Sleeping Beauty on Monty Hall

We present a game show that we claim can serve as a proxy for the notorious Sleeping Beauty Problem. This problem has divided commentators into two camps, `halfers' and `thirders'. In our game show, the potential awakenings of Sleeping Beauty, during which she will be asked about the outcome of the coin toss that determined earlier how many times she is awakened and asked, are replaced by potential contestants, deciding whether to choose heads or tails in a bet they will get to place if chosen as contestants on the outcome of the coin toss that determined earlier how many of them are chosen as contestants. This game show bears out the basic intuition of the thirders. Our goal in this paper, however, is not to settle the dispute between halfers and thirders but to draw attention to our game-show proxy itself, which realizes a version of the Sleeping Beauty Problem without the ambiguities plaguing the original. In this spirit, we design similar game-show proxies for variations on the Sleeping Beauty Problem with stochastic experiments other than a coin toss. We do the same for a variation in which Sleeping Beauty must decide upon being awakened whether or not to switch doors in the famous Monty Hall Problem and have the number of awakenings during which she gets to make that decision depend on the door she picked before she was put to sleep.