What drives failure to maximize payoffs in the lab? A test of the inequality aversion hypothesis

Abstract

Experiments based on the Beard and Beil (Manag Sci 40(2):252–262, 1994) two-player coordination game robustly show that coordination failures arise as a result of two puzzling behaviors: (i) subjects are not willing to rely on others’ self-interested maximization, and (ii) self-interested maximization is not ubiquitous. Such behavior is often considered to challenge the relevance of subgame perfectness as an equilibrium selection criterion, since weakly dominated strategies are actually used. We report on new experiments investigating whether inequality in payoffs between players, maintained in most lab implementations of this game, drives such behavior. Our data clearly show that the failure to maximize personal payoffs, as well as the fear that others might act this way, do not stem from inequality aversion. This result is robust to varying the saliency of decisions, repetition-based learning and cultural differences between France and Poland.

Appendix

1.1 Parametric test for equality of proportions

The experimental design raises the issue of two kinds of correlation in the data. First, since players make a sequence of decisions, each subject’s choices might be serially correlated. Second, interaction partners change after each round of the experiment, which might result in an inter-subject correlation. To account for this structure of the data, we perform statistical tests for the comparisons of means through parametric regressions that assume clustered standard errors at the session level. This specification is asymptotically robust to any misspecification of the OLS residuals Williams (2000); Wooldridge (2003). We moreover apply a delete-one jackknife correction to take into account a potential small sample bias.

1.1.1 Standard errors estimation

The data are split into clusters (at the session level) and we denote $i$ each observation in each cluster $s$, with $i=\{1, \ldots , N_s\}$ and $s=\{1, \ldots , S\}$ so that the total number of observations is $N=\sum _{s=1}^{S}N_s$. We perform statistical tests for differences between means through linear probability models of the form:

$$\begin{aligned} y_{is}=\sum ^{K}_{k=0} \beta _k x_{is,k}+\epsilon _{is} \end{aligned}$$

in which $y_{is}$ is a dummy dependent variable, $X_{is} = \{1, x_{is1}, \ldots , x_{isK}\}$ is the set of explanatory variables including the intercept, $\{\beta _0,$...$, \beta _K\}$ is the set of unknown parameters, and $\epsilon _{is}$ is the error term. We consider regressions on dummy variables reflecting changes in the environment (for instance, experimental treatments). Because the endogenous variable is itself binary, we also have that: $E(y|X) = P(y = 1 | X)$. In this specification, the parameters thus reflect the mean change in the probability of the outcome induced by the change in the environment. In the case of one explanatory variable, $y_{is}= \beta _0 + \beta _1 I_{is} +\epsilon _{is}$, for instance: $E(y_{is}|I_{is} =1) - E(y_{is}|I_{is} =0) = Pr(y_{is} = 1 | I_{is} =1) - Pr(y_{is} = 1 | I_{is} =0) = \beta _1$, so that the parameter measures the mean variation in the probability of $y$. The $t$-tests on each parameter thus provide significance levels on the differences.

To compute the standard errors, we allow for dependence inside clusters as well as unspecified heteroscedasticity across observations,²⁴ i.e. we assume that any two error terms $i$ and $j$ are independent between clusters, $Cov (\epsilon _{ig},\epsilon _{jh})=0~ \forall g \ne h$, but allow for any type of dependence within a cluster, $Cov(\epsilon _{ig},\epsilon _{jg}) = \sigma ^{2}_{ijg}~\forall i,j,g$. To that end, we correct the estimated covariance matrix at the cluster level using the following procedure, in which the model is written at the cluster level, $Y_s = X_s \beta + \epsilon _s$, where $Y_s$ and $\epsilon _s$ are [$N_s$ $\times$ 1] vectors, $X_s$ is a [$N_s$ $\times$ (K+1)] matrix, $\beta$ is a [(K+1) $\times$ 1] vector:

• 1. Using the parameters estimated on pooled data, $\hat{\beta }_{OLS} = (X' X)^{-1} (X' Y)$, we calculate the vector of error terms in each cluster:

  $$\begin{aligned} \hat{\epsilon }_s=Y_s - X_s \hat{\beta }_{OLS} \end{aligned}$$

• 2. We then estimate the cluster robust covariance matrix (CRCME):

  $$\begin{aligned} \hat{V}_{CRCME}=\left( X'X \right) ^{-1} \left( \sum _{s=1}^{S} X_{s}' \hat{\epsilon }_s \hat{\epsilon }_{s}' X_s\right) \left( X'X \right) ^{-1} \end{aligned}$$

  (2)

1.1.2 Correction for small sample bias in standard errors

The procedure described above provides a consistent estimator of the covariance matrix which can typically be biased in small samples. What is more, the bias is generally found to be negative, so that significance tests reject the null hypothesis too often. A first way to deal with this issue is to correct for the degrees of freedom by substituting $\tilde{\epsilon }_{s}=\sqrt{C_{df}}\hat{\epsilon }_{s}$, with $C_{df}=\frac{S (N-1)}{(S-1) (N-K)}$, in (2)—a procedure known in the literature as HC1. Bell and McCaffrey (2002); Cameron et al. (2008) propose a more accurate correction, called HC3, which estimates the residuals as $\tilde{\epsilon }_{s}=\sqrt{\frac{S-1}{S}}[I_{N_s}-H_{ss}]^{-1}\hat{\epsilon }_{s}$, where $I_{N_s}$ is a [$N_s \times N_s$] identity matrix, and $H_{ss}= X_s \left( X'X \right) ^{-1} X_{s}'$. For an OLS regression, this corrected variance-covariance matrix amounts to implement a delete-one jackknife procedure:

$$\begin{aligned} \tilde{V}_{jackknife}=\frac{S-1}{S} \sum _{s=1}^{S} \left( \tilde{\beta }_{-s}-\hat{\beta } \right) \left( \tilde{\beta }_{-s}-\hat{\beta } \right) ' \end{aligned}$$

(3)

where $\tilde{\beta }_{-s}$ is the vector of coefficients estimated after leaving out the $s$th cluster.²⁵