1
The General Will, Group Decision Theory, and Indeterminacy1
Mathias Risse
John F. Kennedy School of Government
Harvard University
April 11, 2003
Abstract: Suppose group deliberation ends without unanimity. Is there a theory of group
decision making that specifies for any set of circumstances a uniquely most reasonable
decision rule? If not, some people will be "losers" though they would not have been had
another, equally reasonable rule been adopted. I argue that this uniqueness claim is false
by showing that for neither preference aggregation nor Bayesian aggregation does there
exist a uniquely most reasonable decision rule.
1. Rousseau and Contemporary Group Decision Theory
1.1 Rousseau's Social Contract asks how individuals can live together without losing
their freedom: Is there "a form of association which defends and protects with all
common forces the person and goods of each associate," Rousseau wonders, "and by
1
Prepared for a special session on the work of Richard Jeffrey during the 26th International Wittgenstein
Symposium in Kirchberg (Austria), August 3-9, 2003. My assignment is to address Jeffrey's group decision
theory. He did not work much on those areas. Around 1970, Jeffrey worked on a book tentatively called
"The New Utilitarianism," but later abandoned the project. The result were two papers on interpersonal
comparisons of utility, reprinted in The Art of Judgment , a collection of his papers. He returned to group
decision theory towards the end of his life, mostly in joint work with Matthias Hild and myself about the
epistemic foundations of game theory and Bayesian aggregation. My concern in this paper is to put our
work on Bayesian aggregation into the context of a larger question about group rationality.
2
means of which each one, while uniting with all, nevertheless obeys only himself and
remains as free as before" (I.6)? Rousseau wrote at the dawn of a golden age of
reflection on group rationality. The last 50 years also witnessed a dramatic increase in
understanding of formal aspects of group rationality. Yet curiously, we understand the
mathematics of social choice better than its philosophy. But first let me continue with
Rousseau for a bit.
Rousseau's solution to his puzzle is "an act of association" that "produces a moral
and collective body composed of as many members as there are voices in the assembly,
which receives from this same act its unity, its common self, its life and its will" (I.6). As
I understand Rousseau, this common self belongs to each individual, but is also associated
with the political community in the same way in which selves are associated with
persons. So Rousseau's problem admits of a solution if that self is each individual's true
self. For then each true self is identical to the self that belongs to the body politic. The
general will of the community, then, is the will associated with this common self in the
same way in which wills are associated with individual selves. Therefore, whatever the
community does in accordance with the general will expresses the will of each true self,
and so is binding on each individual while leaving him free. To complete the answer, we
must show that the general will is constructed from individual beliefs and values by a
process rendering it the will associated with each individual's true self. To this end, I
interpret the general will as the decisions made by the community with its deliberation
subject to suitable constraints.
However, even constrained deliberation is not guaranteed to reach unanimity.
Decisions must often be made under circumstances of radical and persistent
disagreement; conflicts of values that cannot be realized in single lives, single decisions
of deliberating bodies, or single constitutions require decisions in spite of irresolvable
disagreement. So we need an account of what to do when deliberation ends
inconclusively. We might then still reach a general will, provided that for any set of
circumstances under which a group might find itself in disagreement there is a uniquely
reasonable rule which delivers that will. So we need to investigate the following thesis:
Uniqueness : for any given set of circumstances, there is a uniquely reasonable
decision rule the group should adopt if deliberation fails to deliver a decision.
3
If Uniqueness fails, there will be multiple inconsistent but reasonable methods under the
same conditions: some will Alose" because one method was adopted, though they may
have Awon@ had another, equally reasonable, rule been used. I argue that Uniqueness is
false. Frequently, we cannot know the general will if deliberation ends inconclusively,
and may also not have any reason to think that there actually is a general will in such
situations.
1.2 We are now in the midst of contemporary group decision theory. Group decision
theory constitutes no unified field. Instead, there are separate literatures on areas like fair
division, game-theoretic scenarios, and types of aggregation: aggregation of preference
rankings, expected utilities with shared probability functions, and, recently, Bayesian
aggregation of utilities and subjective probabilities. My concern is with aggregation.
Thinking about aggregation leads to puzzles. Suppose a department must make a
hiring decision. All members rank the candidates. How should they determine a group
ranking? One way of doing so is by aggregating ordinal rankings, using proposals to be
discussed in section 2. But suppose somebody suggests a 100-point system: each voter
assigns from 0 to 100 points to each applicant, and then they average. This proposal uses
more than ordinal information. Should we adopt this rule instead? Can we distinguish
conditions under which ordinal rankings are appropriate from conditions under which
other rankings are? Such questions are hard to answer and call for a theory that assesses,
first, the conditions under which particular kinds of rankings are appropriate (e.g., ordinal
or point systems); second, what specific rule(s) is (are) appropriate for the specific kinds
of rankings; and third, what the criteria for "appropriateness" are, respectively. While we
do not possess any such theory, I use this framework to refute Uniqueness. Section 2
argues that Uniqueness is false if we aggregate ordinal rankings, and section 3 argues this
for Bayesian aggregation. Or, to put the claim in terms from the 18th century, we have no
particular reason to think there is a general will unless deliberation ends with unanimity.2
2
Due to the length constraints, the argument in this paper is sketchy. Some of the arguments can be found
at other places as well, although I have never brought them together in a unified argument opposing
Uniqueness. Section 2 draws on my "Arrow's Theorem, Indeterminacy, and Multiplicity Reconsidered," in
4
2. Preference Aggregation: Borda vs. Condorcet
2.1 Suppose we must rank m candidates. The Condorcet proposal does so by looking at
all m(m-1)/2 pairs among m candidates, selecting one or more of the m! rankings in light
of these pariwise votes. We consider all pairwise votes as "data" and ask which ranking
of the candidates these data support best. The Condorcet proposal selects a ranking
supported by a maximal number of votes in all pairwise elections . For each of the m!
rankings, we look at the m(m-1)/2 pairs of candidates and count the number of voters that
rank the respective candidates in the same way as that pair and thus support the ranking.
To illustrate, suppose a group of 48 must rank A, B, and C. 10 rank them (A, B, C), 12
(A, C, B), 5 (B, A, C), 7 (B, C, A), 3 (C, B, A) and 11 (C, A, B). So here we have 3! = 6
rankings and 3(3-1)/2 = 3 pairwise votes. The ranking with the highest support in
pairwise elections is (A, C, B): In A vs. B, 33 people support it (33 people rank A over
B), in B vs. C 26 people, and in A vs. C 27. Thus 86 votes support (A, C, B) in pairwise
votes, compared to 82 for (A, B, C), 64 for (B, A, C), 58 for (B, C, A), 62 for (C, B, A),
and 80 for (C, A, B). So (A, C, B) is a ranking with maximal support .
Consider now the Borda count . Again a group must rank m candidates. First each
person ranks them, assigning 0 to her last-ranked, 1 to her second to the last ranked, etc.,
until she assigns m-1 to her first ranked. Then a sum over these numbers is formed for
each candidate, which is that candidate's Borda count . The group ranks the candidates by
decreasing Borda counts. Suppose we have three people (1, 2, 3) and four candidates (A,
B, C, D). Person 1 ranks them (A, B, C, D), 2 (B, C, D, A), 3 (A, B, D, C). The social
ranking is (B, A, C, D) because A obtains six points, B seven, C three and D two.
Another characterization of the Borda count is that it ranks candidates by their average
position across rankings. Like the Condorcet proposal, the Borda count uses as "input" all
m(m-1)/2 pairwise votes. But Borda asks about the support for each of the m candidates
in all rankings , whereas Condorcet asks about the support for each of the m! rankings in
all pairwise elections .
Ethics 111 (2001), pp 706-734; similarly, section 3 draws on my article "Bayesian Group Agents and Two
Modes of Aggregation," forthcoming in Synthese (2003). That article, in turn, draws on joint work with
Richard C. Jeffrey and Matthias Hild: "Preference Aggregation after Harsanyi," forthcoming in Social
Choice and Welfare (2003).
5
2.2 How to decide between Condorcet and Borda? On earlier occasions I argued for three
claims: first, arguments for majority rule tend to be limited to the case of two candidates;
second, generalizations of these arguments support the Condorcet proposal, and thus that
proposal is what we should mean in general by majoritarian decision making; and third,
none of these arguments is decisive against the Borda count. I illustrate this for one
claim. One argument for majority rule is
Condorcet's Jury Theorem: Supposes it makes sense to speak of being right or
wrong about political decisions. Suppose n agents choose between two options;
that each has a probability of p>_ of being right; and that their probabilities are
independent of each other (i.e., they make up their minds for themselves). Then,
as n grows, the probability of a majority's being right approaches 1.
Formulated like this, the theorem only applies to the case of two candidates. However,
there is a generalization, and this generalization picks out the rankings with maximal
support. The procedure is to go through all m! rankings and calculate the conditional
probability of the pairwise voting results given that the respective ranking is the correct
one. The ranking that bestows maximum likelihood to the voting result is chosen.3
According to the theorem, rankings selected by Condorcet bestow the highest
likelihood on the election result. Yet the option ranked highest by Borda is the single
option that, if the best, bestows highest probability upon the voting results, provided the
voters' competence p is close to _. Borda is concerned with ranking options in terms of
their rightness, and Condorcet with finding the right ranking . For values of p close to _,
this difference carries over to the epistemic scenario, with Condorcet searching for the
ranking with maximal likelihood, and Borda ranking the options in terms of their
maximal likelihood. The Borda count has its counterpart to this theorem reflecting this
difference and regards arguments drawing on it as non-starters. Other arguments in
support of the Condorcet proposal fare similarly, and mutatis mutandis for Borda. As far
as rules for aggregating rankings are concerned, Condorcet and Borda are on a par:
3
This paragraph draws on Peyton Young, "Condorcet's Theory of Voting", American Political Science
Review 82 (1988): 1231-1244,
6
neither has conclusive arguments against the other, whereas both are reasonable rules.
Thus we have sketched an argument against Uniqueness if we are aggregating ordinal
rankings.
3. Bayesian Aggregation: Ex Post vs. Ex Ante
3.1 Our second task is to show that Uniqueness fails for Bayesian aggregation. A
Bayesian agent is an agent described by theories of expected utility theory that take
probabilities to be subjective. Suppose we have a group of such agents, and suppose they
would like for their group as such also to be a Bayesian agent. Moreover, they also would
like for group decisions, and thus for group preferences, probabilities, and utilities, to be
aggregated from the respective individual entities in such a way that at least unanimous
agreements are preserved.
Mongin (1995), using the Savage framework, shows that Bayesian aggregation
satisfying some reasonable condition is available only to fairly homogenous groups.4
Mongin's result holds in ex ante frameworks, where restrictions on group expected
utilities are formulated in terms of individual expected utilities. A typical ex ante rule
may include a restriction of the kind "If each individual's expected utility is higher for a
than for b, the group should rank a higher than b." Mongin's result fails in the ex post
framework, which aggregates probabilities and utilities separately. Typical ex post
aggregation rules may satisfy the restriction "If each individual's utility for a is higher
than the utility for b , then the group utility for a should be higher than the group utility
for b ." A parallel condition for probabilities may hold.
Yet ex post models come with trouble of their own: They display a peculiar
dependence on the level of detail used in describing the decision problem. Suppose a
group of Bayesians aggregate probabilities and utilities according to some ex post rule.
Suppose they analyze their situation in more detail without anybody changing her mind
about the ranking of the actions. Then the group as a whole should not change "its" mind
about that ranking when the more fine-grained utilities and probabilities are aggregated.
However, such a reversal may happen under ex post rules. So we must choose between
two modes of aggregation each of which has its problems.
4
Philippe Mongin, "Consistent Bayesian Aggregation." Journal of Economic Theory (1995); 66, 313-351
7
3.2 On what grounds can we choose? Suppose Ante and Post defend those approaches.
Both need, first, a positive argument for their positions; second, a negative argument
attacking the opponent's view; and third, an argument why the aforementioned worries
about their account are not troublesome. Let us sketch these positions.
Ante's positive argument is that for agents to be taken seriously as participants in
the decision process means being accepted as decision makers , rather than probability
and utility providers. Agents' decisions, however, are based on preferences, or
expectations. Thus restrictions on the decision process should be formulated in terms of
expectations, that is, in the ex ante fashion. Ante's complaint against Post is that the
instability result shows that the ex post approach is futile. Since we never know for sure
whether a more fine-grained look at the same situation would reverse group preferences,
decisions based on the ex post approach are ill-founded. As far as Mongin's theorem is
concerned, Ante may bite the bullet by saying that Mongin's theorems teach us a lesson
about group decision making. Aristotle was right when, in the Politics , he thought of a
political community as a community of like-minded persons. Rational decision making is
unavailable to groups with little in common.
Post proceeds as follows: It is in accordance with the Bayesian credo to
distinguish facts from values. If the group is the decision maker, its method must be to
aggregate probabilities and utilities separately. Agents are taken seriously if their
epistemic views and their values are taken seriously. This view takes account of the
reality of the decision structure while assigning individuals the appropriate place as
members of the group within that structure. Post's complaint against Ante is based on
Mongin's theorem. Post thinks of this result as a reductio of the ex ante approach. For
rational social choice cannot be restricted to homogeneous groups, leaving
inhomogeneous groups without any advice. Post has two replies to the instability result:
On the one hand, he may deny its relevance and argue that in all decision situations that
we encounter in our lives, even in strongly idealized ones, there will be a point when we
have reached the most fine-grained relevant refinement (or at least, the most fine-grained
refinement we are capable of considering). Though it is hard to discuss this any further at
the general level, Post might continue, it will be clear enough in each given scenario. On
8
the other hand, it is also open to Post to bite the bullet and argue that groups are fragile
decision makers, but that this insight, far from refuting the ex post approach, teaches us
something about the nature of groups as decision makers. While, once again, this
argument is sketchy, I claim that neither position has resources to refute the other. This
establishes the falsity of Uniqueness of Bayesian aggregation.
4. Conclusion
Rousseau's general will, I suggested, should be understood as the decisions of the
respective group if its deliberation is subject to suitable constraints. Since even such
deliberation cannot ensure unanimity, we must ask how a group should reach a decision
if deliberation ends inconclusively. In many cases, aggregation of individual views is
needed. For us to have access to the general will, there would have to be a unique
aggregation rule under all conditions under which aggregation is required. I have argued
for two kinds of settings that there is no such uniqueness: there is no uniquely reasonable
decision rule if ordinal rankings are aggregated, and there is no uniquely reasonable rule
if both utilities and subjective probability are aggregated. Thus in many cases some
individuals will be "losers" in the decision process although they would not have been if
had another, equally reasonable rule been adopted. This, I claim, is an inescapable feature
of group decision making.
download The General Will, Group Decision Theory, and Indeterminacy1
