126 The UMAP Journal 28. 2(2007) concept of a separable set. Meanwhile, the factor b(t- to)accounts for the decrease in survival during the time(t-to )spent with the new kidney. Consequently, we assume that our survival functions are separable. This will lead us to an explicit heuristic for lifespan-maximizing decisions, which is the goal of this section. For A; and A, two future alternatives in a separable set, we assign an order accor Ing b()≤/b()d←4≤4 We turn to the derivation of an lifespan-maximizing strategy Such a strat- egy, when presented with alternative Ai at time to, will either accept or wait for other alternatives. In fact: Theorem. If a patient's alternatives form a separable set, then the optimal strategy is either to accept an alternative Ai at all times to or to decline it at all times to. If the patient declines Ai, then the patient must decline all alternatives less than or equal to Ai in the order relation defined above. Similarly, if the patient accepts Aj, then the patient must accept all alternatives greater than or equal to A i Proof: The patient will accept the alternative or probabilistic bundle of alterna tives that the patient' s survival functions indicate gives the greatest lifespan For alternative Ai, the expected lifespan beyond time to is 8i(to, t)dt Suppose that a patient at time 0 declines this alternativein favor of some optimal set of future alternatives. Furthermore, suppose that this set includes some alternative Ak such that Ak S Ai. Then the expected lifespan from this set is (+)(+叫)间 p b(t)+pb (t) dt dto ∑+pk j ranges over all alternatives Ai in the optimalsetexcept A. This double of hartal does not mix integration variables and is therefore equal to a product (+叫)厂(+叫)间-厂 ∑P()+pkb() ∑+Pk Since Ak is less than or equal to Ai, and Ai was declined in favor of the set of alternatives that we are examining, the presence of the h term in the weighted verage under therightintegrand lowers the value of the average. The previous expression is thus less than (2+)(+时间.2购a126 The UMAP Journal 28.2 (2007) concept of a separable set. Meanwhile, the factor b(t - to) accounts for the decrease in survival during the time (t - to) spent with the new kidney. Consequently, we assume that our survival functions are separable. This will lead us to an explicit heuristic for lifespan-maximizing decisions, which is the goal of this section. For .A and A•4 two future alternatives in a separable set, we assign an order according to: 1 0000 0 O bi(t) dt <_1 bj (t) dit A-i4 <_ Aj We turn to the derivation of an lifespan-maximizing strategy. Such a strat￾egy, when presented with alternative Ai4 at time to, will either accept or wait for other alternatives. In fact: Theorem. If a patient's alternatives form a separable set, then the optimal strategy is either to accept an alternative Ai at all times to or to decline it at all times to. If the patient declines Ai, then the patient must decline all alternatives less than or equal to Ai in the order relation defined above. Similarly, if the patient accepts Aj, then the patient must accept all alternatives greater than or equal to Ai. Proof. The patient will accept the alternative or probabilistic bundle of alterna￾tives that the patient's survival functions indicate gives the greatest lifespan. For alternative Aj, the expected lifespan beyond time to is j si(to,t) dt. Suppose that a patient at time 0 declines this alternative in favor of some optimal set of future alternatives. Furthermore, suppose that this set includes some alternative A,k such that Ak <• Aj. Then the expected lifespan from this set is (Z3jPj + Pk) 1f00 eXP [- (Fj 0 Pj + Pk) to] a (to) j* 0 (' 1 pj Ej b' (t) Pi+pA + Pkbk.'(t) dtdo wherej ranges over all alternatives Aj in the optimal set except Ak. This double integral does not mix integration variables and is therefore equal to a product of two integrals: (r,j Pj + pk) f exp [- (rj Pj + Pk) to] a(to)dto Ej Zpjbj(t)+ pkbk(t) dt. 0 0 • j j + P/. Since Ak is less than or equal to Aj, and Ai was declined in favor of the set of alternatives that we are examining, the presence of the k term in the weighted average under the right integrand lowers the value of the average. The previous expression is thus less than (Fjpj + pk) j 0exp [-( 1pj + pk) to] a (to) cito -j >~pj bj(t) + pk.blk(t)dt fo Ej P1