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Discerning Solution Concepts
In Game Theory, solution concepts impose restrictions on the behavior of players given their payoffs. The most popular solution concept is Nash equilibrium (NE) nash51. Solution concepts are often used to establish theoretical results, to identify payoff parameters, and to derive policy and welfare implications from counterfactual analyses. \footnote{ The classic revealed-preference approach to the identification of payoff parameters in discrete games of complete information assumes that the choice of each player is a best response to the observed choices of other players bjorn84,jovanovic89,bresnahan90. This is tantamount to assuming that the players' choices constitute pure strategy NE. The approach can be generalized to allow mixed strategy NE tamer03,BHR, rationalizable strategies aradillas08,kline15, or general convex solution concepts BMM,galichon11. See dePaula13 for a review of the literature.} However, there may exist different solution concepts that are observationally equivalent but yield different payoff parameters, theoretical implications, and counterfactual predictions (see Section (ref) for an example). Thus, it is important to understand when one can tell a solution concept apart from other solution concepts. In such cases, we sat that the solution concept is discernible.
We consider multiplayer binary-action games of complete information, similar to the classic entry game from bresnahan90. We maintain the assumption that the players' choices can display any form of rationalizable behavior in the sense of bernheim1984 and pearce84. We provide a set of conditions that are sufficient to establish discernibility of any solution concept stronger than rationalizability. For instance, it is possible to determine whether the players' decisions arise from NE. Moreover, if they do arise from NE, then they cannot be consistent with any other form of rationalizable behavior. We also identify all the payoff parameters, including those governing the correlation structure of the unobserved heterogeneity. To the best of our knowledge, this is the first formal result that identifies the correlation parameters using a solution concept weaker than pure-strategy NE.
Usually, the testable implications of a solution concept (e.g., NE) can be used to determine whether it is consistent with or could have generated the observed data. \footnote{For example, one can construct a models specification test based on the results from BMM or galichon11.} Our results allow the researcher to answer the question of whether the solution concept actually generated the data. This question is important because of several reasons. A solution concept can be consistent with the data and, at the same time, yield misleading counterfactual predictions. For example, this could happen if an alternative solution concept generated the data, and the two solution concepts are observationally equivalent. We provide an example in Section (ref). Establishing discernibility of NE precludes this possibility and helps to establish the validity of counterfactual analysis and policy implications.
Discernibility is also useful in making sharper counterfactual predictions. One can always assume a less restrictive solution concept (in our case rationalizabilty), build the confidence set for the payoff parameters, and then construct robust confidence bands for the counterfactual of interest. However, these bands can be uninformative because of the weakness of the restrictions imposed on behavior. If one shows that a stronger solution concept (e.g., NE) generated the data, and this solution concept is discernible, then one can build more informative bounds for the counterfactual predictions. In other words, our methodology allows the researcher to determine the strongest restrictions on behavior that are still consistent with the observed data.
Discernibility also has practical implications that may reduce the computational burden. Suppose that the researcher is considering different solution concepts, and establishes discernibility of all of them. Discernibility implies that at most one of the solution concepts under consideration can explain the data. Hence, if a given solution concept explains the data, then the researcher can automatically rule out the other alternatives.
Our strategy to establish discernibility relies on two assumptions. First, we assume that the researcher observes covariates with full support satisfying an exclusion restriction. Second, we assume that the excluded covariates generate enough variation in the conditional distribution of payoffs conditional on covariates. In particular, we require the family of these conditional distributions to be boundedly complete. \footnote{ Completeness of a family distribution is a well-known concept both in Statistical and Econometrics literature. See andrews11. newey03 and darolles11 use a completeness assumption to establish non-parametric identification for conditional moment restrictions. blundell07 use it to achieve identification of Engel curves. hoderlein12 impose bounded completeness in the context of structural models with random coefficients.} Using these assumptions, we identify the distribution of payoffs and the distribution of outcomes conditional on both the observed and unobserved characteristics of the environment. Knowing these distributions allows us to establish discernibility of solution concepts.
We are not the first to exploit the power of completeness assumptions coupled with exclusion restrictions to discriminate between behavior patterns. berry14 apply a strategy similar to ours to a model of oligopolistic competition that allows, among other things, to discriminate between different models of competition. A significant difference between their setting and ours is that they consider continuous games, while we consider discrete games. They crucially rely on having an uncountable set of outcomes to relax the completeness assumption to some extent.
Identification of the payoff parameters is not necessary for discernibility. In Section (ref), we relax rationalizability and allow for some forms of collusive behavior and ambiguity aversion in the sense of gilboa89. This comes at the expense that some of the payoff parameters are no longer point identified. However, we still can establish discernibility of a large class of solution concepts.
We begin with a simple example to motivate the meaning and the importance of discernibility of solution concepts. First, we show that two different solution concepts ---pure strategy Nash equilibrium (PNE) and a behavioral solution concept called strategic ambiguity aversion (SAA)---can be observationally equivalent. That is, they can generate the same distribution over observables. Hence, it is impossible to discern PNE and SAA in our example. Next, we show that PNE and SAA not being discernible can lead to incorrect quantitative and qualitative policy recommendations.
Two firms $i\in \{1,2\}$ simultaneously choose whether to enter a market $(y_i=1)$ or not $(y_i=0)$. Firm $i$'s profit is given by \[ y_{i} \cdot\big[\eta_0 (1- y_{-i}) - \mathbf{e}_{i} \big], \footnote{Throughout the paper, we use boldface font to denote random variables and vectors, and regular font for deterministic ones.} \] where ({i}) $y_{-i}$ is the choice of $i$'s competitor; ({ii}) $\eta_0 \ge0$ is a fixed parameter that measures the effect of competition and is unknown by the researcher; and ({iii}) $\mathbf{e} = (\mathbf{e}_1,\mathbf{e}_2)$ is a vector of payoff shocks unobserved by the researcher. We assume that $\mathbf{e}$ is supported on ${\mathds{R}}^2$ and admits a probability density function that is symmetric around the $45$-degree line (e.g., $\mathbf{e}_1$ and $\mathbf{e}_2$ are independent standard normal random variables). The firms observe both $\eta_0$ and $\mathbf{e}$. That is, the game is of complete information. The researcher observes (can consistently estimate) the distribution of outcomes $\mathbf{y} = (\mathbf{y}_1,\mathbf{y}_2)$. Many of the simplifications we make in this example are for exposition purpose only and are relaxed in subsequent sections.
Following bresnahan90, the classic approach to analyze entry games is to assume that the firms' choices always constitute PNE. Suppose that a researcher wants to test this assumption under the milder assumption that firm behavior is rationalizable. In our example, rationalizability is equivalent to assuming that the firms can choose any action that survives two rounds of elimination of strictly dominated strategies. When $e_i < 0$, entering the market is strictly dominant for firm $i$. When $e_i > \eta_0$, staying out of the market is strictly dominant for firm $i$. This results in four regions of the payoff space in which the game has a unique rationalizable outcome. In the remaining region---the multiplicity region---rationalizability imposes no restrictions on behavior.
Consider the following solution concept. Firm behavior is rationalizable, but firms never enter when there are multiple rationalizable outcomes. This could happen, for instance, if the firms were ambiguity averse and used maxmin strategies when facing strategic uncertainty. This solution concept is called SAA and is analyzed in mass19. The predictions of SAA with $\eta_0=\eta$ for different realizations of $\mathbf{e}$ are illustrated in the left panel of Figure (ref).
We will show that PNE and SAA can produce the exact same distributions over observables. According to PNE, only one firm enters in the multiplicity region. Since either of the two firms could be the one that enters in equilibrium, we need to specify an equilibrium selection rule. We assume that firms always play the equilibria according to which the most profitable firm is the one that enters. The predictions of PNE with such selection rule and $\eta_0=\eta'$ are illustrated in the right panel of Figure (ref).
The duopoly region is the same under both solution concepts. For a fixed value of the competition effect, the no-entry region is smaller under PNE. However, since $\eta_0$ is unknown to the researcher, it is possible to set $\eta'<\eta$ so that both models assign the same probability to no entry. Since both models imply the same probability of both duopoly and no entry, they also imply the same probability of having a monopoly. Note that both the distribution of shocks and each of the two models are symmetric around the 45-degree line. Hence, each of the two monopolies are equally likely under both solution concepts. The formal proof is in Appendix (ref) in the online supplement.
Despite being very different, SAA and PNE can imply identical distributions over outcomes. Hence, in this example, it is impossible to determine whether the data is generated by PNE. At best, the researcher can tell whether the data can be explained by PNE. That is, PNE is not discernible. Exactly for the same reason, SAA is also not discernible. The fact that we use different parameter values for each of the two models is unimportant. For instance, Proposition (ref) in Section (ref) establishes a general nondiscernibility result that can be applied even if the payoff parameters are assumed to be the same under different solution concepts.
Next, we show that PNE and SAA can generate opposite counterfactual predictions in our example. Therefore, the failure of discernibility can lead to incorrect policy recommendations. Suppose that a policymaker wants to increase the number of markets that are served by at least one firm. As a policy instrument, she can choose to offer a subsidy $\tau>0$ to one firm, say Firm $1$, for entering markets in which Firm $2$ does not enter. Entry subsidies are commonly used to incentivize the provision of strategic infrastructure such as broadband internet access goolsbee02. The specific subsidy scheme we analyze allows for a stark and simple exposition. Appendix (ref) in the online supplement presents a similar result with a more realistic subsidy scheme.
Suppose that the data is generated by SAA, but the policymaker evaluates the policy assuming that firms always play PNE. We have already established that the policymaker cannot refute her assumption because PNE and SAA are observationally equivalent. However, under PNE the size of the competition effect must be smaller than the one under SAA (i.e., $\eta'<\eta$). Thus, assuming the incorrect solution concept would lead to inconsistent estimates of $\eta_0$. This, in turn, might lead to flawed welfare evaluations. Since both models are examples of rationalizable behavior, one may expect that an inconsistent estimator of the competition effect will not affect the qualitative implications of different policy interventions. Here, this is not the case.
PNE in the multiplicity region always predict monopolies. Hence, under the PNE hypothesis, all markets are served except for those in which not entering is dominant for both firms. The policy being evaluated decreases the probability of the latter region (see Figure (ref)). Therefore, under the policymaker's assumptions, the policy unambiguously reduces the number of markets without service independently of the parameter values.
However, under SAA, the effect of the policy is always smaller than under PNE, and it can even have the opposite direction for some parameter values. This can happen because the policy also increases the probability of the multiplicity region and, under strategic ambiguity aversion, firms never enter in this region. A firm might be willing to forego the subsidy for fear of another firm entering the market, which would result in negative profits. The net effect of the policy on the probability of monopolies is given by the difference between the probabilities of regions $E^+$ and $E^-$ in Figure (ref). The adverse effect can actually dominate and the policy can increase the probability that a market is not served. For example, one can verify that this is the case whenever $\mathbf{e}_1$ and $\mathbf{e}_2$ are independent standard normal random variables and $\Phi(\eta)>3/4$, where $\Phi({\,\cdot\,})$ is the standard normal cumulative distribution function.
The motivating example from Secion (ref) shows that it is possible for very different solution concepts to be observationally equivalent while implying different policy recommendations. In what follows, we introduce a framework that rules out that possibility. Our assumptions on the data generating process and the covariates observed by the researcher guarantee that a large class of solution concepts are discernible, in a formal sense to be defined.
There are $d_I<+\infty$ players (firms) indexed by $i \in I = \{1,\ldots, d_I\}$. Each player chooses an action $y_i\in Y_i = \{0,1\}$. Appendix (ref) presents a generalization to games with many actions. The set of outcomes is $Y = \times_{i\in I} Y_i$. Let $y_{-i}=(y_j)_{j\neq i}$ denote the vector of actions from $i$'s opponents. Player $i$'s payoffs from outcome $y$ are given by \[ y_{i} \cdot\big( \alpha^{0}_{i,y_{-i}}(\mathbf{w}) + \beta^{0}_i(\mathbf{w}) \mathbf{z}_i - \mathbf{e}_i \big), \footnote{We use boldface font to denote random variables and vectors.} \] where ({i}) $\mathbf{w}$ is a vector of observed player and market characteristics with support $W\subseteq {\mathds{R}}^{d_W}$; ({ii}) $\mathbf{z} = (\mathbf{z}_{i})_{i\in I}$ is a vector of player-specific covariates; ({iii}) $\mathbf{e} = (\mathbf{e}_{i})_{i\in I}$ is a vector of payoff shocks unobserved by the researcher; and ({iv}) $\beta_i^{0},\alpha^{0}_{i,y_{-i}}:W\to{\mathds{R}}$ are unknown functions. We assume that all the parameters and payoff shocks are common knowledge among the players. That is, the game is of complete information.
\setcounter{egentry}{\value{example}}
We impose the following standard assumptions on payoffs (see, for instance, jia2008happens, ciliberto09, BHR, and ciliberto2018market).
Assumption (ref) requires the player-specific covariates to have full support and be relevant. In entry games, examples of continuous firm-specific covariates could be the logarithm of the distance of the market to the existing network of each firm, or to the firms' headquarters. These distances have been used by ciliberto09 and ciliberto2018market to analyze the airline industry. While there is a restriction on $\beta_i^{0}({\,\cdot\,})$, we do not impose any restrictions on $\alpha^{0}_{i,-y}({\,\cdot\,})$.
The normality assumption is common in applied work and helps to simplify the exposition. In Appendix (ref), we replace it with two weaker assumptions. The first one imposes restrictions on the tails of the distribution of $\mathbf{e}$. The second one requires the distribution of $\mathbf{e}$ to constitute a boundedly complete family of distributions. The normality assumption also implies that the probability that a player obtains the same payoffs from different outcomes is zero. Thus, we do need to worry about situations when players may be indifferent between actions. The requirement $\Sigma^{0}_{ii}(w)=1$ is a scale normalization. Note that we allow the payoff shocks to be correlated across players.
To keep the notation tractable, we group covariates and payoff parameters as follows. Let $\mathbf{x}=(\mathbf{z},\mathbf{w})$ be the vector of all observed covariates. Let $\alpha=(\alpha_{i,y_{-i}}({\,\cdot\,}))_{i\in Y,y\in Y}$, $\beta=(\beta_i({\,\cdot\,}))_{i\in I}$, and $\theta=(\alpha,\beta,\Sigma({\,\cdot\,}))\in\Theta$. Hence, we can define the payoff indices $\pi(x,e,\theta)=(\pi_{i,y}(x,e,\theta))_{i\in I,y\in Y}$ by \[ \pi_{i,y}(x,e,\theta) = y_i \cdot \left(\alpha_{i,y_{-i}}({w}) + \beta_{i}({w}) {z}_{i} - {e}_{i}\right). \] The true value of the payoff parameters is denoted by $\theta_0=(\alpha_0,\beta_0,\Sigma^{0}({\,\cdot\,}))$.
An important object for our analysis is the distribution of play $h_{0}$, defined as the conditional distribution of $\mathbf{y}$ given $\mathbf{x}$ and $\mathbf{e}$. That is, \[ h_{0}(y,x,e) = \Pr(\mathbf{y} = y|\mathbf{x}=x,\mathbf{e}=e). \] The distribution of play describes the joint behavior of the players as a function of market and player characteristics. It is a nonparametric latent parameter. Let $h_{0}(x,e)=(h_{0}(y,x,e))_{y\in Y}$, and let $H$ be the set of all possible distributions of play. Given $h,h'\in H$, we say that $h=h'$ if and only if $h(\mathbf{x},\mathbf{e})=h'(\mathbf{x},\mathbf{e})\ \ensuremath{\mathrm{a.s.}}$. Note that, by construction, $h_{0}(x, e)$ belongs to the $d_Y$-dimensional simplex.
We impose the following restriction on the distribution of play. It limits the way the player-specific covariates and shocks can affect the behavior of the players.
Assumption (ref) is a joint assumption on $h_{0}$ and $\theta_{0}$. It requires that the player-specific covariates and shocks can affect choices only via the index $\mathbf{v}$, whose distribution depends on the value of $\theta_{0}$. It says that $\mathbf{y}$ is independent of $\mathbf{z}$ and $\mathbf{e}$ conditional on $\mathbf{v}$ and $\mathbf{w}$. Note that, given $\theta_{0}$, the realizations of $\mathbf{v}$ and $\mathbf{w}$ are sufficient to pin down the payoff indices. Hence, Assumption (ref) can be interpreted as requiring that, after conditioning on the realization of $\mathbf{w}$, the players are payoff driven. If there are two markets with the exact same payoff indices and the same realization of $\mathbf{w}$, then the distribution over outcomes should be the same. Assumption (ref) is implied by the assumptions made in BHR. \footnote{ More specifically, BHR assume that players make choices randomizing among the different NE of the game. Their Assumption 6 requires the selection probabilities to be measurable with respect to the latent utility indices, in our notation. An analogous assumption could be imposed on the selection mechanisms of any model satisfying Assumptions 2.2--2.4 in BMM. Doing so would imply our Assumption (ref).}
Under Assumption (ref) the distribution of play can be robust to policy interventions. For example, if one wants to evaluate policies that only affect firms indirectly through the prices of inputs. The firms might care about the changes in prices, but not about the source of these changes. In situations where this assumption is reasonable, knowing $\theta_0$ and $h_0$ is sufficient to analyze policies that only operate through the payoff indices.
Although Assumption (ref) imposes some structural restrictions to the behavior of players, one may still want to impose additional economic restrictions. These restrictions come in the form of solution concepts such as rationalizability or NE. Solution concepts often depend on the characteristics of the environment. Hence, we allow for the restrictions arising from solution concepts to depend on the payoff parameters.
For example, suppose that players choose actions simultaneously and only use rationalizable strategies, i.e., strategies that survive the iterated elimination of strictly dominated strategies. Let $S_{R}(\theta)$ be the set of $h$ such that, given the payoff indices $\pi(x,e,\theta)$, $h(x, e)$ assigns positive probability only to rationalizable outcomes for all $x$ and $e$.
The Nash hypothesis is that the behavior of the players always constitutes NE of the simultaneous-move game in pure or mixed strategies. There can be multiple NE and, in such cases, there is no consensus on which equilibria are more likely to arise. In order to assume as little as possible about the equilibrium selection, one must allow for arbitrary mixtures of equilibria. The actual distribution of outcomes could be any point in the convex hull of the set of the distribution over outcomes implied by NE. Let $S_{N}(\theta)$ be the set of $h$ such that, for all $x$ and $e$, $h(x,e)$ belongs to the convex hull of the set of distributions over outcomes implied by NE of the game given the payoff indices $\pi(x, e,\theta)$. $S_N(\theta)$ exactly captures the predictions of the Nash hypothesis.
Note that the NE solution concept is nested into rationalizability. That is, $S_{N}(\theta)\subseteq S_{R}(\theta)$ for all $\theta$. Also $S_{N}$ is a convex solution concept in that $S_{N}(\theta)$ is a convex set for all $\theta$.
The distribution of play completely characterizes behavior. However, in general, there are at least two reasons to work with restrictions that are coming from Economic Theory, that is, solution concepts. First, solution concepts might make for more credible counterfactual analyses because they are also supported by nonempirical arguments (dawid16). For instance, one may argue that the policy intervention considered in Section (ref) would not affect whether firms play PNE. However, since only one firm is subsidized, it is possible that the subsidized firm will be more likely to enter in markets with multiple PNE. In this case, the distribution of play would not be policy invariant, but the predictions based on the solution concept would remain accurate.
A second reason to focus on solution concept is portability. A solution concept can make predictions related to changes in some fundamental characteristics of the environment. For example, NE is well defined for two-player and three-player games. In contrast, the distribution of play cannot be easily extrapolated to make predictions if the number of players changes.
Moreover, a solution concept might be relevant beyond the specific application being considered. Many solution concepts from Economic Theory are general theories of behavior. Finding evidence in support for a solution concept in one setting, provides support for its use in other settings. For instance, walker01, chiappori02, and wooders16 have tested the implications of the Nash hypothesis in the context of penalty kicks and tennis serves. \footnote{ These papers consider only zero-sum games. It is not entirely clear whether it is possible to generalize their methodology to general-sum games. } And their analysis is often used to justify the use of NE in general settings unrelated to sports.
For most of the text, we assume that the players' behavior is rationalizable. This assumption is common in the literature (see, for instance, aradillas08 and kline15). Section (ref) relaxes this assumption.
This section clarifies the relation between solution concepts, the distribution of play, and selection mechanisms. Our distribution of play is a complete econometric model in the sense of tamer03 and manski88 in that it “asserts that a random variable $\mathbf{y}$ is a function of a random pair $(\mathbf{x}, $[$\mathbf{e}$]$)$ where $\mathbf{x}$ is observable and [$\mathbf{e}$] is not” (tamer03, pp.\ 150). In other words, $h_0$ is an “empirical” solution concept that completely describes behavior of players without any economic restrictions. In contrast, many solution concepts arising from Economic Theory are generally incomplete in that, even knowing the value of the parameters and all the characteristics of the environment, there can be multiple solutions. Both NE and rationalizability fall under this category. Our approach is to take the distribution of play as a primitive, and model incomplete solution concepts as sets of complete models that depend on the parameters of the environment.
An alternative approach to ours is to define a solution concept as a random set $\mathrm{Sol}(\mathbf{x},\mathbf{e},\theta)$ consisting of possible distributions $p$ over outcomes, which depend on the characteristic of the environment and the payoffs (e.g., BMM and BHR). Then, one would complete the model with a selection mechanism that assigns probabilities $\mathrm{sel}({\,\cdot\,}|x, e,\theta)$ to the different possible distributions emerging from the solution concept. The distribution of play could be defined as a weighted average of the different distributions with weights determined by the selection mechanism. For instance, if $\mathrm{Sol}(x, e,\theta)$ is finite, \[ h(y,x,e) = \sum_{p\in \mathrm{Sol}(x, e,\theta)} p(y) \cdot \mathrm{sel}(p|x, e,\theta). \]
Under some technical measurability assumptions, both approaches are mathematically equivalent in terms of the relation between solution concepts, distributions of play, and the data (see Section 2 in BMM). The difference between the two approaches is that our approach emphasizes the distribution of play, which is well defined independently of any solution concept. In contrast, selection mechanisms can only be defined relative to a specific solution concept. The following example demonstrates the relation between them.
\setcounter{aux}{\value{example}} \setcounter{example}{\value{egentry}}
\setcounter{example}{\value{aux}}
Recall that $\theta_0$ and $h_0$ denote the true payoff parameters and the true distribution of play. Let $\Psi\subseteq\Theta\times H$ denote a set of possible values that $(\theta_0,h_0)$ can take. We are interested in whether solution concepts, in particular NE, are discernible according to the following definition:
Note that \[ \mathds{E}_{\!}\left[\,h_0(y,\mathbf{x},\mathbf{e})|\mathbf{x};\theta_0\,\right] = \Pr(\mathbf{y}=y|\mathbf{x})\ \ensuremath{\mathrm{a.s.}}, \] for all $y\in Y$. Hence, $\mathds{E}_{\!}\left[\,h_0(\mathbf{x},\mathbf{e})|\mathbf{x};\theta_0\,\right]$ is identified (can be consistently estimated from observed data on outcomes and covariates). Thus, the definition of discernibility leads to two properties that fully characterize the relationship between the observed (estimable) distribution of $\mathbf{y}$ conditional on $\mathbf{x}$ and any given solution concept $S$:
In order to establish discernibility of solution concepts, we first establish identification of the payoff parameters and the distribution of play. Later on, we also consider environments where identification of payoff parameters fails to hold.
The proof of the proposition is in Appendix (ref). The identification of the payoff parameters $\beta_{0}$ and $\alpha_{0}$ follows standard arguments that exploit the large support of our player-specific covariates. However, to the best of our knowledge, Proposition (ref) is the first result in the literature that identifies the unknown correlation structure of payoffs in games of complete information under solution concepts weaker than PNE. \footnote{ See kline15 for an identification result assuming either PNE or independent shocks.} The proof involves looking at the limits of the partial derivative of $\Pr(\mathbf{y}=y|\mathbf{x}=x)$ with respect to one of the excluded covariates for a specific outcome vector $y$ along specific rays in the support of excluded covariates. \footnote{ Although the rays themselves have probability zero, the partial derivatives use information in a neighborhood of those rays. Because the covariates are continuous, these open neighborhoods have positive probability and thus observable implications. }
To identify $h_0$ note that, under Assumption (ref), the excluded covariates $\mathbf{z}$ generate variation in the observed conditional distribution over outcomes without changing $h_0$. This is because $h_0$ is affected by $\mathbf{z}$ only via the conditional distribution of the index $\mathbf{v}$ conditional on $\mathbf{z}$. This exogenous variation yields the following identification result.
The proof of Proposition (ref) uses bounded completeness of the family of normal distributions. This property is not unique to normal distributions and is satisfied by many other parametric families. See Appendix (ref) for more details.
Note that the definition of discernibility has the form “there do not exist $(\theta,h)$, $(\theta',h')\in\Psi$ such that\ldots.” Hence, the smaller $\Psi$ is (the more restrictions are imposed), the easier it is to establish discernibility of a solution concept. In particular, if $\Psi$ is a singleton, then every solution concept is trivially discernible. On the other hand, if $\Psi$ is very big (very few restrictions are imposed), then one should not expect many solution concepts to be discernible. In this section, we provide two results. The first one establishes discernibility of solution concepts under a small number of restrictions (Theorem (ref)). The second one shows absence of discernibility despite $\Psi$ being very small (Proposition (ref)).
Under our assumptions, Theorem (ref) implies that we can discern {any} solution concept which implies rationalizable behavior, including $S_{N}$.
Strengthening Assumptions (ref)--(ref) would not change the conclusion of Corollary (ref), because it would only shrink the set $\Psi_R$. In contrast, the following proposition establishes the importance of our exclusion restrictions (Assumption (ref)).
It is important to note that the payoff parameters are known under $\Psi^*$. That is, without Assumption (ref), the NE solution concept is not discernible even in settings with fully known payoff structure (e.g., laboratory experiments).
The sole purpose of Assumption (ref) is to establish identification of the payoff parameters. However, point identification of $\theta_0$ is {not} necessary for discernibility of solution concepts. In this section, we consider two departures from rationalizability that lead to partial identification of the payoff parameters.
Suppose that firms are ambiguity averse in the sense of gilboa89. That is, suppose that each firm ranks its actions in terms of its {minimum} possible payoff, and then chooses the action that maximizes this minimum. This action is called the maxmin action, and is generically unique (indifference between actions is ruled out by continuity of the distribution of the utility shocks). Let $S_{M}(\theta_{0})$ be the set of distributions of play that assign full probability to maxmin actions given the payoff parameter $\theta_0$. The middle panel of Figure (ref) illustrates the predictions of this solution concept for the two-firm entry-game from Example (ref).
Another possibility is that the firms are colluding. Suppose that firms can compensate each other via transfers that are not observed by the researcher. In this case, the firms could agree to choose collusive outcomes that maximize the sum of their individual profits, even if doing so does not maximize the individual profits of some of them. Let $S_{C}(\theta_{0})$ be the set of distributions of play that assign full probability to collusive outcomes. The predictions of the collusive solution concept are illustrated in the left panel of Figure (ref) given the parametrization from Example (ref).
Formally, we assume that the behavior of players is consistent with either rationalizability, maxmin, or collusive behavior.
The proof of Proposition (ref) is in Appendix (ref). Under collusive behavior, we can also identify some linear combinations of $\alpha^{0}$ parameters. But, without imposing assumptions that would reduce the dimensionality, we cannot identify all of them. See Proposition (ref) in Appendix (ref) to get a sense of which linear combinations can be identified.
We conclude this section by noting that some of out results establish discernibility of solution concepts without point identification of either $\theta_0$ or $h_0$. In particular Proposition (ref) does not require point identification $\theta_0$, and Proposition (ref) in the appendix establishes discernibility of $S_{R}$, $S_{M}$, and $S_C$ without point identification of $h_0$.
We have defined discernibility of a given solution concept to mean that the solution concept can explain observed data if and only if the data was generated by it. And we have shown that any solution concept stronger than rationalizability is discernible under commonly imposed assumptions. We have also established identification of payoff parameters, including the correlation between unobserved payoff shocks, allowing for any form of rationalizable behavior. Our results are robust to some departures from rationalizability including ambiguity aversion and collusive behavior. Our exclusion restriction is necessary for discernibility of the NE solution concept in some settings, even when the payoff parameters are known.
It is possible to determine whether the data can be generated by any given convex \footnote{A solution concept is convex if $S(\theta)$ is a convex set for all $\theta$.} solution concept (e.g., NE). For instance, one can construct the sets of conditional moment inequalities characterizing the solution concept (see BMM or galichon11). Then, the identified set (i.e., the set of parameters that satisfy these moment inequalities) is empty if and only if the data can be generated by the solution concept. Our results imply that one can substantially strengthen this conclusion. The identified set of payoff parameters is empty if and only if the data is in fact generated by the solution concept.