The Annals of Probability

The Optimal Reward Operator in Dynamic Programming

D. Blackwell, D. Freedman, and M. Orkin

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Consider a dynamic programming problem with analytic state space $S$, analytic constraint set $A$, and semi-analytic reward function $r(x, P, y)$ for $(x, P)\in A$ and $y\in S$: namely, $\{r > a\}$ is an analytic set for all $a$. Let $Tf$ be the optimal reward in one move, with the modified reward function $r(x, P, y) + f(y)$. The optimal reward in $n$ moves is shown to be $T^n0$, a semi-analytic function on $S$. It is also shown that for any $n$ and positive $\varepsilon$, there is an $\varepsilon$-optimal strategy for the $n$-move game, measurable on the $\sigma$-field generated by the analytic sets.

Article information

Ann. Probab. Volume 2, Number 5 (1974), 926-941.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 49C99
Secondary: 60K99: None of the above, but in this section 90C99: None of the above, but in this section 28A05: Classes of sets (Borel fields, $\sigma$-rings, etc.), measurable sets, Suslin sets, analytic sets [See also 03E15, 26A21, 54H05]

Dynamic programming optimal reward optimal strategy analytic sets gambling


Blackwell, D.; Freedman, D.; Orkin, M. The Optimal Reward Operator in Dynamic Programming. Ann. Probab. 2 (1974), no. 5, 926--941. doi:10.1214/aop/1176996558.

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