In this paper, we consider the pricing problem of American options under an irrational exercise strategy with a rationality parameter. Irrational behavior of option holders as reactions to market movements can lead to an exercising option strategy at a time that might not be an optimal time. Under the irrational exercising time strategy, the pricing problem of the American-style option results in the overvalued option price. A common way to study the irrational behavior of option holders and its impact on the American option pricing problem is to consider intensity-based models with stochastic intensity parameters. Under these models, the option pricing problem leads to a nonlinear parabolic partial differential equation (PDE) with an additional term to the PDE of the American option under rational strategy (classical American option with optimal exercise strategy) due to the intensity functions of models. Although the solution converges to the solution of the classical American option price when the parameter tends to infinity, the classical boundary conditions cannot apply to finite values of the parameter. For this, we propose a finite element method to solve the resulted PDE with a numerical approach. We also present numerical results to show the accuracy of the proposed method.