The relationships between discounted and average criteria of stochastic games with prospect theory

This paper considers nonzero-sum discrete-time stochastic games with finite state and action spaces, and focuses on the performance criteria under prospect theory. Based on the average criterion of stochastic games with prospect theory established firstly in 2018, we first present the discounted criterion of stochastic games with prospect theory and study the relationships between them. Both the criterion/value function and the Nash equilibrium are studied. We derive the equality of discounted and average criterion functions for any fixed strategy by the Abelian theorem, and discuss the relationships of value function. Moreover, since the probability is distorted in prospect theory, there is no optimality equation which is the commonly used tool in the existing literature of Markov decision processes and stochastic games. In this work, the bilateral relationships of equilibria between discounted and average criterion are investigated by the performance criterion (instead of the optimality equation), with the convergence of strategy sequences.