In a recent paper, Avinash explores a decision problem drawn from Seinfeld. When the contraceptive sponge that Elaine favors is taken off the market, she scours the city buying up as many sponges as she can; but with finite supply, she faces a difficult decision as to which men are worthy of her precious sponges.
Avinash uses the tools of stochastic dynamic programming to help Elaine solve her problem.
But when she has a finite stock and cannot buy any more, her optimal decision will be based on a “sponge worthiness threshold” of quality, Qm, such that her decision will be yes if Q > Qm. The threshold depends on the number m of sponges she has: the fewer sponges left, the higher the threshold needed to justify using up one of them.Let Vm denote Elaine’s expected present value of utility when she has a stock of m sponges. She meets a man and observes his quality Q. If she decides to use one of her sponges, she gets the immediate payo Q and has continuation value Vm-1 on the second day,which has present value Vm-1. If she decides not to, there is no immediate payo, only the present value of continuation with m sponges, namely Vm. Therefore her decision rule is:Spongeworthy if Q + Vm-1 > Vmthat is, if Q > Qm (Vm – Vm-1) ; (1)
This is a paper about nothing.