A Rational Logit Dynamic for Decision-Making Under Uncertainty: Well-Posedness, Vanishing-Noise Limit, and Numerical Approximation

The classical logit dynamic on a continuous action space for decision-making under uncertainty is generalized to the dynamic where the exponential function for the softmax part has been replaced by a rational one that includes the former as a special case. We call the new dynamic as the rational logit dynamic. The use of the rational logit function implies that the uncertainties have a longer tail than that assumed in the classical one. We show that the rational logit dynamic admits a unique measure-valued solution and the solution can be approximated using a finite difference discretization. We also show that the vanishing-noise limit of the rational logit dynamic exists and is different from the best-response one, demonstrating that influences of the uncertainty tail persist in the rational logit dynamic. We finally apply the rational logit dynamic to a unique fishing competition data that has been recently acquired by the authors.