Grasping Reality on TypePad, by Brad DeLong

Note to Self: Apropos of the Sleeping Beauty Problem: Adam Elga: Self-Locating Belief and the Sleeping Beauty Problem...

There are three π here:

1. π₁: a parameter of nature that describes the probability that the coin, when tossed, will come up heads: π₁ = 1/2
2. π₂: a number that Sleeping Beauty should keep in mind when deciding at what odds to bet that the coin came up heads if she wants to make money: π₁ = 1/3
3. π₃: a number that Sleeping Beauty should tell a third party who asks her: "I'm going to make one and only one bet this week as to how this coin came out. What number should I have in mind in deciding at what odds to take heads if I want to make money?": π₃ = 1/2

Which of these π₁, π₂, π₃ is the "right" answer Sleeping Beauty should give?