Groupthink

Jeffrey Sanford Russell; John Hawthorne; Lara Buchak

doi:10.1007/s11098-014-0350-8

Philosophical Studies

May 2015, Volume 172, Issue 5, pp 1287–1309 | Cite as

Groupthink

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Jeffrey Sanford Russell
John Hawthorne
Lara Buchak

Article

First Online: 22 July 2014

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Abstract

How should a group with different opinions (but the same values) make decisions? In a Bayesian setting, the natural question is how to aggregate credences: how to use a single credence function to naturally represent a collection of different credence functions. An extension of the standard Dutch-book arguments that apply to individual decision-makers recommends that group credences should be updated by conditionalization. This imposes a constraint on what aggregation rules can be like. Taking conditionalization as a basic constraint, we gather lessons from the established work on credence aggregation, and extend this work with two new impossibility results. We then explore contrasting features of two kinds of rules that satisfy the constraints we articulate: one kind uses fixed prior credences, and the other uses geometric averaging, as opposed to arithmetic averaging. We also prove a new characterisation result for geometric averaging. Finally we consider applications to neighboring philosophical issues, including the epistemology of disagreement.

Keywords

Credence aggregation Formal epistemology Social epistemology Conditionalization Disagreement

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Appendix: Proof of Fact 4

Let W be a countable set of worlds. In this context, a credence function is given by a function from W to $[0,1]$ that sums to one (i.e., a probability mass function). Let n be the number of individuals. Call a sequence of n credence functions admissible iff there is some world that each function gives positive probability. (This restriction goes with the idea discussed in Sect. 3 that credence one is factive.) Then the aggregation rule $\hbox{ag}$ is a function that takes each admissible sequence to a single credence function. In this setting, Conditionalization says that for any sequence $C = \langle C_{1},\,\ldots , C_{n}\rangle$ and set of worlds E, if $C \mid E = \langle C_{1} \mid E,\,\ldots , C_{n} \mid E\rangle$ is defined and admissible, then $\hbox {ag}\,(C \mid E) = \hbox {ag}\,C \mid E$. In what follows C and $C'$ are admissible sequences of credence functions.

For a credence function $C_{i}$, let $C_{i}(v : w)$ stand for the ratio $C_{i}(v)/C_{i}(w)$ (if it is defined).

Pointwise Ratios: For any $v, w \in W$, if $C_{i}(v : w)$ and $C'_{i}(v : w)$ are defined and equal for each i, then $\hbox {ag}\,C(v : w) = \hbox {ag}\,C'(v : w)$.

Lemma 1

Conditionalization is equivalent to Pointwise Ratios.

Proof

Let $E = \{v, w\}$. For each i, if $C_{i}(v : w) = C'_{i}(v : w)$ then $C_{i} \mid E = C'_{i} \mid E$. So by Conditionalization, $\hbox {ag}\,C \mid E = \hbox {ag}\,C' \mid E$. This implies that $\hbox {ag}\,C(v : w) = \hbox {ag}\,C'(v : w)$ as well. So Conditionalization implies Pointwise Ratios.

Conversely, let E be any proposition such that $C \mid E$ is defined and admissible, and let v be a world in E that each individual gives positive probability. Then for each $w \in E, C$ and $C \mid E$ both have the same well-defined ratios between w and v. So $\hbox {ag}\,C$ and $\hbox {ag}\,(C \mid E)$ also have the same ratios for each $w \in E$. So $\hbox {ag}\,C \mid E$ and $\hbox {ag}\,(C \mid E)$ are proportional, and since each of them adds up to one they are identical. So Pointwise Ratios implies Conditionalization. $\square$

Recall that Neutrality means that $\hbox {ag}$ commutes with permutations of $W$, and Continuity means that $\hbox{ag}$ is a continuous function (with respect to the product topology of $[0, 1]^{W \times n}$).

Fact 4:

If ag obeys Conditionalization, Continuity, and Neutrality, ag is a Weighted Geometric Rule.

Proof

Pointwise Ratios and Neutrality together imply that there is some function $F: [0, \infty )^{n} \rightarrow [0, \infty )$ such that for each pair of distinct worlds $v$ and $w$, if the ratios $C_{1}(v :\,w), \ldots , C_{n}(v :\,w)$ are all defined, then

$$\hbox {ag}\,C(v : w) = F(C_{1}(v :\,w),\,\ldots ,\,C_n(v : w))$$

(Pointwise Ratios guarantees that for each v and w there is some function $F_{v, w}$ that determines the group ratio for $v$ and $w$ in terms of the individual ratios, when they are defined. Neutrality guarantees that $F_{v, w}$ is the same for each $v$ and $w$.) Furthermore, if $\hbox{ag}$ is continuous then $F$ is continuous as well.

Ratios have the following property: if $C(u : v)$ and $C(v : w)$ are both defined, then $C(u : w) = C(u : v) \cdot C(v : w)$. This implies that F is multiplicative for positive arguments:

$$F(r_{1}\cdot s_{1},\,\ldots ,\,r_{n}\cdot s_{n})=F(r_{1},\,\ldots ,\,r_{n})\cdot F(s_{1},\,\ldots ,\,s_{n})$$

(where each $r_{i}$ and $s_{i}$ is positive). It’s helpful to map this onto a logarithmic scale: there is a continuous function $G : \mathbb{R}^{n} \rightarrow \mathbb{R}$ such that

$$G (\log r_{1}, \ldots , \log r_{n}) = \log F(r_{1}, \ldots , r_{n})$$

(for positive $r_{i}$). It follows from F’s multiplicative property that G is additive:

$$G(x_{1}+y_{1},\,\ldots ,\,x_{n}+y_{n})=G(x_{1}, \ldots , x_{n})+G(y_{1}, \ldots , y_{n})$$

But any continuous additive function from $\mathbb{R}^{n}$ to $\mathbb{R}$ is linear. (This fact was noted by Cauchy in 1821.) So G is linear, and thus there are weights $a_{1},\,\ldots ,\,a_{n}$, such that

$$G(x_{1}, \ldots , x_{n})=a_{1}\cdot x_{1}+\cdots +a_{n}\cdot x_{n}$$

Undoing the transformation to the logarithmic scale, then,

$$F(r)=r_{1}^{a_{1}}\cdot \cdots \cdot r_n^{a_{n}}$$

This fixes the value of F for positive ratios to be a weighted geometric mean. When $r_{i} = 0$, continuity forces F’s value be the limit value as $r_{i}$ approaches zero. Accordingly, for any i, if $a_{i} > 0$, then $F(r_{1}, \ldots , r_{n})$ must be zero when $r_{i} = 0$, which is consistent with geometric averaging. For $a_{i} = 0$, if we consider $0^{0} = 1$ then again the geometric average extends F continuously to $r_{i} = 0$. For $a_{i} < 0$ there is no finite limit at zero, so that case is impossible for continuous F. So F—the rule for group ratios—is a weighted geometric mean with non-negative weights.

This implies that $\hbox {ag}$ is a Weighted Geometric Rule. Let v be a world with positive individual credences $p_{1},\,\ldots ,\,p_{n}$, and say the group credence in that world is p. Then for any other world w with individual credences $q_{1},\,\ldots ,\,q_{n}$ the ratios $q_{i}/p_{i}$ are defined, and the group ratio is a weighted geometric mean of those ratios. So the group credence in w is

$$p \cdot \left( \frac{q_{1}}{p_{1}}\right) ^{a_{1}} \cdot \cdots \cdot \left( \frac{q_{n}}{p_{n}}\right) ^{a_{n}}$$

which, redistributing parentheses, is just the weighted geometric mean $q_{1}^{a_{1}}\cdot \cdots \cdot q_{n}^{a_{n}}$ multiplied by a constant normalization factor. $\square$

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Jeffrey Sanford Russell
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John Hawthorne
- 1
Lara Buchak
- 2

1.Magdalen CollegeUniversity of OxfordOxfordUK
2.University of California, BerkeleyBerkeleyUSA

Groupthink

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