*3.1. The Game*

The two-person guessing game is an asymmetric, two-player game. Each player has a lower limit, *ai* > 0, an upper limit, *bi* > 0, and a target *pi* ∈ (0, 2). Players are required to input a guess that is within their lower and upper limit. However, their actual choice is not restricted by the limit. Denote player *i*'s input by *xi*. If a player guesses a number *xi* that falls outside the limit interval, then their guess will be adjusted to the closest bound. For example, if *xi* < *ai*, then the adjusted guess *yi* will be *yi* = *ai*. If *xi* > *bi*, then the adjusted guess *yi* is *yi* = *bi*. However, any guess falling within the limit interval will not be adjusted; i.e., *yi* = *xi*.

The goal of the game is to make a guess that minimizes the difference between the player's own guess and the product of their target and his opponent's guess. Denote the difference by *ei* =| *yi* − *pi* · *yj* |. The payoff is a quasi-concave function minimized at zero. Player *i* receives *ui* <sup>=</sup> max{0, 200 <sup>−</sup> *ei*} <sup>+</sup> max{0, 100 <sup>−</sup> *ei* <sup>100</sup> }. Since a player's guesses that have the same adjusted inputs will yield the same outcome for the subject, I use the adjusted guess *yi* as a proxy of how players perform in the game.

In this game, the level-0 player is assumed to play randomly according to a uniform distribution over the action space. Denote the theoretical predicted guess made by a *k*-level player as *x<sup>k</sup> <sup>i</sup>* . Given the assumption imposed on the level-0 player's strategy, level-1 players will best respond to the expected value of level-0 player's guess, i.e., *x*<sup>1</sup> *<sup>i</sup>* <sup>=</sup> *pi* · <sup>E</sup>{*yj* <sup>|</sup> *yj* <sup>∈</sup> [*aj*, *bj*]}. The level-2 player's strategy will then be *x*<sup>2</sup> *<sup>i</sup>* <sup>=</sup> *pi* · {(*x*<sup>1</sup> *<sup>j</sup>* <sup>∈</sup> [*aj*, *bj*]) · *<sup>x</sup>*<sup>1</sup> *<sup>j</sup>* <sup>+</sup> (*x*<sup>1</sup> *<sup>j</sup>* <sup>&</sup>lt; *aj*) · *aj* <sup>+</sup> (*x*<sup>1</sup> *<sup>j</sup>* > *bj*) · *bj*}. The reasoning process follows iterative best responses. It converges to the Nash equilibrium after finite rounds of iterations.

In this paper, I adopt 14 two-person guessing games used by CGC06 and 4 two-person guessing games used by Georganas et al. [14,22]. The parameters of each game are given in Table 1. All the players survive at least two rounds of iterative best responses before reaching the equilibrium (as stated in Table 1 "steps to eqm" column). Since in CGC06, only a few number of subjects reached level 3 in the reasoning process, the choice of parameters in this paper should be sufficient to identify a player's strategic levels in the game.
