*5.3. Result 1: Increasing Cost of Reasoning*

As mentioned in Section 2.2, the first testable prediction involved fixing the subject's first- and second-order beliefs and examining the effect of the changing cost of reasoning on the subject's behavioral levels. There were essentially two comparisons in this case: a comparison between treatment [LL-] and treatment [HL-], and between treatment [LH-] and [HH-]. Note that in both comparisons, the cost of reasoning for the subject varied from low to high; therefore, it was crucial to have partial revelation of the subject's (role A) memorization task. In the partial revelation treatment, role B (the opponent) only knew the probability distribution of the subject's memorization task (0.5, 0.5; L, H); therefore, even with the subject's own tasks varying between two treatments, the subject's second-order belief was controlled to be the same. There were 222 pairs of comparison in total. The summary statistics of the comparisons are presented in Table 7. The plotted distribution of behavioral levels is presented in Figure 5. To aid with the interpretation of the results, the behavioral

levels in the figure are presented in a reverse order (i.e., higher level on the left and lower level on the right).


**Table 7.** The frequency of changing behavioral levels with increasing cost of reasoning.

**Figure 5.** Cumulative level distribution for increasing cost of reasoning.

When the opponent's cognitive load was controlled to be high and with partial revelation, subjects weakly decreased their behavioral levels 89.19% of the time (39.64% strict decrease). In Figure 5b, the [LH-] treatment is first-order stochastic dominant over the [HH-] treatment. The Wilcoxon test (Table 8) was significant at the 1% level for the comparison of the distributions of behavioral levels between these two strategic environments. When regressing the behavioral level on the treatment dummy, the result (Table 9) suggested that the coefficient for treatment dummy was 0.77, which was significant at the 1% level. This implied that the estimated behavioral level weakly decreased when the subject's own cognitive load increased when facing an opponent with high cognitive load. The finding is consistent with the EDR model. The relatively large proportion (49.55%) of constant levels may seem quite surprising at first look. One possible explanation is that these subjects may have had different cognitive bounds in the two treatments. In the [LH-] treatment, subjects may have adjusted their behavioral levels downward from their cognitive bound in that treatment due to some belief they formed when facing opponents with high cognitive loads. In the [HH-] treatment, subjects who had a lower cognitive bound (as they had a high cognitive load) may have displayed a lower behavioral level. When the two behavioral levels from two treatments coincided, I observed no changes in the behavioral levels in the treatment comparison.

The result for the comparison between [LL-] and [HL-] is less clear. As shown in Table 7, 63.97% of the comparisons had weakly decreasing behavioral levels (20.72% strict decrease), and a noticeable percentage (36.03%) of the comparisons had increasing levels. The Wilcoxon test statistic rejected the null hypothesis that the two strategic environments have the same distribution of behavioral levels at the 5% level. However, upon further checking using a one-tail Wilcoxon test, the distribution of behavioral levels shifted rightward when cognitive load changed from low to high when facing an opponent with a low cognitive load. When conducting the standard GLS random effect regression, the coefficient on the treatment dummy was positive and significant at the 10% level.


**Table 8.** Test results for equality of distribution.

∗ indicates < 10% significance, ∗∗ indicates < 5% significance, and ∗∗∗ indicates < 1% significance.


**Table 9.** Regression results for treatment effects.

∗ indicates < 10% significance, ∗∗ indicates < 5% significance, and ∗∗∗ indicates < 1% significance. Standard errors in parenthesis.

In this analysis, I treated equilibrium level as the highest level, since it requires the subjects to perform multiple steps of iterative best responses. However, since many games have equilibrium at the boundary of the limit interval (games 2, 4, 6, 10, 15, and 16 have the equilibrium at the lower limit; games 3, 5, 8, and 12 have the equilibrium at the upper limit), if the subject chooses an equilibrium action by naïvely playing at the boundary, then this behavioral level should not be considered as a higher level than any of the *k* levels. This was not the case for this comparison pair. Although game 9 ([HL-]) had 7.21% equilibrium guesses, those guesses were not at the boundary. However, upon further checking of games 16 ([LL-]) and 9 ([HL-]), I found that the level-5 type in game 16 had the same strategy as the equilibrium strategy of that game. Therefore, some of the equilibrium strategies in game 16 were pooled into level-5 type, which may be one possible explanation for the significant positive coefficient on the treatment dummy. Another explanation may be that the subjects felt more motivated to reason at higher strategic levels when they saw the opponents had easier strategic environments (memorizing three letters) as opposed to their own difficult strategic environments (memorizing seven letters). As a result, they displayed higher behavioral levels. This explanation suggests that other factors, such as motivation factor, may also play a role in determining a subject's behavioral levels.
