How many possible preference orderings

Bartholdi, Tovey, and Trick showed that, due to computational complexity, some social choice rules are resistant to strategic manipulation: it may be an NP-hard problem for a voter to determine how to vote strategically. Dowding and van Hees have argued that not all forms of strategic voting are normatively problematic. Sincere manipulation occurs when a voter i votes for a compromise alternative whose chances of winning are thereby increased and ii genuinely prefers that compromise alternative to the alternative that would otherwise win.

For example, in the US presidential election, supporters of Ralph Nader a third-party candidate with little chance of winning who voted for Al Gore to increase his chances of beating George W.

Bush engaged in sincere manipulation in the sense of i and ii. Plurality rule is susceptible to sincere manipulation, but not vulnerable to insincere manipulation. An implicit assumption so far has been that preferences are ordinal and not interpersonally comparable: preference orderings contain no information about each individual's strength of preference or about how to compare different individuals' preferences with one another.

In voting contexts, this assumption may be plausible, but in welfare-evaluation contexts—when a social planner seeks to rank different social alternatives in an order of social welfare—the use of richer information may be justified. Sen b generalized Arrow's model to incorporate such richer information.

Any welfare function on X induces an ordering on X , but the converse is not true: welfare functions encode more information. The output of a SWFL is similar to that of a preference aggregation rule again, we do not build the completeness or transitivity of R into the definition [ 9 ] , but its input is richer. What we gain from this depends on how much of the enriched informational input we allow ourselves to use in determining society's preferences: technically, it depends on our assumption about measurability and interpersonal comparability of welfare.

By assigning real numbers to alternatives, welfare profiles contain a lot of information over and above the profiles of orderings on X they induce. In particular, many different assignments of numbers to alternatives can give rise to the same orderings. But we may not consider all this information meaningful. Some of it could be an artifact of the numerical representation.

The two profiles might be seen as alternative representations of the exact same information, just on different scales. To express different assumptions about which information is truly encoded by a profile of welfare functions and which information is not and should thus be seen, at best, as an artifact of the numerical representation , it is helpful to introduce the notion of meaningful statements.

Some examples of statements about individual welfare that are candidates for meaningful statements are the following List b; see also Bossert and Weymark Section 5 :. Arrow's view, as noted, is that only intrapersonal level comparisons are meaningful, while all other kinds of comparisons are not.

Of the three kinds of comparison statements introduced above, the meaningful ones are those that are invariant in each equivalence class. Arrow's ordinalist assumption can be expressed as follows:. This renders intrapersonal level and unit comparisons meaningful, but rules out interpersonal comparisons and zero comparisons.

Interpersonal level comparability is achieved under the following enriched variant of ordinal measurability:. Interpersonal unit comparability is achieved under the following enriched variant of cardinal measurability:.

Here, the welfare functions in each profile can be re-scaled and shifted without informational loss, but the same scalar multiple though not necessarily the same shifting constant must be used for all individuals, thereby rendering interpersonal unit comparisons meaningful.

Zero comparisons, finally, become meaningful under the following enriched variant of ordinal measurability List :. This allows arbitrary stretching and squeezing of individual welfare functions without informational loss, provided the welfare level of zero remains fixed , thereby ensuring zero comparability.

Several other measurability and interpersonal comparability assumptions have been discussed in the literature. The following ensures the meaningfulness of interpersonal comparisons of both levels and units:.

Lastly, intra- and interpersonal comparisons of all three kinds level, unit, and zero are meaningful if we accept the following:. Which assumption is warranted depends on how welfare is interpreted. If welfare is hedonic utility , which can be experienced only from a first-person perspective, interpersonal comparisons are harder to justify than if welfare is the objective satisfaction of subjective preferences or desires the desire-satisfaction view or an objective good or state an objective-list view e.

The desire-satisfaction view may render interpersonal comparisons empirically meaningful by relating the interpersonally significant maximal and minimal levels of welfare for each individual to the attainment of his or her most and least preferred alternatives , but at the expense of running into problems of expensive tastes or adaptive preferences Hausman Resource-based, functioning-based, or primary-goods-based currencies of welfare, by contrast, may allow empirically meaningful and less morally problematic interpersonal comparisons.

Once we introduce interpersonal comparisons of welfare levels or units, or zero comparisons, there exist possible SWFLs satisfying the analogues of Arrow's conditions as well as stronger desiderata. In a welfare-aggregation context, Arrow's impossibility can therefore be traced to a lack of interpersonal comparability. Arrow's conditions and theorem can be restated as follows:. Universal domain : The domain of F is the set of all logically possible profiles of individual welfare functions.

Some examples of such SWFLs come from political philosophy and welfare economics. While maximin rank-orders social alternatives in terms of the welfare level of the worst-off individual alone, its lexicographic extension leximin , which was endorsed by Rawls himself, uses the welfare level of the second-worst-off individual as a tie-breaker when there is tie at the level of the worst off, the welfare level of the third-worst-off individual as a tie-breaker when there is a tie at the second stage, and so on.

While substantively less compelling than maximin or utilitarian rules, head-count rules require only zero-comparability of welfare List An important conclusion, therefore, is that Rawls's difference principle, the classical utilitarian principle, and even the head-count method of poverty measurement can all be seen as solutions to Arrow's aggregation problem that become possible once we go beyond Arrow's framework of ordinal, interpersonally non-comparable preferences.

Under CFC, one can provide a simultaneous characterization of Rawlsian maximin and utilitarianism Deschamps and Gevers It uses two additional axioms.

Theorem Deschamps and Gevers : Under CFC, any SWFL satisfying universal domain, ordering, the strong Pareto principle, independence of irrelevant alternatives, anonymity as in May's theorem , minimal equity, and separability is either leximin or of a utilitarian type meaning that, except possibly when there are ties in sum-total welfare, it coincides with the utilitarian SWFL defined above.

Prioritarianism requires RFC and not merely CFC because, by design, the prioritarian social ordering for any welfare profile is not invariant under changes in welfare levels shifting. The present welfare-aggregation framework has been applied to several further areas. It has been generalized to variable-population choice problems, so as to formalize population ethics in the tradition of Parfit Here, we must rank-order social alternatives e.

Let N x denote the set of individuals existing under alternative x. For example, the set N x could differ from the set N y , when x and y are distinct alternatives this generalizes our previous assumption of a fixed set N. The variable-population case raises questions such as whether a world with a smaller number of better-off individuals is better than, equally good as, or worse than a world with a larger number of worse-off individuals.

The focus here is on axiological questions about the relative goodness of such worlds, not normative questions about the rightness or wrongness of bringing them about.

Parfit and others argued that classical utilitarianism is subject to the repugnant conclusion : a world with a very large number of individuals whose welfare levels are barely above zero could have a larger sum-total of welfare, and therefore count as better, than a world with a smaller number of very well-off individuals.

Blackorby, Donaldson, and Bossert e. One solution is the following:. Critical-level utilitarianism avoids the repugnant conclusion when the parameter c is set sufficiently large. It requires stronger measurability of welfare than classical utilitarianism, since it generates a social ordering R that is not generally invariant under re-scaling of welfare units or shifts in welfare levels.

Even the rich framework of RFC would force the critical level c to be zero, thereby collapsing critical-level utilitarianism into classical utilitarianism and making it vulnerable to the repugnant conclusion again.

As Blackorby, Bossert, and Donaldson note,. Thus, in the variable-population case, a more significant departure from the limited informational framework of Arrow's original model is needed to avoid impossibility results.

The SWFL approach has been generalized to the case in which each individual has multiple welfare functions e. In this case, we are faced not only with issues of measurability and interpersonal comparability, but also with issues of inter-opinion or inter-dimensional comparability.

A related literature addresses multidimensional inequality measurement for an introductory review, see Weymark Finally, in the philosophy of biology, the one-dimensional and multi-dimensional SWFL frameworks have been used by Okasha and Bossert, Qi, and Weymark to analyse the notion of group fitness, defined as a function of individual fitness indicators.

A more recent branch of social choice theory is the theory of judgment aggregation. It can be motivated by observing that votes, orderings, or welfare functions over multiple alternatives are not the only objects we may wish to aggregate from an individual to a collective level. In particular, they may have to aggregate individual sets of judgments on multiple, logically connected propositions into collective sets of judgments.

A court may have to judge whether a defendant is liable for breach of contract on the basis of whether there was a valid contract in place and whether there was a breach. An expert panel may have to judge whether atmospheric greenhouse-gas concentrations will exceed a particular threshold by , whether there is a causal chain from greater greenhouse-gas concentrations to temperature increases, and whether the temperature will increase.

Legislators may have to judge whether a particular end is socially desirable, whether a proposed policy is the best means for achieving that end, and whether to pursue that policy.

These problems cannot be formalized in standard preference-aggregation models, since the aggreganda are not orderings but sets of judgments on multiple propositions. The theory of judgment aggregation represents these aggreganda in propositional logic or another suitable logic. Kornhauser and Sager described the following problem.

A structurally similar problem was discovered by Vacca and, as Elster points out, by Poisson A three-judge court has to make judgments on the following propositions:. According to legal doctrine, the premises p and q are jointly necessary and sufficient for the conclusion r. Suppose the individual judges hold the views shown in Table 5. Although each individual judge respects the relevant legal doctrine, there is a majority for p , a majority for q , and yet a majority against r —in breach of legal doctrine.

We can learn another lesson from this example. Relative to the legal doctrine, the majority judgments are logically inconsistent. This observation was the starting point of the more recent, formal-logic-based literature on judgment aggregation beginning with a model and impossibility result in List and Pettit Suppose, for example, an expert panel has to make judgments on three propositions and their negations :.

Here, the set of majority-accepted propositions is inconsistent relative to the constraint of transitivity. A general combinatorial result subsumes all these phenomena. Call a set of propositions minimally inconsistent if it is a logically inconsistent set, but all its proper subsets are consistent.

Proposition Dietrich and List a; Nehring and Puppe : Propositionwise majority voting may generate inconsistent collective judgments if and only if the set of propositions and their negations on which judgments are to be made has a minimally inconsistent subset of three or more propositions. The basic model of judgment aggregation can be defined as follows List and Pettit The propositions on which judgments are to be made are represented by sentences from propositional logic or some other, expressively richer logic, such as a predicate, modal, or conditional logic; see Dietrich We define the agenda , X , as a finite set of propositions, closed under single negation.

Again, for generality, we build no rationality requirement on J such as consistency or completeness into the definition of a judgment aggregation rule. The simplest example of a judgment aggregation rule is propositionwise majority voting. As we have seen, this may produce inconsistent collective judgments. Universal domain : The domain of F is the set of all logically possible profiles of consistent and complete individual judgment sets.

The first three conditions are analogous to universal domain, ordering, and anonymity in preference aggregation. Propositionwise majority voting satisfies all these conditions, except the consistency part of collective rationality. Like other impossibility theorems, this result is best interpreted as describing the trade-offs between different conditions on an aggregation rule. The result has been generalized and strengthened in various ways, beginning with Pauly and van Hees's proof that the impossibility persists if anonymity is weakened to non-dictatorship for other generalizations, see Dietrich and Mongin In judgment aggregation, by contrast, the picture is more complicated.

What matters is not the number of propositions in X but the nature of the logical interconnections between them.

Impossibility results in judgment aggregation have the following generic form: for a given class of agendas, the aggregation rules satisfying a particular set of conditions usually, a domain condition, a rationality condition, and some responsiveness conditions are non-existent or degenerate e. Different kinds of agendas trigger different instances of this scheme, with stronger or weaker conditions imposed on the aggregation rule depending on the properties of those agendas for a more detailed review, see List The significance of combinatorial properties of the agenda was first discovered by Nehring and Puppe in a mathematically related but interpretationally distinct framework strategy-proof social choice over so-called property spaces.

Three kinds of agenda stand out:. A non-simple agenda : X has a minimally inconsistent subset of three or more propositions. A pair-negatable agenda : X has a minimally inconsistent subset Y that can be rendered consistent by negating a pair of propositions in it.

Some agendas have two or more of these properties. The following result holds:. Applied to the preference agenda, this result yields Arrow's theorem for strict preference orderings as a corollary predecessors of this result can be found in List and Pettit and Nehring The literature contains several variants of this theorem.

One variant drops the agenda property of path-connectedness and strengthens independence to systematicity. A second variant drops the agenda property of pair-negatability and imposes a monotonicity condition on the aggregation rule requiring that additional support never hurt an accepted proposition Nehring and Puppe ; the latter result was first proved in the above-mentioned mathematically related framework by Nehring and Puppe A final variant drops both path-connectedness and pair-negatability while imposing both systematicity and monotonicity ibid.

Note also that path-connectedness implies non-simplicity. Therefore, non-simplicity need not be listed among the theorem's conditions, though it is needed in the variants dropping path-connectedness. As in preference aggregation, one way to avoid the present impossibility results is to relax universal domain.

The simplest cohesion condition is unidimensional alignment List c. For any such profile, the majority judgments are consistent: the judgment set of the median individual relative to the left-right ordering will prevail where n is odd. This judgment set will inherit its consistency from the median individual, assuming individual judgments are consistent.

By implication, on unidimensionally aligned domains, propositionwise majority voting will satisfy the rest of the conditions on judgment aggregation rules reviewed above. In analogy with the case of single-peakedness in preference aggregation, several less restrictive conditions already suffice for consistent majority judgments. One such condition introduced in Dietrich and List a, where a survey is provided generalizes Sen's triple-wise value-restriction.

Value-restriction prevents any minimally inconsistent subset of X from becoming majority-accepted, and hence ensures consistent majority judgments. Applied to the preference agenda, value-restriction reduces to Sen's equally named condition.

While the requirement that collective judgments be consistent is widely accepted, the requirement that collective judgments be complete in X is more contentious. In support of completeness, one might say that a given proposition would not be included in X unless it is supposed to be collectively adjudicated. Against completeness, one might say that there are circumstances in which the level of disagreement on a particular proposition or set of propositions is so great that forming a collective view on it is undesirable or counterproductive.

Several papers offer possibility or impossibility results on completeness relaxations e. Given consistent individual judgment sets, unanimity rule guarantees consistent collective judgment sets, because the intersection of several consistent sets of propositions is always consistent.

The reason is combinatorial: any k distinct supermajorities of the relevant size will always have at least one individual in common. So, for any minimally inconsistent set of propositions which is at most of size k to be majority-accepted, at least one individual would have to accept all the propositions in the set, contradicting this individual's consistency Dietrich and List a; List and Pettit Conclusion-based rules, finally, produce consistent collective judgment sets by construction, but always leave non-conclusions undecided.

For the same agendas that lead to the impossibility result reviewed in Section 5. The downside of oligarchic aggregation rules is that they either lapse into dictatorship or lead to stalemate, with the slightest disagreements between oligarchs resulting in indecision since every oligarch has veto power on every proposition.

Recall that systematicity combines an independence and a neutrality requirement. Relaxing only neutrality does not get us very far, since for many agendas there are impossibility results with independence alone, as illustrated in Section 5. One much-discussed class of aggregation rules violating independence is given by the premise-based rules. Informally, majority votes are taken on the premises, and the collective judgments on all other propositions are determined by logical implication.

If the premises constitute a logical basis for the entire agenda, a premise-based rule guarantees consistent and absent ties complete collective judgment sets. The present definition follows List and Pettit ; for generalizations, see Dietrich and Mongin The procedural and epistemic properties of premise-based rules are discussed in Pettit ; Chapman ; Bovens and Rabinowicz ; Dietrich ; List A generalization is given by the sequential priority rules List b; Dietrich and List a.

If the majority judgment on p is consistent with collective judgments on prior propositions, this majority judgment prevails; otherwise the collective judgment on p is determined by the implications of prior judgments. By construction, this guarantees consistent and absent ties complete collective judgments. However, it is path-dependent : the order in which propositions are considered may affect the outcome, specifically when the underlying majority judgments are inconsistent.

For example, when this aggregation rule is applied to the profiles in Tables 5, 6, and 7 but not 8 , the collective judgments depend on the order in which the propositions are considered.

Thus sequential priority rules are vulnerable to agenda manipulation. Similar phenomena occur in sequential pairwise majority voting in preference aggregation e. Distance-based rules can be interpreted as capturing the idea of identifying compromise judgments. Unlike premise-based or sequential priority rules, they do not require a distinction between premises and conclusions or any other order of priority among the propositions. As in preference aggregation, the cost of relaxing independence is the loss of strategy-proofness.

The conjunction of independence and monotonicity is necessary and sufficient for the non-manipulability of a judgment aggregation rule by strategic voting Dietrich and List c; for related results, see Nehring and Puppe Thus we cannot generally achieve strategy-proofness without relaxing either universal domain, or collective rationality, or unanimity preservation, or non-dictatorship.

In practice, we must therefore look for ways of rendering opportunities for strategic manipulation less of a threat. As should be evident, social choice theory is a vast field. Areas not covered in this entry, or mentioned only in passing, include: theories of fair division how to divide one or several divisible or indivisible goods, such as cakes or houses, between several claimants; e.

History of social choice theory 1. Three formal arguments for majority rule 2. Preference aggregation 3. Welfare aggregation 4. Judgment aggregation 5. Arrow himself held the view that interpersonal comparison of utilities has no meaning and … that there is no meaning relevant to welfare comparisons in the measurability of individual utility. Three formal arguments for majority rule To introduce social choice theory formally, it helps to consider a simple decision problem: a collective choice between two alternatives.

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