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Motivated by uncertain parameters encountered in Markov decision processes(MDPs) and stochastic games, we study the effect of parameter uncertainty onBellman operator-based algorithms under a set-based framework. Specifically, wefirst consider a family of MDPs where the cost parameters are in a givencompact set; we then define a Bellman operator acting on a set of valuefunctions to produce a new set of value functions as the output under allpossible variations in the cost parameter. We prove the existence of a fixedpoint of this set-based Bellman operator by showing that it is contractive on acomplete metric space, and explore its relationship with the correspondingfamily of MDPs and stochastic games. Additionally, we show that given intervalset bounded cost parameters, we can form exact bounds on the set of optimalvalue functions. Finally, we utilize our results to bound the value functiontrajectory of a player in a stochastic game.

Bounding Fixed Points of Set-Based Bellman Operator and Nash Equilibria of Stochastic Games