Sunday, 14 March 2021

Permissivism and social choice: a response to Blessenohl

In a recent paper discussing Lara Buchak's risk-weighted expected utility theory, Simon Blessenohl notes that the objection he raises there to Buchak's theory might also tell against permissivism about rational credence. I offer a response to the objection here.

In his objection, Blessenohl suggests that credal permissivism gives rise to an unacceptable tension between the individual preferences of agents and the collective preferences of the groups to which those agents belong. He argues that, whatever brand of permissivism about credences you tolerate, there will be a pair of agents and a pair of options between which they must choose such that both agents will prefer the first to the second, but collectively they will prefer the second to the first. He argues that this consequence tells against permissivism. I respond that this objection relies on an equivocation between two different understandings of collective preferences: on the first, they are an attempt to summarise the collective view of the group; on the second, they are the preferences of a third-party social chooser tasked with making decisions on behalf of the group. I claim that, on the first understanding, Blessenohl's conclusion does not follow; and, on the second, it follows but is not problematic.

It is well known that, if two people have difference credences in a given proposition, there is a sense in which the pair of them, taken together, is vulnerable to a sure loss set of bets.* That is, there is a bet that the first will accept and a bet that the second will accept such that, however the world turns out, they'll end up collectively losing money. Suppose, for instance, that Harb is 90% confident that Ladybug will win the horse race that is about to begin, while Jay is only 60% confident. Then Harb's credences should lead him to buy  a bet for £80 that will pay out £100 if Ladybug wins and nothing if she loses, while Jay's credences should lead him to sell that same bet for £70 (assuming, as we will throughout, that the utility of £$n$ is $n$). If Ladybug wins, Harb ends up £20 up and Jay ends up £30 down, so they end up £10 down  collectively. And if Ladybug loses, Harb ends up £80 down while Jay ends up £70 up, so they end up £10 down as a pair.

So, for individuals with different credences in a proposition, there seems to be a tension between how they would choose as individuals and how they would choose as a group. Suppose they are presented with a choice between two options: on the first, $A$, both of them enter into the bets just described; on the second, $B$, neither of them do. We might represent these two options as follows, where we assume that Harb's utility for receiving £$n$ is $n$, and the same for Jay:$$A = \begin{pmatrix}
20 & -80 \\
-30 & 70
\end{pmatrix}\ \ \
B = \begin{pmatrix}
0 & 0 \\
0 & 0
\end{pmatrix}$$The top left entry is Harb's winnings if Ladybug wins, the top right is Harb's winnings if she loses; the bottom left is Jay's winnings if she wins, and the bottom left is Jay's winnings if she loses. So, given a matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, each row represents a gamble---that is, an assignment of utilities to each state of the world---and each column represents a utility distribution---that is, an assignment of utilities  to each individual. So $\begin{pmatrix} a & b \end{pmatrix}$ represents the gamble that the option bequeaths to Harb---$a$ if Ladybug wins, $b$ if she loses---while $\begin{pmatrix} c & d \end{pmatrix}$ represents the gamble bequeathed to Jay---$c$ if she wins, $d$ if she loses. And $\begin{pmatrix} a \\ c \end{pmatrix}$ represents the utility distribution if Ladybug wins---$a$ to Harb, $c$ to Jay---while $\begin{pmatrix} b \\ d \end{pmatrix}$ represents the utility distribution if she loses---$b$ to Harb, $d$ to Jay. Summing the entries in the first column gives the group's collective utility if Ladybug wins, and summing the entries in the second column gives their collective utility if she loses.

Now, suppose that Harb cares only for the utility that he will gain, and Jay cares only his own utility; neither cares at all about the other's welfare. Then each prefers $A$ to $B$. Yet, considered collectively, $B$ results in greater total utility for sure: for each column, the sum of the entries in that column in $B$ (that is, $0$) exceeds the sum in that column in $A$ (that is, $-10$). So there is a tension between what the members of the group unanimously prefer and what the group prefers.

Now, to create this tension, I assumed that the group prefers one option to another if the total utility of the first is sure to exceed the total utility of the second. But this is quite a strong claim. And, as Blessenohl notes, we can create a similar tension by assuming something much weaker.

Suppose again that Harb is 90% confident that Ladybug will win while Jay is only 60% confident that she will. Now consider the following two options:$$A' = \begin{pmatrix}
20 & -80 \\
0 & 0
\end{pmatrix}\ \ \
B' = \begin{pmatrix}
5 & 5 \\
25 & -75
\end{pmatrix}$$In $A'$, Harb pays £$80$ for a £$100$ bet on Ladybug, while in $B'$ he receives £$5$ for sure. Given his credences, he should prefer $A'$ to $B'$, since the expected utility of $A'$ is $10$, while for $B'$ it is $5$. And in $A'$, Jay receives £0 for sure, while in $B'$ he pays £$75$ for a £$100$ bet on Ladybug. Given his credences, he should prefer $A'$ to $B'$, since the expected utility of $A'$ is $0$, while for $B'$ it is $-15$. But again we see that $B'$ will nonetheless end up producing greater total utility for the pair---$30$ vs $20$ if Ladybug wins, and $-70$ vs $-80$ if Ladybug loses. But we can argue in a different way that the group should prefer $B'$ to $A'$. This different way of arguing for this conclusion is the heart of Blessenohl's result.

In what follows, we write $\preceq_H$ for Harb's preference ordering, $\preceq_J$ for Jay's, and $\preceq$ for the group's. First, we assume that, when one option gives a particular utility $a$ to Harb for sure and a particular utility $c$ to Jay for sure, then the group should be indifferent between that and the option that gives $c$ to Harb for sure and $a$ to Jay for sure. That is, the group should be indifferent between an option that gives the utility distribution  $\begin{pmatrix} a \\ c\end{pmatrix}$ for sure and an option that gives $\begin{pmatrix} c \\ a\end{pmatrix}$ for sure.  Blessenohl calls this Constant Anonymity:

Constant Anonymity  For any $a, c$,$$\begin{pmatrix}
a & a \\
c & c
\end{pmatrix} \sim
\begin{pmatrix}
c & c \\
a & a
\end{pmatrix}$$This allows us to derive the following:$$\begin{pmatrix}
20 & 20 \\
0 & 0
\end{pmatrix} \sim
\begin{pmatrix}
0 & 0 \\
20 & 20
\end{pmatrix}\ \ \ \text{and}\ \ \
\begin{pmatrix}
-80 & -80 \\
0 & 0
\end{pmatrix} \sim
\begin{pmatrix}
0 & 0 \\
-80 & -80
\end{pmatrix}$$And now we can introduce our second principle:

Preference Dominance  For any $a, b, c, d, a', b', c', d'$, if$$\begin{pmatrix}
a & a \\
c & c
\end{pmatrix} \preceq
\begin{pmatrix}
a' & a' \\
c' & c'
\end{pmatrix}\ \ \ \text{and}\ \ \
\begin{pmatrix}
b & b \\
d & d
\end{pmatrix} \preceq
\begin{pmatrix}
b' & b' \\
d' & d'
\end{pmatrix}$$then$$\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} \preceq
\begin{pmatrix}
a' & b' \\
c' & d'
\end{pmatrix}$$Preference Dominance says that, if the group prefers obtaining the utility distribution  $\begin{pmatrix} a \\ c\end{pmatrix}$ for sure to obtaining the utility distribution  $\begin{pmatrix} a' \\ c'\end{pmatrix}$ for sure, and prefers obtaining the utility distribution  $\begin{pmatrix} b \\ d\end{pmatrix}$ for sure to obtaining the utility distribution  $\begin{pmatrix} b' \\ d'\end{pmatrix}$ for sure, then they prefer obtaining $\begin{pmatrix} a \\ c\end{pmatrix}$ if Ladybug wins and $\begin{pmatrix} b \\ d\end{pmatrix}$ if she loses to obtaining $\begin{pmatrix} a' \\ c'\end{pmatrix}$ if Ladybug wins and $\begin{pmatrix} b' \\ d'\end{pmatrix}$ if she loses.

Preference Dominance, combined with the indifferences that we derived from Constant Anonymity, gives$$\begin{pmatrix}
20 & -80 \\
0 & 0
\end{pmatrix} \sim
\begin{pmatrix}
0 & 0 \\
20 & -80
\end{pmatrix}$$And then finally we introduce a closely related principle: 

Utility Dominance  For any $a, b, c, d, a', b', c', d'$, if $a < a'$, $b < b'$, $c < c'$, and $d < d'$, then$$\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} \prec
\begin{pmatrix}
a' & b' \\
c' & d'
\end{pmatrix}$$

This simply says that if one option gives more utility than another to each individual at each world, then the group should prefer the first to the second. So$$\begin{pmatrix}
0 & 0 \\
20 & -80
\end{pmatrix} \prec
\begin{pmatrix}
5 & 5 \\
25 & -75
\end{pmatrix}$$Stringing these together, we have$$A' = \begin{pmatrix}
20 & -80 \\
0 & 0
\end{pmatrix} \sim
\begin{pmatrix}
0 & 0 \\
20 & -80
\end{pmatrix} \prec
\begin{pmatrix}
5 & 5 \\
25 & -75
\end{pmatrix} = B'$$And thus, assuming that $\preceq$ is transitive, while Harb and Jay both prefer $A'$ to $B'$, the group prefers $B'$ to $A'$.

More generally, Blessenohl proves an impossibility result. Add to the principles we have already stated the following:

Ex Ante Pareto  If $A \preceq_H B$ and $A \preceq_J B$, then $A \preceq B$.

And also:

Egoism  For any $a, b, c, d, a', b', c', d'$,$$\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} \sim_H \begin{pmatrix}
a & b \\
c' & d'
\end{pmatrix}\ \ \ \text{and}\ \ \
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} \sim_J \begin{pmatrix}
a' & b' \\
c & d
\end{pmatrix}$$That is, Harb cares only about the utilities he obtains from an option, and Jay cares only about the utilities that he obtains. And finally:

Individual Preference Divergence  There are $a, b, c, d$ such that$$\begin{pmatrix}
a & b \\
a & b
\end{pmatrix} \prec_H \begin{pmatrix}
c & d \\
c & d
\end{pmatrix}\ \ \ \text{and}\ \ \
\begin{pmatrix}
a & b \\
a & b
\end{pmatrix} \succ_J \begin{pmatrix}
c & d \\
c & d
\end{pmatrix}$$Then Blessenohl shows that there are no preferences $\preceq_H$, $\preceq_J$, and $\preceq$ that satisfy Individual Preference Divergence, Egoism, Ex Ante Pareto, Constant Anonymity, Preference Dominance, and Utility Dominance.** And yet, he claims, each of these is plausible. He suggests that we should give up Individual Preference Divergence, and with it permissivism and risk-weighted expected utility theory.

Now, the problem that Blessenohl identifies arises because Harb and Jay have different credences in the same proposition. But of course impermissivists agree that two rational individuals can have different credences in the same proposition. So why is this a problem specifically for permissivism? The reason is that, for the impermissivist, if two rational individuals have different credences in the same proposition, they must have different evidence. And for individuals with different evidence, we wouldn't necessarily want the group preference to preserve unanimous agreement between the individuals. Instead, we'd want the group to choose using whichever credences are rational in the light of the joint evidence obtained by pooling the evidence held by each individual in the group. And those might render one option preferable to the other even though each of the individuals, with their less well informed credences, prefer the second option to the first. So Ex Ante Pareto is not plausible when the individuals have different evidence, so impermissivism is safe.

To see this, consider the following example: There are two medical conditions, $X$ and $Y$, that affect racehorses. If they have $X$, they're 90% likely to win the race; if they have $Y$, they're 60% likely; if they have both, they're 10% likely to win. Suppose Harb knows that Ladybug has $X$, but has no information about whether she has $Y$; and suppose Jay knows Ladybug has $Y$ and no information about $X$. Then both are rational. And both prefer $A$ to $B$ from above. But we wouldn't expect the group to prefer $A$ to $B$, since the group should choose using the credence it's rational to have if you know both that Ladybug has $X$ and that she has $Y$; that is, the group should choose by pooling the individual's evidence to give the group evidence, and then choose using the probabilities relative to that. And, relative to that evidence, $B$ is preferable to $A$.

The permissivist, in contrast, cannot make this move. After all, for them it is possible for two rational individuals to disagree even though they have exactly the same evidence, and therefore the same pooled evidence. Blessenohl considers various ways the permissivist or the risk-weighted expected utility theorist might answer his objection, either by denying Ex Ante Pareto or Preference or Utility Dominance. He considers each response unsuccessful, and I tend to agree with his assessments. However, oddly, he explicitly chooses not to consider the suggestion that we might drop Constant Anonymity. I'd like to suggest that we should consider doing exactly that.

I think Blessenohl's objection relies on an ambiguity in what the group preference ordering $\preceq$ represents. On one understanding, it is no more than an attempt to summarise the collective view of the group; on another, it represents the preferences of a third party brought in to make decisions on behalf of the group---the social chooser, if you will. I will argue that Ex Ante Pareto is plausible on the first understanding, but Constant Anonymity isn't; and Constant Anonymity is plausible on the second understanding, but Ex Ante Pareto isn't.

Let's treat the first understanding of $\preceq$. On this, $\preceq$ represents the group's collective opinions about the options on offer. So just as we might try to summarise the scientific community's view on the future trajectory of Earth's average surface temperate or the mechanisms of transmission for SARS-CoV-2 by looking at the views of individual scientists, so might we try to summarise Harb and Jay's collective view of various options by looking at their individual views. Understood in this way, Constant Anonymity does not look plausible. Its motivation is, of course, straightforward. If $a < b$ and$$\begin{pmatrix}
a & a \\
b & b
\end{pmatrix} \prec
\begin{pmatrix}
b & b \\
a & a
\end{pmatrix}$$then the group's collective view unfairly and without justification favours Harb over Jay. And if$$\begin{pmatrix}
a & a \\
b & b
\end{pmatrix} \succ
\begin{pmatrix}
b & b \\
a & a
\end{pmatrix}$$then it unfairly and without justification favours Jay over Harb. So we should rule out both of these. But this doesn't entail that the group preference should be indifferent between these two options. That is, it doesn't entail that we should have$$\begin{pmatrix}
a & a \\
b & b
\end{pmatrix} \sim
\begin{pmatrix}
b & b \\
a & a
\end{pmatrix}$$After all, when you compare two options $A$ and $B$, there are four possibilities:

  1. $A \preceq B$ and $B \preceq A$---that is, $A \sim B$;
  2. $A \preceq B$ and $B \not \preceq A$---that is, $A \prec B$;
  3. $A \not \preceq B$ and $B \preceq A$---that is, $A \succ B$;
  4. $A \not \preceq B$ and $B \not \preceq A$---that is, $A$ and $B$ and not compatible.

The argument for Constant Anonymity rules out (2) and (3), but it does not rule out (4). What's more, it's easy to see that, if we weaken Constant Anonymity so that it requires (1) or (4) rather than requiring (1), then we see that all of the principles are consistent with it. So introduce Weak Constant Anonymity:

Weak Constant Anonymity  For any $a, c$, then either$$\begin{pmatrix}
a & a \\
c & c
\end{pmatrix} \sim
\begin{pmatrix}
c & c \\
a & a
\end{pmatrix}$$or$$\begin{pmatrix}
a & a \\
c & c
\end{pmatrix}\ \ \text{and}\ \
\begin{pmatrix}
c & c \\
a & a
\end{pmatrix}\ \  \text{are incomparable}$$

Then define the preference ordering $\preceq^*$ as follows:$$A \preceq^* B \Leftrightarrow \left ( A \preceq_H B\ \&\ A \preceq_J B \right )$$Then $\preceq^*$ satisfies Ex Ante Pareto, Weak Constant Anonymity, Preference Dominance, and Utility Dominance. And indeed $\preceq^*$ seems a very plausible candidate for the group preference ordering understood in this first way: where Harb and Jay disagree, it simply has no opinion on the matter; it has opinions only where Harb and Jay agree, and then it shares their shared opinion.

On the understanding of $\preceq$ as summarising the group's collective view, if  $\begin{pmatrix}
a & a \\
c & c
\end{pmatrix} \sim
\begin{pmatrix}
c & c \\
a & a
\end{pmatrix}$ then the group collectively thinks that this option $\begin{pmatrix}
a & a \\
c & c
\end{pmatrix}$ is exactly as good as this option $\begin{pmatrix}
c & c \\
a & a
\end{pmatrix}$. But the group absolutely does not think that. Indeed, Harb and Jay both explicitly deny it, though for opposing reasons. So Constant Anonymity is false.

Let's turn next to the second understanding. On this, $\preceq$ is the preference ordering of the social chooser. Here, the original, stronger version of Constant Anonymity seems more plausible. After all, unlike the group itself, the social chooser should have the sort of positive commitment to equality and fairness that the group definitively does not have. As we noted above, Harb and Jay unanimously reject the egalitarian assessment represented by $\begin{pmatrix}
a & a \\
c & c
\end{pmatrix} \sim
\begin{pmatrix}
c & c \\
a & a
\end{pmatrix}$. They explicitly both think that these two options are not equally good---if $a < c$, then Harb thinks the second is strictly better, while Jay thinks the first is strictly better. So, as we argued above, we take the group view to be that they are incomparable. But the social chooser should not remain so agnostic. She should overrule the unanimous rejection of the indifference relation between them and accept it. But, having thus overruled one unanimous view and taken a different one, it is little surprise that she will reject other unanimous views, such as Harb and Jay's unanimous view that $A'$ is better than $B'$ above. That is, it is little surprise that she should violate Ex Ante Pareto. After all, her preferences are not only informed by a value that Harb and Jay do not endorse; they are informed by a value that Harb and Jay explicitly reject, given our assumption of Egoism. This is the value of fairness, which is embodied in the social chooser's preferences in Constant Anonymity and rejected in Harb's and Jay's preferences by Egoism. If we require of our social chooser that they adhere to this value, we should not expect Ex Ante Pareto to hold.

* See Philippe Mongin's 1995 paper 'Consistent Bayesian Aggregation' for wide-ranging results in this area.

** Here's the trick: if$$\begin{pmatrix}
a & b \\
a & b
\end{pmatrix} \prec_H \begin{pmatrix}
c & d \\
c & d
\end{pmatrix}\ \ \ \text{and}\ \ \
\begin{pmatrix}
a & b \\
a & b
\end{pmatrix} \succ_J \begin{pmatrix}
c & d \\
c & d
\end{pmatrix}$$
Then let$$A' = \begin{pmatrix}
c & d \\
a & b
\end{pmatrix}\ \ \ \text{and}\ \ \
B' = \begin{pmatrix}
a & b \\
c & d
\end{pmatrix}$$Then $A' \succ_H B'$ and $A' \succ_J B'$, but $A' \sim B'$.

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