It’s a good question, and one I considered going into in more detail on in the post (I’ll add a link to this comment). I think it’s helpful to have in mind two types of people: “people who see the exact same evidence you do” (e.g., they look down on the same patterns of wrinkles on your hands, the same exact fading on the jeans they’re wearing, etc) and “people who might, for all you know about a given objective world, see the exact same evidence you do” (an example here would be “the person in room 2”). By “people in your epistemic situation,” I mean the former. The latter I think of as actually a disguised set of objective worlds, which posit different locations (and numbers) of the former-type people. But SIA, importantly, likes them both (though on my gloss, liking the former is more fundamental).
Here are some cases to illustrate. Suppose that God creates either one person in room 1 (if heads) or two people (if tails) in rooms 1 and 2. And suppose that there are two types of people: “Alices” and “Bobs.” Let’s say that any given Alice sees the exact same evidence as the other Alices (the same wrinkles, faded jeans, etc), and that the same holds for Bobs, and that if you’re an Alice or a Bob, you know it. Now consider three cases:
For each person God creates, he flips a second coin. If it’s heads, he creates an Alice. If tails, a Bob.
God flips a second coin. If it’s heads, he makes the person in room 1 Alice; if tails, Bob. But if the first coin was tails and he needs to create a second person, he makes that person different from the first. Thus, if tails-heads, it’s an Alice in room 1, and a Bob in room 2. But if it’s tails-tails, then it’s a Bob in room 1, and an Alice in room 2. (I talk about this case in part 4, XV.)
God creates all Alices no matter what.
Let’s write people’s names with “A” or “B,” in order of room number. And let’s say you wake up as an Alice.
In case one, “coin 1 heads” (I’ll write the coin-1 results in parentheses) corresponds to two objective worlds — A, and B — each with 1⁄4 prior probability. Coin 1 tails corresponds to four objective worlds — AA, AB, BA, and BB — each with 1/8th prior probability. So as Alice, you start by crossing off B and BB, because there are no Alices. So you’re left with 1⁄4 on A, and 1/8th on each of AA, AB, and BA, so an overall odds-ratio of 2:1:1:1. But now, as SIA, you scale the prior in proportion to the number of Alices there are, so AA gets double weight. Now you’re 2:2:1:1. Thus, you end up with 1/3rd on A, 1⁄3 on AA (with 1/6th on each of the corresponding centered worlds), and 1/6th on each of AB and BA. And you’re a “thirder” overall.
Now let’s look at case two. Here, the prior is 1⁄4 on A, 1⁄4 on B, 1⁄4 on AB, and 1⁄4 on BA. So SIA doesn’t actually do any scaling of the prior: there’s a maximum of one A in each world. Rather, it crosses off B, and ends up with 1/3rd on anything else, and stays a “thirder” overall.
Case three is just Sleeping Beauty: SIA scales in proportion to the number of Alices, and ends up a thirder overall.
So in each of these cases, SIA gives the same result, even though the distribution of Alices is in some sense pretty different. And notice, we can redescribe case 1 and 2 in terms of SIA liking “people who, for all you know about a given objective world, might be an Alice” instead of in terms of SIA liking Alices. E.g., in both cases, there are twice as many such people on tails. But importantly, their probability of being an Alice isn’t correlated with coin 1 heads vs. coin 1 tails.
Anthropics cases are sometimes ambiguous about whether they’re talking about cases of type 1 or of type 3. God’s coin toss is closer to case 1: e.g., you wake up as a person in a room, but we didn’t specify that God was literally making exact copies of you in the other rooms—your reasoning, though, treats his probability of giving any particular objective-world person your exact evidence is constant across people. Sleeping Beauty is often treated as more like case 3, but it’s compatible with being more of a case 1 type (e.g., if the experimenters also flip another coin on each waking, and leave it for Beauty to see, this doesn’t make a difference; and in general, the Beauties could have different subjective experiences on each waking, as long as —as far as Beauty knows — these variations in experience are independent of the coin toss outcome). I’m not super careful about these distinctions in the post, partly because actually splitting out all of the possible objective worlds in type-1 cases isn’t really do-able (there’s no well-defined distribution that God is “choosing from” when he creates each person in God’s coin toss—but his choice is treated, from your perspective, as independent from the coin toss outcome); and as noted, SIA’s verdicts end up the same.
It’s a good question, and one I considered going into in more detail on in the post (I’ll add a link to this comment). I think it’s helpful to have in mind two types of people: “people who see the exact same evidence you do” (e.g., they look down on the same patterns of wrinkles on your hands, the same exact fading on the jeans they’re wearing, etc) and “people who might, for all you know about a given objective world, see the exact same evidence you do” (an example here would be “the person in room 2”). By “people in your epistemic situation,” I mean the former. The latter I think of as actually a disguised set of objective worlds, which posit different locations (and numbers) of the former-type people. But SIA, importantly, likes them both (though on my gloss, liking the former is more fundamental).
Here are some cases to illustrate. Suppose that God creates either one person in room 1 (if heads) or two people (if tails) in rooms 1 and 2. And suppose that there are two types of people: “Alices” and “Bobs.” Let’s say that any given Alice sees the exact same evidence as the other Alices (the same wrinkles, faded jeans, etc), and that the same holds for Bobs, and that if you’re an Alice or a Bob, you know it. Now consider three cases:
For each person God creates, he flips a second coin. If it’s heads, he creates an Alice. If tails, a Bob.
God flips a second coin. If it’s heads, he makes the person in room 1 Alice; if tails, Bob. But if the first coin was tails and he needs to create a second person, he makes that person different from the first. Thus, if tails-heads, it’s an Alice in room 1, and a Bob in room 2. But if it’s tails-tails, then it’s a Bob in room 1, and an Alice in room 2. (I talk about this case in part 4, XV.)
God creates all Alices no matter what.
Let’s write people’s names with “A” or “B,” in order of room number. And let’s say you wake up as an Alice.
In case one, “coin 1 heads” (I’ll write the coin-1 results in parentheses) corresponds to two objective worlds — A, and B — each with 1⁄4 prior probability. Coin 1 tails corresponds to four objective worlds — AA, AB, BA, and BB — each with 1/8th prior probability. So as Alice, you start by crossing off B and BB, because there are no Alices. So you’re left with 1⁄4 on A, and 1/8th on each of AA, AB, and BA, so an overall odds-ratio of 2:1:1:1. But now, as SIA, you scale the prior in proportion to the number of Alices there are, so AA gets double weight. Now you’re 2:2:1:1. Thus, you end up with 1/3rd on A, 1⁄3 on AA (with 1/6th on each of the corresponding centered worlds), and 1/6th on each of AB and BA. And you’re a “thirder” overall.
Now let’s look at case two. Here, the prior is 1⁄4 on A, 1⁄4 on B, 1⁄4 on AB, and 1⁄4 on BA. So SIA doesn’t actually do any scaling of the prior: there’s a maximum of one A in each world. Rather, it crosses off B, and ends up with 1/3rd on anything else, and stays a “thirder” overall.
Case three is just Sleeping Beauty: SIA scales in proportion to the number of Alices, and ends up a thirder overall.
So in each of these cases, SIA gives the same result, even though the distribution of Alices is in some sense pretty different. And notice, we can redescribe case 1 and 2 in terms of SIA liking “people who, for all you know about a given objective world, might be an Alice” instead of in terms of SIA liking Alices. E.g., in both cases, there are twice as many such people on tails. But importantly, their probability of being an Alice isn’t correlated with coin 1 heads vs. coin 1 tails.
Anthropics cases are sometimes ambiguous about whether they’re talking about cases of type 1 or of type 3. God’s coin toss is closer to case 1: e.g., you wake up as a person in a room, but we didn’t specify that God was literally making exact copies of you in the other rooms—your reasoning, though, treats his probability of giving any particular objective-world person your exact evidence is constant across people. Sleeping Beauty is often treated as more like case 3, but it’s compatible with being more of a case 1 type (e.g., if the experimenters also flip another coin on each waking, and leave it for Beauty to see, this doesn’t make a difference; and in general, the Beauties could have different subjective experiences on each waking, as long as —as far as Beauty knows — these variations in experience are independent of the coin toss outcome). I’m not super careful about these distinctions in the post, partly because actually splitting out all of the possible objective worlds in type-1 cases isn’t really do-able (there’s no well-defined distribution that God is “choosing from” when he creates each person in God’s coin toss—but his choice is treated, from your perspective, as independent from the coin toss outcome); and as noted, SIA’s verdicts end up the same.