**4. Data Analysis Procedure**

All the subjects played 18 games in total, each against a fixed opponent during the experiment. There were 1998 observations of guesses. Grouping the guesses by games, I looked for level shifts observed with raw guesses. This exercise provided a general view of the effects of cognitive load on the games. I also used density plots of the guesses to visualize the treatment effects.

After the exploration of raw guesses, I estimated the level for each guess using the maximum likelihood method. Instead of assuming the subject's behaviors are determined by a single type across all the games, I assumed the subject's behavior in each game was determined by a single type and the types across games were allowed to be different. This was achievable with the design of my experiment with the variations on cognitive load.

Out of 1998 observations, 831 guesses correspond to a type's exact guesses. As about 40% of the observed guesses were a type's exact guesses, I followed the CGC06 approach in my estimation. Specifically, for each player *i*, game *g*, and level *k*, if player *i* was not making a type's exact guesses in game *g*, then I defined a likelihood function *L*(*yig* | *k*, *λ*) for each level *k* for the player in that game, with beliefs *f <sup>k</sup> <sup>g</sup>* (*y*) and sensitivity parameter *λ*, based on the assumption that they were trying to maximize their expected utility.

Formally, let *xig* be the raw guess observed for player *i* in game *g*. With the specification of lower limits *aig* and upper limits *big*, the adjusted guess is then *yig* = min{max{*aig*, *xig*), *big*}}. The density *f k <sup>g</sup>* (*z*) represents a subject's belief about his opponent's action given their behavioral level being *k*. Although in the literature a subject's belief of the other player's level could follow a certain type of distribution, for example, Poisson distribution as in Camerer et al. (2004), in this study, I followed the standard approach that level-*k* player has point belief about his opponent, that his opponent is level-(*<sup>k</sup>* <sup>−</sup> <sup>1</sup>) with probability 1. *<sup>y</sup>*<sup>0</sup> *<sup>g</sup>* is defined as uniformly spread across the action space. The expected payoff of playing *xig* with behavioral level *k*'s belief is then:

$$\mathcal{U}\_{\rm ig}^k(y\_{\rm ig}) = \int\_1^{1000} \mathcal{U}\_{\rm ig}(y\_{\rm ig'}z) f\_{\mathcal{g}}(z) dz. \tag{4}$$

Let *U<sup>k</sup> ig* = [max(*y<sup>k</sup> <sup>g</sup>* <sup>−</sup> 0.5, *aig*), min(*y<sup>k</sup> <sup>g</sup>* + 0.5, *big*)] be the interval of a type-*k* subject's exact adjusted guesses, allowing an error of 0.5. Any guess for game *g*, subject *i*, who is placed within *U<sup>k</sup> ig*, is then identified as an exact match for *k*-level. Conversely, define *U<sup>k</sup> ig* - = [*aig*, *big*]/*U<sup>k</sup> ig* as the complement of *Uk ig* within the limit interval for subject *i*'s game *g*. The likelihood function is then the following:

$$L(y\_{i\xi}|k,\lambda) = \frac{\exp[\lambda \mathcal{U}\_{i\xi}^k(y\_{i\xi})]}{\int\_{\mathcal{U}\_{i\xi}^k} \mathbb{E}\exp[\lambda \mathcal{U}\_{i\xi}^k(w)] dw}.\tag{5}$$

Since only one observation was used for the estimation, I took the sensitivity parameter (*λ*) as 1.33, which is the averaged estimated value of *λ* in CGC06 with only the subject's guesses. The maximum likelihood estimate of a subject's behavioral level in each game maximizes (5) over *k*, which is:

$$k\_{\rm ig} = \underset{k \in \{1, 2, 3, 4, 5, 6\}}{\arg\max} \ L^\*(y\_{\rm ig}|k). \tag{6}$$

To examine the treatment effects on behavioral levels, I pooled guesses into pairs for comparison. For example, to test the prediction on the changing cost of reasoning, I first identified games with the same first-order belief (either low or high cost of reasoning for opponent) and the same second-order belief (partial revelation), and then they were separated into comparison pairs by the subject's cognitive load tasks. The same selection was performed following the conditions listed in each testable prediction.

For each pair of games, I first conducted a binary comparison on their behavioral levels and I report the summary statistics. Since this is essentially a repeated measure of behavioral level from the same sample, I then conducted the Wilcoxon signed-rank test to check the distribution of behavioral levels. Lastly, I ran a GLS random effect regression to examine the treatment effects on behavioral levels. The regression was run by regressing the estimated level on the treatment variable. A subject's cognitive load was coded as 0 when it was in the low load treatment, and 1 when it was a high-load treatment. The same binary coding was also applied to the opponent's cognitive load treatment. The full revelation of information treatment was coded 0, whereas partial revelation was coded 1.
