An uncentered world is an objective state-trajectory of the material universe; I ignore quantum complications. If you know the uncentered world, it does not follow that you can predict your proximate observations, since you do not know which part of the uncentered world is here and now. A centered world is an uncentered world combined with a "here and now" tag for "where am I / what time is it".

I examine consistent probability assignments over centered worlds, which have relevance to anthropics. The main assumption I make is that these probabilities should not be Dutch-bookable if used by a CDT agent. Dutch book arguments (e.g. diachronic Dutch book arguments for Bayesian updating) typically assume CDT in the background; it is not straightforward to work out which bets EDT will accept in general. CDT Dutch book resistance therefore provides a normative probability framework that generalizes arguments for Bayesian probability.

Thought experiments such as Sleeping Beauty, and variants involving duplication, question how to assign probabilities to centered worlds in situations involving memory loss. One can analogize memory loss to being an individual who is part of a collective with shared goals; such an individual would be motivated to use probabilities over their centered world to take actions towards the shared objective, in a way consistent with how other individuals in the same collective do so. (In the case of AI, it is easier to see how the line between "memory loss" and "member of collective" is not very decision-theoretically important, as the same hardware can run many episodes in sequence with different observations, and both "memory loss" and "member of collective" interpretations are possible.)

What I will show is that the probability assignments that resist Dutch books are those which are "centering-uniform" in that they are arrived at by starting with some fixed measure over uncentered worlds, and weighting them according to number of occurrences of the agent's information state, with uniform probability over centers which have the same world and information state, and with any probabilities allowed conditional on information states with zero -measure. This condition strongly resembles Bostrom's SSSA (strong self-sampling assumption) and/or SSSA+SIA, which only disagree with each other up to a population scaling factor on uncentered worlds. Moreover, as shown in the appendix, when a probability policy resists weak Dutch books, it is centering-uniform with respect to a measure which is non-dogmatic in assigning non-zero measure to all populated uncentered worlds; this "strict centering-uniformity" enforces more extensive agreement with SSSA(+SIA).

Mathematical formulation

Let's set up the finite case formally. Let W be a finite set of (uncentered) worlds. Let C be a finite set of centers (roughly, possible observer-moments, though different time grains can be taken). Let "decenter" a centered world into its uncentered world. Let I be a finite set of information states. Let "observe" a centered world, yielding an information state possessed by the agent with that center. To simplify, we assume o is surjective.

A probability policy assigns to each information state some probability distribution over centers. This represents the probabilities an observer assigns to centered worlds upon making their observation i. We assume all probability policies have whenever ; in other words, is only supported on .

A betting menu gives an agent in information state i the option of buying a contract which pays out in case their center is c (noting, we must have ). A betting menu b is acceptable to a probability policy q iff CDT with q probabilities endorses accepting it in all information states; formally, . (We imagine, for simplicity, that accepting / rejecting the bet makes no difference to the universe other than bank balances, and that the agent's utility is linear in money. This avoids more complicated self-ratification issues; see "On the Interpretation of Decision Problems with Imperfect Recall" and "The Computational Complexity of Single-Player Imperfect-Recall Games" for details on situations where decisions can affect the trajectory. We imagine that all instances of agents with varying centers have the common objective of maximizing the bank balance; we could pretend that this balance is dedicated to an altruistic fund.)

In uncentered world w, the total payoff if all bets are accepted is . Let be the set of populated worlds. A Dutch book for a probability policy q is a betting menu b which is acceptable to q and which results in a sure loss in populated worlds, in the sense that . We wish to characterize probability policies which are not Dutch-bookable.

Say a probability policy q is centering-uniform iff for some probability distribution over with , we have, for all and ,

.

When the sum is zero, is unconstrained. In the case where the sum is non-zero, this is a more familiar SSSA+SIA expression:

A "uniformity" implication is that if have the same uncentered world w with and the same information state i, then , in agreement with SSSA. Note that we can move between SSSA and SSSA+SIA by changing to adjust for population. Hence, centering-uniformity is consistent with both.

The main result: A probability policy is Dutch-bookable iff it is not centering-uniform.

Proof:

A betting menu is a vector . Define the matrix . Define the matrix , .

Now is the CDT expected value of accepting at i, and is the payoff in populated uncentered world w. A Dutch book corresponds to b which satisfies and , componentwise.

Due to Motzkin's transposition theorem, exactly one of the following systems has a solution:

1. Find with and .
2. Find and with .

Problem (1) has a solution iff there is a Dutch book for q. Problem (2) has a solution iff it has a solution with being a probability distribution over populated worlds (by re-scaling); call problem (2) with this extra constraint (2p). Note for all ,

.

First, suppose (2p) has a solution. Then for all , . Pick any and sum this equation over :

Since , we have . So for all ,

.

This shows q to be centering-uniform (extend with for un-populated w). So any non-Dutch-bookable probability policy is centering-uniform.

In the other direction, suppose q is centering-uniform as witnessed by . Then and provide a solution to (2), showing that q is not Dutch-bookable.

Application to Sleeping Beauty

We set up Sleeping Beauty as follows. The centered worlds under consideration are HS (Heads on Sunday), TS (Tails on Sunday), HM (Heads on Monday), TM (Tails on Monday), TT (Tails on Tuesday). It is of course possible to add more (e.g. for after the experiment); this is a minimal setup. The uncentered worlds are H, T (for Heads, Tails). The information states are S, MT (for Sunday and not knowing the coin; and for unknown Monday/Tuesday and not knowing the coin, the awakening state).

A probability policy q will assign to each information state in {S, MT} some probability distribution over centered worlds. By assumption is only supported on {HS, TS} and is only supported on {HM, TM, TT}. Now q is Dutch-book resistant iff it is centering-uniform, i.e. iff for some probability distribution on {H, T}, we have:

Generally, both "halfers" and "thirders" agree that , i.e. on Sunday, Beauty should assign 50% to Heads. Under this supposition, centering-uniformity forces thirding: . However, centering-uniformity is compatible with by setting . This reflects the fact that centering-uniformity is agnostic about the measure on uncentered worlds.

A non-dogmatic probability policy (which has ) must predictably change its probability of Heads from Sunday to Monday, despite the absence of a memory wipe during Sunday night. Many (such as David Lewis) have found this predictable change in probabilities counter-intuitive, but Dutch book analysis suggests that such a predictable change is rationally obligatory. For more on Dutch book arguments applied to Sleeping Beauty, see "Simple Dutch books for Sleeping Beauty halfers".

Conclusion

This post presents a framework for Dutch-book resistant probability assignments in cases of amnesia, which can be applied to anthropic situations. The basic framework is Bostrom's SSSA, and scaling factors such as SIA are optional. The uncentered measure is a representational parameter for Dutch-book resistant probability policies, which are equivalently centering-uniform probability policies. need not be interpreted as the "prior over universes" in standard anthropics (SSA / SIA). Rather, parameterizes the Dutch-book resistant probability policies, analogous to the subjective prior in ordinary Bayesianism, which parameterizes Dutch-book resistant ways of assigning posterior probabilities.

To the degree Dutch book arguments for Bayesianism succeed, so do Dutch book arguments for centering-uniformity in settings with amnesia. Possible escape routes, such as EDT, could also be escape routes from ordinary Dutch book arguments for Bayesianism. Accordingly, centering-uniformity is a normative generalization of Bayesian probability to amnesic situations, since it stands or falls by similar arguments. See "Can de se choice be ex ante reasonable in games of imperfect recall? A complete analysis" for more on the EDT+FNC escape route, which diverges from Bayesian updating.

Philosophically, this post's conclusions are contrary to those of Emily Adlam's "Against Self-Location"; while Adlam argues that there are no rationality constraints for agents' conditional probabilities on their center given their uncentered world ("pure self-locating credences"), I argue that there are, at least under CDT Dutch-book assumptions used to justify Bayesianism; and if anything, pure self-locating credences are more rationally constrained than non-self-locating, -dependent credences over uncentered worlds are. Adlam argues that self-locating uncertainty and indexical caring measure can be ambiguous for agents who are indexically selfish; while I agree in the pure case of indexical selfishness, I argue that these are disambiguated for -altruistic agents who use self-locating uncertainty for decisions that affect the altruistic goals.

The most straightforward mathematical extension possible is to infinite sets of worlds and centers. In particular, and could be restricted (e.g. regular Borel) probability measures over infinite sample spaces. Tools from functional analysis, such as the Riesz-Markov representation theorem, might apply usefully; betting menus could be continuous functions of the space of centers (given some topology), and probability assignments could assign subjective expected values to such menus. I've concentrated on the finite case in this post because it is much more straightforward.

Appendix: weak Dutch books

We examine resistance to weak Dutch books, which guarantee no gain for the agent, and make loss possible. A weak Dutch book for q is a betting menu b which is acceptable to q, for which for all , and for which for some .

Call a probability policy q strictly centering-uniform iff it is centering-uniform for some which assigns non-zero probability to all populated worlds. We show: q is strictly centering-uniform iff there are no weak Dutch books against q.

Proof:

We construct the same matrices as before and consider the two problems:

1. Find with .
2. Find , , and with , , , .

By Motzkin's transposition theorem, exactly one of these has a solution. The first has a solution iff there is a weak Dutch book for q. If the second has a solution, we set and get a solution to the problem (2) of the main proof, showing that q is strictly centering-uniform. Furthermore, if q is strictly centering-uniform (with measure ), we get a solution to (2) with , , and , which shows there are no weak Dutch books against q.

Strict centering-uniformity is analogous to a requirement in ordinary Bayesianism to assign positive probability to every possible world in the sample space, at least in the finite case. It is a non-dogmatism condition.