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}(*g*_{1} + ... + *g*_{n} + *g*)>*E*_{P}(*g*_{1} + ... + *g*_{n}) (≥, respectively) where *g*_{1} + ... + *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*_{p}*s*(*p*, *T*)≤*E*_{p}*s*(*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}(*c**f*)=*c**E*_{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*_{p}*f* ≠ *E*_{q}*f*. (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*_{p}1_{A} = *p*(*A*) where 1_{A} is the function that is 1 on *A* and 0 outside it.) Finally, suppose normalization, namely that *E*_{p}1_{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*_{p}*f* > *E*_{p}*g*) 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*_{p}*f* < *E*_{q}*f*, then we can choose a rational number *r* such that *r* lies between *E*_{p}*f* and *E*_{q}*f*, 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 (*f*_{1}, *g*_{1}),(*f*_{2}, *g*_{2}),... 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*_{p}*s*_{n}(*p*, *T*)>*E*_{p}*s*_{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^{−n}*s*_{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*_{p}*s*(*p*, *T*)≤*E*_{p}*s*(*q*, *T*). By the *E*-propriety of each of the *s*_{n}, we must have *E*_{p}*s*_{n}(*p*, *T*)=*E*_{p}*s*_{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.

## 6 comments:

On the pragmatic reasoning: note that the level set method with inconsistent credences violates something like vNM independence or Savage sure thing. If you consider a choice between two gambles and a choice between the same two gambles with all the payoffs increased by $X, the preferences given by level set method may differ. The $X could be seen as an irrelevant sure thing. Of course, some people reject independence/sure thing.

If I’m thinking straight, the p-expectations of the s functions may be discontinuous in q, making the sum score discontinuous in q. [For an example, think about a coin flip with outcomes H and T. Take the s function derived from the gambles (1 on H, 0 on T) and (0 on H, 1 on T).] Does this worry you?

A detail. In the 5th last para, do you mean something like this: … there is a countable set H of bounded gambles such that if credence functions p and q are distinct, there exist gambles f and g in H such that … ?

Well, we can't expect to have everything sounding reasonable if we have inconsistency.

I think the change in decisions due to a constant increase isn't that surprising when one takes into account risk aversion.

Thanks for catching the lengthy typo.

The level set integral is not in general additive. Is it obvious for the level set previsory method that a sum of proper scoring rules (as used in your construction) is proper?

Ooops! You're quite right: it's quite unlikely to be true that a sum of E-proper scoring rules is proper if E isn't additive! So, everything after "Are there any strongly E-proper scoring rules in such cases?" in my post should be replaced with: "I don't know."

The prospects look poor. No

symmetricscoring rule (as defined below) will work.Call a scoring rule for coin flips

symmetricif, for all p and q, the score on Heads for credence p in Heads and q in Tails equals the score on Tails for credence p in Tails and q in Heads. This is an intuitively reasonable requirement – there can be no purely epistemic reason to treat Heads and Tails differently.Here’s the reasoning. Think about credence functions with equal credences in Heads and Tails. For such a credence function, with a symmetric scoring rule, the score on Heads must equal the score on Tails. So the level set expectation evaluated

using any credence function whateverwill equal the common value. (That’s how the level set method works.) This could be compatible with propriety (if all “equal credence” credence functions have the same scores), but not with strict propriety.The result and the argument extend easily to setups with more outcomes.

Ian:

Brilliant!

More generally, there is no symmetric strictly proper score for any prevision over the monotone credences that has the property that E_p(c) = c for every constant c.

This suggests an interesting philosophical argument that I am not sure of.

First of all, everybody's prevision should satisfy the condition that the expectation of a certain result be that result. The existence of a strictly proper score for a prevision indicates that by one's lights there is always a best credence (namely, one's own!). If there is no strictly proper score, then there is no best credence function. In other words, the defender of inconsistency (at least assuming that they want to allow all the monotone credences) should be a permissivist: in some cases at least, they should think there is another credence just as good as their own.

It would take some work to extend this argument beyond the case of simple coin-flip situations. But it might be quite interesting.

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