1 Introduction

Over the last three decades, the literature on behavioural economics has proposed a number of models to explain various anomalies that can hardly be organised by the standard equilibrium approach. In the context of games, these models consider alternative preferences, traits and/or rationales which relative explanatory powers have been assessed with laboratory experiments (see e.g., McKelvey & Palfrey, Reference McKelvey and Palfrey1995; Selten & Chmura, Reference Selten and Chmura2008; Costa-Gomes et al., Reference Costa-Gomes, Crawford and Iriberri2009; Crawford, Reference Crawford2013). While such horseracing approach documents the models’ relative goodness-of-fit performances and helps determining a ‘best model’, it leaves unanswered the question of whether the estimated models are indeed consistent with the restrictions they impose on individuals’ behaviour. This article presents a novel approach and a specification test to address this question in the context of symmetric binary-choice participation games such as market-entry games, volunteer’s dilemmas, discrete step-level public good and voter participation games. It contributes to the existing literature on this issue in two ways.

First, it provides a useful theory-based selection criterion for models which explanatory powers can hardly be assessed otherwise than by their goodness-of-fit. This is the case with the Quantal Response Equilibrium model (QRE, McKelvey & Palfrey, Reference McKelvey and Palfrey1995), a stochastic version of the Nash equilibrium that assumes players to best-respond to their own and to the others’ payoff disturbances and which predictions hinge upon the distributional properties of these errors (see Goeree et al., Reference Goeree, Holt and Palfrey2016, and the references therein). This model has proven remarkably successful with fitting the data of numerous experiments but its reliance on players’ unobservable payoff disturbances has raised concerns about its falsifiability, see Haile et al. (Reference Haile, Hortaçsu and Kosenok2008). Goeree et al. (Reference Goeree, Holt and Palfrey2005) addressed such concerns by determining restrictions on these disturbances to bracket QRE’s falsifiability (see Goeree et al., Reference Goeree, Holt and Palfrey2016, for further discussion on this topic). Golman (Reference Golman2011) deals with this problem in the context of heterogeneous agents and provides conditions under which the behaviour of the representative agent of a pool of individuals may be rationalised by QRE. These conditions determine whether the aggregation of agents’ payoff disturbances fulfils the i.i.d. assumption on which QRE builds, and they yield useful predictions for asymmetric binary-choice games by restricting the set of QRE-consistent choice frequencies. On the other hand, Melo et al. (Reference Melo, Pogorelskiy and Shum2019) check whether players’ behaviour in multiple games is consistent with the QRE hypothesis. Their procedure exploits a set of restrictions on agents’ choices in different games and on these games’ payoffs. It is also nonparametric in the sense that it does not require the distribution of payoff disturbances in a particular game to be specified.

Unlike these investigations which pertain to QRE settings, ours exploits the behavioural restrictions imposed by a symmetric model on individuals’ participation rates only and therefore allows the comparison of different models, including QRE.

Second, it permits an assessment of a model’s consistency with the assumption of ‘cluster-heterogeneity’, whereby individuals with common characteristics (e.g., their participation rates) are clustered together and share a common model-parameter to be estimated. It thus alleviates the problem of modelling heterogeneity, which typically raises questions about which sort of relaxation of “common knowledge” assumption(s) about what agents believe about others can be used and which still allow one to ‘close’ the model.Footnote ¹ Rogers et al. (Reference Rogers, Palfrey and Camerer2009), for example, develop QRE models where heterogeneity is modelled either in terms of common knowledge beliefs about others’ traits (as in Camerer et al., Reference Camerer, Nunnari and Palfrey2016) or of subjective beliefs, i.e., each player believes that the others’ traits are i.i.d. from the same distribution as her/his own, which is assumed private information (as in Armantier & Treich, Reference Armantier and Treich2009).Footnote ²

Although making these modifications shows that assuming heterogeneity considerably improves the model’s goodness-of-fit, it also heightens the question of the model’s falsifiability since the presumed beliefs about others’ behaviour remain difficult to assess. Our approach does not require additional behavioural assumptions about one’s own or others’ behaviour since it is based on observables, e.g., the players’ participation rates; and it allows one to determine how much cluster-heterogeneity a symmetric model can tolerate to remain consistent with the restrictions it imposes on individual behaviour. While the symmetric assumption provides valuable normative predictions for policy recommendations such as the design of markets, contracts and/or bargaining legislations, it is rather unrealistic and thus restrictive. By considering an observable cluster-heterogeneity rather than a hypothetical heterogeneity in the players’ beliefs, we can better assess a model’s predictions and possibly broaden its range of applications.

We assess our approach with new data on market-entry games of complete information. These games suit well our case since they involve fairly straightforward incentives and may account for a relatively large number of players (which is needed for studying cluster-heterogeneity). These games have also been widely studied in the social sciences and laboratory experiments typically indicate that participants somehow manage to behave almost optimally since their participation rates often even out the expected profits from entry and from no entry (Ochs, Reference Ochs, Budescu, Erev and Zwick1990; Sundali et al., Reference Sundali, Rapoport and Seale1995; Zwick & Rapoport, Reference Zwick and Rapoport2002).Footnote ³ This observation was first coined as ‘magic’ (Kahneman, Reference Kahneman, Tietz, Albers and Selten1988; Meyer et al., Reference Meyer, Van Huyck, Battalio and Saving1992; Rapoport, Reference Rapoport1995), and subsequent experiments have put in perspective the roles of reinforcement learning processes (e.g., Erev & Rapoport, Reference Erev and Rapoport1998; Rapoport et al., Reference Rapoport, Seale, Erev and Sundali1998, Duffy & Hopkins, Reference Duffy and Hopkins2005; Erev et al., Reference Erev, Ert and Roth2010) and other behavioural traits like probability misperception (Rapoport et al., Reference Rapoport, Seale and Ordonez2002) and overconfidence (Camerer & Lovallo, Reference Camerer and Lovallo1999). Goeree and Holt (Reference Goeree and Holt2005) examine these and other participation games from a QRE perspective (with a Logit error structure, i.e., the Logit-QRE) and determine conditions to observe under- or over-participation.

We study these market-entry games through the lens of two stationary behavioural models: the ‘Exploration versus Exploitation’ dilemma (EvE) outlined in Nadal et al. (Reference Nadal, Weisbuch, Chenevez, Kirman, Lesourne and Orléan1998), Weisbuch et al. (Reference Weisbuch, Kirman and Herreiner2000), Kirman (Reference Kirman2011) and Bouchaud (Reference Bouchaud2013) and which essentially entails a trade-off between maximising current and future profits, or through that of Impulse Balance Equilibrium (IBE, Selten et al., Reference Selten, Abbink and Cox2005) which balances off the foregone expected payoffs associated to each possible choice. The details of these models are discussed in the next section and we highlight here two of their properties that motivate our experimental investigation. First, EvE is structurally equivalent to Logit-QRE and thus directly relates to the predictions of Goeree and Holt (Reference Goeree and Holt2005)—in brief, ‘Exploration’ in EvE corresponds to a ‘purely random behaviour’ in QRE, ‘Exploitation’ corresponds to a ‘best-responding behaviour’, and any mix of these two options corresponds to a ‘stochastic best-responding behaviour’. Second, despite their different premises, EvE and IBE fit observed entry probabilities equally well in the range of EvE-consistent choice frequencies. And since the range of IBE-consistent choice frequencies is larger, the usual ‘goodness-of-fit horseracing’ would document nothing more than occurrences where IBE outperforms EvE and is therefore not pursued.Footnote ⁴ Given these properties, we focus analysis on the models’ relative success with consistently organising behaviour in treatments that manipulate payoff levels (i.e., ‘High’ or ‘Low’) and payoff structures (i.e., with payoffs from entry depending on attendance in various ways). In addition, we document the sensitivity of our conclusions to the econometric procedures used, i.e., with(out) imposing symmetry, with(out) assuming homoscedastic errors, and with(out) regularisation of the errors’ variance matrix.

We summarise our experimental findings in the following four points. First, imposing symmetry (as is usually done in the literature) yields significant IBE-estimates that are of similar magnitudes across aggregation levels (i.e., session or pooled data) and EvE-estimates that are either insignificant or that bear little consistency across aggregation levels. Second, relaxing symmetry and using OLS estimation methods leads the specification test to reject EvE with cluster heterogeneity less often than IBE no matter the payoff level or structure (17% vs 42% of all sessions). However, when considering the models’ non-rejected specifications, EvE typically yields insignificant cluster-estimates, and most of its multi-clustered specifications are over-parametrised, i.e., their cluster-estimates are not significantly different from each other. This is not the case for IBE which, in addition, can rationalise the presence or absence of clusters of players with low participation rates. Third, these patterns hardly change when the estimations pertain to the second half of the experiments to account for participants’ experience of play. Fourth, when estimating the models with more efficient econometric procedures with(out) regularisation, the EvE-specifications become more likely to be rejected (25% vs 17% of all sessions) and yield less insignificant cluster-estimates. Yet, most of the non-rejected EvE-specifications are still over-parametrised whereas our conclusions for IBE are hardly affected. In sum, our study indicates that IBE yields more consistent estimates than EvE when symmetry is imposed and that it accommodates cluster-heterogeneity better than EvE when it is relaxed.

The next section presents the EvE and IBE models for market-entry games. Section 3 lays out the econometric procedures and our specification test for this class of binary-choice games. The experimental design and procedures are presented in Sect. 4. Section 5 reports the estimation results when symmetry in the players’ choices is imposed and when it is relaxed. Section 6 concludes.

2 Two stationary models of market-entry games

Assume $n$ agents who independently decide whether to enter a market or not. Agent $i$ 's decision is represented by a variable $d_i$ that takes the value $1$ if she enters and $0$ if not. The payoff from not entering is constant and equal to $H$ , whereas the one from entering is a function $G\left(·\right)$ of the number of entrants $A={\sum }_i{d}_i$ . A congestion problem typically arises if for some integer value $c<n$ , we have $G\left(A\right)\ge H$ if $A\le c$ and $G\left(A\right)<H$ if $A>c$ . With such a reward scheme, any vector of decisions $d$ such that exactly $c$ out of $n$ agents choose to enter constitutes a pure Nash equilibrium. There are exactly $\left(\begin{array}{c}n\\ c\end{array}\right)$ such equilibria, each yielding an aggregate payoff equal to $cG\left(c\right)+\left(n-c\right)H$ .

There may also exist symmetric mixed-equilibrium strategies, i.e., that equalize an agent's expected payoff from entering, ${\pi }^E$ , to that from not entering, ${\pi }^{\mathrm{NE}}=H$ . That is, if $p$ stands for the common probability of entry, then an equilibrium probability $p^{\mathrm{Nash}}$ solves:

(1) ${\pi }^E\left(p\right)\equiv \sum _{k=0}^{n-1}\left(\begin{array}{c}n-1\\ k\end{array}\right)p^k{\left(1-p\right)}^{n-1-k}G\left(k+1\right)=H\equiv {\pi }^{\mathrm{NE}}$

where $k$ is a realization of the random variable $K$ characterizing the number of entrants other than oneself. Note that (1) requires that the $n$ agents behave symmetrically in that they all choose to enter with the same probability $p$ —clearly, one could also consider asymmetric mixed-equilibria in which some agents enter with commonly known probabilities. For reasons that will become clear in Sect. 3, it is convenient to rewrite this expression as being conditional on $p_{-i}$ , the $n-1$ vector of entry probabilities for agents other than agent $i$ Footnote ⁵:

${\pi }_i\left(d=1|p_{-i}\right)=\sum _{k=0}^{n-1}P\left[kgo\right|p_{-i}]G\left(k+1\right).$

2.1 Exploration versus Exploitation: EvE

In this framework, agents aim at finding a compromise between maximizing their current payoff and keeping themselves informed about market conditions to maximize their future payoffs. In our context, we can think of changing market conditions driven by agents’ irregular or stochastic entry behaviour. In this case, agents may find it worthwhile to sometimes explore the alternative option, i.e., entering or not entering the market. While the ‘exploitation’ part of the dilemma, i.e., the maximization of current payoffs, is straightforward, the ‘exploration’ part hinges upon the maximum entropy principle which captures the agent's information seeking behaviour (see Anderson et al., Reference Anderson, de Palma and Thisse1992).Footnote ⁶ In brief, an agent seeking maximal information from her/his decisions would explore each alternative with equal probabilities so that entropy is maximized whereas an agent who does not seek information would clearly avoid exploring and would focus on maximizing current payoffs, so the weight on entropy is minimized. This framework was first used by Nadal et al. (Reference Nadal, Weisbuch, Chenevez, Kirman, Lesourne and Orléan1998) for the study of buyer–seller interactions and we adapt it here for the analysis of market-entry games.

Denote agent $i$ 's probability of entry by $p_i$ and that agent’s expected payoff from entry in terms of the probabilities of entry of the $n-1$ other agents by ${\pi }^E\left(p_{-i}\right)$ . Using Shannon’s measure of entropy $S_i=-p_i$ ln $p_i-\left(1-p_i\right)\text{ln}\left(1-p_i\right)$ with $p_i$ neither 0 nor 1, the agent's objective function to maximise is then given by:

(2) $\begin{array}{cc}{\pi }_i=& p_i{\pi }^E\left(p_{-i}\right)+\left(1-p_i\right)H+\sigma S_i\\ =& p_i{\pi }^E\left(p_{-i}\right)+\left(1-p_i\right)H+\sigma \left(-p_i\text{ln}{p}_i-\left(1-p_i\right)ln\left(1-p_i\right)\right).\\ \end{array}$

where $\sigma \ge 0$ is a parameter capturing the weight that agent $i$ assigns to the preservation of information about market conditions for long term profits. Differentiating this expression with respect to $p_i$ , we obtain the following first-order condition formaximisation:

${\pi }^E\left(p_{-i}\right)-H+\sigma \left(-\text{ln}{p}_i+ln\left(1-p_i\right)\right)=0,$

or equivalently (with $\sigma =1/\lambda$ )

(3) $p_i=\frac{1}{1+\text{exp}\left(-\lambda \left({\pi }^E\left(p_{-i}\right)-H\right)\right)}.$

This yields a system of $n$ equations if there are $n$ agents, and given the homogenous weighting parameter $\lambda$ , this should be solved for the vector $p^{\ast }=\left(p_1,p_2,\dots ,p_n\right)$ . Under the assumption of symmetry, $p_{-i}$ has all its components equal to $p_i$ , which we simply denote by $p$ , and thus $p$ and $\lambda$ are related by:or equivalently

$p=\frac{1}{1+\text{exp}\left(-\lambda \left({\pi }^E\left(p\right)-H\right)\right)}$

(4) $\lambda =\frac{\text{ln}\frac{p}{1-p}}{{\pi }^E\left(p\right)-H}.$

Note that this exactly matches McKelvey and Palfrey's definition of a Logit-QRE (with $\lambda$ standing for the agents’ homogenous ‘best-responsiveness’) so that the models are structurally equivalent if agents’ payoff shocks in QRE are extreme-value i.i.d and if EvE assumes Shannon’s entropy measure.Footnote ⁷ Thus, if rational agents behave symmetrically and do not explore, then $p$ is such that ${\pi }^E\left(p\right)=H$ , i.e., $p=p^{\mathrm{Nash}}$ and $\lambda \to \infty$ . On the other hand, if they maximise exploration, then they choose $p$ such that $p=1-p=0.5$ , so that $\lambda \to 0$ . If $p>0.5$ , $\lambda$ is positive if ${\pi }^E\left(p\right)>H$ and it is negative (theory-inconsistent) otherwise. The Maximum Likelihood estimate of $p$ , assuming independent observations, is the relative frequency of entry, ${\overline{d}}_n$ , and the Maximum Likelihood Estimator (MLE) of $\lambda$ follows from (4). Note that ${\overline{d}}_n$ remains a statistically consistent estimator for $E\left(d\right)=p$ for less restrictive covariance structures of the observations, by various flavours of the weak laws of large numbers.

2.2 Impulse Balance Equilibrium: IBE

IBE basically assumes that if at some stage an alternative option would have yielded a higher payoff, then the agent receives an impulse to use this alternative in the next stage, i.e., agents only take account of foregone payoffs, as in Learning Direction Theory (Selten & Buchta, Reference Selten, Buchta, Budescu, Erev and Zwick1999). It is defined as the long run outcome of such stage-to-stage behaviour. In the context of market-entry games, an agent receives an impulse for entry if the payoff received from not entering is smaller than that from entering. Denoting by $I$ the number of other entrants and by $p$ the common probability of entering the market, the expected magnitude of these impulses for entry is defined as:

$\begin{array}{cc}IM{P}^E\left(p\right)=& E\left(G\left(K+1\right)I_{\{G\left(K+1\right)>H\}}\right)\\ =& \sum _{k=0}^{n-1}\left(\begin{array}{c}n-1\\ k\end{array}\right)p^k{\left(1-p\right)}^{n-1-k}G\left(k+1\right)I_{\{G\left(k+1\right)>H\}}\\ \end{array}$

or equivalently in terms of $p_{-i}$ rather than $p$ :

(5) $IM{P}^E\left(p_{-i}\right)=\sum _{k=0}^{n-1}P\left[k\text{go}\right|p_{-i}]G\left(k+1\right)I_{\left(G\left(k+1\right)>H\right)}.$

Similarly, an agent receives an impulse for no entry if the payoff received from entering is not larger than that from not entering. The expected magnitude of these impulses for no entry is defined as:

$\begin{array}{cc}IM{P}^{\mathrm{NE}}\left(p\right)=& H.P\left(G\left(K+1\right)<H\right)\\ =& H\left(1-\sum _{k=0}^{n-1}\left(\begin{array}{c}n-1\\ k\end{array}\right)p^k{\left(1-p\right)}^{n-1-k}{I}_{\{G\left(k+1\right)>H\}}\right)\end{array}$

or equivalently

(6) $IM{P}^{\mathrm{NE}}\left(p_{-i}\right)=H\left(1-\sum _{k=0}^{n-1}P\left[k\text{go}\right|p_{-i}]I_{\left(G\left(k+1\right)>H\right)}\right).$

Note that these impulses are defined relatively to the game's maximin pure strategy of not entering the market which yields a sure payoff of $H$ . Selten and Chmura (Reference Selten and Chmura2008) further observe that receiving a payoff lower than this sure payoff should be perceived as a loss. To this extent, and in the light of empirical and experimental evidence of loss aversion in agents' preferences (Bernatzi & Thaler, Reference Bernatzi and Thaler1995; Tversky & Kahneman, Reference Tversky and Kahneman1991), we follow Ockenfels and Selten (Reference Ockenfels and Selten2005) and define an IBE for this market-entry game such that agent $i$ is indifferent between ‘receiving $IM{P}^E\left(p_{-i}\right)$ and entering’ and ‘receiving $\kappa IM{P}^{\mathrm{NE}}\left(p_{-i}\right)$ and not entering’, where $\kappa >0$ stands for an impulse weight. That is, agent $i$ would choose to enter the market with probability $p_i$ that equalises her expected weighted impulses:

(7) $p_{i}IM{P}^E\left(p_{-i}\right)=\left(1-p_i\right)\kappa IM{P}^{\mathrm{NE}}\left(p_{-i}\right)$

This impulse balance equation characterizes a long-run IBE in which participants do no more react to the expected impulses they receive. We could of course consider a short-run IBE, i.e., that would solve $IM{P}^E\left(p_{-i}\right)=\kappa IM{P}^{\mathrm{NE}}\left(p_{-i}\right)$ , but the resulting IBE for agent $i$ would then be independent of $p_i$ and, as shown in the next section, this would considerably limit the scope of our study.

Finally, unlike Selten and Chmura (Reference Selten and Chmura2008) who assume $\kappa =2$ , we estimate the impulse weight $\kappa$ by Maximum Likelihood (as for EvE) so the estimator of $p$ is ${\overline{d}}_n$ and the MLE of $\kappa$ (assuming symmetry and $p\ne 1$ ) follows from

(8) $\kappa =\frac{pIM{P}^E\left(p\right)}{\left(1-p\right)IM{P}^{\mathrm{NE}}\left(p\right)}.$

3 A specification test: the $\Sigma$ -test

When we assume symmetry, the models we consider only propose a reparametrization $\lambda \left(p\right)$ for EvE and $\kappa \left(p\right)$ for IBE. Thus, under symmetry, there is no scope for discriminating between these models beyond commenting on implausible values of $\lambda \left(p\right)$ and $\kappa \left(p\right)$ . If we do not impose symmetry, then (3) and (7) can be rewritten as systems of linear restrictions on parameters $\lambda$ and $\kappa$ :

(9) $ln\frac{p_i}{1-p_i}-\lambda \left({\pi }_i^E\left(p_{-i}\right)-H\right)=0\text{for}i=1,\dots ,n$

and

(10) $p_{i}IM{P}^E\left(p_{-i}\right)-\left(1-p_i\right)\kappa IM{P}^{\mathrm{NE}}\left(p_{-i}\right)=0\text{for}i=1,\dots ,n$

Both systems can thus be written in the form $y\left(p\right)-\theta x\left(p\right)=g\left(p,\theta \right)=0$ , with $\theta =\lambda$ or $\kappa$ , and with $y$ , $x$ and $g$ vector functions with values in $R^n$ . The proposed formulation of (9) and (10) in terms of $p_{-i}$ makes it possible to express the EvE or IBE model for homogenous players—in the sense that they share a common single parameter—while still allowing for possibly different individual entry probabilities, and to design a specification test. A further possibility we shall explore is to allow for cluster-heterogeneous players, i.e., players with similar characteristics (e.g., entry-probabilities) whom the model considers identical by assigning them the same parameter. In this case, θ is a vector instead of a scalar and the length of the vector directly affects the power of the $\Sigma$ -test since a vector of length $n$ represents full heterogeneity and leads to never rejecting the null of consistency.

Given the asymptotically normal estimator ${\hat{p}}_T$ of $p$ , the vector of individual entry frequencies, with asymptotic variance $V$ of which we describe a consistent estimator ${\hat{V}}_T$ in Appendix III.A, an optimal asymptotic least squares estimator of $\theta$ isFootnote ⁸:

(11) $\begin{array}{cc}{\hat{\theta }}_T=& {\text{arg min}}_{\theta }{g}^{\prime }\left({\hat{p}}_T,\theta \right){\hat{S}}_T^{-1}g\left({\hat{p}}_T,\theta \right)\\ =& {\left(x^{\prime }\left(p\right){\hat{S}}_T^{-1}x\left(p\right)\right)}^{-1}{x}^{\prime }\left(p\right){\hat{S}}_T^{-1}y\left(p\right)\\ \end{array}$

with ${\hat{S}}_T$ converging to

$S=\frac{\partial g\left(p,\theta \right)}{\partial p^{\prime }}V\frac{\partial g^{\prime }\left(p,\theta \right)}{\partial p}.$

${\hat{\theta }}_T$ is thus the GLS estimator in the regression of $y\left(p\right)$ on $x\left(p\right)$ , the variance of the error term being $S$ .

Given a preliminary estimate of $\theta$ , say ${\tilde{\theta }}_T$ obtained by replacing ${\hat{S}}_T$ in (11) with the identity matrix, i.e., ${\tilde{\theta }}_T$ is the OLS estimator in the regression of $y\left(p\right)$ on $x\left(p\right)$ , a consistent estimator of $S$ is:

${\hat{S}}_T\left({\hat{p}}_T,{\tilde{\theta }}_T\right)=\frac{\partial g\left({\hat{p}}_T,{\tilde{\theta }}_T\right)}{\partial p^{\prime }}{\hat{V}}_T\frac{\partial g^{\prime }\left({\hat{p}}_T,{\tilde{\theta }}_T\right)}{\partial p}.$

The asymptotic variance of ${\hat{\theta }}_T$ is given by $V_{\mathrm{asy}}\left({\hat{\theta }}_T\right)={\left(x^{\prime }\left(p\right)S^{-1}x\left(p\right)\right)}^{-1}$ and a consistent estimator is $\widehat{V_{\mathrm{asy}}\left({\hat{\theta }}_T\right)}={\left(x^{\prime }\left({\hat{p}}_T\right){\hat{S}}_T^{-1}x\left({\hat{p}}_T\right)\right)}^{-1}$ . Under the null that there exists $\theta$ such that $g\left(p,\theta \right)=0$ for the true $p$ , or in other words that the restrictions on entry probabilities embodied by the model are valid,

(12) $T{g}^{\prime }\left({\hat{p}}_T,{\hat{\theta }}_T\right){\hat{S}}_T^{-1}{\left({\hat{S}}_T\left({\hat{p}}_T,{\tilde{\theta }}_T\right)\right)}^{-1}g\left({\hat{p}}_T,{\hat{\theta }}_T\right)\approx {\chi }^2\left(n-1\right),$

and this over-identification test can be used to test the underlying theory. All we need for the implementation of this specification test, for short the $\Sigma$ -test, are thus ${\hat{V}}_T$ and the derivatives $\partial g_i\left(p,\theta \right)/\partial p_i$ . The technical details for the determination of these expressions are given in Appendix III.B. The number of degrees of freedom is $n-1$ when assuming homogeneity [i.e., the length of vector $\theta$ is 1, cf. (12)] and it is at most $n-K$ when assuming heterogeneous players sorted in $K$ clusters (i.e., the length of vector $\theta$ is $K$ ), as discussed in Appendix III.C.

Note finally that since this test exploits the game’s probabilistic structure by rewriting agents’ probabilities of entry as a function of $p_{-i}$ , it can be tailored for the assessment of behaviour in other binary-choice participation games like the volunteer’s game, the (discrete) step-level public good game and voter participation games. This, of course, remains conditional on having well-defined predictions to test, as is the case for EvE and QRE in general but not necessarily for IBE since its long-run equilibrium may not always be defined.Footnote ⁹

4 Experimental design and procedures

The experiments involve groups of 10 participants and a 2 × 3 factorial design which assumes two payoff levels, High and Low, and three payoff structures: one two-step payoff function (DISC) yielding a positive payoff $G$ from entering if attendance $A<c$ and 0 otherwise, and two non-monotone ones (NOM1 and NOM2) in which payoffs first increase and then decrease with $A$ . The binary payoff structure of DISC implies that the players’ choices are strict substitutes whereas the non-monotone structures introduce both strategic complementarity and strategic substitutability in the players’ actions that have been theoretically studied in the context of global congestion games (see e.g., Karp et al., Reference Karp, Lee and Mason2007) but which effects in complete information settings have not yet been investigated experimentally.Footnote ¹⁰

These payoff structures are displayed in Fig. 1, and the models’ equilibrium relationships between $p$ and $\lambda >0$ or $\kappa >0$ for the treatments considered are shown in Fig. 2. For each payoff level, both DISC and NOM1 yield $\left(\begin{array}{c}10\\ 6\end{array}\right)=210$ Nash equilibria in pure strategies, unique mixed-equilibrium strategies and unique IBE strategies whereas NOM2 has one more equilibrium in pure strategies (where all agents choose not to enter), two mixed-equilibrium strategies and two IBE strategies (one with a low entry-probability and one with a high entry-probability).Footnote ¹¹

Fig. 1 Payoff levels and structures of market-entry games. No filling stands for ‘No Entry’, light gray (dark gray) stands for ‘Entry’ when payoffs are Low (High). Payoffs expressed in Experimental Currency Units—see Appendix IV.D for exact figures

Fig. 2 Relationship between $p$ and EvE’s $\lambda$ or IBE’s $\kappa$ . Thick (Thin) lines stand for High (Low) payoff levels. For EvE, the plots report the $p^{\mathrm{Nash}}$ predictions for each payoff structure and level (cf. coloured horizontal lines). For IBE, the plots display the $p^{\mathrm{Nash}}$ predictions (cf. dots) for each payoff structure and level. As $\kappa \to \infty$ , $p\to 0$ in DISC and NOM1

We are interested in checking if and how behaviour is affected by these payoff structures and to what extent it is consistent with EvE and/or IBE when allowing for cluster-heterogeneity. In this regard, since the ranges of probabilities for which EvE and IBE yield model-consistent estimates in DISC and NOM1 are $\left(0.5,p^{\mathrm{Nash}}\right)$ for EvE and $\left(0,1\right)$ for IBE, the models’ cluster-estimates should lie within these ranges and be significantly different from each other for cluster-heterogeneity to be model-consistent and significant. It thus follows that the scope for IBE to accommodate the latter in these treatments is considerably larger than that for EvE.Footnote ¹²

A similar argument holds for NOM2 since the mixed-equilibria have different loci of consistent choice frequencies (defined either on $\left(0.5,p_1^{\mathrm{Nash}}\right)$ or on $\left(0,p_2^{\mathrm{Nash}}\right)$ with $p_2^{\mathrm{Nash}}<0.5$ ) whereas the IBE equilibria have a unique locus (because both equilibria depend on a common $\kappa$ ), so the identification of model-consistent clusters of participants playing in such different (Nash or IBE) equilibria can be achieved with IBE but not with EvE.

Our motivation to consider different payoff levels is to check whether the payoffs’ magnitude affects the presence of model-consistent clusters of players, and thus to possibly complement the findings of McKelvey et al. (Reference McKelvey, Palfrey and Weber2000) who report no significant payoff-magnitude effect on the participants’ QRE best-responsiveness in $2\times 2$ games and evidence of a heterogeneous play.

The experiments were conducted at the Laboratory for Experimental Economics of the University of Jaume I (Spain). Participants were undergraduate students in Business Administration, Law or Engineering and were recruited by public advertisement on campus. We conducted eight sessions per payoff structure (DISC, NOM1, NOM2) with 10 participants per session, totalling 240 individuals. For each payoff structure, we conducted four sessions with Low payoffs and four sessions with High payoffs. The experiments were conducted with a between-subject matching protocol and participants could play in only one session. Upon arriving in the laboratory, they were randomly assigned to cubicles equipped with computer terminals and were given instructions that were read aloud.Footnote ¹³ To avoid framing effects, we presented the game in neutral language by asking participants to choose between actions A and B. Each session involved 150 rounds of play, and at the end of each round, participants were only informed about the total number of players in their group who chose B ("No entry"), their own payoff in that round and their cumulated payoff. This information was appended to a "History" window that could be seen at any time during the experiment. Although participants played in fixed groups of 10, we believe that the provision of a sparse end-of-round information feedback combined with the relatively large number of players (10), and a relatively large ‘market size-to-capacity’ ratio (60%) renders entry-coordination very difficult to achieve. Each session lasted a maximum of 1 h, including the time needed to read the instructions. Participants were rewarded for each round of play at the rate of 0.02 € per 100 points and individual average earnings were €12.77 (i.e., €11.94 in the Low payoff sessions and €13.60 in High payoff ones).