3.3.1. Against Human

I adopted a 2 × 2 × 2 design. For ease of explanation, I specify the two players in the guessing game as having role A and role B in this section. However, subjects were not aware of their role during the experiment. Each subject was given the role of A or B for each treatment exactly once. I used a within-subject design.

To examine the effects of changing the cost of thinking on a subject's level of reasoning, I varied the cognitive load for role A, holding role B's cognitive load constant. As mentioned in the previous section, role A needed to memorize a string of either three or seven random letters when playing the guessing game. To test the effects of changing the opponent's cost of thinking on a player's level of strategic sophistication revealed in the game, I also varied role B's cognitive load by two levels. Changing the cost of thinking of role B essentially tests the effects of changing the first-order belief for role A. Denote the cognitive load of three letters as low load (L) and seven letters as high load (H).

Lastly, I varied the disclosure of information on the cognitive load for role B. The exact cognitive load implemented on role A was either fully revealed to role B or partially revealed as a probability distribution. Denote full revelation as [+] and the counterpart as [-]. In the partial revelation treatment, role B was told that role A has a 0.5 probability of memorizing a string of three letters and a 0.5 probability of memorizing a string of seven letters. The full and partial revelations of the cognitive load information on role B were a method of measuring the effects of changing the second-order belief for role A. In the full revelation treatment, both roles A and B were aware that role A's memorization task is common knowledge. However, in the partial revelation treatment, role A knew their exact memorization task was hidden to role B; therefore, their second-order belief (i.e., their belief about role B's belief of their own cost of thinking) may not coincide with their actual cost of reasoning. A summary of treatments is provided in Table 2. In later sections, I used role A's label to identify the

treatments, as I was essentially examining the treatment effects for role A only. The first letter in the label indicates role A's cognitive load (either L or H). The second letter indicates role B's cognitive load (opponent's cognitive load, either L or H), and the last element of the label indicates full or partial revelation (role A's second order belief, either [+] or [-]). Role B served as a supporting role to complete the information required for each treatment. The information presented to role B for each treatment is also presented in Table 2. However, when later discussing the experimental results, I only refer to each treatment using role A's label. Table 1 provides a summary of treatments and assignments of roles for each game. Each subject played as either role A or role B exactly once for each treatment. There are in total 16 games. For each treatment, the pair of games are symmetric in game parameters and cognitive load realizations. The games were played in two random orders (the first order was as game numbers listed in Table 1; the second order was: 2, 13, 14, 4, 3, 1, 16, 6, 11, 8, 12, 5, 10, 15, 7, 9, 18, 17. Since for each game, there were two players assigned with different cognitive loads, considering player 2's order of play, there were essentially four sequences. The number of subjects in each order was roughly balanced. After dropping subjects with missing data, there were 28 subjects playing the first order as player 1, 29 subjects playing the first order as player 2, and 27 subjects playing the second order as player 1 and player 2 respectively.). Before the start of each session, one of the two was randomly selected.


**Table 2.** The eight treatments.
