Sleeping Kelly is a Thirder

Abstract: The Sleeping Beauty problem was presented by Elga and highlights the role of probabilities in situations with imperfect recall. 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, I argue instead that it is rational for Sleeping Beauty to maximize the growth rate of her wealth using the Kelly Criterion, which leads us to the "thirder" position. Furthermore, this position is shown to be "rational" by Dutch book arguments. If Sleeping Kelly only accepts bets that have a growth rate greater than 1 as a "thirder" then she is not vulnerable to Dutch books. By contrast, if Sleeping Beauty takes the "halfer" position, she is vulnerable to Dutch books. If the bets offered to Sleeping Beauty were to be structured differently and lead to non-multiplicative wealth dynamics, she may no longer be a "thirder".