Product logic and probabilistic Ulam games

There is a well-known game semantics for Lukasiewicz logic, introduced by Daniele Mundici, namely the Rényi–Ulam game. Records in a Rény–Ulam game are coded by functions, which constitute an MV-algebra, and it is possible to prove a completeness theorem with respect to this semantics. In this paper we investigate some probabilistic variants of the Rényi–Ulam game, and we prove that some of them constitute a complete game semantics for product logic, whilst some other constitute a game semantics for a logic between ΠMTL and product logic.