Bound on Difference Between Sum of Probabilities

Question

I have the following probability equation: $$\hat{T}(x,a,x')=T(x,a,x') + \epsilon(x,a,x')$$ for all $x,x'\in X$ and $a \in A$, and $0 \leq T(x,a,x') \leq 1$. and $$P(x) = \max_{a \in A} \sum_{x' \in X}T(x,a,x')P(x').$$ and $$\hat{P}(x) = \max_{a \in A} \sum_{x' \in X}\hat{T}(x,a,x') \hat{P}(x').$$ for all $x \in X$ where $0 \leq P(x,a,x') \leq 1$. I want to find a bound $\delta(x)$ on the absolute difference $|\hat{P}(x) - P(x)|$ such that $|\hat{P}(x) - P(x)| \leq \delta(x)$. Obviously, $\delta(x) \leq 1$ but how can I proceed to find a tighter bound and express $\delta(x)$ in terms of $\epsilon(x,a,x')$? Please assume all the sets are finite.