  In domains where agents interact strategically, game theory is applied widely
to predict how agents would behave. However, game-theoretic predictions are
based on the assumption that agents are fully rational and believe in
equilibrium plays, which unfortunately are mostly not true when human decision
makers are involved. To address this limitation, a number of behavioral
game-theoretic models are defined to account for the limited rationality of
human decision makers. The "quantal cognitive hierarchy" (QCH) model, which is
one of the more recent models, is demonstrated to be the state-of-art model for
predicting human behaviors in normal-form games. The QCH model assumes that
agents in games can be both non-strategic (level-0) and strategic (level-$k$).
For level-0 agents, they choose their strategies irrespective of other agents.
For level-$k$ agents, they assume that other agents would be behaving at levels
less than $k$ and best respond against them. However, an important assumption
of the QCH model is that the distribution of agents' levels follows a Poisson
distribution. In this paper, we relax this assumption and design a
learning-based method at the population level to iteratively estimate the
empirical distribution of agents' reasoning levels. By using a real-world
dataset from the Swedish lowest unique positive integer game, we demonstrate
how our refined QCH model and the iterative solution-seeking process can be
used in providing a more accurate behavioral model for agents. This leads to
better performance in fitting the real data and allows us to track an agent's
progress in learning to play strategically over multiple rounds.
