a new policy, and this is no small convenience in practice. Finally, if $f(X) \in \mathbb{S}_X$, this result strengthens to point-identification as \begin{align} \mathsf{AEP} = \mathsf{ADP}. \end{align} In this case, we can benefit from the computational simplicity of the ADP, while preserving the equilibrium-based nature of the AEP. \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Invariance Condition on Equilibrium Selection Rules} As mentioned before, the sufficient conditions for the bounds in ((ref)) are that (i) the counterfactual policy $f$ affects a publicly observable part of the payoff state $X$, and (ii) an invariance condition on the equilibrium selection rules is satisfied. Condition (i) accommodates many applications in the empirical literature, although not all. Section (ref) below describes its limitations. As for (ii), the condition says that \textit{if the set of equilibrium action profiles remains the same after the policy for each fixed $w=(x,\varepsilon)$, their selection probability given $w$ remains the same as well.} For example, suppose that we model the equilibrium selection rule as a function: \begin{align*} e(g \mid x, \varepsilon) = h_g(x, \varepsilon), \quad g \in \mathcal{E}, \end{align*} for some function $h_g$. Suppose further that under the counterfactual policy $f$, the equilibrium selection rule becomes \begin{align*} e(g \mid f(x), \varepsilon) = h_g(f(x), \varepsilon), \quad g \in \mathcal{E}. \end{align*} Then, the invariance condition holds.\footnote{Equation (5) in Bajari/Hong/Ryan:10:Eca, for instance, sets $h_g(x, \varepsilon)$ as a logistic function. Here, we do not specify the functional form of the equilibrium selection rule or require it to be identified or partially identified from data. Instead, we clarify the precise form of the invariance condition that is required for the decomposition-based prediction in this paper.} As mentioned in the introduction, some empirical applications have already used some versions of the invariance conditions as we discuss below. \textbf{Example 1 (Games with a Unique Equilibrium):} The literature of empirical auctions often assumes the existence of a unique equilibrium bidding strategy (Guerre/Perrigne/Vuong:09:ECMA). In this case, the invariance condition for the equilibrium selection rule is vacuously satisfied. If the counterfactual prediction is with regards to a change of a payoff component that belongs to public information, our result shows that we can apply the decomposition method and recover the counterfactual prediction without making assumptions required for the identification of the value distribution of the bidders. Details are found in Section (ref) below. \textbf{Example 2 (The Same Equilibrium in the Data and in the Counterfactual):} A common assumption in the empirical literature is that the same equilibrium is played in the data and in the counterfactual (see Aguirregabiria/Mira:10:JOE, for example). In our notation, this means that $e(g \mid x,\varepsilon) = e_f(g \mid x, \varepsilon) = 1$ for some $g \in \mathcal{E} \cap \mathcal{E}_f$, whenever $(x,\varepsilon)$ realizes in the common support of $X$ and $f(X)$. In this case, the invariance condition for the equilibrium selection rule is satisfied. \textbf{Example 3 (Focusing on Specific Equilibria):} Some researchers prefer to focus on specific equilibria for their counterfactual analysis. This includes the highest profit equilibrium action for a certain firm (Jia:08:Eca) or the Pareto efficient equilibrium (see DePaula:13:ARE), among many others. For example, the highest profit equilibrium action does not change \textit{once the payoff state is fixed}. If we let the highest profit equilibrium action profile at the payoff state $(X,\varepsilon)$ be denoted by $g(x,\varepsilon)$, the policy does not change $g(x,\varepsilon)$ at \textit{the same payoff state} $(x,\varepsilon)$. Our focusing on this equilibrium action profile means that $e(g \mid x, \varepsilon) = e_f(g \mid x,\varepsilon) = 1$. Hence, the invariance condition for the equilibrium selection rule is satisfied. \@startsection{subsection}{2} \z@{.5\linespacing\@plus.7\linespacing}{.5\linespacing}{A Finite-Player Bayesian Game} We turn to presenting our main results in a general set-up of strategic interactions. For this, we first introduce a finite player Bayesian game following Bergemann/Morris:16:TE (BM, hereafter).\footnote{Bergemann/Morris:16:TE considered the case where the state space and the signal space are finite sets for simplicity. In this paper, we consider more general spaces for the state and signal spaces because the researcher's models often involve both discrete and continuous variables.} In our model, the Bayesian game is populated by a finite set of players indexed by $N=\{1,...,n\}$. Let $\mathbb{W}$ denote the set from which the payoff state $W$ takes value, and $\mathbb{Y} = \mathbb{Y}_1 \times ... \times \mathbb{Y}_n$ the set from which the action profile $Y = (Y_1,...,Y_n)$ takes values. Each player's action space $\mathbb{Y}_i$ can be a continuum or a countable set. Each player $i$ is endowed with a payoff function $u_i: \mathbb{Y} \times \mathbb{W} \rightarrow \mathbf{R}$, and chooses an action from $\mathbb{Y}_i$. Let the payoff state $W$ be drawn from a distribution $\mu_W$ on $\mathbb{W}$.\footnote{We assume that $\mathbb{Y} \times \mathbb{W}$ is endowed with a topology and a Borel $\sigma$-field. Throughout the section, we suppress measure-theoretic qualifiers, such as Borel sets, measurable functions, or a statement holding almost everywhere. Details of the mathematical set-up are found in the Online Appendix.} Following BM, we call $B = (\mathbb{Y},\mathbb{W},u,\mu_W)$ a \bi{basic game}, where $u = (u_i)_{i \in N}$ denotes the payoff profile. Each player $i$ observes a signal vector $T_i$ taking values in the space $\mathbb{T}_i\subset \mathbf{R}^{d_T}$, $d_T \ge 1$. Define the signal profile $T=(T_1,...,T_n) \in \mathbb{T}$, where $\mathbb{T} = \mathbb{T}_1 \times...\times \mathbb{T}_n$. Once the payoff state is realized to be $w \in \mathbb{W}$, the signal profile $T$ is drawn from a distribution $\alpha(\cdot \mid w)$ on $\mathbb{T}$. Define the collection of distributions $\mathcal{S} = \{\alpha(\cdot \mid w): w \in \mathbb{W}\}$. As in BM, we call $I = (\mathbb{W},\mathbb{T},\mathcal{S})$ an \bi{information structure}. The information structure in combination with $\mu_W$ induces the joint distribution of $(W,T)$ as follows: $A \subset \mathbb{W}$ and $B \subset \mathbb{T}$, \begin{align*} \mu_{W,T}(A \times B) = \int_A \alpha(B \mid w) d\mu_W(w). \end{align*} A \bi{Bayesian game} $G$ consists of the basic game and information structure, i.e., $G = (B,I)$. A Bayesian game can be a complete information game or an incomplete information game depending on the information structure $I$. We will give examples of various information structures later. Let us introduce strategies. First let $\sigma(\cdot \mid w,t)$ be a conditional distribution on $\mathbb{Y}$, when the payoff state and the signal profile are realized to be $(w,t) \in \mathbb{W}\times \mathbb{T}$. Let $\Sigma$ be a collection of such conditional distributions. For each $i=1,...,n$, let $\sigma_i(\cdot \mid w,t)$ be the $i$-th coordinate marginal conditional distribution of $\sigma(\cdot \mid w,t)$. Following BM, we call each $\sigma \in \Sigma$ a \bi{decision rule}. For each $i=1,...,n$, $t_i \in \mathbb{T}_i$ and $\sigma \in \Sigma$ and a transform $\tau_i: \mathbb{Y}_i \rightarrow \mathbb{Y}_i$, we write the expected payoff of player $i$ as \begin{eqnarray} U_i(\tau_i,t_i;\sigma) = \int \int u_i(\tau_i(y_i),y_{-i},w)d\sigma(y_i,y_{-i} \mid w,t_i,t_{-i}) d\mu_{W,T_{-i}|T_i}(w,t_{-i} \mid t_i), \end{eqnarray} where $\mu_{W,T_{-i}|T_i}(\cdot,\cdot \mid t_i)$ denotes the conditional distribution of $(W,T_{-i})$ given $T_i = t_i$ under the joint distribution $\mu_{W,T}$ of $(W,T)$. The quantity $U_i(\tau_i,t_i;\sigma)$ denotes the conditional expected payoff of player $i$ given her signal $T_i = t_i$ when the player $i$ deviates from the action $y_i$ recommended to her according to the decision rule $\sigma$ and chooses $\tau_i(y_i)$ instead. We say that a decision rule $\sigma \in \Sigma$ is a \bi{Bayes Correlated Equilibrium (BCE)} for $G$ if for each $i=1,...,n$, and each $\tau_i: \mathbb{Y}_i \rightarrow \mathbb{Y}_i$ and $t_i$ in the support of $T_i$, \begin{align} U_i(\mathsf{Id},t_i;\sigma) \ge U_i(\tau_i,t_i;\sigma), \end{align} where $\mathsf{Id}$ is the identity map on $\mathbb{Y}_i$. Denote by $\Sigma_{\mathsf{BCE}}(G)$ the set of BCE's of game $G$. BCE generalizes other solution concepts, such as Bayes Nash equilibria. Later we show how our result carries over to these other solution concepts. Concrete examples of the set-up and our results are provided in Section (ref). As anticipated above, they include widely used models, including those applied in entry games and auctions, among others. \@startsection{subsection}{2} \z@{.5\linespacing\@plus.7\linespacing}{.5\linespacing}{Counterfactual Predictions from a Game} \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Predictions from a Bayesian Game} Predictions from a game $G$ are generated from the distribution of the observed action profile $Y$ conditional on the payoff state $W$, when such an action profile is generated by an equilibrium of the game $G$. Given $\sigma \in \Sigma$ and the information structure $I = (\mathbb{W},\mathbb{T},\mathcal{S})$, we define a probability measure $\rho_\sigma(A\mid w)$ on $\mathbb{Y}$ for each $w \in \mathbb{W}$ as follows:\footnote{The conditional probability $\rho_\sigma$ corresponds to what Bergemann/Morris:16:TE called an “outcome induced by the decision rule $\sigma$”.} for all $A \subset \mathbb{Y}$, \begin{eqnarray} \rho_\sigma(A\mid w) \equiv \int \sigma(A \mid w,t)d\alpha(t|w). \end{eqnarray} Hence, $\rho_\sigma(A\mid w)$ indicates the probability of the action profile realizing in the set $A$ when the payoff state is $W = w$, and the actions are drawn by the decision rule $\sigma$. For example, in an entry game where firms $i=1,...,n$ are deciding whether to enter ($Y_i = 1$) or not ($Y_i = 0$), and this depends on market and firm-level characteristics $W = w$, then $\rho_\sigma(A \mid w)$ is the probability of the entry profile $(Y_1,...,Y_n)$ being realized in $A$ given market conditions $w$. The researcher observes only one action profile $Y$ from the game $G$, even when there are multiple equilibria in this game. In order to complete the description of how $Y$ is generated, we introduce a generic form of the equilibrium selection rule. For a game $G$, we define the \bi{equilibrium selection rule} (denoted by $e(\cdot;G)$) to be a distribution on $\Sigma_{\mathsf{BCE}}(G)$. Thus, the generation of $Y$ is described as follows: Step 1: An equilibrium $\sigma \in \Sigma_{\mathsf{BCE}}$ is drawn from the distribution $e(\cdot;G)$. Step 2: The value of the payoff state $W=w$ is drawn from the distribution $\mu_W$. Step 3: The action profile $Y$ is drawn from the distribution $\rho_\sigma(\cdot\mid w)$. The three steps summarize the causal structure of the model for observed actions $Y$ and the payoff state $W$. Given a game $G$ and $w \in \mathbb{W}$, we define a \bi{(randomized) reduced form of game $G$} as\footnote{The integral with respect to the equilibrium selection rule is an integral of a real function on the space of conditional distributions which we topologize appropriately. Details can be found in the Online Appendix.}, for $A \subset \mathbb{Y}$, \begin{align} \rho_G(A\mid w) \equiv \int_{\Sigma_{\mathsf{BCE}}(G)} \rho_\sigma(A\mid w) d e(\sigma;G). \end{align} The term randomized reduced form is due to the generation of $Y$ being completely described by the couple $(\rho_G,\mu_W)$. It can be represented by first drawing $W=w$ from $\mu_W$, then drawing $Y$ from the distribution $\rho_G(\cdot\mid w)$. Thus, $\rho_G(A \mid w)$ gives the probability of $Y$ taking a value in a set $A$ when $W$ is \textit{fixed} to be $w$. Hence the reduced form $\rho_G$ gives the prediction rule of the game in counterfactual analysis. From here on, any probability statements (including expectation and conditional expectation) involving $(Y,W)$ are with respect to the joint distribution defined as follows: \begin{align*} P\{Y \in A, W \in S\} = \int_{S} \rho_G(A \mid w) d\mu_W(w), \quad A \subset \mathbb{Y}, \quad S \subset \mathbb{W}. \end{align*} \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Counterfactual Predictions} Our counterfactual experiment involves a new policy that changes the payoff state $W$ into $f(W)$ for some map $f:\mathbb{W} \rightarrow \mathbb{W}$. The policy changes the basic game $B$ into \begin{eqnarray} B_f = (\mathbb{Y},\mathbb{W},u,\mu_W \circ f^{-1}), \end{eqnarray} where $\mu_W \circ f^{-1}$ is the distribution of $f(W)$ when $W$ is drawn from $\mu_W$. Thus our counterfactual analysis involves a policy that transforms the pre-policy game $G = (B,I)$ into a post-policy game\footnote{It is important to note that here we regard the information structure $I$ as invariant to the policy. This does not mean that the signals remain the same after the policy. Note that the post-policy signals are generated from the distribution $\alpha(\cdot \mid f(w))$. Hence if the payoff states are drawn differently, this affects realized signals.} \begin{align*} G_f = (B_f,I). \end{align*} As we describe in Section (ref), this can include changes to market characteristics (population size, or changes to existing market policies) or changes to observable reserve prices in auctions. It further extends to many other settings where the policy affects a random variable with sampling variation, such as individual income, taxes, etc. A counterfactual prediction at $f(w)$ in game $G_f$ can be made from the reduced form $\rho_{\sigma}$ induced by $\sigma \in \Sigma_{\mathsf{BCE}}(G_f)$, i.e., \begin{eqnarray} \rho_{\sigma}(A \mid f(w)) = \int \sigma(A \mid f(w),t)d\alpha(t \mid f(w)). \end{eqnarray} Let $e(\cdot;G_f)$ be the equilibrium selection rule of the game $G_f$. Then the \bi{equilibrium-based prediction of the post-policy game $G_f$} is given by \begin{align} \rho_{G_f}(A \mid f(w)) \equiv \int_{\Sigma_{\mathsf{BCE}}(G_f)} \rho_{\sigma}(A \mid f(w)) d e(\sigma;G_f), \quad A \subset \mathbb{Y}. \end{align} The quantity $\rho_{G_f}(A \mid f(w))$ represents the probability of the action profile from the post-policy game $G_f$ realizing in $A$ when the payoff state is $f(w)$. An alternative way of generating a prediction is to use the following: \begin{align} \rho_G(A \mid f(w)) \equiv \int_{\Sigma_{\mathsf{BCE}}(G)} \rho_{\sigma}(A \mid f(w)) d e(\sigma;G). \end{align} We call $\rho_G(A \mid f(w))$ the \bi{decomposition-based prediction from game $G$}. When $f(w)$ is in the support of $W$, the last integral is equal to \begin{align*} P\{Y \in A \mid W = f(w)\}. \end{align*} Hence, this prediction extrapolates the relation between $Y$ and $W$ in the pre-policy game to the post-policy game. In general, decomposition-based predictions do not coincide with equilibrium-based predictions in ((ref)), when $\rho_G$ is not policy-invariant. Our main result below presents a set of sufficient conditions under which the equilibrium-based predictions have upper and lower bounds that can be identified using only decomposition-based predictions. The bounds coincide when $f(W)$ is in the support of $W$. \@startsection{subsection}{2} \z@{.5\linespacing\@plus.7\linespacing}{.5\linespacing}{Interval-Identification of Equilibrium-Based Predictions} \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{The Main Result} The main result of this paper shows that the equilibrium-based prediction is interval-identified by the decomposition-based prediction under two sufficient conditions. The first condition is concerned with the information structure and requires that the policy affects only part of the payoff states that are commonly observed by all the players. The second condition is an invariance condition on the equilibrium selection rules. \begin{assumption}[Information Structure Condition] (i) $\mathbb{W} = \mathbb{W}_C \times \mathbb{W}_I$, for some sets $\mathbb{W}_C$ and $\mathbb{W}_I$,\footnote{The subscript $C$ is mnemonic for “common” and $I$ for “idiosyncratic”. These subscripts are used in place of $1$ and $2$ to avoid a conflict with subscripts $i$ for individual players we use later.} and \begin{align*} f(w) = (\tilde f(w_C),w_I), \quad w = (w_C, w_I) \in \mathbb{W}, \end{align*} for some map $\tilde f: \mathbb{W}_C \rightarrow \mathbb{W}_C$. (ii) For each $i=1,...,n$, $\mathbb{T}_i = \mathbb{T}_{C,i} \times \mathbb{T}_{I,i}$, where $\mathbb{T}_{C,i} = \mathbb{W}_C$, and \begin{align*} \alpha(A \mid w) = 1\left\{s(w) \in A \right\}, \end{align*} for $A \subset \mathbb{T}$, for a map $s = (s_1,...,s_n): \mathbb{W} \rightarrow \mathbb{T}$, where for each $i =1,...,n$, \begin{align*} s_i(w) = (w_C,\varphi_i(w_I)), \quad w = (w_C,w_I) \in \mathbb{W}, \end{align*} for some map $\varphi_i: \mathbb{W}_I \rightarrow \mathbb{T}_{I,i}$. \end{assumption} Assumption (ref)(i) requires that the change of the payoff state by the policy $f$ is restricted to the first component $W_C$ of the payoff state $W = (W_C,W_I)$. This assumption guarantees that the counterfactual payoff state is changing a publicly observed payoff state, rather than one that is private information. This is used to ensure that players' posterior beliefs remain invariant after the policy. The Online Appendix provides a simple example of a private information entry game which shows that when the policy changes the private information component of the payoff state, decomposition-based predictions fail. The failure is because a change in a private component of players' payoff state leads to changes to posterior beliefs and creates incentives to deviate from the strategies used in the pre-policy game. Assumption (ref)(ii) says that, in terms of the relation between signals and payoff state vector, we have \begin{align*} T = s(W) = (s_1(W),...,s_n(W)), \quad s_i(W) = (W_C, \varphi_i(W_I)), \end{align*} for some maps, $s$ and $\varphi_i$, $i=1,...,n$. Hence, every player observes $W_C$ that is subject to a change by the policy in (i). Assumption (ref) allows for a setting where the signals include payoff irrelevant signals, by payoff functions $u_i$ specified to depend on only part of $W$. Assumption (ref) admits a broad class of information structures, as we see through examples below. In these examples, we specify the payoff state vector as $W = (W_1,...,W_n)$, where \begin{align} W_i = (X_i, \varepsilon_i), \end{align} so that each individual player $i$'s payoff depends on $W$ only through $W_i$. Assumption (ref) in this setting does not put any restrictions on the joint dependence of $W_i$ across players. Hence we can accommodate the information structures considered by Magnolfi/Roncoroni:22:ReStud in the entry game setting. \textbf{Example 1: Complete Information: } In the case of a complete information game, we simply specify $T_i = W$, so that each map $s_i$ is an identity map. In this case, any policy $f$ that changes $W$ satisfies Condition (i) of Assumption (ref) because we can simply take $W_C$ to be $W$ here. \textbf{Example 2: Public-Private Dichotomy of Signals: } In the case of a public-private information structure where $X_i$ is publicly observed while $\varepsilon_i$ belongs to private information, we specify $W = (W_1,...,W_n)$, $W_i = (X_i, \varepsilon_i)$, with $T_i = s_i(W)$ such that $s_i(W) = (X,\varepsilon_i)$, where $X = (X_1,...,X_n)$. In this case, Condition (i) of Assumption (ref) is satisfied, whenever the policy $f$ is one that alters $X$, not $\varepsilon_i$. \textbf{Example 3: Observable Private Signals: } The researcher may observe signals that are part of the private information. To reflect this setting, we specify $W = (W_1,...,W_n)$, $W_i = (X_i, \varepsilon_i)$, with $X_i = (X_{1,i},X_{2,i})$, and take $T_i = s_i(W)$, with $s_i(W) = (X_1, X_{2,i}, \varepsilon_{i})$, $X_1 = (X_{1,i})_{i =1}^n$, so that $X_{2,i}$ belongs to private information and is observed by the researcher. In this case, as long as $f$ alters $X_1$, not $(X_{2,i}, \varepsilon_i)$, Condition (i) of Assumption (ref) is satisfied. Let us introduce the invariance condition on equilibrium selection rules. For each Bayesian game $G$ and $w \in \mathbb{W}$, let \begin{align*} \Sigma_{\mathsf{BCE},w}(G) &=\left\{\sigma(\cdot \mid w,s(w)): \sigma \in \Sigma_{\mathsf{BCE}}(G) \right\}. \end{align*} This is the $w$-section of the set of BCE for game $G$, i.e., each member of the set $\Sigma_{\mathsf{BCE},w}(G)$ is an equilibrium probability over the action profiles given the payoff state $w$ and the signal $s(w)$. Similarly, we denote the $w$-section of the equilibrium selection rule $e(\cdot; G)$ as $e_w(\cdot;G)$: for each $w \in \mathbb{W}$ and for each $A \subset \Sigma_{\mathsf{BCE},w}(G)$, \begin{align*} e_w(A;G) = e(\{\sigma \in \Sigma_{\mathsf{BCE}}(G): \sigma(\cdot \mid w,s(w)) \in A\};G). \end{align*} We also define $e_w(\cdot;G_f)$ to be the $w$-section of $e(\cdot;G_f)$. Now, the invariance condition is stated as follows. \begin{assumption}[Invariance of Equilibrium Selection Rules] For each $w \in \mathbb{S}_{W} \cap \mathbb{S}_{f(W)}$, if $\Sigma_{\mathsf{BCE},w}(G) = \Sigma_{\mathsf{BCE},w}(G_f)$, then, we have \begin{align*} e_w(\cdot;G) = e_w(\cdot;G_f), \end{align*} where $\mathbb{S}_{W}$ and $\mathbb{S}_{f(W)}$ are the supports of $W$ and $f(W)$ under $\mu_W$ respectively. \end{assumption} The invariance condition requires that, if the equilibrium action profiles remain the same after the policy \textit{at the same payoff state $w$}, their selection probability remains the same at the payoff state as well. In other words, once the payoff state is realized, and the set of action profiles in equilibrium in the post-policy game is the same as that in the pre-policy game, the action profile in the post-policy game is selected with the same probability as in the pre-policy game.\footnote{This assumption speaks only to the case where $\Sigma_{\mathsf{BCE},w}(G) = \Sigma_{\mathsf{BCE},w}(G_f)$. Recall that the payoff state $w$ includes both observable and unobservable states.} The invariance condition naturally arises from a \textit{consistency condition} among equilibrium selection rules across different games. The consistency condition requires that, for each Bayesian game $G$, $e_w(\cdot;G)$ is derived from a primitive probability measure on $\Sigma$ as a conditional probability concentrated on $\Sigma_{\mathsf{BCE},w}(G)$. More specifically, let $\lambda$ be a probability distribution over $\Sigma$, and we define $\lambda_w$ from $\lambda$ in the same way as we defined $e_w$ from $e$. Then, for any set $A$ of probability measures on $\mathbb{Y}$, the consistency condition states that, for each $w \in \mathbb{W}$, we have \begin{align} e_w(A;G) = \frac{\lambda_w(A \cap \Sigma_{\mathsf{BCE},w}(G))}{\lambda_w(\Sigma_{\mathsf{BCE},w}(G))}. \end{align} Then, Assumption (ref) is satisfied, because $e_w(\cdot;G)$ depends on $G$ only through $\Sigma_{\mathsf{BCE},w}(G)$.\footnote{We call this condition consistency of equilibrium selection rules, as it is analogous to the consistency of beliefs in a Bayesian game where individual players' beliefs are derived from a common prior on the true state of the game.} The invariance condition is already satisfied by many equilibrium selection rules used in the literature of empirical research. This includes the common assumption that the same equilibrium is played in the data and in the counterfactual (see Aguirregabiria/Mira:10:JOE). Relatedly, many papers choose a specific equilibrium which is analyzed in both the data and the counterfactual. This includes the highest profit equilibrium for a certain firm (Jia:08:Eca), the Pareto superior equilibrium (see DePaula:13:ARE), or the equilibrium studied by Milgrom/Weber:82:ECMA in common value auctions. Furthermore, this invariance condition already holds when an equilibrium selection rule is kept the same for counterfactual predictions (e.g., in Bajari/Hong/Ryan:10:Eca, or as assumed in Berry/Haile:14:Ecma). It is also satisfied if it is deemed constant across the equilibrium in the data and counterfactual (e.g., Aguirregabiria:12:EL, applied in Aguirregabiria/Ho:12:JOE). Our invariance condition is closely related to the one used in Aguirregabiria/Mira:13:WP. Their invariance condition requires that the equilibrium selection rule depend on the payoff state $w$ and the structural parameters $\theta$ only through the payoff function evaluated at $(w,\theta)$. On the other hand, our invariance condition is satisfied when the equilibrium selection rule is determined by the payoff state $w$ only. Sometimes the invariance condition may be reasonable in a setting with small changes to the payoff state (e.g., small increases in minimum wages), because small changes can be well approximated by keeping the same game environment. In fact, many of the policy experiments in entry games in empirical research are “small policies” which change payoff states (or the set of affected players) by only a small amount, such as 10-15% changes in observed policy variables (as opposed to, say, doubling). For example, Jia:08:Eca considered the increase in market size by a small amount, 10%, and Magnolfi/Roncoroni:22:ReStud considered a change in a market characteristic affecting only 13% of markets. This local argument is also used in Aguirregabiria:12:EL and Aguirregabiria/Ho:12:JOE, since they explore local approximations of the counterfactual values around the data. In order to use the invariance condition, we need to ensure that, for any $w \in \mathbb{S}_W \cap \mathbb{S}_{f(W)}$, we have $\Sigma_{\mathsf{BCE},w}(G) = \Sigma_{\mathsf{BCE},w}(G_f)$. The essential role of Assumption (ref) is to ensure this by maintaining each player's posterior after the policy. For a real map $h$ on $\mathbb{Y} \times \mathbb{W}$, let us define \begin{align} \mathsf{AEP}(h) &\equiv \int \int_{\mathbb{Y}} h(y,w) d \rho_{G_f}(y \mid w)d(\mu_W \circ f^{-1})(w), \text{ and }\\ \notag \mathsf{ADP}(h) &\equiv \int \mathbf{E}[h(Y,W) \mid W = f(w)]1\{f(w) \in \mathbb{S}_W\}d\mu_W(w). \end{align} The quantity $\mathsf{AEP}(h)$ represents our target parameter which is the equilibrium-based expectation of $h(Y,f(W))$ after the policy\footnote{Note the change of variables within the integral, so that we are integrating $f(w)$.}, whereas $\mathsf{ADP}(h)$ represents its decomposition-based counterpart. Depending on the choice of $h$, which can also be vector-valued, we can express various objects of interest as listed below. \textbf{Example 1: Expected Actions} We simply take $h(y,w) = y$. Then $\mathsf{AEP}(h)$ represents the expected action profile after the policy. \textbf{Example 2: Distribution of the Action Profile} We take $h(y,w) = 1\{y \le t\}$, $t \in \mathbf{R}^n$. Then $\mathsf{AEP}(h)$ is the CDF of the action profile at $t$, after the policy. \textbf{Example 3: Distribution of Maximum Actions} We take $h(y,w) = 1\{\max_{1 \le i \le n: w_i \in S} y_i \le t\}$, $t \in \mathbf{R}$, $w = (w_1,...,w_n)$, for some set $S \subset \mathbb{W}$. Then $\mathsf{AEP}(h)$ represents the CDF at $t$ of the maximum action among those players $i$ with $W_i \in S$, after the policy. We are prepared to present our main result. \begin{theorem} Suppose that Assumptions (ref) and (ref) hold for the pre-policy game $G = (B,I)$ and the post-policy game $G_f = (B_f,I)$. Let a map $h:\mathbb{Y} \times \mathbb{W} \rightarrow \mathbf{R}$ be such that, for all $w \in \mathbb{W}$ such that $f(w) \notin \mathbb{S}_W$, \begin{align} \underline h(w) \le \inf_{y \in \mathbb{Y}} h(y,w) \le \sup_{y \in \mathbb{Y}} h(y,w) \le \overline h(w), \end{align} for some maps $\underline h, \overline h: \mathbb{W} \rightarrow \mathbf{R}$. Then, \begin{align} \mathsf{ADP}(h) + \Delta(\underline h) \le \mathsf{AEP}(h) \le \mathsf{ADP}(h) + \Delta(\overline h), \end{align} where \begin{align*} \Delta(\overline h) &\equiv \mathbf{E}\left[\overline h(f(W))1\left\{f(W) \notin \mathbb{S}_W\right\} \right], \text{ and }\\ \Delta(\underline h) &\equiv \mathbf{E}\left[\underline h(f(W))1\left\{f(W) \notin \mathbb{S}_W\right\} \right]. \end{align*} \end{theorem} When $Y_i \in [h_L,h_U]$, $h_L < h_U$, the theorem yields bounds for the conditional average predicted outcome of player $i$. More specifically, for $i=1,...,n$, define $h_i: \mathbb{Y} \times \mathbb{W} \rightarrow \mathbf{R}$ as $h_i(Y,W) = Y_i$. Then the theorem above yields that, for each $i = 1,...,n$, \begin{align*} \mu_i(f) + h_L \cdot P\left\{f(W) \notin \mathbb{S}_W\right\} \le \mathsf{AEP}(h_i) \le \mu_i(f) + h_U \cdot P\left\{f(W) \notin \mathbb{S}_W\right\}, \end{align*} where \begin{align*} \mu_i(f) \equiv \int \mathbf{E}[Y_i \mid W = f(w)]1\{f(w) \in \mathbb{S}_W\} d\mu_W(w). \end{align*} In an entry game where $Y_i \in \{0,1\}$, the bounds above with $h_U = 1$ and $h_L = 0$ are bounds for the predicted entry probability of firm $i$ after the policy. We can use the results to obtain bounds for the average effect of the policy $f$ on the player $i$'s outcome. The average treatment effect (denoted by $\mathsf{ATE}_i$) is defined as \begin{align*} \mathsf{ATE}_i \equiv \mathsf{AEP}(h_i) - \mathbf{E}[Y_i]. \end{align*} Then we obtain the following bounds for the average treatment effect: \begin{align*} \mu_i(f) - \mathbf{E}[Y_i] + h_L \cdot P\left\{f(W) \notin \mathbb{S}_W\right\} \le \mathsf{ATE}_i \le \mu_i(f) - \mathbf{E}[Y_i] + h_U \cdot P\left\{f(W) \notin \mathbb{S}_W\right\}. \end{align*} One might ask whether the bounds in Theorem (ref) are sharp. The following proposition gives a sense in which the answer is affirmative. \begin{proposition} Suppose that a policy $f: \mathbb{W} \rightarrow \mathbb{W}$ and $\mu_W$ are given such that the support of $W$ overlaps that of $f(W)$. Suppose further that maps $h: \mathbb{Y} \times \mathbb{W} \rightarrow \mathbf{R}$, $\underline h, \overline h: \mathbb{W} \rightarrow \mathbf{R}$, are given as in Theorem (ref), where $\mathbb{Y}$ is a countable set and for all $w \in \mathbb{W}$, $\overline h(w) = \sup_{y \in \mathbb{Y}} h(y,w)$ and $\underline h(w) = \inf_{y \in \mathbb{Y}} h(y,w)$.\footnote{The condition of countability of $\mathbb{Y}$ can be removed, for example, if $h$ is a continuous map and $\mathbb{Y}$ is compact.} Then, there exists a Bayesian game $G$ such that Assumptions (ref) and (ref) hold and either of the two inequalities in ((ref)) holds with equality. \end{proposition} Note that simple shape constraints such as $\rho_{G_f}( \cdot \mid w)$ being monotone or concave in $w$ in all the BCEs do not help improve the bounds because a constant map also satisfies such constraints trivially. However, shape constraints may help estimate $\mathsf{ADP}(h)$ more accurately. The next section overviews the heuristics of the proof of Theorem (ref), while the subsequent section overviews identification of the bounds in the theorem. Concrete examples applying those results to entry games and auctions are then provided in Section (ref). \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Heuristics behind Theorem (ref)} To understand how Theorem (ref) follows from Assumptions (ref) and (ref), let us assume for simplicity that $h(Y,W) = Y$ and $Y \in \{0,1\}$. First, note that \begin{align*} \mathbf{E}[Y \mid W=f(w)] = \int y d\rho_G(y \mid f(w)). \end{align*} Suppose that we have proved that, for $w \in \mathbb{S}_{W} \cap \mathbb{S}_{f(W)}$, \begin{align} \rho_G(\cdot \mid f(w)) = \rho_{G_f}(\cdot \mid f(w)). \end{align} Then, \begin{align*} \mathbf{E}[Y \mid W=f(w)]1\{f(w) \in \mathbb{S}_{W} \cap \mathbb{S}_{f(W)} \} &= \int y d\rho_G(y \mid f(w)) 1\{f(w) \in \mathbb{S}_{W} \cap \mathbb{S}_{f(W)} \}\\ \notag &= \int y d\rho_{G_f}(y \mid f(w)) 1\{f(w) \in \mathbb{S}_{W} \cap \mathbb{S}_{f(W)} \}\\ \notag &\le \int y d\rho_{G_f}(y \mid f(w)). \end{align*} The last integral is bounded by \begin{align*} &\int y d\rho_{G_f}(y \mid f(w)) 1\{f(w) \in \mathbb{S}_{W} \cap \mathbb{S}_{f(W)} \} + 1\{f(w) \notin \mathbb{S}_{W} \cap \mathbb{S}_{f(W)} \}\\ &=\int y d\rho_{G}(y \mid f(w)) 1\{f(w) \in \mathbb{S}_{W} \cap \mathbb{S}_{f(W)} \} + 1\{f(w) \notin \mathbb{S}_{W} \cap \mathbb{S}_{f(W)} \}. \end{align*} Integrating out $w$ in the terms using $\mu_W$, we obtain the desired bounds in Theorem (ref). Thus, the crucial step is to show ((ref)). By the invariance condition in Assumption (ref), it suffices for ((ref)) to show that, for all $w \in \mathbb{S}_W \cap \mathbb{S}_{f(W)}$, we have \begin{align*} \Sigma_{\mathsf{BCE},w}(G) = \Sigma_{\mathsf{BCE},w}(G_f). \end{align*} This can be shown under the conditions for the information structure in Assumption (ref). In fact, since the policy changes only the publicly observable payoff component, it can be shown that the sets $\Sigma_{\mathsf{BCE}}(G)$ and $\Sigma_{\mathsf{BCE}}(G_f)$ coincide when we restrict the payoff state to $\mathbb{S}_W \cap \mathbb{S}_{f(W)}$. We refer the reader to the Online Appendix for details. \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Identification of the Bounds} If we observe the actions $Y$ and the payoff state $W$ of the pre-policy game, then we can recover the bounds in Theorem (ref) from data without specifying the details of the game. However, in practice, we often do not observe the whole vector $W$. Suppose that $W_i = (X_i, \varepsilon_i)$, where $X_i$ is observed and $\varepsilon_i$ unobserved by the researcher. Let the policy $f$ be of the form $f(W) = (f_1(W_1),...,f_n(W_n))$, where \begin{align} f_i(W_i) = (f_i^*(X_i),\varepsilon_i), \quad i=1,...,n, \end{align} for some map $f_i^*$. As for $h$, we consider $h(y,w) = h^*(y,x)$, $\overline h(w) = \overline h^*(x)$, and $\underline h(w) = \underline h^*(x)$, for some maps $h^*$, $\overline h^*$, and $\underline h^*$, which depend only on observable states. The identification of the bounds in Theorem (ref) can be derived in this setting as follows. First, by the choice of $f$ in ((ref)), we can identify \begin{align} \Delta(\underline h) &= \mathbf{E}\left[\underline h^*(f^*(X))1\left\{f^*(X) \notin \mathbb{S}_X\right\} \right] \text{ and }\\ \notag \Delta(\overline h) &=\mathbf{E}\left[\overline h^*(f^*(X))1\left\{f^*(X) \notin \mathbb{S}_X\right\} \right], \end{align} where $\mathbb{S}_X$ denotes the support of $X$, and $f^*(X) = (f_1^*(X_1),...,f_n^*(X_n))$. Thus, for the interval identification of $\mathsf{AEP}(h)$, it suffices to identify $\mathsf{ADP}(h)$. However, the identification strategy of $\mathsf{ADP}(h)$ depends on whether $X$ and $\varepsilon$ are independent or not. The following proposition considers a setting where $X$ and $\varepsilon$ are independent. \begin{proposition} Suppose that $X$ and $\varepsilon$ are independent. Then, \begin{align} \mathsf{ADP}(h) = \int \mathbf{E}\left[h^*(Y,X) \mid X = f^*(x)\right] 1\{f^*(x) \in \mathbb{S}_X\}d\mu_X(x), \end{align} where $\mu_X$ denotes the distribution of $X$. \end{proposition} Note that the identification result allows for $\varepsilon_1,...,\varepsilon_n$ to be correlated; this correlation can come from some unobserved characteristics of the game. Suppose that $X$ and $\varepsilon$ are potentially correlated. In this case, the decomposition-based approach may still be implemented using a control function approach (Blundell/Powell:03:Adv and Imbens/Newey:09:Eca).\footnote{Game-theoretic models often involve a simultaneous system of equations. Note that we exclude the setting where the policy variable is part of the endogenous outcomes in such equations. For example, in a two-player game, with outcomes, $Y = (Y_1,Y_2)$, we focus on a policy that changes the payoff state $X$ which is not one of the two outcomes. When one of the endogenous outcomes is a policy variable, the structural equations need to be transformed into a triangular system of equations to apply a control function approach. Blundell/Matzkin:14:QE present precise conditions for such a transform to exist. These conditions may be implausible in some empirical applications.} More specifically, suppose that $X_i = (X_{i,a},X_{i,b})$ and \begin{align*} f_i^*(X_i) = (g_i(X_{i,a}),X_{i,b}), \end{align*} for some map $g_i$ so that the policy alters only $X_{i,a}$. Define \begin{align*} X_a = (X_{1,a},...,X_{n,a}) \text{ and } X_b = (X_{1,b},...,X_{n,b}), \end{align*} and let $g(x_a) = (g_1(x_{1,a}),...,g_n(x_{n,a}))$, $x_a = (x_{1,a},...,x_{n,a})$. Then, we obtain the following identification result. \begin{proposition} Suppose that $X_a$ and $\varepsilon$ are conditionally independent given $X_b$. Then, \begin{align} \mathsf{ADP}(h) = \int \mathbf{E}\left[ h^*(Y,X) \mid (X_a,X_b) = (g(x_a),x_b)\right] 1\left\{g(x_a) \in \mathbb{S}_{X_a}\right\} d\mu_{X_a,X_b}(x_a,x_b), \end{align} where $\mu_{X_a,X_b}$ is the distribution of $(X_a,X_b)$, and $\mathbb{S}_{X_a}$ denotes the support of $X_a$. \end{proposition} In general, when $X_a$ and $\varepsilon$ are dependent due to some unobserved game-specific characteristics, we may consider $X_b$ as an observed vector including game characteristics such that conditioning on $X_b$ removes the stochastic dependence between $X_a$ and $\varepsilon$. In many examples, the payoff state $W_i$ of each player $i$ enters the payoff as a partial index form: $W_i = (X_{i,a},V_i,\varepsilon_i)$, where $V_i = X_{i,b}'\theta_i$, with coefficient $\theta_i$. If $X_a$ and $\varepsilon$ are conditionally independent given $V = (V_1,...,V_n)$, which is an assumption weaker than the previous assumption that $X$ and $\varepsilon$ are independent, we can rewrite ((ref)) as: \begin{align} \mathsf{ADP}(h) = \int \mathbf{E}\left[ h^*(Y,X) \mid (X_a,V) = (g(x_a),v)\right] 1\left\{g(x_a) \in \mathbb{S}_{X_a} \right\} d\mu_{X_a,V}(x_a,v), \end{align} where $\mu_{X_a,V}$ is the distribution of $(X_a,V)$, and $V = (V_1,...,V_n)$. We can identify and estimate $\theta_1,...,\theta_n$ (up to a scale) following the literature of multi-index models (see Ichimura/Lee:91:NSEM, Lee:95:JOE, Donkers/Schafghans:08:ET, Xia:08:JASA and Ahn/Ichimura/Powell/Ruud:18:JBES and references therein.) The main difference here is that we do \textit{not} introduce the multi-index structure as a semiparametric restriction on a nonparametric function; it naturally follows from the index structure of the payoff function in the game. The multi-index models are useful for dimension reduction when the game involves only a few players, and the dimension of $X_i$ is large. We provide some implementation details in the Online Appendix. \@startsection{subsection}{2} \z@{.5\linespacing\@plus.7\linespacing}{.5\linespacing}{Extensions} \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Extension to Other Solution Concepts} As mentioned in the main text, our results extend to other solution concepts beyond BCE. In particular, we can accommodate any refinement to BCE, which includes Bayes-Nash Equilibria, among others. To see this, we let $\Sigma' \subset \Sigma$ be a given subcollection of decision rules $\sigma$ and consider the restricted BCE: \begin{align} \Sigma_{\mathsf{BCE}}'(G) = \Sigma_{\mathsf{BCE}}(G) \cap \Sigma'. \end{align} We call this set the set of \bi{Bayes Correlated Equilibria (BCE) restricted to} $\Sigma'$. For example, suppose that $\Sigma'$ is the collection of decision rules $\sigma$ of the following form: for any $A = A_1 \times ... \times A_n$, \begin{eqnarray} \sigma(A \mid w,t) = \prod_{i=1}^n \beta_i(A_i \mid w,t_i), \end{eqnarray} where $\beta_i(\cdot \mid w,t_i)$ is a conditional distribution on $Y_i$ given $(W,T_i) = (w,t_i)$. Then a BCE restricted to $\Sigma'$ is the set of \bi{Bayes Nash Equilibria (BNE)}. We can add further restrictions such as symmetry or differentiability depending on the application. Let us turn to the interval-identification of equilibrium-based predictions in terms of a BCE restricted to $\Sigma'$. First, similarly as before, define the equilibrium selection rules $e'(\cdot;G)$, and $e'(\cdot;G_f)$ as a distribution on $\Sigma_{\mathsf{BCE}}'(G)$ and $\Sigma_{\mathsf{BCE}}'(G_f)$ respectively. Similarly, we define $\rho_G'$ and $\rho_{G_f}'$ using $\Sigma_{\mathsf{BCE}}'(G)$, $\Sigma_{\mathsf{BCE}}'(G_f)$, $e'(\cdot;G)$, and $e'(\cdot;G_f)$. Let $\mathsf{ADP}'(h)$ and $\mathsf{AEP}'(h)$ be the same as $\mathsf{ADP}(h)$ and $\mathsf{AEP}(h)$ except that we substitute $\rho_G'$ and $\rho_{G_f}'$ for $\rho_G$ and $\rho_{G_f}$ in the definition in ((ref)). Similarly as in the case of BCE, we assume that the invariance condition on $e'(\cdot;G)$ and $e'(\cdot;G_f)$ holds in terms of the BCEs restricted to $\Sigma'$. Then, we obtain an analogue of Theorem (ref) as follows. \begin{corollary} Suppose that Assumptions (ref) and (ref) (in terms of the BCEs restricted to $\Sigma'$) hold for the pre-policy game $G = (B,I)$ and the post-policy game $G_f = (B_f,I)$. Suppose further that maps $h: \mathbb{Y} \times \mathbb{W} \rightarrow \mathbf{R}$, $\underline h, \overline h: \mathbb{W} \rightarrow \mathbf{R}$, are given as in Theorem (ref). Then, \begin{align} \mathsf{ADP}'(h) + \Delta(\underline h) \le \mathsf{AEP}'(h) \le \mathsf{ADP}'(h) + \Delta(\overline h). \end{align} \end{corollary} Hence, decomposition-based predictions can still be used for counterfactual analysis in a setting with various other solution concepts such as BNE or its further restricted versions. They coincide with the equilibrium-based predictions when $\mathbb{S}_{f(W)} \subset \mathbb{S}_W$. \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{When the Policy Also Changes the Information Structure} When a government policy is announced in advance, additional signals are often created through various reports of analysis on the policy and may be used by agents. Thus, the policy may change the information structure of the players as well. In the Online Appendix, we extend our main results to such cases. In particular, we use the connection between information structures and the set of equilibrium reduced-forms in BM and show that the bounds in Theorem (ref) accommodate scenarios where the policy also introduces new signals, as long as the latter do not reveal other players' pre-policy signals and the payoff state beyond what has been known to the player. This includes the cases where the policy may be used as a coordination device by players (e.g., sunspots, as in Shell:89:GE,Peck/Shell:91:ReStud), or when this signal is about future policy implications (e.g., in one empirical application below, government reports discuss market structure following the policy repeal, but are unlikely to reveal pre-policy signals of individual agents.) \@startsection{subsection}{2} \z@{.5\linespacing\@plus.7\linespacing}{.5\linespacing}{Examples} \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Entry Games} Consider an entry game which received a great deal of attention in the literature (e.g., Ciliberto/Tamer:09:Eca, Jia:08:Eca, Grieco:14:RAND and Magnolfi/Roncoroni:22:ReStud). Suppose that there are $n$ firms, $i=1,...,n$, who choose a binary action $Y_i \in \{0,1\}$, $Y = (Y_1,...,Y_n)$, where $Y_i = 1$ represents entry in the market and $Y_i=0$ staying out of the market. We specify the payoff generically as $u_i(y, W)$, $W = (X, \varepsilon)$, where $X = (X_1,...,X_n) \in \mathbf{R}^{n d_X}$ is observed and $\varepsilon = (\varepsilon_1,...,\varepsilon_n) \in \mathbf{R}^{n d_\varepsilon}$ unobserved by the researcher. As for the information structure, we focus on two information structures that are often used in the literature of empirical research. \begin{assumption} Either of the following two conditions is satisfied: (i) The game is of complete information, i.e., $T_i = (X, \varepsilon)$, for $i=1,...,n$. (ii) The game has a public-private dichotomy of signals, i.e., $T_i = (X, \varepsilon_i)$, for $i=1,...,n$. \end{assumption} For example, Assumption (ref)(i) is satisfied in the complete information environments of Bresnahan/Reiss:91:JOE, Jia:08:Eca and Ciliberto/Tamer:09:Eca. The public-private dichotomy case is studied in depth by Grieco:14:RAND. As for the policy, we assume that $X$ is subject to a change by a policy whereas $\varepsilon$ is not. \begin{assumption} $f(X,\varepsilon) = (f^*(X),\varepsilon)$, for some map $f^*$. \end{assumption} A counterfactual policy $f$ in the above assumption was considered by all the papers cited at the beginning of this section. For example, Ciliberto/Tamer:09:Eca and Grieco:14:RAND set the values of a dummy variable in $X_i$ to 0. Meanwhile, Jia:08:Eca changes market size, a variable in $X_i$. Let us describe the invariance condition for the equilibrium selection rules in this setting. Let $G$ denote the pre-policy game and $G_f$ the post-policy game. We define $\Sigma_{\mathsf{BCE},w}(G)$ and $\Sigma_{\mathsf{BCE},w}(G_f)$ as in Assumption (ref) with $w = (x,\varepsilon)$. Then, the invariance condition we require takes the following form: \begin{assumption} For each $w = (x,\varepsilon)$ in the intersection of the supports of $(X,\varepsilon)$ and $(f^*(X), \varepsilon)$ such that $\Sigma_{\mathsf{BCE},w}(G) = \Sigma_{\mathsf{BCE},w}(G_f)$, we have $e_w(\cdot; G) = e_w(\cdot; G_f)$. \end{assumption} Our target parameter is the conditional probability of $Y = a$, $a \in \{0,1\}^n$, after the policy, given $f^*(X) \in C$ for some set $C$, which is defined as follows: \begin{align*} p_f(Y=a \mid C) \equiv \frac{1}{P\left\{f^*(X) \in C\right\}}\int_{C \times \mathbf{R}^{d_\varepsilon}} \int_{\Sigma_{\mathsf{BCE}}(G_f)} \sigma(a \mid w,s(w)) de(\sigma;G_f) d(\mu_W \circ f^{-1})(w), \end{align*} where $w = (x,\varepsilon)$. For example, $p_f(Y = (1,...,1) \mid C)$ denotes the conditional probability of all firms entering the market given $f^*(X) \in C$, after the policy. The result below shows how this probability is bounded by decomposition-based predictions. \begin{corollary} Suppose that Assumptions (ref)-(ref) hold. Then for each $a \in \{0,1\}^n$ and $C \subset \mathbf{R}^{nd_X}$, \begin{align*} \mathbf{E}\left[\pi\left(a \mid f^*(X)\right) \mid f^*(X) \in C\right] &\le p_f(Y=a \mid C) \\ &\le \mathbf{E}\left[\pi\left(a \mid f^*(X)\right) \mid f^*(X) \in C\right] + P\{f^*(X) \notin \mathbb{S}_X \mid f^*(X) \in C\}, \end{align*} where $\pi(a \mid x) = P\{Y= a \mid X = x\}1\{x \in \mathbb{S}_X\}$ and $\mathbb{S}_X$ denotes the support of $X$. \end{corollary} The results do not rely on any further specification of the payoff function, or a parametric assumption for the distribution of $\varepsilon_i$. The conditional expectation $\mathbf{E}\left[\pi(a \mid f^*(X)) \mid f^*(X) \in C\right]$ is identified using the data from the pre-policy game. We provide a step-by-step empirical implementation of this result in Section (ref) and the Online Appendix. For example, suppose that the researcher would like to obtain the predicted joint entry probability of the firms under a policy that changes $X$ into $f^*(X)$ such that $f^*(X)$ lies in the support of $X$. Then, Corollary (ref) says that the counterfactual prediction is point-identified as \begin{align*} \int P\{Y= (1,...,1) \mid X = f^*(x)\}dP_X(x), \end{align*} where $P_X$ denotes the distribution of $X$. It is quite simple to obtain this prediction: we use the nonparametric regression of $1\{Y=(1,...,1)\}$ on $X$, and integrate the regression function after transforming the regressors by the map $f^*$. More importantly, this prediction is obtained without specifying further details of the game such as a functional form restrictions or the parametric specification of the payoff function or a parametric distribution of unobserved heterogeneity. \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Auctions} A common approach for counterfactual analysis in the empirical auction literature is to first nonparametrically identify and estimate the distribution of valuations from the distribution of bids, and use those estimates for counterfactual analysis (see Athey/Haile:07:Handbook for a survey.) One may wonder if we can use the decomposition approach to generate counterfactual predictions without identifying the valuation distribution from data. While we can in a more general setting, for the sake of concreteness, we focus on the setting of the first-price sealed bid, independent private value auction of Guerre/Perrigne/Vuong:09:ECMA, and consider a counterfactual policy that alters the reserve price (see Paarsch:97:JOE,Haile/Tamer:03:JPE for two examples of such a policy).\footnote{Reserve prices are set by the seller before the auction takes place. They are the minimum value for which the seller is willing to sell the good: if no bid is higher than the reserve price during the auction, then the good remains unsold. As a result, in empirical work, they are usually considered as an auction characteristic (primitive). The reserve price is often observed by the bidders and by the researcher. If the focus is on auctions with unobserved reserve prices (as in Elyakime/Laffont/Loisel/Vuong:97:JBES) our results do not apply. If the researchers are interested in predicting bids after setting the reserve price beyond its support in the data (which is suggested in Table 4 of Haile/Tamer:03:JPE), they may use the bounds approach outlined above.} The reserve price is often a publicly observed state variable. In this model, there are potential bidders $i=1,...,n$, who observe both private valuation $V_i$ drawn from a distribution with common support, $\mathbb{S}_V$, and commonly observe the vector of auction specific characteristics $(X,\eta)$, where $X=(X_1,R)$, with $R$ denoting the reserve price, which is observed by the researcher, while $\eta$ represents unobserved auction heterogeneity. Each potential bidder $i$ chooses to enter or not depending on the value of $(V_i,X,\eta)$. The entry rule is modeled as a reduced form $I: \mathbb{S}_V \times \mathbb{S}_X \times \mathbb{S}_\eta \rightarrow \{0,1\}$, where $\mathbb{S}_X$ and $\mathbb{S}_\eta$ denote the support of $X$ and $\eta$. Each participating bidder $i$ uses a bidding strategy (bid) $s_i: \mathbb{S}_V \times \mathbb{S}_X \times \mathbb{S}_\eta \rightarrow \mathbf{R}_+$, and wins if they submit the highest bid which is higher than the reserve price. The policy of interest is a change in the reserve price: a change of $R$ into $f(R)$ for some map $f$. We denote $\mathbb{S}_R$ and $\mathbb{S}_{f(R)}$ the supports of $R$ and $f(R)$. For each auction, let $\tilde N =\{i: I(V_i,X,\eta) = 1\}$ be the set of participants, and $s^* = (s_i^*)_{i \in \tilde N}$ a symmetric pure strategy BNE in the post-entry auction game. The researcher observes $(X,B)$, where $B = (B_i)_{i \in \tilde N}$ and $B_i = s_i^*(V_i,X,\eta)$, $i \in \tilde N$. The following assumption summarizes the features of this game relevant to the decomposition approach. \begin{assumption} (i) $(X,\eta)$ is publicly known, but valuations, $V_i$, are private information. (ii) The policy changes the reserve price $R$ into $f^*(R)$ for some map $f^*$. (iii) The auction has a unique symmetric pure strategy BNE. \end{assumption} The uniqueness of the pure strategy symmetric BNE in this auction is well studied in the literature. (See Guerre/Perrigne/Vuong:09:ECMA.) Since the equilibrium is unique, the invariance condition for equilibrium selection rules is trivially satisfied. As we saw before, the decomposition-based approach applies to a setting where the researcher focuses on a subset of BCE satisfying restrictions such as symmetry or differentiability. We introduce an additional assumption that is used to identify the bounds in the decomposition approach. \begin{assumption} $(V,\eta)$ is conditionally independent of $R$ given $X_1$. \end{assumption} This assumption requires selection on observables, i.e., the source of dependence between $(V,\eta)$ and the reserve price $R$ is fully captured by observed auction characteristics $X_1$. Our object of interest is the conditional distribution of the auction revenue under the counterfactual reserve price $f^*(R)$ given $X_1 \in C$ for some set $C$ in the support of $X_1$: \begin{align*} p_f(A \mid C) \equiv P\left\{\max_{i \in \tilde N^f} B_i^f \in A \mid X_1 \in C \right\}, \quad A \subset \mathbf{R}_+, \end{align*} where $B_i^f = s_i^*(V_i, X_1,f^*(R),\eta)$ and $\tilde N^f$ denotes the set of participants after the policy. We define \begin{align*} p(A \mid x_1,r) \equiv P\left\{ \max_{i \in \tilde N} B_i \in A \mid (X_1,R) = (x_1,r) \right\}. \end{align*} The conditional probability $p(A \mid x_1,r)$ is identified for all $(x_1,r)$ in the support of $(X_1,R)$, and can be estimated from the pre-policy auction data without recovering the value distribution from the data. The following result is a corollary to Theorem (ref) and Proposition (ref). \begin{corollary} Suppose that Assumptions (ref)-(ref) hold, and let $C$ be a subset of the support of $X_1$. Then, for each $A \subset \mathbf{R}$, \begin{align*} &\mathbf{E}\left[p( A \mid X_1,f^*(R))1\{ f^*(R) \in \mathbb{S}_R \} \mid X_1 \in C \right]\\ &\quad \quad \quad \leq p_f(A \mid C) \leq \mathbf{E}\left[p( A \mid X_1,f^*(R))1\{ f^*(R) \in \mathbb{S}_R\} \mid X_1 \in C \right] + P\left\{f^*(R) \notin \mathbb{S}_R \mid X_1 \in C \right\}. \end{align*} Furthermore, if the support of the reserve price after the policy is within the support of the reserve price before the policy (i.e., $\mathbb{S}_{f^*(R)} \subset \mathbb{S}_R$), for each $A \subset \mathbf{R}$, \begin{align} p_f(A \mid C) = \mathbf{E}\left[p( A \mid X_1,f^*(R))1\{ f^*(R) \in \mathbb{S}_R\} \mid X_1 \in C \right]. \end{align} \end{corollary} The corollary says that when $\mathbb{S}_{f^*(R)} \subset \mathbb{S}_R$, the counterfactual quantity $p_f(A \mid C)$ can be directly recovered from data, without first recovering the valuation distribution. Hence, we can obtain point-identification of the counterfactual prediction without relying on the conditions invoked in the literature to ensure the nonparametric identification of the valuation functions. This also simplifies the estimation procedure, as there is no need to estimate the latter from data. For example, Haile/Tamer:03:JPE are interested in the reserve price $r$ that maximizes expected auction revenue. We can apply the decomposition method in finding the optimal reserve price. First, from ((ref)), the expected auction revenue when the reserve price $R$ is \textit{counterfactually fixed at $r \in \mathbb{S}_R$} is given by \begin{align*} \int \mathbf{E}\left[ \max_{i \in \tilde N} B_i \mid X_1 = x_1, R = r \right] dP_{X_1}(x_1). \end{align*} Hence, the optimal reserve price is identified as one that maximizes this expected revenue, as long as the optimal reserve price is within the support of the reserve price in the data. In this case, we do not need to recover the valuation function from data. \@startsection{subsection}{2} \z@{.5\linespacing\@plus.7\linespacing}{.5\linespacing}{The Scope of Decomposition-Based Predictions} While our result is widely applicable to many settings of counterfactual predictions from game-theoretic models, there are important examples that its scope does not cover. First and foremost, our main result restricts the counterfactual policies to those that affect the observed payoff state. In doing so, we keep other aspects of the environment unchanged after the policy. For instance, the result does not generally apply when the counterfactual policy alters the functional form of the payoff, the action space, or the set of (possible) players (e.g., mergers). The latter cases include Hortacsu/McAdams:10:JPE, who study the effects of different auction formats for selling treasuries on bidder expected surplus, and Roberts/Sweeting:13:AER who study the effect of changing the mechanism by which a good is sold (from an auction set-up to a sequential bidding design) on expected revenues and payoffs in an incomplete information game. A change in the mechanism alters (unobserved) payoffs and expected revenues. In other cases, researchers are interested in counterfactuals involving a change in the agents' choice sets. For example, Keane/Wolpin:10:IER present and estimate a model where women choose labor supply, fertility and welfare participation, among other outcomes. One of their counterfactuals eliminates a welfare program (and hence, the agents' possibility to choose to participate in it) to study the program's effects on labor supply across racial groups. The results on decomposition methods in this paper do not apply in this context either, as the probability of playing such (deleted) actions in the data cannot be extrapolated to the counterfactual environment. See Kalouptsidi/Scott/Souza:20:QE for further examples and some identification results when the policy changes agents' choice sets. Our paper's framework takes a policy variable among the payoff states $W$, not among the endogenous outcomes $Y$ in the game. However, in some applications, we may be interested in the counterfactual analysis which involves a policy that changes an endogenous outcome variable, such as a policy that changes the price in a simultaneous system of equations for price and quantity. Our framework excludes such counterfactual analysis. Another key conceptual feature of this restriction is that our policy variable only changes (exogenous) variables or their index (such as $X_i'\beta$) with sampling variation, thereby excluding policies that affect structural parameters or those that cannot be mapped to observable random variables. Thus our framework excludes counterfactual analysis of a policy's effect on the welfare or the profits of the agents in the game, where the identification of the welfare or the profits require identification of structural parameters in the first place. \@startsection{section}{1} \z@{0.6\linespacing\@plus\linespacing}{.6\linespacing}{Empirical Applications} \@startsection{subsection}{2} \z@{.5\linespacing\@plus.7\linespacing}{.5\linespacing}{Ciliberto and Tamer (2009) Revisited} We revisit the counterfactual analysis in Ciliberto/Tamer:09:Eca using our results from Section (ref). They investigated the effect of the repeal of the Wright amendment on airline entry in markets out of Dallas Love Field Airport. The Wright amendment had been implemented in 1979 to stimulate the use of the newer (and not as central) Dallas Fort Worth (DFW) Airport. As of the early 2000's, it restricted the flights out of the central Dallas Love Field to other cities in Texas or those from some neighboring states. A full repeal of the amendment was agreed by the major airlines and DFW Airport in 2008\footnote{The agreement involved, most notably, decreasing the number of gates in Dallas Love Field to restrict its impact on Dallas Fort Worth.} and was to be fully implemented in 2014. This repeal could have led to significant changes in market entry and, hence, on consumer welfare. Ciliberto/Tamer:09:Eca produced a counterfactual prediction of the outcomes after the repeal of the Wright amendment, after estimating the identified set from a complete information entry game permitting multiple equilibria. This is the game presented in Example (ref).\footnote{We provide extensive details on the empirical application, including the description of the covariates, estimators and inference procedures in the Online Appendix. Following Ciliberto/Tamer:09:Eca, we assume that unobservable payoff components $\varepsilon_i$ are i.i.d. and independent of all covariates, so we do not need to use a control function approach.} A market was defined by a route between two airports, irrespective of directions or layovers. Thus, this included connecting flights through a third airport. They modeled the Wright amendment as a dummy variable covariate, $X_{i,m}^{\text{Wright}}$, which equaled 1 if market $m$ was affected by the Wright amendment (affecting all the firms in the market) and 0 otherwise. For a counterfactual analysis, they considered the counterfactual experiment of repealing the Wright amendment, setting $X^{\text{Wright}}_{i,m}$ to 0, and studied its effects on market entry. The support of $X_{i,m}^{\text{Wright}}$ in the data is $\{0,1\}$, and hence contains its post-policy support that is $\{0\}$. Hence, Corollary (ref) says that the decomposition-based prediction coincides with the equilibrium-based prediction, where the latter prediction can be obtained from estimating all structural parameters under the equilibrium invariance condition. As discussed in Section (ref), this result holds even if the policy introduces new signals to agents which may be used as coordination devices. (In this complete information setting, such new signals cannot reveal private information unbeknownst to the firms.) For example, the results still hold if the congressional hearings associated with the repeal of the Wright amendment led firms to coordinate towards equilibria more desirable to specific airlines (e.g., Southwest), or to equilibria more desirable to the regulator (e.g., with more entry). Further discussion is provided in the Supplemental Note. \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Our Set-up} For this application, we follow their work and focus on the decisions of the four main airlines in their analysis (American Airlines, Delta Airlines, Southwest Airlines and United Airlines) and use their same dataset. We perform a dimension reduction to resolve near multicollinearity between covariates. This reduction is useful because our decomposition-based prediction must include all firm-level covariates in $X_m$ (the covariates $X_{j,m}$ for every $j \neq i$ impact $i$'s entry decision in equilibrium through affecting $j$'s decision to enter).\footnote{In this context there are 8 market-level covariates and 2 variables at the firm-market level. They are described in detail in the Online Appendix. This generates a total of 16 covariates to be included in the analysis. While the parametric estimator is robust to including all 16 covariates due to its additional structure, the performance of the nonparametric estimator is improved with a smaller subset. In general, although our identification results are nonparametric, nonparametric estimation can be challenging when the dimensionality of $X$ is large. Discretization can aid in estimation, but our identification results do not depend on it. Insights from micro theory can be utilized to guide the choice of variables.} We drop three out of eight market level variables (market size, per capita income growth and Dallas market) that lack variation for nonparametric estimation. This is motivated both by data considerations, as well as by the theoretical construction of the dropped variables.\footnote{Both market size and per capita income growth appear well predicted by income per capita and market presence (variables that already capture economic performance at the market level and included in the analysis). Meanwhile the binary Dallas market variable is highly correlated with the Wright amendment variable - by construction, any market in Dallas that does not use Dallas Love Field Airport must be using Dallas Fort Worth Airport instead. However, Dallas Fort Worth is the hub for American Airlines - and American Airlines' market presence is already included as a covariate. Details are provided in the Online Appendix.} We drop one additional firm-market level covariate (a proxy for the firm's cost) using the causal structure of the game, because this variable is a function of other covariates in the analysis, such as route distance, by construction. \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Results} We compare the results from the decomposition approach to the original results in Ciliberto/Tamer:09:Eca. The results of this exercise are shown in Columns 1 and 2 of Table (ref) using two different estimators (a linear/parametric and a nonparametric estimator), while their original results are shown in Column 3. In the first column, we assume that the expected entry of $i$ in market $m$ is given by the linear form $\mathbf{E}[Y_{i,m} \mid X_m=x_m] = x_m'\gamma_i$ and we estimate it using Ordinary Least Squares in a linear regression framework, where $Y_{i,m}$ denotes the indicator of entry by firm $i$ in market $m$. We present this specification as a simple benchmark. We then present the counterfactual estimate which is the estimated change in entry in the post-policy game relative to the data for the markets previously affected by the policy. This estimated change can be written as \begin{align*} \frac{1}{|\mathcal{M}|} \sum_{m \in \mathcal{M}} f^*(X_m)'\hat{\gamma}_i - \overline{Y}_i, \end{align*} where $\hat \gamma_i$ is the estimator of $\gamma_i$, $\mathcal{M}$ represents the set of markets previously affected by the Wright amendment, $f^*(X_m)$ represents the values of the covariates for market $m$ after the policy, and $\overline{Y}_i$ is the average outcome in the data for firm $i$ in those markets in $\mathcal{M}$. In the second column, we estimate the conditional expectation $g_i(x) = \mathbf{E}[Y_{i,m} \mid X_m=x]$ nonparametrically. We use a leave one out kernel estimator with a bandwidth chosen by cross-validation. (See the Online Appendix for details.) While Column 1's results for Southwest Airlines and United Airlines are very similar to those in Column 2, we prefer the latter as our main specification. This is because a linear reduced form (Column 1) cannot be induced from equilibria in the entry game specification. Meanwhile, Column 2 is consistent with equilibria of the entry game specification and, by virtue of being nonparametric, illustrates further benefits of the decomposition approach (i.e., not requiring parametrizations of the utility function, unobserved heterogeneity, etc.). In the third column, we restate the results of counterfactual predictions from the main specification in Ciliberto/Tamer:09:Eca (Table VII, Column 1). These are the maximum predicted increase in the share of Dallas Love Field markets that are served by each airline following the 2014 repeal of the Wright amendment according to their estimates. We find that our results across specifications are broadly consistent with theirs, as they are below their estimated maximum entry. Now we take this exercise one step further. The Wright amendment was actually repealed in 2014. This means that we can observe how airlines entered the markets after the repeal of the Wright Amendment and after any new information arose during its implementation. We compile 2015 data from the DB1B Market and Ticket Origin and Destination Dataset of the U.S. Department of Transportation (the same source as the original dataset), and treat it the same way as the original authors - see the Online Appendix for details. We focus on the same 81 markets from the original paper. The actual change in entry in 2015 in the data relative to the original data is shown in Column 4 of Table (ref). We then compare the counterfactual estimates from Ciliberto/Tamer:09:Eca and our decomposition approach to the actual changes. \begin{table}[t] \begin{centering} \caption{ Ciliberto/Tamer:09:Eca Revisited: Model Predicted and Empirical Counterfactuals of the Repeal of the Wright Amendment} \resizebox{\columnwidth}{!}{ \begin{tabular}{cccccc} \hline \hline \tabularnewline & &\multicolumn{4}{c}{Outcome: Change in Probability of Entry in Dallas-Love Markets} \\ \\ \cline{2-6} \tabularnewline & & Decomposition Method & Decomposition Method & Ciliberto & Tamer (2009) & Empirical \\ & & Linear Model & Nonparametric Model & Maximum Predicted Entry & \\ \tabularnewline \hline \multicolumn{1}{c} & & & & & \\ American Airlines & & -0.030 & 0.128 & 0.463 & -0.04 \\ & & (0.037) & (0.036) & &\\ \tabularnewline Delta Airlines & & -0.023 & 0.174 & 0.499 & 0.46 \\ & & (0.043)& (0.039) & &\\ \tabularnewline Southwest Airlines & & 0.508 & 0.451 & 0.474 & 0.471 \\ & & (0.037)&(0.056) & &\\ \tabularnewline United Airlines & & -0.009 & 0.043 & 0.147 & 0 \\ & &(0.031) &(0.016) & &\\ \\ \hline \multicolumn{1}{c} & & & & &\\ \end{tabular} } \end{centering} \parbox{6.2in}{ Notes: We report the estimated counterfactual changes to the entry of major airlines into Dallas Love Field markets following the repeal of the Wright Amendment. In Columns 1-2, we use our decomposition approach to provide point estimates of this counterfactual effect, using the same pre-2014 dataset of Ciliberto/Tamer:09:Eca. Column 1 uses a linear model, while Column 2 reports a nonparametric estimate. Standard errors for these columns are computed by the bootstrap, following the approach in the Online Appendix with $B=999$ replications. In the third column, we restate the results in Table VII, Column 1 of Ciliberto/Tamer:09:Eca, who presented the maximum change in entry of those airlines. Finally, the Wright Amendment was fully repealed in 2014, allowing all airlines to enter those markets. The final column shows the realized values of the change in entry for those airlines in affected markets in 2015, after the repeal.} \end{table} The results show that the decomposition method (Columns 1-2) using pre-repeal data performs well relative to the empirical outcomes in Column 4. Both the parametric and nonparametric estimates of the decomposition approach capture the large increase in entry by Southwest Airlines, and the negligible post-repeal entry by American Airlines and United Airlines. This lack of entry by American and United post-repeal is broadly consistent with the multiple equilibria in an entry model: Southwest and Delta entered frequently after the repeal, but American and United stayed out of those markets. The empirical values are also within the maximum bounds reported in Ciliberto/Tamer:09:Eca. However, as the authors only reported the maximum predicted entry, their results appear further apart from the realized values for American and United. While the lack of entry results for these firms is consistent with Ciliberto/Tamer:09:Eca's results of maximum predicted entry, this would imply that their counterfactual analysis predicted a range of 0 to 50% of markets entered by those airlines, a large range for policy analysis. While the decomposition-based results perform well for American, United and Southwest, the method performs worse in predicting entry by Delta Airlines. This could simply be a feature of out-of-sample prediction, possibly due to changes to Delta between 2008-2014 (including the acquisition of Northwest Airlines, which was completed in 2010), and/or due to the definition of markets in this dataset.\footnote{Delta Airlines only operates from Dallas Love Field to Atlanta, but there are multiple connecting flights from Atlanta. Routes that include layovers are considered as separate markets by Ciliberto/Tamer:09:Eca.} Nevertheless, we consider that the decomposition-based prediction performed well overall in this out-of-sample exercise, particularly as it used data from years before the policy was implemented and matched well with multiple observed outcomes.\footnote{In the Online Appendix, we provide results for the other component in the typical aggregate decomposition (e.g., Oaxaca-Blinder): the average difference in entry across markets that are not subject/subject to the Wright amendment due to observable characteristics alone: after all, airlines would be less likely to enter smaller markets, even if the Wright amendment was not present/repealed. We show that a naive comparison of entry across different markets (ignoring the difference in characteristics) would overstate the effects of the policy. } \@startsection{subsection}{2} \z@{.5\linespacing\@plus.7\linespacing}{.5\linespacing}{The Threat of Southwest Airlines on Competitors' Entry Behavior} We can pursue further counterfactual exercises beyond those in Ciliberto/Tamer:09:Eca. One salient example is to study competition effects in the airline industry. This includes the role of Southwest Airlines' rise on its competitors' behavior, which has received attention due to the latter's status as a new, lower-cost entry with new organizational strategies (see Knorr/Arndt, for example). Goolsbee/Syverson:08:QJE studied one angle to this question, focusing on whether Southwest Airlines' threat of entry affected established airlines' (e.g., American, Delta, United) pricing behavior. In this section, we use the decomposition-based prediction within the same entry game above to extend their analysis beyond pricing. We use the same dataset and the same model specification from the last section. This includes the entry game environment with complete information. It also includes the definition of market entry/operation (i.e., a route between two airports, irrespective of directions or layovers). Our only change is regarding the policy of interest: we replace `Wright Amendment' by a binary variable equaling 1 if Southwest operates in both endpoints of a market $m$, and 0 otherwise. This means that the variable is 1 if Southwest has entered markets that include airports in both endpoints of $m$. This is the same variable as in Goolsbee/Syverson:08:QJE, which they interpret as a “discontinuous” threat of Southwest entry. However, despite its name as a “threat of entry”, it is actually interpretable as a cost shock.\footnote{Note that entry decisions in other markets are independent from those in market $m$ due to the maintained assumption of independence of $\varepsilon_{i,m}$. Hence, it can be used as a policy variable in this exercise. However, we cannot condition on actual entry in a certain market, whether by Southwest or another airline, since that is a simultaneous decision in this environment.} By operating in markets that share both endpoints as market $m$, Southwest has a lower cost of operating market $m$ (since it has established facilities, labor, etc. on both endpoints of a route). However, a smaller cost for Southwest to operate in market $m$ does not restrict Southwest - or any other airline - from entering market $m$ or any other market. Our outcomes of interest are whether the threat of Southwest entry changes established airlines' actual entry behavior (as opposed to pricing, as in Goolsbee/Syverson:08:QJE) and whether such effects vary across smaller or larger markets (as found in Ellison/Ellison:11:AEJ and Tenn/Wendling:14:ReStat in the pharmaceutical industry).\footnote{Since our framework is static, the interpretation of our results also differs from those cited above. In a static environment, there is no dynamic choice of capacity and there are no incumbents, so the latter cannot “accommodate” entry over time. Nevertheless, we think the present exercise is still informative about whether a state in which Southwest is likely to enter induces differential (strategic) behavior by established airlines.} \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Results} Our counterfactual policy sets the Southwest Threat of Entry variable to 0 for all markets. Therefore, we identify whether American, Delta and United are more/less likely to enter markets when Southwest is no longer a threat to entry (i.e., when it no longer has a lower cost of operating in such markets). The arguments for the validity of our decomposition-based prediction are analogous to those in the previous exercise: (i) the policy is observable to players, since Southwest's routes are observable and (ii) the policy is within the support of the data, as there are many markets where Southwest does not operate in both endpoints. Hence, under the invariance condition on the equilibrium selection rules, Corollary (ref) can be applied. Estimation and inference on these effects follow those in the previous section, detailed in Appendix (ref). The results are presented below. \begin{table}[t] \begin{centering} \caption{ Goolsbee/Syverson:08:QJE Revisited: Effects of Removing the Threat of Southwest Airlines Entry on Other Airlines' Probability of Entry} \resizebox{\columnwidth}{!}{ \begin{tabular}{ccccccccc} \hline \hline \tabularnewline & &\multicolumn{6}{c}{Outcome: Change in the Entry Prob. after Removing Southwest Threat of Entry} \\ \\ \cline{2-8} \tabularnewline & & \multicolumn{2}{c}{All Markets} & \multicolumn{2}{c}{Small Markets} & \multicolumn{2}{c}{Large Markets} \\ & & Linear & Nonparametric & Linear & Nonparametric& Linear & Nonparametric \\ \tabularnewline \hline \multicolumn{1}{c} & & & & & & & \\ American Airlines & & -0.080 & -0.074 & -0.153 & -0.189 & -0.006 & 0.052 \\ & & (0.023) & (0.021) & (0.034) & (0.031) & (0.033) & (0.027)\\ \tabularnewline Delta Airlines & & -0.079 & -0.078 & -0.109 & -0.134 & -0.037 & -0.015 \\ & & (0.024)& (0.021) & (0.037) & (0.028) & (0.034) & (0.028) \\ \tabularnewline United Airlines & & -0.056 & -0.047 & -0.073 & -0.079 & -0.056 & 0.006 \\ & &(0.023) &(0.020) & (0.029) & (0.025) & (0.032) & (0.026) \\ \\ \hline \multicolumn{1}{c} & & & & &\\ \end{tabular} } \end{centering} \parbox{6.2in}{ Notes: We report the estimated counterfactual changes to the entry of major airlines (American, Delta, United) after the removal of the threat of Southwest Airlines entry. The effect is estimated on the markets originally subject to that threat, as defined in Goolsbee/Syverson:08:QJE. The first two columns show the effects for all such markets, for both linear and nonparametric estimates. Columns 3-4 show the results for markets affected by such a threat, but below the median market size, while the last shows the effects for markets larger than the median. Standard errors for these columns are computed by the bootstrap, following the approach in the Online Appendix with $B=999$ replications. A negative coefficient represents a decrease in entry if there were no threat of Southwest entry, relative to there being such a threat.} \end{table} As we can see from Table (ref), a threat of Southwest entry \textit{increases} the average entry probability of American Airlines (7.4-8%), Delta Airlines (7.8-7.9%) and United Airlines (4.7-5.6%) in such markets (i.e., removing a threat of Southwest entry, thereby increasing Southwest's cost, decreases competitors' average entry probabilities). The results suggest that in markets where a Southwest entry is likely, the established firms (with larger market presence) will be more likely to enter. While our results cannot be strictly interpreted as endogenous deterrence or accommodation, they are consistent with the multiple equilibria in the game, together with an equilibrium selection mechanism whereby larger and established firms are more likely to enter when many firms are willing to do so. This is an additional outcome affected by competition, beyond pricing (Goolsbee/Syverson:08:QJE and Tenn/Wendling:14:ReStat). To further understand our results, we follow Ellison/Ellison:11:AEJ and Tenn/Wendling:14:ReStat and check whether such effects depend on market size. To do so, we re-estimate the model for markets below and above the median market size. Consistent with those papers, we find nonmonotonic effects of the Southwest threat of entry on its competitors' decisions. In particular, we find that the increased entry due to Southwest's threat is driven by small markets. In small markets (i.e., below the median market size), where profits are more limited, the threat of Southwest entry induces other firms to do so. This is consistent with such airlines coordinating on entry as an equilibrium “deterrence” to Southwest (beyond pricing). In contrast, the threat of Southwest does not induce entry in larger markets. This is consistent with (a static interpretation of) “accommodation” in large markets: when profit is large enough, all firms enter for that state even if others are likely to enter, as suggested in Tenn/Wendling:14:ReStat.\footnote{Table (ref) in the Online Appendix shows the results for the second term in the average aggregate decomposition (i.e., the role of market characteristics, keeping the same threat of Southwest Airlines entry).} \@startsection{subsubsection}{3} \z@{.5\linespacing\@plus.7\linespacing}{-.5em}{\normalfont}{Heterogeneous Effects depending on Number of Airlines Threatening Entry} We can extend the previous exercise beyond Southwest Airlines. For instance, researchers may be interested whether the competition effects differ across the number of potential entrants (i.e., number of airlines with lower costs). This can also be answered within our framework. Indeed, we can redefine our policy variable as the number of airline $i$'s competitors that threaten entry in market $m$. We keep the same definition of threat of entry as in the last section: i.e., a competitor operates flights out of each endpoint of a route -- thereby lowering its costs, but not the route itself. Since our emphasis is on the same four airlines (American, Delta, Southwest, United), each firm $i$ may face $\{0,1,2,3\}$ competitors threatening entry in each market. We conduct three separate counterfactuals exercises to study how a decrease in the number of airlines threatening entry affect $i$'s choice to enter. Each exercise decreases the number of potential entrants by one (i.e., making markets with three entrants have only two entrants, etc.). The results are shown in Table (ref) in the Online Appendix.\footnote{We note that the different exercises are not generally comparable, because the effects are calculated over different markets (i.e., those markets with two competitors threatening entry for American are likely to be very different than those with only one threat).} As we can see, airlines would generally increase entry if they had no potential competitors rather than one. After all, the airline is more likely to benefit from the market's profits and less likely to compete in such a market. However, there is a net decrease in entry for American, Delta and United in markets facing higher threat of entry. Indeed, decreasing the number of $i$'s competitors from three to two decreases average entry in such markets - consistent with the previous section's results. \@startsection{section}{1} \z@{0.6\linespacing\@plus\linespacing}{.6\linespacing}{Conclusion} Decomposition methods are appealing in counterfactual analysis for their computational tractability and simplicity. However, in strategic settings, predictions generated by those methods may fail to be incentive compatible after the policy or to account for additional coordination possibilities induced by the policy. In this paper, we have presented a set of core conditions that validate the use of the methods in strategic settings. Most importantly, we have provided a precise formulation of the invariance condition on the equilibrium selection rules that is required for the approach. Essentially, under the invariance condition, the agents can be viewed as playing “the same game” after the policy. As demonstrated in this paper, this condition already encompasses many existing assumptions on equilibrium selection in empirical research. Our result opens up a new approach of counterfactual analysis in a strategic setting, where we do not need to recover the structural parameters and the set of equilibria for the analysis. The result's primary contribution is to clarify conditions for such an approach to work. There are several extensions from this paper's proposal that look promising to us. A most prominent extension is to explore a set of conditions for the decomposition-based approach in a dynamic game setting. A policy in these games generally induces a change of the agents' posteriors through a change of a future path of the payoff states, and it seems nontrivial to maintain the invariance of the posterior after the policy. It appears interesting in this regard to note the approach of Kocherlakota:19:JME who introduced independent shocks to the policy so that the posterior of the private sector for future policies remains invariant. Future work can expand this insight and explore the validity of decomposition-based predictions in a dynamic setting. Our proposal can generate a wide range of intermediate approaches, where the target of the prediction is of the form $h(Y,W;\theta)$, and $\theta$ is part of the structural parameters in the game. Then, the message of our paper is that under the conditions stated in this paper, we can perform a counterfactual analysis using the decomposition method, after identifying $\theta$. The main point here is that we do not need to recover the full set of structural parameters of the game or the set of equilibria from data. For example, such an intermediate approach can be used to extend this paper's framework to counterfactual analysis where the target of the prediction is welfare or profits. \@startsection{section}{1} \z@{0.6\linespacing\@plus\linespacing}{.6\linespacing}{Acknowledgements} We thank Aimee Chin, Sukjin Han, Chinhui Juhn, Arvind Magesan, Daniela Scur, Eduardo Souza-Rodrigues, Ko Sugiura, Andrea Szabo, Xun Tang, and participants at many seminars and conferences for their helpful comments. All errors are ours. Song acknowledges that this research was supported by Social Sciences and Humanities Research Council of Canada. Corresponding address: Kyungchul Song, Vancouver School of Economics, University of British Columbia, Vancouver, BC, Canada. Email address: [email removed]. \putbib[counterfactual2]

bibunit[econometrica]