3.4 EWA Learning Model
3.4.2 Estimation Strategy
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3.4. EWA LEARNING MODEL
Table 3.1: Summary of Estimated Learning Models Parameters Strategy N(0), φ, ρ, δ,
λ
Aj(0) d
Model I Number Estimated Estimated
Model II Number Estimated Initialized
Model III Number Estimated Initialized Estimated Model IV Level-k rule Estimated Estimated
Model V Level-k rule Estimated Initialized
Model I is referred to Camerer and Ho (1999). Model II and Model III are referred to Camerer, Ho and Chong(2002)
Summary of estimated learning models Table3.1summarize the mod-els which were estimated in this study. Model I, II, III tried to replicate EWA learning proposed by Camerer and Ho (1999) and Camerer, Ho and Chong (2002). The common feature of these Camerer’s EWA learning is that sub-jects try to learn the attractiveness of all guessing numbers. On the contrary, Model IV and V assumed that subjects try to learn the attractiveness of level-k rules. We also tried some perturbations and modifications, including that initializing initial attractions Aj(0) by observed probability of strategy in the first period and introducing additional parameter d to replace the unrealistic assumption about the knowledge of winning number.
3.4.2 Estimation Strategy
Attractions are transformed to the choice probability by the logit function which is given by
Pij(t + 1) = eλ·Aji(t−1) Pmi
k=1eλ·Aki(t−1)
where mi denotes the number of choices, and mi = 101 in Camerer’s EWA and mi = 6 in our EWA rule learning. Define player i’s initial attractions as a vector Ai(0) ≡ (A1i(0), A2i(0), ....Ami i(0)). We assume a representative agent and calibrate his learning behavior, so Ai(0) = A(0) ∀i. The number of subjects is denoted by N and the total sample size, 10 · N is denoted by M. Then the log-likelihood function LL(A(0), N(0), φ, ρ, δ, λ), is
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We also imposed some restrictions of parameters,
0 ≤ Aj(0) ≤ 1000,∀j
The parameters are first estimated by the choices data of all 108 subjects in all ten rounds. In this way, our results can be compared with the original work done by Camerer and Ho (1999) and a modified version by Camerer, Ho and Chong(2002). We then separated our data into two groups, high and low WMC, and obtained two sets of parameter estimates, to see how cogni-tive capacity affect learning behaviors. We tried several numerical nonlinear global optimization methods in Mathematica, including simulated anneal-ing, Nelder-Mead, random search and differential evolution, to maximize the likelihood function. When a specific method derived superior results, we fur-ther explored some options of this method, such as number of search points, number of random seeds and post process for local search, to avoid reporting local optima.
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Chapter 4 Results
We conducted 6 experimental sessions with 108 subjects. In what follows, we will first present the results of the effects of cognitive capacity on guessing performance (Section 4.1), i.e., the lower left edge of Figure 1.1. We then move to the lower right edge of Figure1.1and present the effects of cognitive capacity on cognitive hierarchy (Section 4.2). In both cases, the effects of cognitive capacity are not just studied in a static manner, but also in a dynamic fashion by taking learning into account. This can be illustrated by our examination of not just the reasoning level distribution conditional on different WMCs (Section 4.2.1), but also the evolution of this distribution over time (Sections4.2.2and4.2.3). Finally, we shall come to the upper edge of Figure 1.1 and present our findings on the connection between guessing performance and the cognitive hierarchy (Section 4.3).
4.1 Cognitive Capacity and Guessing Perfor-mances
Do subjects with higher working memory capacity (WMC) perform better than subjects with lower WMC? To answer this question, we have to decide a performance measure. In this context, the subjects’ performance can be measured by their behavior (guessing accuracy) or the consequence of their behavior (payoffs). The two measures will be equally good if the latter is a strict monotone transformation of the former. However, by the usual payoff design of the beauty contest experiment, this is not the case; we, therefore, consider the guessing accuracy to be a more direct and informative measure for performance than the payoffs.
To measure subject i’s guessing accuracy in period t, we consider the
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4.1. WMC AND GUESSING PERFORMANCE
Table 4.1: Correlation coefficients between WMC scores and guessing differ-ences in the beauty contest experiment
Kendall’s τb Spearman’s ρ
Period Estimates p-value Estimates p-value
1 −0.2368∗∗ 0.0003 −0.3415∗∗ 0.0003
2 −0.1600∗ 0.0144 −0.2339∗ 0.0148
3 −0.0957 0.1441 −0.1349 0.1639
4 −0.0040 0.9513 −0.0265 0.7854
5 −0.1601∗ 0.0156 −0.2094∗ 0.0297
6 −0.0875 0.1871 −0.1333 0.1691
7 0.0041 0.9512 −0.0004 0.9967
8 −0.0401 0.5444 −0.0617 0.5255
9 −0.0797 0.2285 −0.1119 0.2488
10 −0.1366 0.0537 −0.2057∗ 0.0327
1-10 −0.2016∗∗ 0.0020 −0.2944∗∗ 0.0020
2-10 −0.1606∗ 0.0138 −0.2321∗ 0.0158
3-10 −0.1433∗ 0.0279 −0.2059∗ 0.0325
4-10 −0.1231 0.0591 −0.1762 0.0681
5-10 −0.0982 0.1322 −0.1503 0.1206
6-10 −0.0627 0.3365 −0.0913 0.3475
7-10 −0.03 0.646 −0.0447 0.6458
8-10 −0.0776 0.2353 −0.1098 0.258
9-10 −0.1663∗ 0.0113 −0.2450∗ 0.0106
∗and∗∗ denote the significance of estimates at the 5 percent and 1 percent levels, respec-tively.
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4.1. WMC AND GUESSING PERFORMANCE
absolute error:
εi(t) = |gi(t) − target(t)|,
where target(t) = 23m(t). With this performance measure, we are curious to know whether there is any correlation existing between the cognitive ca-pacity (WMC) and the performance measure. To find out, we run the rank correlation between WMC and εi(t) for each t (t = 1, 2, ..., 10). Table 4.1 gives the Kendall rank coefficient (the 2nd column) and the Spearman rank coefficient (the 4th column). The table is presented in two parts: the upper panel shows the coefficient for each single period, whereas the lower panel shows the coefficient over the consecutive periods from the specified begin-ning period to the final period. Therefore, the first row of the lower panel,
“1-10”, gives the aggregate result, i.e., the rank coefficient by taking into ac-count all subjects over all periods. Table4.1shows that this result is negative for both the Kendall rank coefficient (τb) and Spearman rank coefficient (ρ), and the minus sign is statistically significant (columns 3 and 5). This result suggests that, as a whole, subjects with higher WMC perform better, and hence supports a positive effect of cognitive capacity.
However, if we break down this aggregate result into individual periods (the upper panel of Table 4.1), negative correlations are also observed in all periods, while only a few of them are statistically significant.1 A closer look indicates that these few results occur mainly in the initial periods, say, periods 1 and 2 (the upper panel) and in those windows which begin with these initial periods, such as windows “1-10”, “2-10” and “3-10” (the lower panel). This suggests that the WMC effect in the initial periods substantially influences the overall significance.
Then, as time goes on, subjects may learn with experience, and their learning may help to narrow the original gap in cognitive capacity. However, the question is: will the effect of learning completely annihilate the effect of cognitive capacity, and will the subjects with lower WMC eventually catch up in their performance compared to the subjects with higher WMC? Would the effect of cognitive capacity persistently matter being conditional on the effect of learning, or will the effect of learning eventually dominate and make up for the deficit in talents? This question is a little intriguing because it is difficult to know in general what is the minimum time required for the learning to work, and all experiments must inevitably be based on a pre-chosen limited duration to avoid human fatigue. However, even though only ten iterations are run for each experiment, we believe that, in the context of BCG, it will be reasonably long enough to examine the above catching-up or convergence hypothesis.
1The only exception is period 7 (τb), but it is also insignificant.
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4.1. WMC AND GUESSING PERFORMANCE
Figure 4.1: Guessing Errors between the High WMC groups and the Low WMC groups
Subjects are divided into the high WMC groups and Low WMC groups using different thresholds. The six curves shown above are the mean guessing errors of the six groups, the bottom one-fourth (WMC < P25), the bottom one-third (WMC < P33), the second half (WMC < mean), the top one-fourth (WMC < P75), the top one-third (WMC > P67), and the first half (WMC > mean). The numbers inside the brackets are the number of subjects belonging to the respective group.
To better trace the effect of learning, a more apparent way to present the result is to draw the guessing error over time and over groups with different WMCs. This is done in Figure4.1. Since there are 108 subjects possibly with 108 possible learning curves, drawing all of them in one figure will not help us see anything. We, therefore, present the learning curves in groups. We first divide the subjects into two groups, one with higher WMCs, and one with lower WMCs. We consider three different thresholds to group them; they are the mean, one third and one fourth. Based on these three thresholds, subjects or subsets of subjects are divided into two groups, the bottom (the bottom one-half, the bottom one-third, and the bottom one-fourth) and the top (the top one-half, the top one-third, and the top one-fourth). With a little abuse of the notations, we can denote these groups by percentiles; hence, P25 (the 25th percentile) denotes the bottom one-fourth, P33 the bottom one-third, P67 the top one-third, and P75 the top one-fourth.2
We then figure out the mean guessing error of each group in each period,
2All the thresholds or percentiles are computed based on our Experimental Subject Database (ESD) which includes the WMC scores of 740 subjects.