Wednesday, 18 May 2022

Should we agree? III: the rationality of groups

In the previous two posts in this series (here and here), I described two arguments for the conclusion that the members of a group should agree. One was an epistemic argument and one a pragmatic argument. Suppose you have a group of individuals. Given an individual, we call the set of propositions to which they assign a credence their agenda. The group's agenda is the union of its member's agendas; that is, it includes any proposition to which some member of the group assigns a credence. The precise conclusion of the two arguments we describe is this: the group is irrational if there no single probability function defined on the group's agenda that gives the credences of each member of the group when restricted to their agenda. Following Matt Kopec, I called this norm Consensus. 

Cats showing a frankly concerning degree of consensus
 

Both arguments use the same piece of mathematics, but they interpret it differently. Both appeal to mathematical functions that measure how well our credences achieve the goals that we have when we set them. There are (at least) two such goals: we aim to have credences that will guide our actions well and we aim to have credences that will represent the world accurately. In the pragmatic argument, the mathematical function measures how well our credences achieve the first goal. In particular, they measure the utility we can expect to gain by having the credences we have and choosing in line with them when faced with whatever decision problems life throws at us. In the epistemic argument, the mathematical function measures how well our credences achieve the second goal. In particular, they measure the accuracy of our credences. As we noted in the second post on this, work by Mark Schervish and Ben Levinstein shows that the functions that measure these goals have the same properties: they are both strictly proper scoring rules. The arguments then appeal to the following fact: given a strictly proper scoring rule, if the members of a group do not agree on the credences they assign in the way required by Consensus, then there are some alternative credences they might assign instead that are guaranteed to be better according to that scoring rule.

I'd like to turn now to assessing these arguments. My first question is this: In the norm of Probabilism, rationality requires something of an individual, but in the norm of Consensus, rationality requires something of a group of individuals. We understand what it means to say that an individual is irrational, but what could it mean to say that a group is irrational?

Here, I follow Kenny Easwaran's suggestion that collective entities---in his case, cities; in my case, groups---can be said quite literally to be rational or irrational. For Easwaran, a city is rational "to the extent that the collective practices of its people enable diverse inhabitants to simultaneously live the kinds of lives they are each trying to live." As I interpret him, the idea is this: a city, no less than its individual inhabitants, has an end or goal or telos. For Easwaran, for instance, the end of a city is enabling its inhabitants to live as they wish to. And a city is irrational if it does not provide---in its physical and technological infrastructure, its byelaws and governing institutions---the best means to that end among those that are available. Now, we might disagree with Easwaran's account of a city's ends. But the template he provides by which we might understand group rationality is nonetheless helpful. Following his lead, we might say that a group, no less than its individual members, has an end. For instance, its end might be maximising the total utility of its members, or it might be maximizing the total epistemic value of their credences. And it is then irrational if it does not provide the best means to that end among those available. So, for instance, as long as agreement between members is available, our pragmatic and epistemic arguments for Consensus seem to show that a group whose ends are as I just described does not provide the best means to its ends if it does not deliver such agreement.

Understanding group rationality as Easwaran does helps considerably. As well as making sense of the claim that the group itself can be assessed for rationality, it also helps us circumscribe the scope of the two arguments we've been exploring, and so the scope of the version of Consensus that they justify. After all, it's clear on this conception that these arguments will only justify Consensus for a group if

  1. that group has the end of maximising total expected pragmatic utility or total epistemic utility, i.e., maximising the quantities measured by the mathematical functions described above;
  2. there are means available to it to achieve Consensus.

So, for instance, a group of sworn enemies hellbent of thwarting each other's plans is unlikely to have as its end maximising total utility, while a group composed of randomly selected individuals from across the globe is unlikely to have as its end maximising total epistemic utility, and indeed a group so disparate might lack any ends at all.

And we can easily imagine situations in which there are no available means by which the group could achieve Consensus, perhaps because it would be impossible to set up reliable lines of communication.

This allows us to make sense of two of the conditions that Donald Gillies places on the groups to which he takes his sure loss argument to apply (this is the first version of the pragmatic argument for Consensus; the one I presented in the first post and then abandoned in favour of the second version in the second post). He says (i) the members of the group must have a shared purpose, and (ii) there must be good lines of communication between them. Let me take these in turn to understand their status more precisely.

It's natural to think that, if a group has a shared purpose, it will have as its end maximising the total utility of the members of the group. And indeed in some cases this is almost certainly true. Suppose, for instance, that every member of a group cares only about the amount of biodiversity in a particular ecosystem that is close to their hearts. Then they will have the same utility function, and it is natural to say that maximising that shared utility is the group's end. But of course maximising that shared utility is equivalent to maximising the group's total utility, since the total utility is simply the shared utility scaled up by the number of members of the group.

However, it is also possible for a group to have a shared purpose without its end being to maximise total utility. After all, a group can have a shared purpose without each member taking that purpose to be the one and only valuable end. Imagine a different group: each member cares primarily about the level of biodiversity in their preferred area, but each also cares deeply about the welfare of their family. In this case, you might take the group's end to be maximising biodiversity in the area in question, particularly if it was this shared interest that brought them together as a group in the first place, but maximising this good might require the group not to maximise total utility, perhaps because some members of the group have family who are farmers and who will be adversely affected by whatever is the best means to the end of greater biodiversity.

What's more, it's possible for a group to have as its end maximising total utility without having any shared purpose at all. For instance, a certain sort of utilitarian might say that the group of all sentient beings has as its end the maximisation of the total utility of its members. But that group does not have any shared purpose.

So I think we can use the pragmatic and epistemic arguments to determine the groups to which the norm of Consensus applies, or at least the groups for which our pragmatic and epistemic arguments can justify its application. It is those groups that have as their end either maximising the total pragmatic utility of the group, or maximising their total epistemic utility, or maximising some weighted average of the two---after all, the weighted average of two strictly proper scoring rules, one measuring epistemic utility and one measuring pragmatic utility, is itself a strictly proper scoring rule. Of course, this requires an account of when a group has a particular end. This, like all questions about when collectives have certain attitudes, is delicate. I won't say anything more about it here.

Let's turn next to Gillies' claim that Consensus applies only to groups between whose members there are reliable lines of communication. In fact, I think our versions of the arguments show that this condition lives a strange double life. On the one hand, if such lines of communcation are necessary to achieve agreement across the group, then the norm of Consensus simply does not apply to a group when these lines of communication are impossible, perhaps because of geographical, social, or technological barriers. A group cannot be judged irrational for failing to achieve something it could not possibly achieve, however much closer it would get to its goal if it could achieve that. 

On the other hand, if such lines of communication are available, and if they increase the chance of agreement among members of the group, then our two arguments for Consensus are equally arguments for establishing such lines of communication, providing that the cost of doing so is outweighed by the gain in pragmatic or epistemic utility that comes from achieving agreement.

But these arguments do something else as well. They lend nuance to Consensus. In some cases in which some lines of communication are available but others aren't, or are too costly, our arguments still provide norms. Take, for instance, a case in which some central planner is able to communicate a single set of prior credences that each member of the group should have, but after the members start receiving evidence, this central planner can no longer coordinate their credences. And suppose we know that the members will receive different evidence: they'll be situated in different places, and so they'll see different things, have access to different information sources, and so on. So we know that, if they update on the evidence they receive in the standard way, they'll end up having different credences from one another and therefore violating Consensus. You might think, from looking at Consensus, that the group would do better, both pragmatically and epistemically, if each of its members were to ignore whatever evidence were to come in and to stick with their prior regardless in order to be sure that they remain in agreement and satisfy Consensus both in their priors and their posteriors.

In fact, however, this isn't the case. Let's take an extremely simple example. The group has just two members, Ada and Baz. Each has opinions only about the outcomes of two independent tosses of a fair coin. So the possible worlds are HH, HT, TH, TT. Ada will learn the outcome of the first, and Baz will learn the outcome of the second. A central planner can communicate to them a prior they should adopt, but that central planner can't receive information from them, and so can't receive their evidence and pool it and communicate a shared posterior to them. How should Ada and Baz proceed? How should they pick their priors, and what strategies should each adopt for updating when the evidence comes in? The entity we're assessing for rationality is the quadruple that contains Ada's prior together with her plan for updating, and Baz's prior together with his plan for updating. Which of these are available? Well, nothing constrains Ada's priors and nothing constrain's Baz's. But there are constraints on their updating rules. Ada's updating rule must give the same recommendation at any two worlds at which her evidence is the same---so, for instance, it must give the same recommendation at HH as at HT, since all she learns at both is that the first coin landed heads. And Baz's updating rule must give the same recommendation at any two worlds at which his evidence is the same---so, for instance, it must give the same recommendation at HH as at TH. Then consider the following norm:

Prior Consensus Ada and Baz should have the same prior and both should plan to update on their private evidence by conditioning on it.

And the argument for this is that, if they don't, there's a quadruple of their priors and plans that (i) satisfy the constraint outlined above and (ii) together have greater total epistemic utility at each possible world; and there's a quadruple of their priors and plans that (i) satisfy the constraint outlined above and (ii) together have greater total expected pragmatic utility at each possible world. This is a corollary of an argument that Ray Briggs and I gave, and that Michael Nielsen corrected and improved on. So, if Ada and Baz are in agreement on their prior, and plan to stick with it rather than update on their evidence because that way they'll retain agreement, then they're be accuracy dominated and pragmatically dominated.

You might wonder how this is possible. After all, whatever evidence Ada and Baz each receive, Prior Consensus requires them to update on it in a way that leads them to disagree, and we know that they are then accuracy and pragmatically dominated. This is true, and it would tell against the priors + updating plans recommended by Prior Consensus if there were some way for Ada and Baz to communicate after their evidence came in. It's true that, for each possible world, there is some credence function such that if, at each world, Ada and Baz were to have that credence function rather than the ones they obtain by updating their shared prior on their private evidence, then they'd end up with greater total accuracy and pragmatic utility. But, without the lines of communication, they can't have that.

So, by looking in some detail at the arguments for Consensus, we come to understand better the groups to which it applies and the norms that apply to those groups to which it doesn't apply in its full force.

Friday, 6 May 2022

Should we agree? II: a new pragmatic argument for consensus

There is a PDF version of this blogpost available here.

In the previous post, I introduced the norm of Consensus. This is a claim about the rationality of groups. Suppose you've got a group of individuals. For each individual, call the set of propositions to which they assign a credence their agenda. They might all have quite different agendas, some of them might overlap, others might not. We might say that the credal states of these individual members cohere with one another if there is a some probability function that is defined for any proposition that appears in any member's agenda, and the credences each member assigns to the propositions in their agenda match those assigned by this probability function to those propositions. Then Consensus says that a group is irrational if it does not cohere.

A group coming to consensus

In that post, I noted that there are two sorts of argument for this norm: a pragmatic argument and an epistemic argument. The pragmatic argument is a sure loss argument. It is based on the fact that, if the individuals in the group don't agree, there is a series of bets that their credences require them to accept that will, when taken together, lose the group money for sure. In this post, I want to argue that there is a problem with the sure loss argument for Consensus. It isn't peculiar to this argument, and indeed applies equally to any argument that tries to establish a rational requirement by showing that someone who violates it is exploitable. Indeed, I've raised it elsewhere against the sure loss argument for Probabilism (Section 6.2, Pettigrew 2020) and the money pump argument against non-exponential discounting and changing preferences in general (Section 13.7.4, Pettigrew 2019). I'll describe the argument here, and then offer a solution based on work by Mark Schervish (1989) and Ben Levinstein (2017). I've described this sort of solution before (Section 6.3, Pettigrew 2020), and Jason Konek (ta) has recently put it to interesting work addressing an issue with Julia Staffel's (2020) account of degrees of incoherence.

Sure loss and money pump arguments judge the rationality of attitudes, whether credences or preferences, by looking at the quality of the choices they require us to make. As Bishop Butler said, probability is the very guide of life. These arguments evaluate credences by exactly how well they provide that guide. So they are teleological arguments: they attempt to derive facts about the epistemic right---namely, what is rationally permissible---from facts about the epistemic good---namely, leading to pragmatically good choices.

Say that one sequence of choices dominates another if, taken together, the first leads to better outcomes for sure. Say that a collection of attitudes is exploitable if there is a sequence of decision problems you might face such that, if faced with them, these attitudes will require you to make a dominated sequence of choices.

For instance, take the sure loss argument for Probabilism: if you violate Probabilism because you believe $A\ \&\ B$ more strongly than you believe $A$, your credence in the former will require you to pay some amount of money for a bet that pays out a pound if $A\ \&\ B$ true and nothing if it's false, and your credence in the latter will require you to sell for less money a bet that pays out a pound if $A$ is true and nothing if it's false; yet you'd be better off for sure rejecting both bets. So rejecting both bets dominates accepting both; your credences require you to accept both; so your credences are exploitable. Or take the money pump argument against cyclical preferences: if you prefer $A$ to $B$ and $B$ to $C$ and $C$ to $A$, then you'll choose $B$ when offered a choice between $B$ and $C$, you'll then pay some amount to swap to $A$, and you'll then pay some further amount to swap to $C$; yet you'd be better off for sure simply choosing $C$ in the first place and not swapping either time that possibility was offered. So choosing $C$ and sticking with it dominates the sequence of choices your preferences require; so your preferences are exploitable.

But, I contend, the existence of a sequence of decision problems in response to which your attitudes require you to make a dominated series of choices does not on its own render those attitudes irrational. After all, it is just one possible sequence of decision problems you might face. And there are many other sequences you might face instead. The argument does not consider how your attitudes will require you to choose when faced with those alternative sequences, and yet surely that is relevant to assessing those attitudes, for it might be that however bad is the dominated sequences of choices the attitudes require you to make when faced with the sequence of decision problems described in the argument for exploitability, there is another sequence of decision problems where those same attitudes require you to make a series of choices that are very good; indeed, they might be so good that they somehow outweigh the badness of the dominated sequence. So, instead of judging your attitudes by looking only at the outcome of choosing in line with them when faced with a single sequence of decision problems, we should rather judge them by looking at the outcome of choosing in line with them when faced with any decision problem that might come your way, weighting each by how likely you are to face it, to give a balanced view of the pragmatic benefits of having those credences. That's the approach I'll present now, and I'll show that it leads to a new and better pragmatic argument for Probabilism and Consensus.

As I presented them, the sure loss arguments for Probabilism and Consensus both begin with a principle that I called Ramsey's Thesis. This is a claim about the prices that an individual's credence in a proposition requires her to pay for a bet on that proposition. It says that, if $p$ is your credence in $A$ and $x < pS$, then you are required to pay $£x$ for a bet that pays out $£S$ if $A$ is true and $£0$ if $A$ is false. Now in fact this is a particular consequence of a more general norm about how our credences require us to choose. Let's call the more general norm Extended Ramsey's Thesis. It says how our credence in a proposition requires us to choose when faced with a series of options, all of whose payoffs depend only on the truth or falsity of that proposition. Given a proposition $A$, let's say that an option is an $A$-option if its payoffs at any two worlds at which $A$ is true are the same, and its payoffs at any two worlds at which $A$ is false are the same. Then, given a credence $p$ in $A$ and an $A$-option $a$, we say that the expected payoff of $a$ by the lights of $p$ is
$$
p \times \text{payoff of $a$ when $A$ is true} + (1-p) \times \text{payoff of $a$ when $A$ is false}
$$Now suppose you face a decision problem in which all of the available options are $A$-options. Then Extended Ramsey's Thesis says that you are required to pick an option whose expected payoff by the lights of your credence in $A$ is maximal.*

Next, we make a move that is reminiscent of the central move in I. J. Good's argument for Carnap's Principle of Total Evidence (Good 1967). We say what we take the payoff to be of having a particular credence in a particular proposition given a particular way the world is and when faced with a particular decision problem. Specifically, we define the payoff of having credence $p$ in the proposition $A$ when that proposition is true, and when you're faced with a decision problem $D$ in which all of the options are $A$-options, to be the payoff when $A$ is true of whichever $A$-option available in $D$ maximises expected payoff by the lights of $p$. And we define the payoff of having credence $p$ in the proposition $A$ when that proposition is false, and when you're faced with a decision problem $D$ in which all of the options are $A$-options, to be the payoff when $A$ is false of whichever $A$-option available in $D$ maximises expected payoff by the lights of $p$. So the payoff of having a credence is the payoff of the option you're required to pick using that credence.

Finally, we make the move that is central to Schervish's and Levinstein's work. We now know the payoff of having a particular credence in propositiojn $A$ when you face a decision problem in which all options are $A$-options. But of course we don't know which such decision problems we'll face. So, when we evaluate the payoff of having a credence in $A$ when $A$ is true, for instance, we look at all the decision problems populated by $A$-options we might face and weight them by how likely we are to face them and then take the payoff of having that credence when $A$ is true to be the expected payoff of the $A$-options it would leave us to choose faced with the decision problems we'll face. And then we note, as Schervish and Levinstein themselves note: if we make certain natural assumptions about how likely we are to face different decisions, then this resulting measure of the pragmatic payoff of having credence $p$ in proposition $A$ is a continuous and strictly proper scoring rule. That is, mathematically, the functions we use to measure the pragmatic value of a credence function are identical to the functions we use to evaluate the epistemic value of a credence that we use in the epistemic utility argument for Probabilism and Consensus.**

With this construction in place, we can piggyback on the theorems stated in the previous post to give new pragmatic arguments for Probabilism and Consensus. First: Suppose your credences do not obey Probabilism. Then there are alternative ones you might have instead that do obey that norm and, at any world, if we look at each decision problem you might face and ask what payoff you'd receive at that world were you to choose from the options in that decision problem as the two different sets of credences require, and then weight those payoffs by how likely they are to face that decision to give their expected payoff, then the alternatives will always have the greater expected payoff. This gives strong reason to obey Probabilism.

Second: Take a group of individuals. Now suppose the group's credences do not obey Consensus. Then there are alternative credences each member might have instead such that, if they were to have them, the group would obey Consensus and, at any world, if we look at each decision problem each member might face and ask what payoff that individual would receive at that world were they to choose from the options in that decision problem as the two different sets of credences require, and then weight those payoffs by how likely they are to face that decision to give their expected payoff, then the alternatives will always have the greater expected total payoff when this is summed across the whole group.

So that is our new and better pragmatic argument for Consensus. The sure loss argument points out a single downside to a group that violates the norm. Such a group is vulnerable to exploitation. But it remains silent on whether there are upsides that might balance out that downside. The present argument addresses that problem. It finds that, if a group violates the norm, there are alternative credences they might have that are guaranteed to serve them better in expectation as a basis for decision making.

* Notice that, if $x < pS$, then the expected payoff of a bet that pays $S$ if $A$ is true and $0$ if $A$ is false is
$$
p(-x + S) + (1-p)(-x) = pS- x
$$
which is positive. So, if the two options are accept or reject the bet, accepting maximises expected payoff by the lights of $p$, and so it is required, as Ramsey's Thesis says.

** Konek (ta) gives a clear formal treatment of this solution. For those who want the technical details, I'd recommend the Appendix of that paper. I think he presents it better than I did in (Pettigrew 2020).

References

Good, I. J. (1967). On the Principle of Total Evidence. The British Journal for the Philosophy of Science, 17, 319–322.

Konek, J. (ta). Degrees of incoherence, dutch bookability & guidance value. Philosophical Studies.

Levinstein, B. A. (2017). A Pragmatist’s Guide to Epistemic Utility. Philosophy of Science, 84(4), 613–638.

Pettigrew, R. (2019). Choosing for Changing Selves. Oxford, UK: Oxford University Press.

Pettigrew, R. (2020). Dutch Book Arguments. Elements in Decision Theory and Philosophy. Cambridge, UK: Cambridge University Press.

Schervish, M. J. (1989). A general method for comparing probability assessors. The Annals of Statistics, 17, 1856–1879.

Staffel, J. (2020). Unsettled Thoughts. Oxford University Press.


Thursday, 5 May 2022

Should we agree? I: the arguments for consensus

You can find a PDF of this blogpost here.

Should everyone agree with everyone else? Whenever two members of a group have an opinion about the same claim, should they both be equally confident in it? If this is sometimes required of groups, of which ones is it required and when? Whole societies at any time in their existence? Smaller collectives when they're engaged in some joint project?

Of course, you might think these are purely academic questions, since there's no way we could achieve such consensus even if we were to conclude that it is desirable, but that seems too strong. Education systems and the media can be deployed to push a population towards consensus, and indeed this is exactly how authoritarian states often proceed. Similarly, social sanctions can create incentives for conformity. So it seems that a reasonable degree of consensus might be possible.

But is it desirable? In this series of blogposts, I want to explore two formal arguments. They purport to establish that groups should be in perfect agreement; and they explain why getting closer to consensus is better, even if perfect agreement isn't achieved---in this case, a miss is not as good as a mile. It's still a long way from their conclusions to practical conclusions about how to structure a society, but they point sufficiently strongly in a surprising direction that it is worth exploring them. In this first post, I set out the arguments as they have been given in the literature and polish them up a bit so that they are as strong as possible.

Since they're formal arguments, they require a bit of mathematics, both in their official statement and in the results on which they rely. But I want to make the discussion as accessible as possible, so, in the main body of the blogpost, I state the arguments almost entirely without formalism. Then, in the technical appendix, I sketch some of the formal detail for those who are interested.

Two sorts of argument for credal norms

There are two sorts of argument we most often use to justify the norms we take to govern our credences: there are pragmatic arguments, of which the betting arguments are the most famous; and there are epistemic arguments, of which the epistemic utility arguments are the most well known.

Take the norm of Probabilism, for instance, which says that your credences should obey the axioms of the probability calculus. The betting argument for Probabilism is sometimes known as the Dutch Book or sure loss argument.* It begins by claiming that the maximum amount you are willing to pay for a bet on a proposition that pays out a certain amount if the proposition is true and nothing if it is false is proportional to your credence in that proposition. Then it shows that, if your credences do not obey the probability axioms, there is a set of bets each of which they require you to accept, but which when taken together lose you money for sure; and if your credences do obey those axioms, there is no such set of bets.

The epistemic utility argument for Probabilism, on the other hand, begins by claiming that any measure of the epistemic value of credences must have certain properties.** It then shows that, by the lights of any epistemic utility function that does have those properties, if your credences do not obey the probability axioms, then there are alternatives that are guaranteed to be have greater epistemic utility than yours; and if they do obey those axioms, there are no such alternatives.

Bearing all of this in mind, consider the following two facts.

(I) Suppose we make the same assumptions about which bets an individual's credences require them to accept that we make in the betting argument for Probabilism. Then, if two members of a group assign different credences to the same proposition, there is a bet the first should accept and a bet the second should accept that, taken together, leave the group poorer for sure (Ryder 1981, Gillies 1991). 

(II) Suppose we measure the epistemic value of credences using an epistemic utility function that boasts the properties required of it by the epistemic utility argument for Probabilism. Then, if two members of a group assign different credences to the same proposition, there is a single credence such that the group is guaranteed to have greater total epistemic utility if every member adopts that single credence in that proposition (Kopec 2012).

Given the epistemic utility and betting arguments for Probabilism, neither (I) nor (II) is very surprising. After all, one consequence of Probabilism is that an individual must assign the same credence to two propositions that have the same truth value as a matter of logic. But from the point of view of the betting argument or the epistemic utility argument, this is structurally identical to the requirement that two different people assign the same credence to the same proposition, since obviously a single proposition necessarily has the same truth value as itself! However we construct the sure loss bets against the individual who violates the consequence of Probabilism, we can use an analogous strategy to construct the sure loss bets against the pair who disagree in the credences they assign. And however we construct the alternative credences that are guaranteed to be more accurate than the ones that violate the consequence of Probabilism, we can use an analogous strategy to construct the alternative credence that, if adopted by all members of the group that contains two individuals who currently disagree, would increase their total epistemic utility for sure.

Just as a betting argument and an epistemic utility argument aim to establish the individual norm of Probabilism, we might ask whether there is a group norm for which we can give a betting argument and an epistemic utility argument by appealing to (I) and (II)? That is the question I'd like to explore in these posts. In the remainder of this post, I'll spell out the details of the epistemic utility argument and the betting argument for Probabilism, and then adapt those to give analogous arguments for Consensus.

The Epistemic Utility Argument for Probabilism

Two small bits of terminology first:

  • Your agenda is the set of propositions about which you have an opinion. We'll assume throughout that all individuals have finite agendas.
  • Your credence function takes each proposition in your agenda and returns your credence in that proposition.

With those in hand, we can state Probabilism

Probabilism Rationality requires of an individual that their credence function is a probability function. 

What does it mean to say that a credence function is a probability function? There are two cases to consider.

First, suppose that, whenever a proposition is in your agenda, its negation is as well; and whenever two propositions are in your agenda, their conjunction and their disjunction are as well. When this holds, we say that your agenda is a Boolean algebra. And in that case your credence function is a probability function if two conditions hold: first, you assign the minimum possible credence, namely 0, to any contradiction and the maximum possible credence, namely 1, to any tautology; second, your credence in a disjunction is the sum of your credences in the disjuncts less your credence in their conjunction (just like the number of people in two groups is the number in the first plus the number in the second less the number in both).

Second, suppose that your agenda is not a Boolean algebra. In that case, your credence function is a probability function if it is possible to extend it to a probability function on the smallest Boolean algebra that contains your agenda. That is, it's possible to fill out your agenda so that it's closed under negation, conjunction, and disjunction, and then extend your credence function so that it assign credences to those new propositions in such a way that the result is a probability function on the expanded agenda. Defining probability functions on agendas that are not Boolean algebras allows us to say, for instance, that, if your agenda is just It will be windy tomorrow and It will be windy and rainy tomorrow, and you assign credence 0.6 to It will be windy and 0.8 to It will be windy and rainy, then you violate Probabilism because there's no way to assign credences to It won't be windy, It will be windy or rainy, It won't be rainy, etc in such a way that the result is a probability function.

The Epistemic Utility Argument for Probabilism begins with three claims about how to measure the epistemic value of a whole credence function. The first is Individual Additivity, which says that the epistemic utility of a whole credence function is simply the sum of the epistemic utilities of the individual credences it assigns. The second is Continuity, which says that, for any proposition, the epistemic utility of a credence in that proposition is a continuous function of that credence. And the third is Strict Propriety, which says that, for any proposition, each credence in that proposition should expect itself to be have greater epistemic utility than it expects any alternative credence in that proposition to have. With this account in hand, the argument then appeals to a mathematical theorem, which tells us two consequences of measuring epistemic value using an epistemic utility function that has the three properties just described, namely, Individual Additivity, Continuity, and Strict Propriety.

(i) For any credence function that violates Probabilism, there is a credence function defined on the same agenda that satisfies it and that has greater epistemic utility regardless of how the world turns out. In this case, we say that the alternative credence function dominates the original one. 

(ii) For any credence function that is a probability function, there is no credence function that dominates it. Indeed, there is no alternative credence function that is even as good as it at every world. For any alternative, there will be some world where that alternative is strictly worse.

The argument concludes by claiming that an option is irrational if there is some alternative that is guaranteed to be better and no option that is guaranteed to be better than that alternative.

The Epistemic Utility Argument for Consensus

As I stated it above, and as it is usually stated in the literature, Consensus says that, whenever two members of a group assign credences to the same proposition, they should assign the same credence. But in fact the epistemic argument in its favour establishes something stronger. Here it is: 

Consensus Rationality requires of a group that there is a single probability function defined on the union of the agendas of all of the members of the group such that the credence function of each member assigns the same credence to any proposition in their agenda as this probability function does.

This goes further than simply requiring that all agents agree on the credence they assign to any proposition to which they all assign credences. Indeed, it would place constraints even on a group whose members' agendas do not overlap at all. For instance, if you have credence 0.6 that it will be rainy tomorrow, while I have credence 0.8 that it will be rainy and windy, the pair of us will jointly violate Consensus, even though we don't assign credences to any of the same propositions, since no probability function assigns 0.6 to one proposition and 0.8 to the conjunction of that proposition with another one. In these cases, we say that the group's credences don't cohere.

One notable feature of Consensus is that it purports to govern groups, not individuals, and we might wonder what it could mean to say that a group is irrational. I'll return to that in a later post. It will be useful to have the epistemic utility and betting arguments for Consensus to hand first.

The Epistemic Utility Argument for Consensus begins, as the epistemic argument for Probabilism does, with Individual Additivity, Continuity, and Strictly Propriety. And it adds to those Group Additivity, which says that group's epistemic utility is the sum of the epistemic utilities of the credence functions of its members. With this account of group epistemic value in hand, the argument then appeals again to a mathematical theorem, but a different one, which tells us two consequences of Group and Individual Additivity, Continuity, and Strict Propriety:***

(i) For any group that violates Consensus, there is, for each individual, an alternative credence function defined on their agenda that they might adopt such that, if all were to adopt these, the group would satisfy Consensus and it would be more accurate regardless of how the world turns out. In this case, we say that the alternative credence functions collectively dominate the original ones.

(ii) For any group that satisfies Consensus, there are no credence functions the group might adopt that collectively dominate it.

The argument concludes by assuming again the norm that an option is irrational if there is some alternative that is guaranteed to be better.

The Sure Loss Argument for Probabilism

The Sure Loss Argument for Probabilism begins with a claim that I call Ramsey's Thesis. It tells you the prices at which your credences require you to buy and sell bets. It says that, if your credence in $A$ is $p$, and $£x < £pS$, then you should be prepared to pay $£x$ for a bet that pays out $£S$ if $A$ is true and $£0$ if $A$ is false. And this is true for any stakes $S$, whether positive, negative, or zero. Then it appeals to a mathematical theorem, which tells us two consequences of Ramsey's Thesis.

(i) For any credence function that violates Probabilism, there is a series of bets, each of which your credences require you to accept, that, taken together, lose you money for sure.

(ii) For any credence function satisfies Probabilism, there is no such series of bets.

The argument concludes by assuming a norm that says that it is irrational to have credences that require you to make a series of choices when there is an alternative series of choices you might have made that would be better regardless of how the world turns out.

The Sure Loss Argument for Consensus

The Sure Loss Argument for Consensus also begins with Ramsey's Thesis.  It appeals to a mathematical theorem that tells us two consequences of Ramsey's Thesis.

(i) For any group that violates Consensus, there is a series of bets, each offered to a  member of the group whose credences require that they accept it, that, taken together, lose the group money for sure.

(ii) For any group that satisfies Consensus, there is no such series of bets.

And it concludes by assuming that it is irrational for the members of a group to have credences that require them to make a series of choices when there is an alternative series of choices they might have made that would be better for the group regardless of how the world turns out.

So now we have the Epistemic Utility and Sure Loss Arguments for Consensus. In fact, I think the Sure Loss Argument doesn't work. So in the next post I'll say why and provide a better alternative based on work by Mark Schervish and Ben Levinstein. But in the meantime, here's the technical appendix.

Technical appendix

First, note that Probabilism is the special case of Consensus when the group has only one member. So we focus on establishing Consensus.

Some definitions to begin:

  • If $c$ is a credence function defined on the agenda $\mathcal{F}_i = \{A^i_1, \ldots, A^i_{k_i}\}$, represent it as a vector as follows:$$c = \langle c(A^i_1), \ldots, c(A^i_{k_i})\rangle$$
  • Let $\mathcal{C}_i$ be the set of credence functions defined on $\mathcal{F}_i$, represented as vectors in this way.
  • If $c_1, \ldots, c_n$ are credence functions defined on $\mathcal{F}_1, \ldots, \mathcal{F}_n$ respectively, represent them collectively as a vector as follows:
    $$
    c_1 \frown \ldots \frown c_n = \langle c_1(A^1_1), \ldots, c_1(A^1_{k_1}), \ldots, c_n(A^n_1), \ldots, c_n(A^n_{k_n}) \rangle
    $$
  • Let $\mathcal{C}$ be the set of sequences of credence functions defined on $\mathcal{F}_1, \ldots, \mathcal{F}_n$ respectively, represented as vectors in this way. 
  • If $w$ is a classically consistent assignment of truth values to the propositions in $\mathcal{F}_i$, represent it as a vector $$w = \langle w(A^i_1), \ldots, w(A^i_{k_i})\rangle$$ where $w(A) = 1$ if $A$ is true according to $w$, and $w(A) = 0$ if $A$ is false according to $w$.
  • Let $\mathcal{W}_i$ be the set of classically consistent assignments of truth values to the propositions in $\mathcal{F}_i$, represented as vectors in this way.
  • If $w$ is a classically consistent assignment of truth values to the propositions in $\mathcal{F} = \bigcup^n_{i=1} \mathcal{F}_i$, represent the restriction of $w$ to $\mathcal{F}_i$ by the vector $$w_i = \langle w(A^i_1), \ldots, w(A^i_{k_i})\rangle$$So $w_i$ is in $\mathcal{W}_i$. And represent $w$ as a vector as follows:
    $$
    w = w_1 \frown \ldots \frown w_n = \langle w(A^1_1), \ldots, w(A^1_{k_1}), \ldots, w(A^n_1), \ldots, w(A^n_{k_n})\rangle
    $$
  • Let $\mathcal{W}$ be the set of classical consistent assignments of truth values to the propositions in $\mathcal{F}$, represented as vectors in this way.

Then we have the following result, which generalizes a result due to de Finetti (1974):

Proposition 1 A group of individuals with credence functions $c_1, \ldots, c_n$ satisfy Consensus iff $c_1 \frown \ldots \frown c_n$ is in the closed convex hull of $\mathcal{W}$.

We then appeal to two sets of results. First, concerning epistemic utility measures, which generalizes a result to Predd, et al. (2009):

Theorem 1

(i) Suppose $\mathfrak{A}_i : \mathcal{C}_i \times \mathcal{W}_i \rightarrow [0, 1]$ is a measure of epistemic utility that satisfies Individual Additivity, Continuity, and Strict Propriety. Then there is a Bregman divergence $\mathfrak{D}_i : \mathcal{C}_i \times \mathcal{C}_i \rightarrow [0, 1]$ such that $\mathfrak{A}_i(c, w) = -\mathfrak{D}_i(w, c)$.

(ii) Suppose $\mathfrak{D}_1, \ldots, \mathfrak{D}_n$ are Bregman divergences defined on $\mathcal{C}_1, \ldots, \mathcal{C}_n$, respectively. And suppose $\mathcal{X}$ is a closed convex subset of $\mathcal{C}$. And suppose $c_1 \frown \ldots \frown c_n$ is not in $\mathcal{X}$. Then there is $c^\star_1 \frown \ldots \frown c^\star_n$ in $\mathcal{Z}$ such that, for all $z_1 \frown \ldots \frown z_n$ in $\mathcal{Z}$,
$$
\sum^n_{i=1} \mathfrak{D}_i(z_i, c^\star_i) < \sum^n_{i=1} \mathfrak{D}_i(z_i, c_i)
$$

So, by Proposition 1, if a group $c_1, \ldots, c_n$ does not satisfy Consensus, then $c_1 \frown \ldots \frown c_n$ is not in the closed convex hull of $\mathcal{W}$, and so by Theorem 1 there is $c^\star_1 \frown \ldots \frown c^\star_n$ in the closed convex hull of $\mathcal{W}$ such that, for all $w$ in $\mathcal{W}$, $$\mathfrak{A}_i(c, w) < \mathfrak{A}(c^\star, w)$$ as required.

Second, concerning bets, which is a consequence of the Separating Hyperplane Theorem:

Theorem 2
Suppose $\mathcal{Z}$ is a closed convex subset of $\mathcal{C}$. And suppose $c_1 \frown \ldots \frown c_n$ is not in $\mathcal{Z}$. Then there are vectors
$$
x = \langle x^1_1, \ldots, x^1_{k_1}, \ldots, x^n_1, \ldots, x^n_{k_n}\rangle
$$
and
$$
S = \langle S^1_1, \ldots, S^1_{k_1}, \ldots, S^n_1, \ldots, S^n_{k_n}\rangle
$$
such that, for all $x^i_j$ and $S^i_j$,
$$
x^i_j < c_i(A^i_j)S^i_j
$$
and, for all $z$ in $\mathcal{Z}$,
$$
\sum^n_{i=1} \sum^{k_i}_{j = 1} x^i_j > \sum^n_{i=1} \sum^{k_i}_{j=1} z^i_jS^i_j
$$

So, by Proposition 1, if a group $c_1, \ldots, c_n$ does not satisfy Consensus, then $c_1 \frown \ldots \frown c_n$ is not in the closed convex hull of $\mathcal{W}$, and so, by Theorem 2, there is $x = \langle x^1_1, \ldots, x^1_{k_1}, \ldots, x^n_1, \ldots, x^n_{k_n}\rangle$ and $S = \langle S^1_1, \ldots, S^1_{k_1}, \ldots, S^n_1, \ldots, S^n_{k_n}\rangle$ such that (i) $x^i_j < c_i(A^i_j)S^i_j$ and (ii) for all $w$ in $\mathcal{W}$,
$$\sum^n_{i=1} \sum^{k_i}_{j = 1} x^i_j > \sum^n_{i=1} \sum^{k_i}_{j=1} w(A^i_j)S^i_j$$
But then (i) says that the credences of individual $i$ require them to pay $£x^i_j$ for a bet on $A^i_j$ that pays out $£S^i_j$ if $A^i_j$ is true and $£0$ if it is false. And (ii) says that the total price of these bets across all members of the group---namely, $£\sum^n_{i=1} \sum^{k_i}_{j = 1} x^i_j$---is greater than the amount the bets will payout at any world---namely, $£\sum^n_{i=1} \sum^{k_i}_{j=1} w(A^i_j)S^i_j$.

* This was introduced independently by Frank P. Ramsey (1931) and Bruno de Finetti (1937). For overviews, see (Hajek 2008, Vineberg 2016, Pettigrew 2020).

**Much of the discussion of these arguments in the literature focusses on versions on which the epistemic value of a credence is taken to be its accuracy. This literature begins with Rosenkrantz (1981) and Joyce (1998). But, following Joyce (2009) and Predd (2009), it has been appreciated that we need not necessarily assume that accuracy is the only source of epistemic value in order to get the argument going.

*** Matthew Kopec (2012) offers a proof of a slightly weaker result. It doesn't quite work because it assumes that all strictly proper measures of epistemic value are convex, when they are not---the spherical scoring rule is not. I offer an alternative proof of this stronger result in the technical appendix below.

References

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de Finetti, B. (1974). Theory of Probability, vol. I. New York: John Wiley & Sons.

Gillies, D. (1991). Intersubjective probability and confirmation theory. The British Journal for the Philosophy of Science, 42(4), 513–533.

Hájek, A. (2008). Dutch Book Arguments. In P. Anand, P. Pattanaik, & C. Puppe (Eds.) The Oxford Handbook of Rational and Social Choice, (pp. 173–195). Oxford: Oxford University Press.

Joyce, J. M. (1998). A Nonpragmatic Vindication of Probabilism. Philosophy of Science, 65(4), 575–603.

Joyce, J. M. (2009). Accuracy and Coherence: Prospects for an Alethic Epistemology of Partial Belief. In F. Huber, & C. Schmidt-Petri (Eds.) Degrees of Belief. Dordrecht and Heidelberg: Springer.

Kopec, M. (2012). We ought to agree: A consequence of repairing Goldman’s group scoring rule. Episteme, 9(2), 101–114.

Pettigrew, R. (2020). Dutch Book Arguments. Cambridge University Press.

Predd, J., Seiringer, R., Lieb, E. H., Osherson, D., Poor, V., & Kulkarni, S. (2009). Probabilistic Coherence and Proper Scoring Rules. IEEE Transactions of Information Theory, 55(10), 4786–4792.

Ramsey, F. P. (1926 [1931]). Truth and Probability. In R. B. Braithwaite (Ed.) The Foundations of Mathematics and Other Logical Essays, chap. VII, (pp. 156–198). London: Kegan, Paul, Trench, Trubner & Co.

Rosenkrantz, R. D. (1981). Foundations and Applications of Inductive Probability. Atascadero, CA: Ridgeview Press.

Ryder, J. (1981). Consequences of a simple extension of the Dutch Book argument. The British Journal for the Philosophy of Science, 32(2), 164–167.

Vineberg, S. (2016). Dutch Book Arguments. In E. N. Zalta (Ed.) Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University.