Under the framework given by a growth condition, a Lyapunov property and some continuity assumptions, the present work shows the existence of lower semicontinuous solutions to the Shapley equation for zero-sum semi-Markov games with Borel spaces, weakly continuous transition probabilities and possible unbounded payoff. It is also shown the existence of stationary optimal strategies for the minimizing player and stationary ε-optimal strategies for the maximizing player. These results are proved using a fixed-point approach. Moreover, it is shown the existence of a deterministic stationary minimax strategy for a minimax semi-Markov inventory problem under mild assumptions on the demand distribution.
Bibliographical noteFunding Information:
Work partially supported by Consejo Nacional de Ciencia y Tecnología (CONACYT-Mexico) under grant Ciencia Frontera 2019-87787.
© 2022, The Author(s), under exclusive licence to Springer Nature B.V.
- Average payoff
- Fixed-point approach
- Lyapunov conditions
- Semi-Markov games
- Shapley equation