im in cell tails tails heasd heads

These considerations support the initial intuition about Dungeon: that it is a situation in which one should reason in accordance with SSA. One thing worth noting about Dungeon is that we didn’t specify how the prisoners arrived in their cells. The prisoners’ history is irrelevant so long as they don’t know anything about it that gives them clues about the color of their cells. For example, they may have been allocated to their respective cells by some objectively random process such as by drawing balls from an urn (while blindfolded so they couldn’t see where they ended up). Or they may have been allowed to choose cells for themselves, a fortune wheel subsequently being spun to determine which cells should be painted blue and which red. But the thought experiment doesn’t depend on there being a well-defined randomization mechanism. One may just as well imagine that prisoners have been in their cells since the time of their birth, or indeed since the beginning of the universe. If there is a possible world in which the laws of nature determine, without any appeal to initial conditions, which individuals are to appear in which cells and how each cell is painted, then the inmates would still be rational to follow SSA, provided only that they did not have knowledge of the laws or were incapable of deducing what the laws implied about their own situation. Objective chance, therefore, is not an essential ingredient of the thought experiment; it runs on low-octane subjective uncertainty. 4. So far, so good. In Dungeon, the number of observers featuring in the experiment was fixed. Now let us consider a variation where the total number of observers depends on which hypothesis is true. This is where the waters begin to get treacherous. Incubator. Stage (a): The world consists of a dungeon with one hundred cells. The cells are numbered on the outside consecutively from 1 to 100. The numbers cannot be seen 5 from inside the cells. There is also a mechanism called “the incubator”. The incubator first creates one observer in cell #1. It then flips a coin. If the coin lands tails, the incubator does nothing more. If the coin lands heads, the incubator creates one observer in each of the remaining ninety-nine cells as well. It is now a time well after the coin was tossed, and everyone knows all the above. Stage (b): A little later, you are allowed to see the number on your cell door, and you find that you are in cell #1. Question: What credence should you give to tails at stages (a) and (b)? We shall consider three different models for how to reason, each giving a different answer. These three models may appear to exhaust the range of plausible solutions, although we shall later outline a fourth model which is the one that in fact I think points to the way forward. Model 1. At stage (a) you should set your credence of tails equal to 50%, since you know that the coin toss was fair. Now consider the conditional credence you should assign at stage (a) to being in a certain cell given a certain outcome of the coin toss. For example, the conditional probability of being in cell #1 given tails is 1, since that is the only cell you can be in if that happened. And by applying SSA to this situation, we get that the conditional probability of being in cell #1 given heads is 1/100. Plugging these values into the well-known mathematical result known as Bayes’s theorem, we get Pr(tails | I am in cell #1) Pr( #1| ) Pr( ) Pr( #1| ) Pr( )