This is a followup to my previous article, Sleeping Beauty Resolved? Some objected to the solution I presented (building on Radford Neal's analysis) on the grounds that it runs afoul of a Thirder betting argument:
If Beauty uses anything other than a probability of exactly for Heads, she will accept certain bets she should not, and reject others she should accept.
Alas, there is an alternative Halfer betting argument that makes the same claim, but replacing with .
I'll show that both arguments are wrong, as they get the effective payoffs wrong; but if Beauty uses the correct effective payoffs, together with a probability of for Heads, she makes the right betting decisions.
To get there requires addressing some questions of unique identity, so that's where I'll start.
Indexicals and Identity
Thirder arguments often make use of statements such as
today is Monday
today is Tuesday,
treating them as mutually exclusive propositions. Probability theory is based on the classical propositional logic, and deals exclusively with classical propositions; are the above legitimate classical propositions?
I argue that they are not. The word "today" is problematic; it is an indexical, which the article "Demonstratives and Indicatives" in the Internet Encyclopedia of Philosophy defines as
...any expression whose content varies from one context of use to another. The standard list of indexicals includes adverbs such as “now”, “then”, “today”, “yesterday”, “here”, and “actually”.
The article furthermore remarks,
Indexicals and demonstratives raise interesting technical challenges for logicians seeking to provide formal models of correct reasoning in natural language...
and goes on to discuss various efforts to construct logics appropriate for reasoning with indexicals.
Clearly indexicals pose a problem for classical logic, else there would be no interest in constructing alternative logics to deal with them. As Richard Epstein writes in Classical Mathematical Logic: The Semantic Foundations of Logic,
...we cannot take sentence types as propositions if we allow the use of indexicals in our reasoning
(p. 4). This is because the meaning of a classical proposition must be definite and stable:
When we reason together, we assume that words will continue to be used in the same way... We will assume that throughout any particular discussion equiform words will have the same properties of interest to logic. We therefore identify them and treat them as the same word...
...if we accept this agreement, we must avoid words such as 'I', 'my', 'now', or 'this', whose meaning or reference depends on the circumstances of their use. Such words, called indexicals, play an important role in reasoning, yet our demand that words be types requires that they be replaced by words that we can treat as uniform in meaning or reference throughout a discussion.
(p. 3). In short:
Every usage of the same proposition in an argument / analysis must mean the same thing and have the same true/false value in all contexts within the scope of the analysis.
The important point is to ensure that any temporal (or spatial, etc.) reference is uniquely defined. If we are having a face-to-face conversation and I use the word "now," it's clear that means the specific, well-defined point in time at which I utter that word. If I refer to "the day on which Julius Caesar took his first sip of wine," then even though nobody knows what day that was, I have uniquely identified a particular day--there cannot be two such days.
But when Beauty asks, "Is today Monday?", how does she identify which "today" she means? Can she find some uniquely identifying descriptor that unambiguously distinguishes "today" from "the other day"?
Maybe. If the experimenters randomly choose to put a black marble on her night stand one day, and a white marble the other day, and Beauty knows this, then as soon as she glances at the night stand and sees (say) a black marble, she can then say that "today" means "the day on which there is a black marble on the nightstand." In this case the usual Thirder arguments hold, and she gets a probability for Heads of .
But if Beauty is an AI and her entire state of mind and stream of experiences from Monday are exactly reproduced on Tuesday, then the term "today" is inescapably ambiguous--Beauty has no way of uniquely identifying "today". Thirder arguments based on using "today is Monday" and "today is Tuesday" as mutually exclusive propositions are invalid in this case. As shown previously, the Halfer argument of "no new relevant information" applies in this case, and Beauty gets a probability of for Heads.
In the intermediate case, where is the stream of perceptions Beauty has experienced since awakening and , , is the probability of experiencing the identical stream of perceptions at some time on the other day, then "today" is partially identified as "the day in which Beauty experiences stream of perceptions ." (The closer is to zero, the more probable it is that uniquely identifies "today.") We then get a probability for Heads that is intermediate between the Halfer and Thirder positions: .
If You Already Know What Your Conclusion Will Be...
This issue of unique identity answers a question @travisrm89 asked:
How can receiving a random bit cause Beauty to update her probability, as in the case where Beauty is an AI? If Beauty already knows that she will update her probability no matter what bit she receives, then shouldn't she already update her probability before receiving the bit?
This question references the special case where Beauty is an AI whose only sensory input after awakening on Monday/Tuesday is a sequence of random bits. I showed that her probability of Heads before receiving any bits is , and after receiving the first bit this falls to , no matter which bit is received. The above argument is also one a Halfer could use against the Thirder position: if Beauty already knows on Sunday that her probability for Heads is going to be on Monday, why isn't that already her probability for Heads?
To answer this, let's consider where the principle travisrm89 invokes comes from. Let and be two mutually exclusive and exhaustive propositions, with meaning "the next observation is ." If our probability for updates to regardless of what we observe--that is, if for both --then
which says that our probability for should already be .
But the new information Beauty has after receiving one random bit doesn't fit the above pattern. In this special case of the problem her new information is
for some , where means "the first bit Beauty receives on day is y". Significantly, the two propositions and are not mutually exclusive if --both are true if the coin lands Tails and Beauty receives different first bits on Monday and Tuesday. Thus we have an exhaustive but not mutually exclusive set of possibilities, and the sum of their probabilities exceeds 1: .
which is why we can have even though .
It is only in the case of --Beauty's experiences on the two days are identical so the days are entirely indistinguishable--that the sum of observation probabilities is 1, and the prior and posterior probabilities are the same.
I'll start with the part everybody agrees on:
Suppose that Beauty is offered a bet with a payoff of if the coin lands Heads, and a payoff of if the coin lands Tails. Either of these payoffs can be negative, in which case it is a loss. (The interesting cases are where one is positive and the other negative.) If the coin lands Tails she is offered the bet on both Monday and Tuesday. Since Beauty assesses the same probability of Heads on both Monday and Tuesday, she will either accept the bet both times or reject it both times. Thus the "objective" expected payout if she accepts the bet is
Beauty should accept the bet if, and only if, this quantity is positive. This is the case in the standard example where and .
The standard Thirder betting argument goes like this:
If Beauty assesses a probability for Heads after awakening on Monday/Tuesday, then she computes an expected payout of
and will accept the bet only iff this is positive. If then Beauty's expected payout is identical to the "objective" expected payout (up to a constant, positive factor) and so she will make the correct decision to accept or reject the bet, whatever the payoffs used.
The Halfer counterargument [reference?] goes like this:
Beauty knows full well that, if the coin lands Tails, she is going to compute the same probability of Heads on both Monday and Tuesday, and that she will therefore make identical decisions on those two days, and obtain identical outcomes. So in that case she is making a decision for two days, not just one. Therefore she should compute her expected payout as
and accept the bet only if this quantity is positive. If then her subjective expected payout is identical to the objective expected payout, and so she will make the correct decision to accept or reject the bet, whatever the payoffs used.
There are elements of truth to both positions. Any decision rule, including the rule that one should maximize expected gain, is a function that maps the available information to a recommended action. In the SB problem the available information is Beauty's background knowledge about the experimental setup, plus the stream of perceptions she has experienced since awakening. If the same stream of perceptions occurs on both days, Beauty's decision rule must give the same action both days, and so in this circumstance her payout for Tails is . However, if Beauty does not experience the stream of perceptions on the other day, then in principle her decision rule could give different actions for the two days, and her payout for Tails is just