  We consider the problem of designing a control policy for an infinite-horizon
discounted cost Markov decision process $\mathcal{M}$ when we only have access
to an approximate model $\hat{\mathcal{M}}$. How well does an optimal policy
$\hat{\pi}^{\star}$ of the approximate model perform when used in the original
model $\mathcal{M}$? We answer this question by bounding a weighted norm of the
difference between the value function of $\hat{\pi}^\star $ when used in
$\mathcal{M}$ and the optimal value function of $\mathcal{M}$. We then extend
our results and obtain potentially tighter upper bounds by considering affine
transformations of the per-step cost. We further provide upper bounds that
explicitly depend on the weighted distance between cost functions and weighted
distance between transition kernels of the original and approximate models. We
present examples to illustrate our results.
