Monty Hall drives a wedge between Judy Benjamin and the Sleeping Beauty: a reply to Bovens

Abstract

Bovens (2010) points out that there is a structural analogy between the Judy Benjamin problem (JB) and the Sleeping Beauty problem (SB). On grounds of this structural analogy, he argues that both should receive the same solution, viz. the posterior probability of the eastern region of the matrix in Table 1 should equal 1/3. Hence, P*(Red) = 1/3 in the JB and P*(Heads) = 1/3 in the SB. Bovens’s argument rests on a standard error in implementing Bayesian updating, which is spelled out in Shafer 1985. When we are informed of some proposition, we do not only learn the proposition in question, but also that we have learned the proposition as one of the many propositions that we might have learned. The information is generated by a protocol, which determines the various propositions that we might learn. We should then update not on the proposition in question, but rather on the fact that we learned this proposition as one of the many propositions that we might have learned. A well-known application of this insight is the Monty Hall problem (MH) as Speed (1985: 276) points out in a discussion of Shafer 1985. As an illustration, let us apply Shafer’s insight to the MH. In the MH, the contestant in a game show learns that there is a goat behind two of three doors X, Y and Z and a car behind one door. She is asked to pick one of the three doors. The contestant picks door X. Monty will then open one of the remaining doors, which he knows to have a goat behind it. Suppose Monty opens door Y. The contestant is then asked whether she wants to …