There are slices of cake where the consumer gets a utility value log(1 j)beta^(t - 1) value from it where j is the number of slices consumed each day and t is the number of days since t = 1. What would be the OPT recurrence to maximize utility to determine how many slices of cake to eat each day for any certain beta?

CodePudding user response：

Assuming that we’re summing those logs, then letting OPT(n, t) be the maximum utility of eating n slices in the first t days,

OPT(n, 0) = 0
OPT(n, d) = max from j = 0 to n of [OPT(n − j, d − 1)   log(1   j) beta^(t − 1)]

Since utility decreases over time, optimal consumption over time does not increase. Therefore an n-day horizon suffices. Compute OPT(n, n) and follow the argmaxes backward.