*5.5. Result 3: Changing the Second-Order Belief*

In the experiment, I used a (0.5, 0.5) probability distribution on the revelation of cognitive load treatments to control for the subject's second-order belief. In the full revelation treatment, role B knew the exact memorization task that was received by role A (the subject), either three (low load) or seven letters (high load) with a probability of one. Therefore, role A's (the subject) second-order belief was either ((1, 0); (L, H)) or ((0, 1); (L, H)). In the partial revelation treatment, role B knew that the probability of three or seven letters for role A was (0.5, 0.5), which made role A (the subject) have a second-order belief of ((0.5, 0.5); (L, H)). If comparing two games with different second-order beliefs for the subject, with everything else controlled as constant, then a second-order belief of low load with probability of one should be considered as more cognitively capable perceived by role B than a second-order belief of((0.5, 0.5); (L, H). The experiment, as shown in Table 11, supported that most subjects had a clear understanding of their opponent's cognitive load when the load was explicitly elicited, and they almost had uniform beliefs about their opponents' cognitive loads when they were in the partial revelation treatment as role B.


**Table 11.** Subject's belief about his opponent's cognitive load.

In the dataset, I found 888 pairs for comparison that allowed me to examine the effect of changing the second-order belief. I separated them into two groups: a comparison between the full revelation of low load to partial revelation, and a comparison between a partial revelation and a full revelation of high load. Both comparisons were performed in the direction of increasing second-order belief (i.e., *c ij <sup>i</sup>* increases). The detailed comparison pairs and summary statistics are listed in Table 12. The distribution of behavioral levels is plotted in Figure 7.

**Table 12.** The frequency of changing levels with changing second-order belief.


**Figure 7.** Level Distribution for changing second-order belief.

The effect of changing the second-order belief was generally weak, except for the cases where the subjects had high cognitive loads when facing opponents with low cognitive loads. For the treatment where both players had low cognitive loads, about 77.03% of the pairs had weakly decreasing behavioral levels when second-order belief changed from full to partial revelation. Among these comparisons, only 25.68% had strictly decreasing levels. This finding suggested that the changes in second-order belief may not have been strong enough for the subjects to adjust their behavioral level downward, even though both subjects had a low cognitive load and were relatively competent at contemplating over the strategic environment. To examine the effect of the second-order belief, it was first necessary to determine the effect of changing the first-order belief for the same group of subjects. In Table 10, the subject's behavioral responses to the changing opponent's cognitive load were limited when the subject had a high cognitive load. Now, consider the finding in the [LH+] to [LH-] comparison to the [HH-] to [HH+] comparison (Table 12); changing the second-order belief of the subject effectively changed his opponent's first-order belief. If the subject holds the belief about his opponent (who has a high cognitive load treatment) that the changes in his opponent's behavioral level are limited, then the subject should not decrease his behavioral level at all. This partially explains the low frequency of strictly decreasing behavioral levels for subjects who faced opponents with high cognitive loads.

The comparison between [HL-] and [HL+] is consistent with the EDR model. In Figure 7c, the [HL-] treatment is first-order stochastic dominant over the [HL+] treatment. Of the guesses, 87.84% had weakly decreasing behavioral levels, with 35.14% having strict decreases. Changing from partial revelation to full revelation of high cognitive load, the second-order belief decreased the subject's cognitive capability perceived by their opponent. Subjects were responsive to this change in the belief system, and adjusted their behavioral levels downward to best respond to their opponents. Testable prediction 3 suggests that if the subject's behavioral level is binding by their cognitive bound, then they are not able to make further adjustments according to their changing beliefs. The large percentage of constant levels for these comparisons supported this statement.

The Wilcoxon test results showed that the level distribution changed for changing second-order belief. When conducting a one-tailed test, the test result suggested that for [LH+ to LH-] and [HH- to HH+] treatments, the distribution of levels significantly (at the 1% level) shifted rightward (increasing behavioral levels). This may have occurred due to the subject's belief that their opponent with high cognitive load will engage in higher behavioral level. This result seems to comply with the results in Section 5.4, but the underlying reasons need further investigation.

The regression coefficient on the treatment dummy further supported the results. Since the treatment dummy was coded as zero with full revelation and one with partial revelation, the coefficient of 0.57 for [HL-, HL+] comparison suggested that the behavioral level decreased from partial to full revelation. It was significant at the 1% level. Again, the [LH+, LH-] and [HH-, HH+] comparisons were the opposite direction of model predictions, and they were also highly significant. In general, when the subjects faced opponents with high cognitive loads, they were responsive to changing second-order beliefs, but not in the direction that is predicted by the EDR model. However, when they faced more cognitively capable opponents, then they were mostly responsive to this change in the belief system because they thought their opponents were responsive to this information in their strategic environment. This finding is consistent with the EDR model when the opponent has a low cognitive load, which supports the opposite direction when the opponent is in a less cognitively capable situation.
