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Summary

In Section~\ref{continuity} we proved that the discounted value is continuous in the parameters of the game, see Theorem~\ref{theorem7}.

One weakness of this result is that it does not bound the Lipschitz constant of the value function $(\lambda,q,r) \mapsto v_\lambda(s;q,r)$.

In this chapter, we will strengthen Theorem~\ref{theorem7}, and, using the concept of $B$-graphs, develop a bound on the Lipschitz constant of the value function.

Our technique will allow us to study the continuityof the limit $\lim_{\lambda \to 0} v_\lambda(s;q,r)$ as a function of $q$ and $r$.

Summary

In this chapter we define semi-algebraic sets and study their basic properties. We then apply our findings to prove that for every initial state s the limit $\lim_{\lambda \to 0} v_\lambda(s)$ exists.

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7 - B-Graphs and the Continuity of the Limit limλ→0 ʋλ(s;q,r)

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6 - Semi-Algebraic Sets and the Limit of the Discounted Value

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2 results

7 - B-Graphs and the Continuity of the Limit limλ→0 ʋλ(s;q,r)

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6 - Semi-Algebraic Sets and the Limit of the Discounted Value

Summary