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Empirical equilibrium
We study the foundations of empirical equilibrium, a refinement of Nash equilibrium in normal-form games introduced by Velez-Brown-2019-SP. We provide two alternative characterizations of the set of empirical equilibria of a game, and determine its relationship with previous refinements in the literature. The first characterization eases its computation in applications. The second characterization provides a direct connection between one of the most popular models for the analysis of data from economics experiments and this refinement.
It is well known that the set of Nash equilibria of a game may contain elements that are implausible. That is, it is somehow evident that some Nash equilibria are not likely to be observed if the game actually takes place (see Example (ref)). If one is interested in the positive content of this theory, this is problematic. In applications, one often performs extreme case scenario analyses based on the set of outcomes predicted for a certain game by a solution concept. If some of these outcomes are implausible, then a worst case scenario analysis can be unnecessarily pessimistic, and a best case scenario analysis can be unrealistically optimistic.
The response of game theory and economics to this problem has been to advance equilibrium refinements, i.e., selections from the Nash equilibrium solution. With only few exceptions---which have never been used in applications nor further studied (details below)---equilibrium refinements determine as implausible each equilibrium in which an agent plays a weakly dominated action with positive probability (from Selten-1975-IJGT and Myerson-1978-IJGT, to K-Mertens-1986-Eca and the latest iterations in Milgrom-Mollner-2017-SSRN and Fudenberg-He-2018; see VanDamme-1991-Springer for an earlier survey). By contrast, experimental studies have shown that weakly dominated behavior is persistently observed in strategic situations. In particular, a robust body of evidence has emerged from experiments in games that have dominant strategy equilibria induced by strategy-proof mechanisms Velez-Brown-2019-SP.
Motivated by the gap between theory and data, Velez-Brown-2019-SP proposed empirical equilibrium, a refinement of Nash equilibrium that is based solely on observables. This refinement produces a delicate selection from the Nash equilibrium set that does not discard all weakly dominated behavior. It produces sharp predictions for dominant strategy games Velez-Brown-2019-SP and partnership dissolution auctions Velez-Brown-2018-EI. It allows one to come to terms with experimental evidence on these games Brown-Velez-2019-TBB.
Empirical equilibrium is defined by means of the following thought experiment. Consider a researcher who samples behavior in normal-form games and constructs a theory that explains this behavior. The researcher determines the plausibility of Nash equilibria based on the empirical content of the theory by requiring that Nash equilibria be in its closure. That is, if a Nash equilibrium cannot be approximated to an arbitrary degree by the empirical content of the researcher's theory, it is identified as implausible or unlikely to be observed. Empirical equilibrium is based on the non-parametric theory that each agent chooses actions with higher probability only when they are better for her given what the other agents are doing. More precisely, an empirical equilibrium is a Nash equilibrium that is the limit of a sequence of behavior satisfying weak payoff monotonicity, i.e., between two alternative actions for an agent, say $a$ and $b$, if the agent plays $a$ with higher frequency than $b$, it is because given what the other agents are doing, $a$ has higher expected utility than $b$.
One can give empirical equilibrium a static or dynamic interpretation. First, its definition simply articulates the logical implication of the hypothesis that the researcher's theory is well specified, for this hypothesis is refuted by the observation of a Nash equilibrium that is not an empirical equilibrium. Alternatively, suppose that the researcher hypothesizes that behavior will converge to a Nash equilibrium through an unmodeled evolutionary process that produces a path of behavior that is consistent with her theory. Then, the researcher can also conclude that the only Nash equilibria that will be approximated by behavior are empirical equilibria.
Thus, the essential component in the definition of empirical equilibrium is the closure of the empirical content of weak payoff monotonicity, the theory on which its definition is based. In our main results we study the relationship between this empirical content and that of alternative non-parametric and parametric theories.
Our first result is that empirical equilibrium can be alternatively defined by proximity to full-support payoff monotone behavior, i.e., each interior distribution in which probabilities are ordinally equivalent to the expected utility vectors it induces (Theorem (ref)). This result is interesting in two dimensions. On the one hand, since payoff monotone behavior is also weakly payoff monotone, this characterization simplifies the computation of the empirical equilibrium set in applications. For instance, this result is crucial for the computation of the empirical equilibrium set in partnership dissolution auctions Velez-Brown-2018-EI. On the other hand, it shows a robustness property of empirical equilibria. Even though weak payoff monotonicity is intuitive, it may accept behavior in which actions that have different expected utility are played with the same probability. Thus, if one is suspicious of the plausibility of this type of behavior, our theorem guarantees the approximation of an empirical equilibrium can always be done by means of payoff monotone behavior.
Empirical equilibrium is related with firm equilibria, which is defined by means of approximation by exchangeable randomly disturbed payoff models VanDamme-1991-Springer; vanishing control cost approachable equilibria, which is defined by means of approximation in control cost games VanDamme-1991-Springer; and logistic QRE approachable equilibria Mackelvey-Palfrey-1996-JER. Though not considered before in the literature, we could also define a refinement by approximation by means of behavior in structural QRE mckelvey:95geb, and regular QRE Mackelvey-Palfrey-1996-JER,Goeree-Holt-Palfrey-2005-EE, two popular theories for the analysis of data in experiments. Each of these theories generate interior payoff monotone behavior. Theorem (ref) alerts us about the possibility that empirical equilibrium coincides with some of these alternative refinements. The remaining results in the paper determine if this is so.
The empirical content of control cost games is contained in that of the regular QRE theory VanDamme-1991-Springer,Goeree-Holt-Palfrey-2016-Book. It turns out that empirical equilibrium can be defined based on the empirical content of a finitely dimensional subfamily of control cost games (Lemma (ref)). Thus, it can also be defined by means of approximation by regular QRE behavior (Lemma (ref)).
Regular QRE also allows us to articulate the idea of increasingly sophisticated behavior. That is, one can identify sequences of regular QRE that are utility maximizing in the limit, i.e., that generate sequences of behavior that can accumulate only on mutual best responses. Remarkably, each empirical equilibrium in a game is the limit of behavior generated by a sequence of regular QRE that is utility maximizing in the limit (Theorems (ref) and (ref)).\footnote{It is not clear that vanishing control cost approachable equilibrium coincides with empirical equilibrium. This equilibrium refinement requires control cost functions vanish in a particular parametric form that is more restrictive than the general form used to prove Theorems (ref) and (ref).} Regular QRE has been successful generating key comparative statics in data from diverse strategic environments Goeree-Holt-Palfrey-2016-Book.\footnote{Behavior in the empirical implementation of the popular cognitive hierarchy models of Camerer-et-al-QJE-2004 can be arbitrarily approximated by regular QRE with heterogeneous non-common prior beliefs ROGERS-et-al-2009-JET.} Even though it is not universal, estimates of parameters of the most popular form of regular QRE, its logistic form, tend to move toward its infinitely sophisticated extreme in experiments that are repeated multiple rounds mckelvey:95geb,Goeree-Holt-Palfrey-2016-Book. Thus, Theorems (ref) and (ref) provide us with a direct connection between the practice in experimental economics, the hypotheses that these researchers usually adopt and test in their analysis, and empirical equilibrium.
In contrast to the universal possibility of approximation of empirical equilibria by regular QRE, there are empirical equilibria that cannot be approximated by the empirical content of logistic QRE, structural QRE, and exchangeable randomly disturbed payoff models (Sec. (ref)).\footnote{Thus, one should not be surprised if behavior in an experiment moves toward best responses while logistic QRE parameter estimates do not move towards their sophisticated extreme.} Indeed, if at least an agent has at least three actions available, one can always construct a payoff matrix for which there is an empirical equilibrium that cannot be approximated by the empirical content of a general form of monotone randomly disturbed payoff models introduced by GOVINDAN-Reny-Robson-2003-GEB and whose empirical content includes that of these three models (Theorem (ref)). This is surprising. Randomly disturbed payoff models are not refutable if their parameters are left unrestricted in the domain in which they are defined Haile-et-al-2008. Moreover, even if one also disciplines them with convergence to best response operators, they induce no proper refinement of the Nash equilibrium set (Proposition (ref)). However, if they are disciplined by payoff monotonicity, their empirical content is constrained beyond monotonicity itself.
Despite their popularity in experimental research and the central role that randomly disturbed payoff models have in the foundations of Nash equilibrium, very little was known about their empirical content when they are disciplined by monotonicity. In particular, Goeree-Holt-Palfrey-2005-EE posed the question whether the empirical content of structural QRE (for a fixed game in which at least an agent has at least three actions available) coincides with that of the regular QRE models. A byproduct of our results is a sharp answer to this question, i.e., that the empirical contents of these theories differ in fundamental ways. On the one hand, regular QRE, in particular its parametric form in control cost games, generate the same empirical content as payoff monotonicity. Thus, the empirical content of regular QRE, for a fixed game, is not restricted by any of the additional assumptions on the unobservables that define the theory (see Sec. (ref)). On the other hand, monotone randomly disturbed payoff models can be misspecified for the study of behavior that not only is consistent with payoff monotonicity, but is also disciplined by increasing sophistication.
The remainder of the paper is organized as follows. Sec. (ref) introduces the model and definitions. Sec. (ref) presents a series of examples that allow the reader to familiarize with empirical equilibrium and show that this refinement is independent from the tremble based refinements previously defined in the literature. Sec. (ref) presents our results. Sec. (ref) discusses and concludes.
We study the plausibility of Nash equilibria in a finite normal-form game $\Gamma(u):=(N,A,u)$ where $N:=\{1,...,n\}$ is a set of agents; $(A_i)_{i\in N}$ are the corresponding action spaces and $A:= A_1\times\dots\times A_n$ the set of action profiles; and $u:=(u_i)_{i\in N}$ is the profile of expected utility indices, i.e., functions $u_i:A\rightarrow \mathbb{R}$. Let $\mathcal{U}$ be the set of all utility profiles. Our interpretation of the game is standard. Agents simultaneously choose an action. Given that action profile $a:=(a_i)_{i\in N}$ is chosen, agent $i$'s payoff is $u_i(a)$. Our analysis will not involve comparisons of behavior across games with different agent sets or action spaces. Thus, $N$ and $A$ are fixed throughout.
A strategy for agent $i$ is a probability distribution on $A_i$, denoted generically by $\sigma_i\in\Delta(A_i)$. A pure strategy places probability one on a given action. We identify pure strategies with the actions themselves. A strategy is interior if it places positive probability on each possible action. A profile of strategies is denoted by $\sigma:=(\sigma_i)_{i\in N}\in\Sigma(A):=\Delta(A_1)\times\dots\times\Delta(A_n)$. Given $S\subseteq N$, we denote a subprofile of strategies for these agents by $\sigma_S$. When $S=N\setminus\{i\}$, we simply write $\sigma_{-i}\in \Sigma(A)_{-i}:=\times_{j\in N\setminus\{i\}}\Delta(A_j)$. Consistently, we concatenate partial strategy profiles as in $(\sigma_{-i},\mu_i)$. We consistently use this convention when operating with vectors, as with action profiles.
Agent $i$'s expected utility given strategy profile $\sigma$ is \[\textrm{E}_\sigma u_i=\sum_{a\in A}u_i(a)\sigma(a),\] where $\sigma(a)=\sigma_1(a_1)\dots\sigma_n(a_n)$. Following our convention of identifying pure strategies with actions, we write $\textrm{E}_{(\sigma_{-i},a_i)}u_i$ for the utility that agent $i$ gets from playing actions $a_i$ when the other agents play $\sigma_{-i}$. We say that an action $a_i\in A_i$ is weakly dominated by action $\hat a_i \in A_i$ if for each $a_{-i}\in A_{-i}$, $u_i(a_{-i},\hat a_i)\geq u_i(a)$ with strict inequality for at least an element of $A_{-i}$. We say that $a_i\in A_i$ is a weakly dominated action if there is another action that weakly dominates it.
The following are the basic prediction for game $\Gamma(u)$ and three of its most prominent refinements.
Our main objective is to study empirical equilibrium, a refinement of Nash equilibrium that is based on an empirical characterization of behavior. That is, we envision that the researcher samples empirical distributions of behavior in a finite set of normal-form games. Based on the analysis of the data, the researcher constructs a refutable theory that explains this behavior. Then, uses this theory to determine the plausibility of Nash equilibria in all normal-form games. If a Nash equilibrium is not in the closure of the empirical content of the researchers theory, the researcher would be able to reject the specification of the theory were he or she to observe this equilibrium. Thus, under the hypothesis that the researcher's theory is well specified, each Nash equilibrium that does not belong to the closure of the empirical content of the researcher's theory is implausible. Empirical equilibrium is the refinement so defined when the researcher endorses the non-parametric theory that each agent chooses actions with higher probability only when they are better for her given what the other agents are doing.
Intuitively, a profile of strategies is weakly payoff monotone for a game if differences in behavior reveal differences in expected payoffs.
It is well-known that proper equilibria are Nash equilibria. One can easily see that proper equilibria are empirical equilibria. For each finite game the set of Proper equilibria is non-empty Myerson-1978-IJGT. Thus, for each finite game the set of empirical equilibria is also non-empty Velez-Brown-2019-SP.
The following property of a strategy profile, which implies weak payoff monotonicity, will allow us to provide an alternative useful characterization of empirical equilibrium.
In this section we study the relationship between empirical equilibrium and undominated and perfect equilibria. We do so by analyzing a series of examples showing that these equilibrium concepts are independent. The main purpose of this discussion is to provide the reader with clear intuition about empirical equilibrium by contrasting it with these more familiar refinements.\footnote{In Sec. (ref) we discuss two refinements introduced by VanDamme-1991-Springer and mckelvey:95geb that are subrefinements of empirical equilibrium. These studies provide examples showing that these refinements may not be contained in the set of undominated equilibria. Thus, one can conclude from the examples in VanDamme-1991-Springer and mckelvey:95geb that empirical equilibria may involve weakly dominated actions are played with positive probability.}
Each refinement that rules out weakly dominated behavior coincides with empirical equilibrium in game $\Gamma(u^1)$. Undominated equilibria and empirical equilibria are independent, however.
Empirical equilibrium does a subtle selection from the Nash equilibrium set. It determines the plausibility of a strategy based on its relative merits with respect to the alternative actions that the agent may choose. The following example drives this point home. It illustrates it for a parametric family of games. This family is a generalization of a game proposed by Myerson-1978-IJGT to show that it is possible to introduce weakly dominated actions in $\Gamma(u^1)$, and considerably change its set of trembling hand perfect equilibria.
Empirical equilibrium can be equivalently defined by proximity of interior payoff monotone behavior.
The characterization of the set of empirical equilibria by means of approximation of interior payoff monotone distributions simplifies its computation. In applications, the computation of this set usually requires two steps. First, one needs to identify conditions satisfied by each possible empirical equilibrium. Then, one needs to show that each equilibrium satisfying these properties is an empirical equilibrium. It is in the first step of this process that Theorem (ref) proves to be essential. One can identify all the implications of proximity to weakly payoff monotone behavior by assuming only proximity to interior payoff monotone behavior. The gain can be considerable in applications, for this avoids the analysis of corner cases Velez-Brown-2018-EI.
It is worth noting that computing the set of empirical equilibria of a game usually also involves a non-trivial second step in which one proves that the candidates one identified as possible empirical equilibria are actually so. At this point it is more convenient to construct a sequence of weakly payoff monotone behavior that converges to the candidate, that is actually neither interior, nor payoff monotone Velez-Brown-2018-EI. In this sense, if one were to equivalently define empirical equilibrium based on the empirical content of payoff monotonicity, Theorem (ref) would still be a valuable tool in its characterization in applications.
Theorem (ref) indicates a form of stability of empirical equilibria. Think for instance of an equilibrium in a game that is itself a non-interior weakly payoff monotone distribution, e.g., a Nash equilibrium in which each agent plays her unique best response.\footnote{These equilibria are usually referred to as strict Harsanyi-1973-IJGT.} One will always conclude that the equilibrium is an empirical equilibrium by taking the respective constant sequence. Theorem (ref) implies that this is not the only sequence that will sustain the argument. One will always be able to find a sequence of interior payoff monotone distribution that converges to the equilibrium.
Empirical equilibrium can be equivalently defined by approximation by the empirical content of regular Quantal Response Equilibria Mackelvey-Palfrey-1996-JER,Goeree-Holt-Palfrey-2005-EE. This theory assumes that agents are noisy best responders, whose actions are responses to the vectors of expected utility. Formally, the model is parameterized by a quantal response function (QRF), i.e., for agent $i$ a function $p_i:\mathbb{R}^{A_i}\rightarrow \Delta(A_i)$. For each $a_i\in A_i$ and each $x\in\mathbb{R}^{A_i}$, $p_{ia_i}(x)$ denotes the value assigned to $a_i$ by $p_i(x)$. A QRF $p_i$ is regular if it satisfies the following four properties Goeree-Holt-Palfrey-2005-EE:
- Interiority: $p_i>0$.
- Continuity: $p_i$ is a continuous function.
- Responsiveness: for $x\in \mathbb{R}^{A_i}$, $\eta>0$, and $a_i\in A_i$, $p_{ia_i}(x+\eta1_{a_i})>p_{ia_i}(x)$.\footnote{$1_{a_i}$ denotes the vector in $\mathbb{R}^{A_i}$ that has $1$ in component $a_i$ and $0$ otherwise.}
- Monotonicity: for $x\in \mathbb{R}^{A_i}$ and $\{a_i,\hat a_i\}\subseteq A_i$ such that $x_{a_i}>x_{\hat a_i}$, $p_{ia_i}(x)>p_{i\hat a_i}(x)$.
A quantal response equilibrium (QRE) of $\Gamma(u)$ with respect to a profile of QRFs, $p:=(p_i)_{i\in N}$, is a fixed point of the composition of $p$ and the expected payoff operator in $\Gamma(u)$ Goeree-Holt-Palfrey-2005-EE, i.e., a profile of distributions $(\sigma_i)_{i\in N}$ such that for each $i\in N$, $\sigma_i=p_i((\textrm{E}_{(\sigma_{-i},a_i)}u_i)_{a_i\in A_i})$. When $p$ is regular, we refere to the corresponding QRE also as regular.
Because regular QRFs are interior, monotone, and continuous, each regular QRE of $\Gamma(u)$ is an interior payoff monotone distribution for $\Gamma(u)$. The converse also holds. That is, regular QRE can be seen as a basis for interior payoff monotone behavior.\footnote{In a paper that circulated simultaneously with the first version of our paper, Goeree-et-al-2018 study the empirical restrictions imposed by a ranking based form of payoff monotonicity that is satisfied by each Nash equilibrium. Thus, this analysis does not produce a refinement of Nash equilibrium, which is our fundamental contribution. At a technical level, our results overlap with theirs only in that they prove a result equivalent to Lemma (ref). They do not address the approximation of Nash equilibria by means of increasingly sophisticated regular QRE, nor by means of a finite dimensional form of regular QRE.}
Theorem (ref) and Lemma (ref) together imply that empirical equilibrium can be equivalently defined by proximity to the empirical content of the regular QRE theory. This has two significant consequences. First, we are able to connect the empirical equilibrium refinement with the empirical content of a model that is a staple of the analysis of experimental data. Second, it opens the possibility that we provide a foundation of empirical equilibrium in terms of approximation by behavior associated with agents who are best responders in the limit. To make this precise we first need to identify the conditions under which this is so for a sequence of QRFs.
We can then define a refinement of Nash equilibrium in the same spirit as empirical equilibrium, but taking as basis for plausibility of behavior regular QRE for increasingly sophisticated regular QRFs.
Clearly, for each $u\in\mathcal{U}$, $\textrm{R}(\Gamma(u))\subseteq \textrm{N}_{eq}(\Gamma(u))$. The interpretation of this refinement is similar to that of empirical equilibrium. It differs in that it explicitly models approximation of Nash equilibria by behavior of agents who are infinitely sophisticated in the limit. One can argue that empirical equilibrium is a more cautious refinement, for it is based only on observables, however. Indeed, QRFs are not observables, and some of their properties, as continuity, are not refutable with finite data.
Fortunately, we do not have to choose between these notions of plausibility of behavior in games.
Theorem (ref) allows us to alternatively describe empirical equilibrium in a way that speaks closely to the practice in experimental economics. If one expects that as players gain experience their behavior will be fit by increasingly sophisticated regular QRE, the only Nash equilibria that can be approximated by data are the empirical equilibria.
Two predecessors of empirical equilibrium are the Nash equilibria that are limits of behavior in Harsanyi-1973-IJGT's randomly disturbed payoff models for exchangeable perturbations VanDamme-1991-Springer and logistic QRE approachable equilibria Mackelvey-Palfrey-1996-JER. These are subrefinements of empirical equilibrium. In what follows we show that these belong to a general family of refinements of Nash equilibrium that are strict subrefinements of empirical equilibrium. As a by product, which we discuss in Sect. (ref), we advance our understanding of the empirical content of monotone randomly disturbed payoff models, a topic that has received attention due to the popularity of these models for the analysis of data from economics experiments Goeree-Holt-Palfrey-2005-EE,Haile-et-al-2008,GOLMAN-2011-JET.
We follow GOVINDAN-Reny-Robson-2003-GEB's construction of additive randomly disturbed payoff models. It subsumes Harsanyi-1973-IJGT's and all additive randomly disturbed payoff models that guarantee agents' behavior is uniquely determined by utility maximization for almost all realization of payoffs. These include the popular structural QRE of mckelvey:95geb.
Given $\Gamma(u)=(N,A,u)$ and a vector of independent Borel probability measures on $\mathbb{R}^A$, $(\mu_i)_{i\in N}$, let $(\Gamma(u),\mu)$ be the incomplete information game in which $\mu:= \mu_1\times\cdots\times\mu_n$ is a common prior on payoff types. Given type $\eta_i\in \mathbb{R}^A$ for agent $i$, her expected utility index is $a\in A\mapsto u_i(a)+\eta_i(a)$. Whenever convenient, given an agent, say $i$, whose action set is $A_i:=\{a_1,...,a_K\}$, we write a vector $x\in\mathbb{R}^A$ as $(x_{a_l})_{a_l\in A_{i}}$ where $x_{a_l}:=(x_{(a_{-i},a_l)})_{a_{-i}}\in A_{-i}$. The interpretation of these perturbations is that either the observer who models the strategic situation by means of game $\Gamma(u)$ does not observe the real payoffs perceived by the agents Harsanyi-1973-IJGT, or that the agent fails to perfectly recognize the difference of payoffs between the actions and correctly maximize mckelvey:95geb. We require throughout that:
The main two models that fit into our framework are Harsanyi-1973-IJGT's, in which each $\mu_i$ is absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^A$; and mckelvey:95geb's structural QRE, in which perturbations are perfectly correlated across action profiles with the same action for an agent, so they can be identified with measures on $\mathbb{R}^{A_i}$, that are further assumed to be absolutely continuous with respect to the Lebesgue measure on $\mathbb{R}^{A_i}$.
Consider $i\in N$. The assumption of purifying perturbations implies that for each $\sigma_{-i}\in\Sigma(A)_{-i}$ and for $\mu_i$ almost every realization of the perturbation $\eta_i$, agent $i$ with type $\eta_i$ has a unique best response to $\sigma_{-i}$. Thus, given $\mu_i$ and $\sigma_{-i}$, the probability with which agent $i$ is observed playing a given action is uniquely defined by utility maximization. Let \[B^{u_i,\mu_i}_i(\sigma_{-i}):= (B^{u_i,\mu_i}_{ia_i}(\sigma_{-i}))_{a_i\in A_i}\in \Delta(A_i),\] be this distribution. It is easy to see that this function is continuous GOVINDAN-Reny-Robson-2003-GEB, and that the fixed points of $\sigma\in\Sigma(A)\mapsto(B^{u_i,\mu_i}_i(\sigma_{-i}))_{i\in N}\in \Sigma(A)$, which exist by Brouwer's fixed point theorem, are the set of observable strategies in Bayesian Nash equilibria of $(\Gamma,\mu)$. We denote the set of profiles $\sigma\in\Sigma(A)$ that are induced by some Bayesian Nash equilibrium of $(\Gamma(u),\mu)$ by $\textrm{BNE}(\Gamma(u),\mu))$.
In mckelvey:95geb's model the best response operator can be further characterized by the function $x\in\mathbb{R}^{A_i}\mapsto Q_i^{\mu_i}(x)$, where \[\sigma_{-i}\in \Sigma(A)_{-i}\mapsto B_i^{u_i,\mu_i}(\sigma_{-i})=Q_i^{\mu_i}(\textrm{E}_{(\sigma_{-i},\cdot)}u_i)\in \Delta(A_{i}).\] The function $Q^{\mu_i}_i$ which only depends on $N$, $A$, and $\mu_i$, and does not depend on $u$, is referred to as a structural Quantal Response Function (sQRF) mckelvey:95geb,Goeree-Holt-Palfrey-2005-EE. These functions are regular QRFs as defined in Sec. (ref) Goeree-Holt-Palfrey-2005-EE. The sQRF most commonly used in empirical analysis of experimental data is the logistic form, $l^\lambda$, which is associated with the so-called double-exponential i.i.d perturbation Goeree-et-al-2018, and assigns to each $a_i\in A_i$ and each $x\in\mathbb{R}^{A_i}$ the value,
Additive randomly disturbed payoff models also allow us to articulate the idea that agents behavior approximates that of utility maximizers. More precisely, consider a sequence of purifying perturbations $\{\mu^\lambda\}_{\lambda\in \mathbb{N}}$. We say that this sequence vanishes when for each $i\in N$ and each neighborhood of zero, $G$, as $\lambda\rightarrow\infty$, $\mu_i^\lambda(G)\rightarrow1$. If there is a convergent sequence of Bayesian Nash equilibria of the respective games $(\Gamma(u),\mu^\lambda)$ for a sequence of vanishing perturbations, the limit of the corresponding induced observable strategies is necessarily a Nash equilibrium of $\Gamma(u)$ VanDamme-1991-Springer. Thus, any sequence of sQRFs whose corresponding perturbations are vanishing is utility maximizing in the limit.
If additive randomly disturbed payoff models are unrestricted, each possible observable distribution of behavior in a normal-form game is in the empirical content of this theory. More precisely, for each $\sigma\in\Sigma(A)$ and each $u\in\mathcal{U}$, there is a purifying perturbation $\mu$ such that $\sigma\in \textrm{BNE}(\Gamma(u),\mu)$ Haile-et-al-2008. If this theory is further disciplined by convergence to Nash behavior, it produces no strict refinement of the Nash equilibrium set.
Thus, requiring proximity to the empirical content of unrestricted randomly disturbed payoff models does not refine the set of Nash equilibria.
Mackelvey-Palfrey-1996-JER and Goeree-Holt-Palfrey-2005-EE argue that additive randomly disturbed payoff models lack of refutability can be resolved by imposing consistency with payoff monotonicity, a phenomenon for which there is empirical support. Relatedly, two attempts have been made with the purpose of refining the set of Nash equilibria based on monotone randomly disturbed payoff models. First, VanDamme-1991-Springer imposes permutation invariance of perturbations. Second, Mackelvey-Palfrey-1996-JER propose approximation by logistic QRE, i.e., restrict to a particular parametric family of perturbations. Both constructions are implicitly imposing that best responses are ordinally equivalent to expected utility.
The following theorem allows us to identify a sharp difference between these and other possible approaches based on monotone additive randomly disturbed payoff models and empirical equilibrium. It states that for each strategy space in which at least an agent has at least three actions available, one can always construct a payoff matrix so the resulting normal-form game possesses an empirical equilibrium that cannot be approximated by any additive randomly disturbed payoff model whose associated best response correspondences are weakly monotonic.
Theorem (ref) reveals that even though additive randomly disturbed payoff models are not refutable if unrestricted, their empirical content, if restricted by monotonicity of best responses, is restricted beyond weak payoff monotonicity. The weakly monotone behavior that is missed by the empirical content of these models can be that in the neighborhood of a Nash equilibrium. In particular, this implies that the refinements proposed by VanDamme-1991-Springer and Mackelvey-Palfrey-1996-JER depend on their structural form of approximation and are strict subrefinements of empirical equilibrium for some games.
We learn from Theorem (ref) that if behavior is weakly payoff monotone and approaches a Nash equilibrium, there is a regular QRE model that fits this behavior. The family of regular QRFs is infinitely dimensional. Thus, this theorem does not point exactly to a parametric family of models that is well specified for the analysis of experimental data that one expects to be payoff monotone.
By Theorem (ref), the most obvious candidate, the monotone structural QRE model, including all its parametric incarnations, e.g., the logistic form, is not flexible enough to account for some payoff monotone behavior in finite games. This brings the need to find a parametric incarnation of the regular QRE model that spans all possible payoff monotonic behavior, and thus is suitable for empirical analysis. In this section we develop such a parametric model. As a byproduct we provide a constructive proof of our results in Sec. (ref).
An alternative approach to rationalize deviations from utility maximizing behavior in normal-form games is VanDamme-1991-Springer's control costs model. Here agents are parameterized by a function that determines how difficult for the agent is to make no mistake when playing a given strategy.
A control cost function for player $i$ is $f_i:[0,1]\rightarrow\mathbb{R}\cup\{\infty\}$ with the following properties.
- $f_i$ is strictly decreasing with $f_i(0)=\infty$ and $f_i(1)=0$.
- $f_i$ is continuously differentiable on $(0,1]$.\footnote{VanDamme-1991-Springer assumes that each $f_i$ is twice differentiable. One can easily see that Lemmas 4.2.1-4.2.5 and Theorem 4.2.6. in VanDamme-1991-Springer go through with our weaker assumption. One needs continuous differentiability in order to apply Lagrange's theorem in Lemma 4.2.3. All other results follow from convexity and continuity. The greater generality of our model allows us to easily construct control cost functions hitting some specific targets of its derivative without matching the second derivative.}
- $f_i$ is a strictly convex function.
Given a normal-form game $\Gamma(u)=(N,A,u)$ and a profile of control cost functions $f:=(f_i)_{i\in N}$ the associated game with control costs is that in which players are $N$, agent $i$'s action space is $\Delta(A_i)$, and payoff of action profile $\sigma\in\Sigma(A)$ is for each $i\in N$, $\textrm{E}_{\sigma}u_i-\sum_{a_i\in A_i}f_i(\sigma_i(a_i))$. For each $\sigma_{-i}\in\Sigma(A)_{-i}$, there is a unique best response for agent $i$ in each control cost game, which solely depends on the profile of expected utility of the different actions in $\Gamma$ VanDamme-1991-Springer. Let $p^f_i$ be this function. One can easily see that this function is a regular QRF VanDamme-1991-Springer,Goeree-et-al-2018.
The control cost model is less general than the regular QRE model to the extent that control cost functions impose some restrictions of behavior across different extended games, i.e., if one varies $u$. We now show that for a fixed $u$, they are behaviorally equivalent.
Let $\sigma$ be a Nash equilibrium of the control costs game associated with $\Gamma(u)$ and $f$. By interiority of each $p^f_i$ and continuous differentiability and strict convexity of each $f_i$, $\sigma$ can be characterized by the first order conditions VanDamme-1991-Springer: for each $i\in N$ and $\{a_l,a_k\}\subseteq A_i$, \[\textrm{E}_{(\sigma_{-i},a_l)}u_i-\textrm{E}_{(\sigma_{-i},a_k)}u_i=f'_i(\sigma_i(a_l))-f'_i(\sigma_i(a_k)).\] Thus, we can span the whole spectrum of interior payoff monotone behavior with a family of control costs functions whose derivative can be chosen for each $i\in N$ at $|A_i|-1$ given points in $(0,1)$. Indeed, an asymptote at zero stitched to a second order spline is flexible enough to interpolate any increasing slope and at the same time guarantee strict convexity and continuous differentiability.
Formally, let $\sigma$ be interior and payoff monotone for $\Gamma(u)$. Suppose for simplicity that $A_i:=\{a_1,...,a_K\}$ and $\sigma_i(a_1)\leq...\leq\sigma_i(a_K)$. Let $m_{l+1}<0$ and $m_l:= m_{l+1}-(\textrm{E}_{(\sigma_{-i},a_{l+1})}u_i-\textrm{E}_{(\sigma_{-i},a_{l})}u_i)$. Since $\sigma$ is payoff monotone for $\Gamma$, $m_l\leq m_{l+1}$. The following function is strictly convex, strictly decreasing, and interpolates slopes $m_l$ and $m_{l+1}$ at the respective extremes of the interval in which it is defined: for each $y\in [\sigma_i(a_l),\sigma_i(a_{l+1})]$,
Let $\varepsilon>0$. Define $f_i$ on $[\sigma_i(a_K),1]$ as $y\mapsto (\varepsilon/[2(1-\sigma_i(a_K))])(y-1)^2$.\footnote{This is simply a strictly decreasing, strictly convex, function such that $f_i(1)=0$, which parameterizes our construction.} Then, stitch the second degree polynomials on the subsequent intervals $[\sigma_i(a_l),\sigma_i(a_{l+1})]$ for $l=K-1,K-2,...,1$, i.e., define $f_i$ as in ((ref)) in the respective intervals. Finally, stitch a strictly decreasing and strictly convex asymptote defined on $(0,\sigma_i(a_1)]$. With a view towards identifying control costs functions with further properties, one can add a calibration point $y^*_i\in(0,1)$ such that $y^*_i\not\in\sigma_i(A_i):=\{\sigma_i(a_1),...,\sigma_i(a_K)\}$ and guarantee that $f_i(y^*)<f_i(\sigma_i(a_l))+\varepsilon$ where $\sigma_i(a_l)$ is the minimum in $\{\sigma_i(a_1),...,\sigma_i(a_K)\}$ that is greater than $y^*$. Let us refer to $f:=(f_i)_{i\in N}$ so constructed as a spline calibrated by $\sigma$, $y^*:=(y^*_i)$, and $\varepsilon$.
The control costs model also allows us to identify sequences of regular QRFs that are utility maximizing in the limit. We say that a sequence of profiles of control costs functions, $\{f^\lambda\}_{\lambda\in\mathbb{N}}$, vanishes, if for each $i\in N$ and each $x\in(0,1]$, $\lim_{\lambda\rightarrow\infty}f^\lambda_i(x)=0$. It is well known that the behavior in games with vanishing control costs can converge only to Nash equilibria of the underlying game VanDamme-1991-Springer.\footnote{Technically, Theorem 4.3.1 in VanDamme-1991-Springer applies only to vanishing sequences of control cost functions of the form $\varepsilon f$ with $\varepsilon\rightarrow 0$. One can easily see that his argument extends for a general sequence of vanishing control cost functions as we define it because our assumption also implies that for each $x\in(0,1]$, $(f^\lambda_i)'(x)\rightarrow0$. We have included an explicit proof of this result in an online Appendix.} Thus, for a sequence of vanishing control cost functions its associated sequence of regular QRFs are utility maximizing in the limit.
Empirical equilibria can be alternatively characterized as the limits of behavior in games with vanishing control costs.
We have advanced our understanding of the empirical equilibrium refinement. Empirical equilibria are defined as the Nash equilibria that are the limit weakly payoff monotone behavior. Alternatively, they can be characterized as those that are the limits of interior payoff monotone behavior (Sec. (ref)). This characterization facilitates the computation of this set in applications. Empirical equilibria are also the Nash equilibria that are the limits of regular QRE behavior for sequences of noisy best responses that approximate best response operators in the limit (Sec. (ref)). A particular finite dimensional family of regular QRE, second order spline control cost games, actually span the whole empirical content of regular QRE for a fixed game (Sec. (ref)). This characterization provides a clear connection between this equilibrium refinement and one of the most popular structural theories for the analysis of data from economics experiments.
In contrast to regular QRE approximation, monotone best response additive randomly disturbed payoff models do not span the whole empirical content of regular QRE for a fixed game and thus are not a basis to define empirical equilibrium (Sec.(ref)). From the point of view of the foundations of this refinement this result highlights that it is based only on a refutable theory, weak payoff monotonicity, and not on implicit or less understood cardinal or structural assumptions. By revealing this differences we also advance our understanding of randomly disturbed payoff models, in particular, the structural QRE model. As Theorem (ref) is stated, it implies that for each action space in which at least an agent has at least three actions one can construct a payoff matrix that admits an empirical equilibrium that is not in the closure of the empirical content of randomly disturbed payoff models whose best response correspondences are weakly payoff monotone. This result can be generalized for the structural QRE model. Trivially, the later requirement can be stated in terms of the corresponding structural QRFs requiring monotonicity directly on them. More interestingly, one can actually simply require that structural QRFs admit only weakly monotone fixed points.\footnote{More precisely, one can show, by means of a fixed point argument that if a QRF violates weak payoff monotonicity, then there is a payoff matrix for which there is a QRE that violates weak payoff monotonicity.} Thus, for each action space in which at least an agent has at least three actions one can construct a payoff matrix that admits an empirical equilibrium that is not in the closure of the empirical content of structural QRE that generate weakly payoff monotone behavior.
Thus, our results show a clear tradeoff between specification and refutability of structural QRE models. If there is no reason to rule out behavior beyond payoff monotonicity, a data analysis based on a parametric version of the structural QRE model that guarantees monotonicity may be misspecified. As a consequence, the interpretation of structural QRE estimates requires that specification of the model be empirically addressed.