*2.2. Theoretical Predictions*

I formulated the testable predictions following the EDR construction discussed in Section 2.1. For any game *G* = {*Xi*, *ui*}*i*<sup>=</sup>1,2, let *ki*(*xi*) be the reflected behavioral level of player *i* by choosing action *xi*, where *Xi* is the set of actions available for player *i* and *ui* is the payoff function for player *i*.


Player *i* reacts to the change in the cost of reasoning of their opponents. More specifically, if he observes his opponent's cost of reasoning increasing, he will adjust their strategy in the game to best respond to his opponent. That means they may choose to take an action that corresponds to a lower level of strategic sophistication. However, such adjustments of strategies are binding by the cognitive bound when the player believes their opponent has a lower cost of reasoning compared to their own cost. In the context of my experiment, a player should choose a weakly lower level of strategy if he observes his opponent's memorization task becoming more difficult (i.e., from a string of three letters to a string of seven letters).

3. **Changing the second-order belief**: For any *ci* and *c<sup>i</sup> j* , *ki*(*xi*) (weakly) decreases with *c ij <sup>i</sup>* . If *ci* ≥ *c ij i* , then player *i*'s cognitive bound is binding. By fixing player *i*'s own cost of reasoning and his opponent's cost, through only changing player *i* s second-order belief, player *i* should adjust their strategic actions. For example, when a player has a low cost of reasoning in the game, if they believe that their opponent has a wrong belief about themselves, namely, they believe that their opponent thinks the cost of reasoning for them is very high, then they can switch to an action that is associated with a lower level of reasoning. However, this adjustment of strategic actions according to the second-order belief is restricted by player *i*'s own cognitive bound, meaning that they cannot make any adjustments that requires a higher level of reasoning than their cognitive bound. In the context of my experimental design, players should adjust their actions when the information structure shifts from full revelation of cognitive load to partial revelation.

4. **Cognitive bound**: Given *ci*, for any *c<sup>i</sup> <sup>j</sup>* and *c ij <sup>i</sup>* , *ki*(*xi*) never exceeds ¯ *ki*. When fixing player *i*'s own cost of reasoning, their behavioral level should never exceed their cognitive bound. In the context of this experiment, on an individual level, actions observed in games 17 and 18 should correspond to the highest level of reasoning that one player can achieve under the respective cognitive load.
