Committee credences

Suppose the members of a committee individually assign credences or probabilities to a bunch of propositions—maybe propositions about climate change or about whether a particular individual is guilty or innocent of some alleged crimes. What should we take to be “the committee’s credences” on the matter?

Here is one way to think about this. There is a scoring rule s that measures the closeness of a probability assignment to the truth that is appropriate to apply in the epistemic matter at hand. The scoring rule is strictly proper (i.e., such that an individual by their own lights is always prohibited from switching probabilities without evidence). The committee can then be imagined to go through all the infinitely many possible probability assignments q, and for each one, member i calculates the expected value E_pis(q) of the score of q by the lights of the member’s own probability assignment p_i.

We now need a voting procedure between the assignments q. Here is one suggestion: calculate a “committee score estimate” for q in the most straightforward way possible—namely, by adding the individuals’ expected scores, and choose an assignment that maximizes the committee score estimate.

It’s easy to prove that given that the common scoring rule is strictly proper, the probability assignment that wins out in this procedure is precisely the average p̄ = (p₁+...+p_n)/n of the individuals’ probability assignments. So it is natural to think of “the committee’s credence” as the average of the members’ credences, if the above notional procedure is natural, which it seems to be.

But is the above notional voting procedure the right one? I don’t really know. But here are some thoughts.

First, there is a limitation in the above setup: we assumed that each committee member had the same strictly proper scoring rule. But in practice, people don’t. People differ with regard to how important they regard getting different propositions right. I think there is a way of arguing that this doesn’t matter, however. There is a natural “committee scoring rule”: it is just the sum of the individual scoring rules. And then we ask each member i when acting as a committee member to use the committee scoring rule in their voting. Thus, each member calculates the expected committee score of q, still by their own epistemic lights, and these are added, and we maximize, and once again the average will be optimal. (This uses the fact that a sum of strictly proper scoring rules is strictly proper.)

Second, there is another way to arrive at the credence-averaging procedure. Presumably most of the reason why we care about a committee’s credence assignments is practical rather than purely theoretical. In cases where consequentialism works, we can model this by supposing a joint committee utility assignment (which might be the sum of individual utility assignments, or might be consensus utility assignment), and we can imagine the committee to be choosing between wagers so as to maximize the agreed-on committee utility function. It seems natural to imagine doing this as follows. The committee expectations or previsions for different wagers are obtained by summing individual expectations—with the individuals using the agreed-on committee utility function, albeit with their own individual credences to calculate the expectations. And then the committee chooses a wager that maximizes its prevision.

But now it’s easy to see that the above procedure yields exactly the same result as the committee maximizing committee utility calculated with respect to the average of the individuals’ credence assignments.

So there is a rather nice coherence between the committee credences generated by our epistemic “accuracy-first”
procedure and what one gets in a pragmatic approach.

But still all this depends on the plausible, but unjustified, assumption that addition is the right way to go, whether for epistemic or pragmatic utility expectations. But given this assumption, it really does seem like the committee’s credences are reasonably taken to be the average of the members’ credences.

Open-mindedness and propriety

Suppose we have a probability space Ω with algebra F of events, and a distinguished subalgebra H of events on Ω. My interest here is in accuracy H-scoring rules, which take a (finitely-additive) probability assignment p on H and assigns to it an H-measurable score function s(p) on Ω, with values in [−∞,M] for some finite M, subject to the constraint that s(p) is H-measurable. I will take the score of a probability assignment to represent the epistemic utility or accuracy of p.

For a probability p on F, I will take the score of p to be the score of the restriction of p to H. (Note that any finitely-additive probability on H extends to a finitely-additive probability on F by Hahn-Banach theorem, assuming Choice.)

The scoring rule s is proper provided that E_ps(q) ≤ E_ps(p) for all p and q, and strictly so if the inequality is strict whenever p ≠ q. Propriety says that one never expects a different probability from one’s own to have a better score (if one did, wouldn’t one have switched to it?).

Say that the scoring rule s is open-minded provided that for any probability p on F and any finite partition V of Ω into events in F with non-zero p-probability, the p-expected score of finding out where in V we are and conditionalizing on that is at least as big as the current p-expected score. If the scoring rule is open-minded, then a Bayesian conditionalizer is never precluded from accepting free information. Say that the scoring rule s is strictly open-minded provided that the p-expected score increases of finding out where in V we are and conditionalizing increases whenever there is at least one event E in V such that p(⋅|E) differs from p on H and p(E) > 0.

Given a scoring rule s, let the expected score function G_s on the probabilities on H be defined by G_s(p) = E_ps(p), with the same extension to probabilities on F as scores had.

It is well-known that:

1. The (strict) propriety of s entails the (strict) convexity of G_s.

It is easy to see that:

2. The (strict) convexity of G_s implies the (strict) open-mindedness of s.

Neither implication can be reversed. To see this, consider the single-proposition case, where Ω has two points, say 0 and 1, and H and F are the powerset of Ω, and we are interested in the proposition that one of these point, say 1, is the actual truth. The scoring rule s is then equivalent to a pair of functions T and F on [0,1] where T(x) = s(p_x)(1) and F(x) = s(p_x)(0) where p_x is the probability that assigns x to the point 1. Then G_s corresponds to the function xT(x) + (1−x)F(x), and each is convex if and only if the other is.

To see that the non-strict version of (1) cannot be reversed, suppose (T,F) is a non-trivial proper scoring rule with the limit of F(x)/x as x goes to 0 finite. Now form a new scoring rule by letting T * (x) = T(x) + (1−x)F(x)/x. Consider the scoring rule (T*,0). The corresponding function xT * (x) is going to be convex, but (T*,0) isn’t going to be proper unless T* is constant, which isn’t going to be true in general. The strict version is similar.

To see that (2) cannot be reversed, note that the only non-trivial partition is {{0}, {1}}. If our current probability for 1 is x, the expected score upon learning where we are is xT(1) + (1−x)F(0). Strict open-mindedness thus requires precisely that xT(x) + (1−x)F(x) < xT(1) + (1−x)F(0) whenever x is neither 0 nor 1. It is clear that this is not enough for convexity—we can have wild oscillations of T and F on (0,1) as long as T(1) and F(1) are large enough.

Nonetheless, (2) can be reversed (both in the strict and non-strict versions) on the following technical assumption:

3. There is an event Z in F such that Z ∩ A is a non-empty proper subset of A for every non-empty member of H.

This technical assumption basically says that there is a non-trivial event that is logically independent of everything in H. In real life, the technical assumption is always satisfied, because there will always be something independent of the algebra H of events we are evaluating probability assignments to (e.g., in many cases Z can be the event that the next coin toss by the investigator’s niece will be heads). I will prove that (2) can be reversed in the Appendix.

It is easy to see that adding (3) to our assumptions doesn’t help reverse (1).

Since open-mindedness is pretty plausible to people of a Bayesian persuasion, this means that convexity of G_s can be motivated independently of propriety. Perhaps instead of focusing on propriety of s as much as the literature has done, we should focus on the convexity of G_s?

Let’s think about this suggestion. One of the most important uses of scoring rules could be to evaluate the expected value of an experiment prior to doing the experiment, and hence decide which experiment we should do. If we think of an experiment as a finite partition V of the probability space with each cell having non-zero probability by one’s current lights p, then the expected value of the experiment is:

4. ∑_A ∈ Vp(A)E_pAs(p_A) = ∑_A ∈ Vp(A)G_s(p_A),

where p_A is the result of conditionalizing p on A. In other words, to evaluate the expected values of experiments, all we care about is G_s, not s itself, and so the convexity of G_s is a very natural condition: we are never oligated to refuse to know the results of free experiments.

However, at least in the case where Ω is finite, it is known that any (strictly) convex function (maybe subject to some growth conditions?) is equal to G_u for a some (strictly) proper scoring rule u. So we don’t really gain much generality by moving from propriety of s to convexity of G_s. Indeed, the above observations show that for finite Ω, a (strictly) open-minded way of evaluating the expected epistemic values of experiments in a setting rich enough to satisfy (3) is always generatable by a (strictly) proper scoring rule.

In other words, if we have a scoring rule that is open-minded but not proper, we can find a proper scoring rule that generates the same prospective evaluations of the value of experiments (assuming no special growth conditions are needed).

Appendix: We now prove the converse of (2) assuming (3).

Assume open-mindedness. Let p₁ and p₂ be two distinct probabilities on H and let t ∈ (0,1). We must show that if p = tp₁ + (1−t)p₂, then

5. G_s(p) ≤ tG_s(p₁) + (1−t)G_s(p₂)

with the inequality strict if the open-mindedness is strict. Let Z be as in (3). Define

6. p′(A∩Z) = tp₁(A)

7. p′(A∩Z^c) = (1−t)p₂(A)

8. p′(A) = p(A)

for any A ∈ H. Then p′ is a probability on the algebra generated by H and Z extending p. Extend it to a probability on F by Hahn-Banach. By open-mindedness:

9. G_s(p′) ≤ p′(Z)E_p′Zs(p′_Z) + p′(Z^c)E_p′Z^cs(p′_Z^c).

But p′(Z) = p(Ω∩Z) = t and p′(Z^c) = 1 − t. Moreover, p′_Z = p₁ on H and p′_Z^c = p₂ on H. Since H-scores don’t care what the probabilities are doing outside of H, we have s(p′_Z) = s(p₁) and s(p′_Z^c) = s(p₂) and G_s(p′) = G_s(p). Moreover our scores are H-measurable, so E_p′Zs(p₁) = E_p1s(p₁) and E_p′Z^cs(p₂) = E_p2s(p₂). Thus (9) becomes:

10. G_s(p) ≤ tG_s(p₁) + (1−t)G_s(p₂).

Hence we have convexity. And given strict open-mindedness, the inequality will be strict, and we get strict convexity.

Probability, belief and open theism

Here are few plausible theses:

1. A rational being believes anything that they take to have probability bigger than 1 − (1/10¹⁰⁰) given their evidence.

2. Necessarily, God is rational.

3. Necessarily, none of God’s beliefs ever turn out false.

These three theses, together with some auxiliary assumptions, yield a serious problem for open theism.

Consider worlds created by God that contain four hundred people, each of whom has an independent 1/2 chance of freely choosing to eat an orange tomorrow (they love their oranges). Let p be the proposition that at least one of these 400 people will freely choose to eat an orange tomorrow. The chance of not-p in any such world will be (1/2)⁴⁰⁰ < 1/10¹⁰⁰. Assuming open theism, so God doesn’t just directly know whether p is true or not, God will take the probability of p in any such world to be bigger than 1 − (1/10¹⁰⁰) and by (1) God will believe p in these worlds. But in some of these worlds, that belief will turn out to be false—no one will freely eat the orange. And this violates (3).

I suppose the best way out is for the open theist to deny (1).

Discontinuous epistemic utilities

I used to take it for granted that it’s reasonable to make epistemic utilities be continuous functions of credences. But this is not so clear to me right now. Consider a proposition really central to a person’s worldview, such as:

• life has (or does not have) a meaning

• God does (or does not) exist

• we live (or do not live) in a simulation

• morality is (or is not) objective.

I think a case can be made that if a proposition like that is in fact true, then there is a discontinuous upward jump in epistemic utility as one goes from assigning a credence less than 1/2 to assigning a credence more than 1/2.

Misleadingness simpliciter

It is quite routine that learning a truth leads to rationally believing new falsehoods. For we all rationally believe many falsehoods. Suppose I rationally believe a falsehood p and I don’t believe a truth q. Then, presumably, I don’t believe the conjunction of p and q. But suppose I learn q. Then, typically, I will rationally come to believe the conjunction of p and q, a falsehood I did not previously believe.

Thus there is a trivial sense in which every truth I learn is misleading. But a definition of misleadingness on which every truth is misleading doesn’t seem right. Or at least it’s not right to say that every truth is misleading simpliciter. What could misleadingness simpliciter be?

In a pair of papers (see references here) Lewis and Fallis argue that we should assign epistemic utilities to our credences in such a way that conditioning on the truth should never be bad for us epistemically speaking—that it should not decrease our actual epistemic utility.

I think this is an implausible constraint. Suppose a highly beneficial medication has been taken by a billion people. I randomly sample a hundred thousand of these people and see what happened to them in the week after receiving the medication. Now, out of a billion people, we can expect about two hundred thousand to die in any given week. Suppose that my random sampling is really, really unlucky, and I find that fifty thousand of the people in my sample died a week because of the medication. Completely coincidentally, of course, since as I said the medication is highly beneficial.

Based on my data, I rationally come to believe the importantly false claim that the medication is very harmful. I also come to believe the true claim that half of my random sample died a week after taking the medication. But while that claim is true, it is quite unimportant except as misleading evidence for the harmfulness of the medication. It is intuitively very plausible that after learning the truth about half of the people in my sample dying, I am worse off epistemically.

It seems clear that in the medication case, my data is true and misleading in a non-trivial way. This suggests a definition of misleadingness simpliciter:

• A proposition p is misleading simpliciter if and only if one’s overall epistemic utility goes down when one updates on p.

And this account of misleadingness is non-trivial. If we measure epistemic utility using strictly proper scoring rules, and if our credences are consistent, then the expected epistemic value of updating on the outcome of a non-trivial observation is positive. So we should not expect the typical truth to be misleading in the above sense. But some are misleading.

From this point of view, Lewis and Fallis are making a serious mistake: they are trying to measure epistemic utilities in such a way as to rule out the possibility of misleading truths.

By the way, I think I can prove that for any measure of epistemic utility obtained by summing a single strictly proper score across all events, there will be a possibility of misleadingness simpliciter.

Final note: We don’t need to buy into the formal mechanism of epistemic utilities to go with the above definition. We could just say that something is misleading iff coming to believe it would rationally make one worse off epistemically.

Valuing and behavioral tendencies

It is tempting to say that I value a wager W at x provided that I would be willing to pay any amount up to x for W and unwilling to pay an amount larger than x. But that’s not quite right. For often the fact that a wager is being offered to me would itself be relevant information that would affect how I value the wager.

Let’s say that you tossed a fair coin. Then I value a wager that pays ten dollars on heads at five dollars. But if you were to try to sell me that wager for a dollar, I wouldn’t buy it, because your offering it to me at that price would be strong evidence that you saw the coin landing tails.

Thus, if we want to define how much I value a wager at in terms of what I would be willing to pay for it, we have to talk about what I would be willing to pay for it were the fact that the wager is being offered statistically independent of the events in the wager.

But sometimes this conditional does not help. Imagine a wager W that pays $123.45 if p is true, where p is the proposition that at some point in my life I get offered a wager that pays $123.45 on some eventuality. My probability of p is quite low: it is unlikely anybody will offer me such a wager. Consequently, it is right to say that I value the wager at some small amount, maybe a few dollars.

Now consider the question of what I would be willing to pay for W were the fact that the wager is being offered statistically independent of the events in the wager, i.e., independent of p. Since my being offered W entails p, the only way we can have the statistical independence is if my being offered W has credence zero or p has credence one. It is reasonable to say that the closest possible world where one of these two scenarios holds is a world where p has credence one because some wager involving a $123.45 has already been offered to me. In that world, however, I am willing to pay up to $123.45 for W. Yet that is not what I value W at.

Maybe when we ask what we would be willing to pay for a wager, we mean: what we would be willing to pay provided that our credences stayed unchanged despite the offer. But a scenario where our credences stay unchanged despite the offer is a very weird one. Obviously, when an offer is made, your credence that the offer is made goes up, unless you’re seriously irrational. So this new counterfactual question asks us what we would decide in worlds where we are seriously irrational. And that’s not relevant to the question of how we value the wager.

Maybe instead of asking about the prices at which I would accept an offer, I should instead ask about the prices at which I would make an offer. But that doesn't help either. Go back to the fair coin case. I value a wager that pays you ten dollars on heads at negative five dollars. But I might not offer it to you for eight dollars, because it is likely that you would pay eight dollars for this wager only if you actually saw that the coin turned out heads, in which case this would be a losing proposition for me.

The upshot is, I think, that the question of what one values a wager at is not to be defined in terms of simple behavioral tendencies or even simple counterfactualized behavioral tendencies. Perhaps we can do better with a holistic best-fit analysis.

A defense of probabilistic inconsistency

Evidence E is misleading with regard to a hypothesis H provided that Bayesian update on E changes one’s credence in H in the direction opposed to truth. It is known that pretty much any evidence is misleading with regard to some hypothesis or other. That’s no tragedy. But sometimes evidence is misleading with regard to an important hypothesis. That’s no tragedy of the shift in the credence of that important hypothesis is small. But it could be tragic if the shift is significant—think of a quack cure for cancer beating out the best medication in a study due to experimental error or simply chance.

In other words, misleadingness by itself is not a big deal. But significant misleadingness with respect to an important hypothesis can be tragic.

Suppose I am lucky enough to start with consistent credences in a limited algebra F of propositions including q, and suppose I have a low credence in a consistent proposition q. Now two friends, whom I know for sure to speak only truth, speak to me:

• Alice: “Proposition q is actually true.”

• Bob: “She’s right, as always, but the fact that q is true is significantly misleading with respect to a number of quite important hypotheses in F.”

What should I do? If I were a perfect Bayesian agent, my likelihoods would be sufficiently well defined that I would just update on Alice saying her piece and Bob saying her piece, and be done with it. My likelihoods would embody prior probability assignments to hypotheses about the kinds of reasons that Alice and Bob could have for giving me their information, the kinds of important hypotheses in F that q could be misleading about, etc.

But this is too complicated for a more ordinary Bayesian agent like me. Suppose I could, just barely, do a Bayesian update on q, and gain a new consistent credence assignment on F. Even if Bob were not to have said anything, updating on q would not be ideal, because the ideal agent would update not just on q, but on the facts that Alice chose to inform me of q at that very moment, in those very words, in that very tone of voice, etc. But that’s too complicated for me. For one, I don’t have enough clear credences in hypotheses about different informational choices Alice could have made. So if all I heard was Alice’s announcement, updating on q would be a reasonable choice given my limitations.

But with Bob speaking, the consequences of my simply updating on q could be tragic, because Bob has told me that q is significantly misleading in regard to important stuff. What should I do? One possibility is to ignore both statements, and leave my credences unchanging, pretending I didn’t hear Alice. But that’s silly: I did hear her.

But if I accept q on the basis of Alice’s statement (and Bob’s confirmation), what should I do about Bob’s warning? Here is one option: I could raise my credence in q to 1, but leave everything else unchanged. This is a better move than just ignoring what I heard. For it gets me closer to the truth with regard to q (remember that Alice only says the truth), and I don’t get any further from the truth regarding anything else. The result will be an inconsistent probability assignment. But I can actually do a little better. Assuming q is true, it cannot be misleading about propositions entailed by q. For if q is true, then all propositions entailed by q are true, and raising my credences in them to 1 only improves my credences. Thus, I can safely raise my credence in everything entailed by q to 1. Similarly, I can safely lower my credence in anything that entails ∼q to 0.

Here, then, is a compromise: I set my credence in everything in F entailed by q to 1, and in everything in F that entails ∼q to 0, and leave all other credences for things in F unchanged. This has gotten me closer to the truth by any reasonable measure. Moreover, the resulting credences for F satisfy the Zero, Non-negativity, Normalization, Monotonicity, and Binary Non-Disappearance axioms, and as a result I can use a Level-Set Integral prevision to avoid various Dutch Book and domination problems. [Let’s check Monotonicity. Suppose r entails s. We need to show that C(r)≤C(s). Given that my original credences were consistent and hence had Monotonicity, the only way I could lack Monotonicity now would be if q entailed r and s entailed ∼q. Since r entails s, this would mean that q would entail ∼q, which would imply that q is not consistent. But I assumed it was consistent.]

I think this line of reasoning shows that there are indeed times when it can be reasonable to have an inconsistent credence assignment.

By the way, if I continue to trust the propositions I had previously assigned extreme credences to despite Bob’s ominous words, an even better update strategy would be to set my credence to 1 for everything entailed by q conjoined with something that already had credence 1, and to 0 for everything that when conjoined with something that had credence 1 entails ∼q.

Best estimates and credences

Some people think that expected utilities determine credences and some thing that credences determine expected utilities. I think neither is the case, and want to sketch a bit of a third view.

Let’s say that I observe people playing a slot machine. After each game, I make a tickmark on a piece of paper, and if they win, I add the amount of the win to a subtotal on a calculator. After a couple of hours—oddly not having been tossed out by the casino—I divide the subtotal by the number of tickmarks and get the average payout. If I now get an offer to play the slot machine for a certain price, I will use the average payout as an expected utility and see if that expected utility exceeds the price (in a normal casino, it won’t). So, I have an expected utility or prevision. But I don’t have enough credences to determine that expected utility: for every possible payout, I would need a credence in getting that payout, but I simply haven’t kept track of any data other than the sum total of payouts and the number of games. So, here the expected utility is not determined by the credences.

The opposite is also not true: expected utilities do not determine credences.

Now consider another phenomenon. Suppose I step on an analog scale, and it returns a number w₁ for my weight. If that’s all the data I have, then w₁ is my best estimate for the weight. What does that mean? It certainly does not mean that I believe that my weight is exactly w₁. It also does not mean that I believe that my weight is close to w₁—for although I do believe that my weight is close to w₁, I also believe it is close to w₁ + 0.1 lb. If I were an ideal epistemic agent, then for every one of the infinitely many possible intervals of weight, I would have a credence that my weight lies in that interval, and my best estimate would be an integral of the weight function over the probability space with respect to my credence measure. But I am not an ideal epistemic agent. I don’t actually have much of a credence for the hypothesis that my weight lies between w₁ − 0.2 lb and w₁ + 0.1 lb, say. But I do have a best estimate.

This is very much what happened in the slot machine case. So expected values are not the only probabilistic entity not determined by our credences. Rather, they are a special case of best estimates. The expected utility of the slot machine game is simply my best estimate at the actual utility of the slot machine game.

We form and use lots of such best estimates.

Note that the best estimate need not even be a possible value for the thing we are estimating. My best estimate payoff for the slot-machine given my data might be $0.94, even though I might know that in fact all actual payouts are multiples of a dollar.

With this in mind, we can take credences to be nothing else than best estimates at the truth value, where we think of truth value as either 0 (false) or 1 (true). (Here, I think of the fact that the standard Polish word for probability is “prawdopodobieństwo”—truthlikeness, verisimilitude.) Just as in the case above, when my best estimate for the truth is 0.75, I do not think the actual truth value is 0.75: I like classical logic, and think the only two possible values are 0 and 1.

Here, then, is a picture of what one might call our probabilistic representation of the world. We have lots of best estimates. Some of these are best estimates of utilities. Some are best estimates of other quantities, such as weights, lengths, cardinalities, etc. Some are best estimates of truth values. A consistent agent is one such that there exists a probability function such that all of the agent’s best estimates are mathematical expetations of the corresponding values with respect to the probability function. In particular, this probability function would extend the agent’s credences, i.e., the agent’s best estimates for truth values.

On this picture, there is no privileging between expected utilities, credences or other best estimates. It’s just estimates all around.

Previsions for inconsistent credences and arguments for probabilism

Fix a sample space Ω and an algebra F events on Ω. A gamble is an F-measurable real-valued function on Ω. A credence function is a function from a F to the reals. A prevision or price function on a set of set G of gambles is just a function from G to the real numbers. A previsory method E on a set of gambles G and a set of credence functions C assigns to each credence function P ∈ C a prevision E_P on G.

A previsory method on G and C has the weak domination property provided that if f and g are two gambles such as that f ≤ g everywhere on Ω, then E_P(f)≤E_P(g) for every f and g in G and P in C. It has the strong domination property provided that it has the weak domination property and if f < g everywhere on Ω, then E_P(f)<E_P(g). It has the zero property provided that E_P(0)=0.

Mathematical expectation is a previsory method on the set of all bounded gambles and all probability functions. It has the zero and strong domination properties.

The level set integral is a previsory method on the set of all bounded gambles and all monotonic credence functions (P is monotonic iff P(⌀)=0, P(Ω)=1 and P(A)≤P(B) whenever A ⊆ B). It has the zero and weak domination properties.

The level set integral has the strong domination property on the set of weakly countably additive monotonic credence functions, where P is weakly countably additive provided that Ω cannot be written as a countable union of sets each of credence 0. If F (or Ω) is finite, we get weak countable additivity for free from monotonicity.

A previsory method E requires (permits) a gamble f given a credence P provided that E_P(f)>0 (E_P(f)≥0); it requires (permits) it over some set S of gambles provided that E_P(f)>E_P(g) (E_P(f)≥E_p(g)) for every g in S.

A previsory method with the zero and weak domination properties cannot be strongly Dutch-Booked in a single wager: i.e., there is no gamble U such that U < 0 everywhere that the method requires. If it also has the strong domination property, it cannot be weakly Dutch-Booked in a single wager: there is no U such that U < 0 everywhere that the method permits.

Suppose we combine a previsory method with the following method of choosing which gambles to adopt in a sequence of offered gambles: you are required (permitted) to accept gamble g provided that E_P(g₁ + ... + g_n + g)>E_P(g₁ + ... + g_n) (≥, respectively) where g₁ + ... + g_n are the gambles already accepted. Then given the zero and weak domination properties, we cannot be strongly Dutch-Booked by a sequence of wagers, and given additionally the strong domination property, we cannot be weakly Dutch-Booked, either.

Given that level set integrals provide a non-trivial and mathematically natural previsory method with the zero and strong domination properties on a set of credence functions strictly larger than the consistent ones, Dutch-Book arguments for consistency fail.

What about epistemic utility, i.e., scoring-rule, arguments? I think these also fail. A scoring-rule assigns a number s(p, q) to a credence function p and a truth function q (i.e., a probability function whose values are always 0 or 1). Let T be truth, i.e., a function from Ω to truth functions such that T(ω)(A) if and only if ω ∈ A. Thus, T(ω) is the truth function that says “we are at ω” and we can think of s(p, T) as a gamble that measures how far p is from truth.

If E is previsory method on a set of gambles G and a set of credence functions C, then we say that s is an E-proper scoring rule provided that s(p, T) is in G for every p in C and E_ps(p, T)≤E_ps(q, T) for every p and q in C. We say that it is strictly proper if additionally we have strict inequality whenever p and q are different.

If E is mathematical expectation, then E-propriety and strict E-propriety are just propriety and strict propriety.

It is thought (Joyce and others) that one can make use of the concept of strictly propriety to argue for that credence functions should be consistent. This uses a domination theorem that says that if s is a strictly proper additive scoring rule, then for any inconsistent credence function p there is a consistent function q such that s(p, T(ω)) < s(q, T(ω)) for all ω. (Roughly, an additive scoring rule adds up scores point-by-point over Ω.)

However, I think the requirement of additivity is one that someone sceptical of the consistency requirement can reasonably reject. There are mathematical natural previsory methods E that apply to some inconsistent credences, such as the monotonic ones, and these can be used to define (at least under some conditions) strictly E-proper scoring rules. And the domination theory won’t apply to these rules because they won’t be additive. Indeed, that is one of the things the domination theorem shows: if C includes an inconsistent credence function and E has the strong domination property, then no strictly E-proper scoring rule is additive.

So, really, how helpful the domination theorem is for arguing for consistency depends on whether additivity is a reasonable condition to require of a scoring rule. It seems that someone who thinks that it is OK to reason with a broader set of credences than the consistent ones, and who has a natural previsory method E with the strong domination property for these credences, will just say: I think the relevant notion isn’t propriety but E-propriety, and there are no strongly E-proper scoring rules that are additive. So, additiveness is not a reasonable condition.

Are there any strongly E-proper scoring rules in such cases?

[The rest of the post is based on the mistake that E-propriety is additive and should be dismissed. See my discussion with Ian in the comments.]

Sometimes, yes.

Suppose E is previsory method with the weak domination condition on the set of all bounded gambles on Ω. Suppose that E has the scaling property that E_p(cf)=cE_p(f) for any real constant c. (Level Set Integrals have scaling.) Further, assume the separability property that there is a countable set of B of bounded gambles such that for any two distinct credences p and q, there is a bounded gamble f in B such that E_pf ≠ E_qf. (Level Set Integrals on a finite Ω—or on a finite field of events—have separability: just let B be all functions whose values are either 0 or 1, and note that E_p1_A = p(A) where 1_A is the function that is 1 on A and 0 outside it.) Finally, suppose normalization, namely that E_p1_Ω = 1. (Level Set Integrals clearly have that.)

Note that given separability, scaling and normalization, there is a countable set H of bounded gambles such that if p and q are distinct, there exist f and g in H such that E_p requires f over g (i.e., E_pf > E_pg) and E_q does not or vice versa. To see this, let H consist of B together with all constant rational-valued functions, and note that if E_pf < E_qf, then we can choose a rational number r such that r lies between E_pf and E_qf, and then E_p and E_q will disagree on whether f is required over r ⋅ 1_Ω.

Let H be the countable set in the above remark. By scaling, we may assume that all the gambles in H are bounded by 1. Let (f₁, g₁),(f₂, g₂),... be an enumeration of all pairs of members of H. Define s_n(p, T(ω)) for a credence function p in C as follows: if E_p requires f_n over g_n then s_n(p, T(ω)) = −f_n(ω), and otherwise s_n(p, T(ω)) = −g_n(ω).

Note that s_n is an E-proper scoring rule. For suppose that q is a different credence function from p and E_ps_n(p, T)>E_ps_n(q, T). Now there are four possibilities depending on whether E_p and E_q require f_n over g_n and it is easy to see that each possibility leads to a contradiction. So, we have E-propriety.

Now, let s(p, T) be Σ_n = 1^∞ 2⁻ⁿs_n(p, T). The sum of E-proper scoring rules is E-proper, so this is an E-proper scoring rule.

What about strict propriety? Suppose that p and q are credence functions in C and E_ps(p, T)≤E_ps(q, T). By the E-propriety of each of the s_n, we must have E_ps_n(p, T)=E_ps_n(q, T) for all n. Thus, for all pairs of members of H, the requirements of E_p and E_q must agree, and by choice of H, p and q cannot be different.

Probability and normativity

The Born rule is a central part of quantum mechanics that tells us that the probability of a particle detector detecting a particle in a region U is equal to ∫_U|ψ(x)|²dx, where ψ is normalized.

What exactly "probability" in the Born rule means depends on the particular interpretation of quantum mechanics. On some interpretations (e.g., on some interpretations of collapse interpretations) it will be a physical propensity and on others (e.g., Bohm and some versions of Everett) it will be something like a frequency. But any adequate interpretation of quantum mechanics needs to generate predictions, and hence needs to tell us what rational credences there are: what credences agents should assign to outcomes.

This means that a criterion of adequacy on an interpretation of quantum mechanics is that the Born rule must be understood in such a way that if a rational agent believes the Born rule, she should assign credences in accordance with it. There needs, thus, to be a bridge between the "probability" of the Born rule and the credences an agent should have.

The simplest version of the bridge is identity: take "probability" to mean an appropriate conditional rational credence. If that's done, then quantum mechanics is directly a normative theory: it tells us what we should believe.

On other interpretations, however, serious epistemology is needed to move from the probability in the Born rule to the credences an agent should have. For instance, we may need a version of the Principal Principle. This serious epistemology is normative: it is about the credences an agent should have.

Thus, either quantum physics includes normative claims or it needs further normative claims to generate predictions. (And there is nothing special here about quantum physics.)