*5.1. General Examination of Raw Guesses*

There were a total 1998 observations and 831 guesses corresponded to a specific level (levels 1 to 5, and equilibrium). When identifying levels, I assigned the lowest possible level to a guess that matched multiple types. For example, in game 3, equilibrium was reached after three rounds of iterative best responses, and the equilibrium was at the boundary of the target interval. In this case, although levels 3, 4, and 5, and the equilibrium all have corresponding guesses at 900, a subject's guess of 900 only assigned the subject to type level 3. This method of identification restricted over-assignments of the types.

Figure 3 shows the distribution of guesses that matched specific levels. Of the 831 guesses that matched a specific level, 43.92% were level 1 guesses, 31.41% were level 2 guesses, 14.20% were equilibrium guesses, and level 3 and higher corresponded to the remaining 10% of the guesses. To provide a clearer picture of the treatment effect, I used a Markov matrix for some treatments with these exactly matched guesses. Tables 3 and 4 present the level transitions between comparable games. For example, Table 3 consists of all the comparable pairs of changing a subject's own cost of reasoning, fixing the opponent with a high cognitive load (game 7 [LH-] and game 14 [HH-]). There were a total of 111 pairs of comparisons, 24 of which had both guesses that exactly matched a specific level. From games 7 to 14, 12 subjects reached level 1 in game 7 and 83.33% stayed at level 1 in game 14. Eight subjects reached level 2 in game 7, 87.5% of which stayed at level 2 and below in game 14. This result largely complies with the theory prediction that increasing cost of reasoning while fixing first- and second-order belief constant decreases the level of reasoning weakly. Likewise, Table 4 presents all the comparison pairs of changing the subject's first-order belief while fixing their own cost of reasoning and keeping their second-order belief constant (game 1 [LL+], game 8 [LL+], game 4 [LH+], and game 15 [LH+]). There were a total of 444 pairs of comparison, 99 of which had both guesses matched to a specific level. Forty pairs had level 1 guesses in the [LL+] treatment and 62.5% of them remained level 1 in the [LH+] treatment games. Similarly, 27 pairs had level two guesses in the [LL+] treatment. When changing the subject's first-order belief by increasing the cognitive load of their opponents, about 90% of these pairs had level 2 or lower guesses in the [LH+] treatment. These statistics largely coincided with the theoretical prediction—with increasing the cost of reasoning for the opponents, the subjects adjusted by weakly decreasing their behavioral levels of playing the game. Due to the limited number of exact matches, I was not able to conduct the same exercise for all the treatment pairs. However, complete discussion of the treatment effects is provided below with estimated behavioral levels.

**Figure 3.** Distribution of exact matches.

**Table 3.** Markov matrix of level transitions for increasing cost of reasoning, opponent with high load.


**Table 4.** Markov matrix of level transitions for changing first-order belief, subject with low load.


The pattern of subjects' adjustments to the changing strategic environment is also illustrated with density plots of each game. This time, all the raw guesses (after adjustments according to upper and lower limits) were used to plot the graphs. Figure 4 illustrates the treatment effects for the three theoretical predictions. To better compare across games, level 1 guesses were centered, and all the guesses were adjusted accordingly. The colored vertical lines illustrate the level-exact guesses. For example, in Figure 4a, the vertical red dashed line indicates level-1 guesses. Both density plots in the figure show peaks around the red vertical line, which indicate higher proportions of level-1 (or close to level 1) strategy used within the games across all the subjects. Notably, in the density plot for the [LH-] treatment (G7), there is another peak centered right at the level 2 guess for that game (indicated by blue dashed line). The density plot clearly shows that in the game where subjects have a lower cost of reasoning ([LH-]), guesses are congregated at both levels 1 and 2, whereas in the game where subjects have a higher cost of reasoning ([HH-]), only a peak at the level-1 guess is observed. Likewise, in Figure 4b, four games are plotted to illustrate the treatment effects of increasing cost of reasoning for the opponent. In Figure 4c, three games are used to demonstrate changing second-order beliefs. Note that both games 1 and 8 are relevant in both graphs, as the [LL+] treatment is relevant for both comparisons. As illustrated in the figure, in one of the games, the three peaks correspond to level 1, level 2, and equilibrium. When increasing the cost of reasoning for the opponent, the level 1 peak is still observable; however, only one game has a level-2 peak. Similarly, when changing the

second-order belief from low load with probability 1 to (0.5, 0.5; L, H), only the level 1 peak remains, as then the subjects thought that their opponents thought there was a 50% probability that the subject was experiencing a high cognitive load. I omitted other vertical lines that indicated different levels due to the absence of peaks in the density plots.

(**a**) Density plot of changing cost of reasoning (opponent high load).

(**b**) Density plot of changing cost of reasoning for opponent (subject low load).

**Figure 4.** *Cont.*

(**c**) Density plot of changing second-order belief (LL).

**Figure 4.** Density plot of raw guesses.
