2024/03/12 by Ivan Novikov, Novikov, Ivan
Decision Sciences · Economics, Econometrics and Finance · #FOS: Mathematics #Game Theory and Applications #Game Theory and Voting Systems #Optimization and Control (math.OC) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2403.07467
openalex created_date 2024/03/12 · openalex publication_date 2024/03/12 · openalex updated_date 2026/08/01
We study λ-discounted zero-sum games as the discount factor λ approaches 0 (that is, the players are more and more patient), in the context of games with stage duration. In stochastic games with stage duration h, players act at times 0, h, 2h, and so on. The payoff and leaving probabilities are proportional to h. When h tends to 0, such discrete-time games approximate games played in continuous time. The asymptotic behavior of the values (when both λ and h tend to 0) has already been studied for stochastic games with full state observation and for state-blind games. We consider the same question for the case of stochastic games with deterministic public signals on the state. We construct a stochastic game with public signals, with no asymptotic value (as the discount factor λ goes to 0) if the stage duration is 1, but with an asymptotic value when the stage duration h and the discount factor λ both tend to 0. Informally, this means that the asymptotic value in discrete time does not exist, whereas it does exist in continuous time. This situation cannot occur in stochastic games with full state observation.