# Can the rewards be stochastic when the transition model is deterministic?

Suppose we have a deterministic environment where knowing $$s,a$$ determines $$s'$$. Is it possible to get two different rewards $$r\neq r'$$ in some state $$s_{\text{fixed}}$$? Assume that $$s_{\text{fixed}}$$ is a fixed state I get to after taking the action $$a$$. Note that we can have situations where in multiple iterations we have: $$(s,a) \to (s_1, r_1) \\ (s,a) \to (s_{\text{fixed}}, r_1) \\ (s,a) \to (s_{\text{fixed}}, r_2) \\ (s,a) \to (s_3, r_3) \\ \vdots$$

My question is, would $$r_1 =r_2$$?

My question is, would $$r_1 =r_2$$?
Usually when you declare that you have "a deterministic environment", you imply that both $$s'$$ and $$r$$ are fixed values depending on $$(s,a)$$. So in your examples, you would expect your observations to also have $$r_1 = r_2$$
However, it is possible to define a MDP where transition to state $$s'$$ is deterministic, but $$r$$ is not. For instance, you could define reward in a game equal to the sum of a number of dice rolled, with better rewards (on average) resulting in more dice. This is still a valid MDP and can be solved using RL techniques.