The Role of the Protocol in Anthropic Reasoning

It is shown how thinking in terms of the protocol used can help clarify problems related to anthropic reasoning, such as the Doomsday Argument and the Sleeping Beauty Problem. There is a complex of problems related to anthropic reasoning, the Doomsday Argument [Leslie 1996] and the Sleeping Beauty Problem [Elga 2000] perhaps chief among them, that have generated a great deal of heat in the philosophy literature. Solutions have been proposed and then disposed of (see http://http://www.anthropic-principle.com/?q=resources/preprints for an annotated list of references). Here I discuss another approach to resolving these problems, which has applicability far beyond the scope of these problems. The point is to make precise exactly what protocol is being followed in all these puzzles. This point was already made by Shafer [1985], and is discussed at some length in [Halpern 2003][Chapter 6]. Before discussing the Doomsday Argument, I consider the second-ace puzzle [Freund 1965], since it illustrates the role of the protocol particularly well. The following discussion is largely taken from [Halpern 2003]. Example 0.1: [The second-ace puzzle] A deck has four cards: the ace and deuce of hearts, and the ace and deuce of spades. After a fair shuffle of the deck, two cards are dealt to Alice. It is easy to see that, at this point, there is a probability of 1/6 that Alice has both aces, a probability of 5/6 that Alice has at least one ace, a probability of 1/2 that Alice has the ace of spades, and a probability of 1/2 that Alice has the ace of hearts: of the six possible deals of two cards out of four, Alice has both aces in one of them, at least one ace in five of them, the ace of hearts in three of them, and the ace of spades in three of them. Alice then says, “I have an ace.” Conditioning on this information (by discarding the possibility that Alice was dealt no aces), Bob computes the probability that Alice holds both aces to be 1/5. This seems reasonable. The probability, according to Bob, of Alice having two aces goes up if he learns that she has an ace. Next, Alice says, “I have the ace of spades.” Conditioning on this new information, Bob now computes the probability that Alice holds both aces to be 1/3. Of the three deals in which Alice holds the ace of spades, she holds both aces in one of them. As a result of learning not only that Alice holds at least one ace, but that the ace is actually the ace of spades, the conditional probability that Alice holds both aces goes up from