Bayesian deterministic decision making: A normative account of the operant matching law and heavy-tailed reward history dependency of choices

The decision making behaviors of humans and animals adapt and then satisfy an ``operant matching law'' in certain type of tasks. This was first pointed out by Herrnstein in his foraging experiments on pigeons. The matching law has been one landmark for elucidating the underlying processes of decision making and its learning in the brain. An interesting question is whether decisions are made deterministically or probabilistically. Conventional learning models of the matching law are based on the latter idea; they assume that subjects learn choice probabilities of respective alternatives and decide stochastically with the probabilities. However, it is unknown whether the matching law can be accounted for by a deterministic strategy or not. To answer this question, we propose several deterministic Bayesian decision making models that have certain incorrect beliefs about an environment. We claim that a simple model produces behavior satisfying the matching law in static settings of a foraging task but not in dynamic settings. We found that the model that has a belief that the environment is volatile works well in the dynamic foraging task and exhibits undermatching, which is a slight deviation from the matching law observed in many experiments. This model also demonstrates the double-exponential reward history dependency of a choice and a heavier-tailed run-length distribution, as has recently been reported in experiments on monkeys