4.2 WMC and Cognitive Hierarchy
4.2.2 Dynamic Analysis: Evolution of Distributions
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4.2. WMC AND COGNITIVE HIERARCHY Table 4.3: Do High and Low WMC subjects exhibit similar level/guess dis-tributions?
Higher vs. Lower Mean P67 vs. P33 P75 vs. P25
Period Levela Guessb Levela Guessb Levela Guessb
1 0.0368∗ 0.0688 0.0417∗ 0.1051 0.0004∗∗ 0.0087∗∗
2 0.0038∗∗ 0.0145∗ 0.0076∗∗ 0.0720 0.0041∗∗ 0.0243∗
3 0.7212 0.4045 0.2848 0.1701 0.0502 0.083
4 0.2452 0.0452∗ 0.1379 0.0679 0.1172 0.0894
5 0.0098∗ 0.3119 0.0010∗∗ 0.1251 0.0016∗∗ 0.2249
6 0.3044 0.2565 0.2598 0.3703 0.0630 0.0799
7 0.4686 0.1789 0.5990 0.1462 0.4283 0.2249
8 0.6579 0.9999 0.9394 0.9999 0.9767 0.6459
9 0.3492 0.6050 0.6362 0.6912 0.7346 0.5464
10 0.1156 0.6890 0.2270 0.8493 0.1406 0.6207
∗ and ∗∗ denote the significance of the estimates at the 5 percent and 1 percent levels, respectively.
a p-value from Pearson’s chi-squared test with simulated p-value. The p-values are com-puted by running Monte Carlo simulations ten times, each with 107 replicates, and then taking the average of the ten simulated p.
b p-value from the Kolmogorov-Smirnov test.
while gender is found to be significant in many experimental studies10, it is not influential in the behavior of sophisticated reasoning. This result is also consistent withBurnham et al. (2009).
4.2.2 Dynamic Analysis: Evolution of Distributions
Earlier in Section4.1, we have shown that the effect of cognitive capacity on guessing performance may become weaker as subjects learn over time. Hence, the role of learning should be properly taken into account in our analysis. To trace the effect of learning, in this section, we shall study the evolution of the conditional distribution of the reasoning level. This is equivalent to putting a time axis to the static distribution analyzed above. What concerns us is whether the conditional distributions between the high WMC groups and
close to zero in the initial stage.
10For example, it is found that women give more than men in the dictator game (Eckel and Grossman,2008a), and women are more risk averse than men, although the latter is only evidenced in field experiments and is less conclusive in laboratory experiments (Eckel and Grossman,2008b).
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4.2. WMC AND COGNITIVE HIERARCHY
d<0 d=0 d=1 d=2 d=3 d>3
Figure 4.3: Level-k distribution conditional on WMC percentiles in period 1
low WMC groups convere when subjects learn over time. With this in mind, Figures4.3,4.4, and C.1toC.8(see AppendixC) give the level distributions of the high and low groups from the initial period to the last period.
The evolution of these distributions will help us see the effect of learning on cognitive hierarchy. One can start this analysis by first testing whether the level distribution between the high group and the low group are the same. This result is presented in Table 4.3, where we can see that the two distributions are significantly different only in the initial periods (periods 1 and 2), and this result is consistent in all three thresholds (columns 2, 4 and 6). After the initial periods, the null of having identical distributions cannot be rejected, except in period 5. 11
Let us then focus on the distributions in the initial periods when the distributions of the two groups are still heterogeneous, and see how the two groups differ in cognitive hierarchies. In period 1 (Fig. 4.3), the high WMC group exhibits the majority level (the peak) at level one and the second majority level at level two, regardless of the threshold being applied (panels
11While the subsequent level distributions between the two groups are less significant, for the two-tailed groups, the bottom and top one-fourth (P25 and P75), the difference in their level distribution is still quite noticeable in some periods, say, periods 6 and 10 (FiguresC.4andC.8).
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4.2. WMC AND COGNITIVE HIERARCHY
d<0 d=0 d=1 d=2 d=3 d>3
Figure 4.4: Level-k distribution conditional on WMC percentiles in period 2
(a), (b), and (c), Figure4.3). On the other hand, subjects with WMCs “lower than the mean” (panel (d), Figure4.3) exhibit the largest peak at level zero and the second largest peak at level one. However, for the subjects with lower WMCs, the groups P33and P25(panels (e) and (f)), the level distribution not only loses its focus at level zero or level one, but also tends to be distributed more evenly over different levels. This is particularly so for the case of P25. It suggests that the P25 group is behaviorally distinct from the others. These results provide a plausible correspondence between cognitive capacity and cognitive hierarchy, at least before learning takes effect. Roughly speaking, from the higher WMC groups to the lower ones, we can see the heterogeneity of the typical behavior (the majority) moving in a spectrum from level-1 reasoning, to level-0 reasoning, and ending up with random behavior.12
When coming to period 2, subjects’ levels of reasoning demonstrate
in-12 From an aggregate viewpoint, the uniform-like distribution, the distribution with al-most no modes, is interpreted as random behavior. In other words, we view the subjects with WMCs in the bottom one-fourth guessing as if they randomly picked a level in the pos-sible domain. This interpretation is only for the convenience of understanding aggregate behavior, and certainly does not exclude the possibility of the sophisticated thinking of some individuals in this group. What is also worth mentioning is that randomly-behaving agents have been coupled with zero-intelligence agents in the agent-based economic liter-ature (Chen,2012).
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4.2. WMC AND COGNITIVE HIERARCHY
teresting changes in becoming more sophisticated which shows evidence of learning. This can be seen from the mode shifting from level one to level two for the high WMC groups, and also the mode shifting from level zero to level one for the two low WMC groups (the second half and the bottom one-third). Even for the bottom one-fourth there is the mode at zero and one being developed. These together show that with one-period of experi-ence the subjects, regardless of their WMCs, all become sophisticated, or learn to think more deliberately. This observation enables us to see that the distributions of the high and the low groups become statistically insignificant in later periods (Figures C.1 to C.8).
While in the subsequent periods the level distribution of the high and the low WMC groups tend to have a distribution with a mode at level 2, deviations from this ideal type can still be found. In fact, a quick eye-browsing from Figures C.1 to C.8 shows that level 2 as a modal state may be constantly perturbed, if not unstable; for example, in period 6, level-3 reasoning becomes dominant in all high WMC groups (Figure C.4). Hence, to better harness the learning dynamics of individual subjects, it is necessary to augment the previous dynamic analysis based on conditional distributions with another domain, i.e., time correlation, to which we now turn.