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By Martin Gardner

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Is non-positive. It equals zero if and only if k is in D^. In that some R. are negative and others are zero, the model resembles a stopping problem. LEMMA 2. Let \ be a bias-optimal policy, and let Π = ( . . ,π , IT*) be a time-optimal decision procedure. Then (51) λ P e(n) < e(n + 1) < P πη+1 e(n) . PROOF : By definition , _ π η+1 π η+1 π η+1 e(n+l) = R +P e(n) < P e(n) since R < 0 for any π . Similarly, since R = 0 , e(n+l) > RX + Ρ λ e(n) = Ρ λ e(n) . m Since Ρ δ is stochastic for each policy δ , it is a simple consequence of (51) that min^ e(n).

If the simplex algorithm calls for entry of row A. , remove row l AfW. THEOREM 7. The following procedures make the same sequence of pivots. (i) The simplex routine, initiated with basis B and applied to Program II, using the modified exit rule. (ii) The policy iteration routine, initiated with policy δ, and, at each policy improvement step, changing only the one decision for which t. is most positive. Remark: Program II involves variables {x^}, while policy iteration involves the dual variables {9^ ν ^}· It is widely known (cf.

Every policy corresponds in this way to a basic f e a s ible solution to Program II. One might then hope that if Program II is initiated with a basis corresponding to a policy it executes a series of pivot s t e p s , with each successive basis corresponding to a policy. To see what happens, we first rewrite the constraints as row vectors. (The simplex routine is normally described in terms of column vectors, but row vectors meld better with our scheme of notation. ) Define the 1 by N vector Ak = (1, - P k l ' i2' .