The Sleeping Beauty Problem: Sleeping Kelly is a Thirder

The Sleeping Beauty problem is a problem of imperfect recall that has received considerable attention. One approach to solving the Sleeping Beauty problem is to allow Sleeping Beauty to make decisions based on her beliefs, and then characterize what it takes for her decisions to be "rational". In particular, she can be allowed to make monetary bets based on her beliefs, with the assumption that she wants to gain wealth rather than lose it. However, this approach is often coupled with the assumption that Sleeping Beauty should maximize the expected value of her bets. Here, show that Sleeping Beauty maximizes the expected growth rate of her wealth as a "thirder" sizing bets using the Kelly Criterion under multiplicative dynamics. Furthermore, this position is shown to be impervious to Dutch books. By contrast, the "halfer" position is shown to be vulnerable to Dutch books under similar circumstances.