*2.1. Model*

I adopted Alaoui and Penta's EDR model for theoretical predictions [13]. In this model, players follow an endogenous reasoning process that determines the strategic bound in a particular context. With added structure on beliefs, the model is able to predict a player's actual level of play in any game that could use a *k*-level iterative best response reasoning process. The main benefit of using this model is that the structure of the model allowed me to conduct a comparative statics exercise on a player's reasoning process. One of the main goals of this study was to conduct a comparative static exercise on the cost side. Below, I provide more detailed descriptions of some key features of this model. These features are relevant to the experimental design and predictions for this paper.

A player's cognitive bound is a mapping from the incremental cost of reasoning (*c*(*k*)) and the incremental value of reasoning (*v*(*k*)) at each level to the intersection of the two terms.

$$\kappa(\upsilon, \mathfrak{c}) = \min \{ k \in \mathbb{N} \mid \upsilon(k) \ge \mathfrak{c}(k) \text{ and } \upsilon(k+1) < \mathfrak{c}(k+1) \} \tag{1}$$

A player reaches their cognitive bound at the *k*th level by having a value of reasoning for that level exceeds cost of reasoning, but their cost–benefit analysis no longer supports the one-higher level (i.e., *k* + 1) of reasoning. Further denote the cognitive bound of player *i* as ¯ *ki*, where:

$$
\vec{k\_i} = \kappa (v\_{i\prime} c\_i). \tag{2}
$$

According to Alaoui and Penta, the value of reasoning is affected by the payoff of the game [13]. The cost of reasoning is an endogenous characteristic of an individual, which is largely related to their cognitive or reasoning ability. In this paper, I take their assumption on the value of reasoning and continue to assume that the payoff is the only incentive for players to apply logical reasoning in the games. I provide a further discussion on the cost of reasoning. Beyond an individual's endogenous ability, the strategic environment (such as cognitive load) provides many challenges for a person in applying strategic reasoning, which alters the cost of reasoning.

A player's belief is represented as a tuple. Since the game in my design is symmetric in payoffs, a player's belief can be restricted to the beliefs about the cost of reasoning. Therefore, the first element of the tuple, *ci*, represents player *i*'s own cost of reasoning. The second element is player *i*'s beliefs of his opponent's (player *j*) cost of reasoning, denoted as *c<sup>i</sup> j* . The last element *c ij <sup>i</sup>* is player *i*'s second-order belief, which is their belief about player *j*'s belief of themselves. Any higher-order beliefs could be nested to the first- and second-order beliefs; therefore, a player's belief is represented as:

$$t\_i = (c\_{i\prime}c\_{j\prime}^i c\_i^{lj}).\tag{3}$$
