124 The UMAP Journal 28.2(2007) Max Wait Timo(Dond 02 一 Donavon Rate10 Figure 3. Wait time (in days) for various values of donation rate r, with list paired donation, applied over time to the current waitlist. We assume that patients want to maximize expected years of life. Let there be a current transplant available to the patient; we call this the immediate alternative and denote it by Ao. The patient and doctor have some estimate of how this transplant will affect survival; we assume that they have a survival function so(0, t)that describes chance of being alive at timet after the transplant. We further assume that this survival function is continuous and has limit zero at infinity: In other words, the patient is neither strangely prone to die in some infinitesimal instant nor capable of living forever. The patient also has a set of possible future transplants, which we callfil- ture alternatives and write as(Al, A2, .. An). Each future alternative Ai also has a corresponding survival function si(to, t), where to is the starting tim of transplant and t is the current time. We assume that there is a constant robability pi that alternative A will become available at any time. While this is not completely true, we include it to make the problem manageable: More complicated derivations would incorporate outside factors whose complexity would overwhelm our current framework. Finally, if the patient opts for a fu- ture alternative and delays transplant, survival is governed by a default survival ction sd124 The UMAP Journal 28.2 (2007) "•-0- Donation Rate 2.01 1.5 15 6 7 8 9 10 Time (Years) Figure 3. Wait time (in days) for various values of donation rate r, with list paired donation, applied over time to the current waitlist. We assume that patients want to maximize expected years of life. Let there be a current transplant available to the patient; we call this the immediate alternative and denote it by Ao. The patient and doctor have some estimate of how this transplant will affect survival; we assume that they have a survivalfunction so (0, t) that describes chance of being alive at time t after the transplant. We further assume that this survival function is continuous and has limit zero at infinity: In other words, the patient is neither strangely prone to die in some infinitesimal instant nor capable of living forever. The patient also has a set of possible future transplants, which we callfit￾ture alternatives and write as (A1, A 2,... , .A,,). Each future alternative .A- also has a corresponding survival function si(to, t), where to is the starting time of transplant and t is the current time. We assume that there is a constant probability pi that alternative Ai4 will become available at any time. While this is not completely true, we include it to make the problem manageable: More complicated derivations would incorporate outside factors whose complexity would overwhelm our current framework. Finally, if the patient opts for a fu￾ture alternative and delays transplant, survival is governed by a default survival function Sd