Quantal Response Equilibrium Predictions

Once the quantal response functions are obtained by numerical methods, it is possible make predictions about the mean and variance of pledges by low and high payoff players. It is also possible to construct the distribution of total con- tributions and make predictions about the mean and variance of this distribu- tion. Moreover, one can measure the probability mass of pledged contributions above an individual’s valuation for low types, or the probability mass of total contributions in excess of the threshold, which predict the over-contribution rate and the provision rate, respectively.14 _{Using the parameterization of the thresh-}

old game discussed in section 1.2.1, Figures 1.1 and 1.2 plot three points along the principal branch correspondence across all four experimental treatments for λ ∈ {0.2, 0.4, 0.6}. Figure 1.1 plots the individual quantal response functions by payoff-type, while Figure 1.2 presents the distribution of total contributions and the threshold. Notice that the predicted values of all variables of interest, in- cluding the mean and variance of individual and total contributions, as well as average over-contribution and provision rates, vary in a non-linear fashion across all treatments for any given value of λ.

14_{Note that these predictions are averaged across states, as many of the estimated moments}

in the experimental data must be averaged across states as well. For now, these means are constructed using the binomial probabilities that apply ex-ante. I re-weight these predictions using the frequencies found in the experiment for the predictions provided in section 1.3.3.

Figure 1.1: QRE: Individual Contributions

As a stochastic model of choice, the response functions of QRE may be used to construct a structural maximum likelihood estimator for the precision parameter, λ. While this value has the same units as payoffs, it does not have an absolute interpretation. Rather, it is a relative measure of responsiveness to expected pay- offs, i.e. it is a relative measure of how well incentives predict choices. A note of

caution is appropriate here: λ is implicitly embedded inside of σ−i, so the preci-

sion parameter is only a relative measure of responsiveness to the endogenously determined expected payoff functions, πi(·). This is because the logit equilibrium

is a closed model where quantal responses and expected payoffs are simultane- ously determined. An increase in relative responsiveness may sharpen or dull incentives, so a rise in λ can potentially increase or decrease the mean or variance of individual or total pledged contributions. In certain cases, increased payoff responsiveness can increase the noisiness in decision-making for some individuals.

Figure 1.2: QRE: Total Contributions

Typically, economic models track the evolution of an endogenous parameter as a function of exogenously specified shocks. This model works in reverse, by track- ing the evolution of endogenous shocks as a function of an exogenously varying parameter. As such, there is little a priori reason to expect the exogenous pa- rameter to be constant across games or treatments. For example, McKelvey and

Palfrey (1995) and Rogers et al. (2009) find parameter estimates that vary signif- icantly across games, while Sheremeta (2011) finds similar differences within four implementations of simultaneous contests. Allowing additional degrees of free- dom for each treatment risks the criticism of over-fitting, while fixing one degree of freedom across treatments risks misspecification error. With no prior reason to impose the restriction, I allow the logit parameter to vary across treatments; although, this has been criticized as post-hoc and empirically vacuous (Haile et al., 2008).

To preempt such doubts or concerns, I will first lay out some properties of the model found across a range of parameter values. The model makes a host of directional predictions for values of λ ∈ [0, 0.75].15 These predictions correspond to some simple economic intuition of decision errors, and they provide the primary insights of the model; although, the specific magnitudes of such effects ultimately require estimation.

In response to the criticism from Haile et al. (2008) that QRE lacks empirical content, Goeree et al. (2005b) define a list of regularity conditions necessary for QRE to place testable restrictions upon the data: interiority, continuity, respon- siveness, and monotonicity. All of these are satisfied by the logit specification

15_{It would be more satisfying to make claims regarding every λ ∈ [0, ∞). Unfortunately, the}

bifurcations mentioned in section 1.2.2 make numerical calculations difficult when all players share the same payoff type under complete information. Nonetheless, the interval used here is sufficiently large as to nest all of the parameter estimates found in the data. I discuss this issue in greater detail towards the end of this section.

with independent preference shocks. Indeed, the directional predictions I present below are genuinely falsifiable restrictions on the data, with only one degree of freedom required to predict the level of eight separate variables. This reasoning is similar in spirit to the analysis presented by Goeree et al. (2005b) that yields testable hypotheses for the matching pennies game imposed by all regular quantal response equilibria.

Figures 1.3 and 1.4 plot the group-level and individual-level predictions, re- spectively. Note that the group-level variables considered in Figure 1.3 are state- contingent. Hence, averages must be taken across all states to collapse these predictions into a single figure, and these averages correspond to the observed mo- ments in the experimental data where states are rotating round-to-round. Figure 1.3(a) plots the average probability that total contributions reach the threshold, and Figure 1.3(b) plots the average probability that a low payoff player pledges more than her value to the public good. Figures 1.3(c) and 1.3(d) plot the aver- age mean and variance of total contributions normalized by the threshold. This normalization is useful for making clean comparisons across group-size, where the threshold and the range of total contributions scale proportionally.

Several clear patterns emerge regarding the relevant treatment effects for any value of the precision parameter. First, the provision rate and average contribu- tions decrease when information is reduced or the number of players is increased. Moreover, the effects of group-size appear to dominate the effects of information. For low values of λ, information effects are negligible, since players who do not respond to payoffs must necessarily be ignoring the information contained in their incentives. As precision increases, payoff uncertainty has a larger impact on out- comes; though, even large groups with complete information cannot catch up to small groups with incomplete information for the parameter values presented here.

Unsurprisingly, the average variance of contributions is decreasing in lambda, im- plying that increased sensitivity to payoffs is associated with an overall reduction in noise. This finding is highly pronounced for complete information, though the dissipation of noise is noticeably slower under incomplete information. Lastly, the over-contribution rate of low payoff players decreases in tandem with this reduc- tion in noisiness. Therefore, the over-contribution predicted by QRE represents a pure decision-error that vanishes as behavior approaches best response.

Figure 1.4: Individual-Level QRE Predictions

Figure 1.4 presents the findings at the individual level. Figures 1.4(a) and 1.4(c) plot the mean of individuals contributions of low and high payoff players, respectively, while Figures 1.4(b) and 1.4(d) do the same for the variance of indi- vidual contributions. While the standard deviation would be a more natural for interpreting the units of noisiness, I focus attention on the variance for the sake

of statistical tests. When examining the data, it will be necessary to statistically detect significant changes in variability, and the sample variance estimator has a well-known distribution for testing such hypotheses. Each player conditions her quantal response upon the full state of the world under complete information, so the mean and variance of individual contributions may vary state-by-state; while the opposite is true of incomplete information, by definition. The values presented here take averages of these distributional moments across all states of nature for this reason.

To begin, notice that average contributions start at twenty for both payoff types in Figures 1.4(a) and 1.4(c). This is because all players have the same choice set, [0, 40], and uniform mixing implies a mean right in the center. For low payoff players, this means an average benefit of zero, so it is no surprise that their average contributions are decreasing as they become less sloppy in decision- making. Such a reduction in average contributions would significantly reduce expected payoffs via a reduced probability of success, if it were not for the fact that high types increase their contributions to compensate. Thus, I am left with the weak inequality that the average contribution of high payoff players is at least as large as that of low types. These changes are greater under complete information and for smaller groups, though such differences are only minor. Likewise, increased payoff responsiveness is associated with a reduction in individual noisiness in

Figures 1.4(b) and 1.4(d), and that these changes are more pronounced for smaller groups. For low payoff players, this reduction in variance is faster under complete information, but the difference across information treatments is imperceptible for high types.

Before proceeding to the experiments, it is important to say something about the limiting logit equilibria in the threshold game. Recall that there are uncount- ably many efficient Nash equilibria that meet the threshold, and there are also uncountably many dominated Nash equilibria that fall well below the threshold. It is quite possible that there are uncountably many logit quantal response equilibria for λ > λ∗, and all of these equilibria may even be a part of the principal branch. These equilibria may connect to the efficient outcomes in the limit, or they may connect to the dominated outcomes (or both). If the principal branch exclusively nests the efficient outcomes at the threshold, then we might expect certain prop- erties of QRE to carry through in the limit. For example, the comparative static described above that average contributions be greater for high payoff players than low types may restrict the limiting equilibria to contributions (cl, ch) such that

cl≤ ch.

On the other hand, all bets are off if the limiting QRE nests the dominated Nash equilibria, where high and low payoff players will both be weakly indifferent across all actions below their private valuations. This limits the usefulness of logit

QRE since the model cannot solve the underlying problem of equilibrium selec- tion. Indeed, some parameterizations may admit multiple equilibria, regardless of whether behavior is assumed to follow best response or quantal response. Little more can be said on the matter, except that these issues of indeterminacy do not arise in the range of parameter values typically estimated using experimental data. Moreover, the thesis of this research is that quantal response behavior is a sensible and useful assumption for modeling coordination failure, so the limiting properties that lead back to pure best-response are of little interest.