Conditions and Theorem
A constitution might seem to be a promising alternative to dictatorship and vote-immune social convention or external control. Arrow describes the connectedness of a social ordering as requiring only that some social choice be made from any environment of available social states. Since some social state will prevail, this is hard to deny (especially with no place on the ballot for abstention). The transitivity of a social ordering has an advantage over requiring unanimity (or much less) to change between social states if there is a maladapted status quo (that is, one subject to "democratic paralysis"). Absent deadlock, transitivity crowds out any reference to the status quo as a privileged default blocking the path to a social choice (p. 120).
Arrow proposes the following "apparently reasonable" conditions to constrain the social ordering(s) of the constitution (pp. 25, 96-97).
- 1. Universal (Unrestricted) Domain U (subsequently so called): Every logically possible set of orderings maps to its own social ordering.
Each voter is permitted by the constitution to rank the set of social states in any order, though with only one ordering per voter for a given set of orderings.
| Arrow refers to a constitution satisfying this condition as collective rationality. It can be compared to the rationality of a voter ordering. But the prescription of collective rationality, which Arrow proposes, is distinct from the descriptive use of a voter ordering, which he deploys. Hence, his denial at the end of the book that collective rationality is "merely an illegitimate transfer from the individual to society." (p. 120) |
- 2. Independence of Irrelevant Alternatives I: Let , ..., and ', ..., ' be 2 sets of orderings in the constitution. Let S be a subset of hypothetically available (not merely conceivable) social states, say x and y, from the set of social states. For each voter i, let the ranking of x and y be the same for and for ' . (Different voters could still have different rankings of the 2 social states.) Then the social orderings for the 2 respective sets of orderings select the same state(s) from the subset as the social choice.
| Condition I: Let , ..., and ', ..., ' be 2 sets of orderings in the constitution. Let S be any subset of hypothetically available social states from the set of social states. For each voter i and for each pair of x and y in S, let x y if and only if x ' y. Then the social choice functions for the 2 respective sets of orderings map to an identical social choice set: C(S) = C'(S). |
| This identical mapping happens even with differences in rankings of any voter between the two sets of orderings outside that subset of social states. Consider a hypothetical “run-off vote” between say only 2 available social states. The social choice is associated with the sets of rankings for that subset, not with rankings of unavailable social states beyond the subset. Thus, that social choice for the subset is unaffected by say a change in orderings only beyond the subset. |
Arrow describes this condition as an extension of ordinalism with its emphasis on prospectively observable behavior (for the subset in question). He ascribes practical advantage to the condition from "every known electoral system" satisfying it (p. 110).
- 3. The (weak) Pareto Principle P: For any x and y in the set of social states, if all prefer x over y, then x is socially selected over y.#
| Condition P: For any x and y in the set of social states, if, for every voter i, x y, then x P y. |
| As Sen suggests, Pareto unanimity (with universal domain) overrides any social convention selecting some social state. |
The conditions, particularly the second and third, may seem minimal, but jointly they are harsh, as may be represented in either of two ways.
- Arrow’s Theorem : The 3 conditions of the constitution imply a dictator who prevails as to the social choice whatever that individual's preference and those of all else.
An alternate statement of the theorem adds the following condition to the above:
- 4. Nondictatorship D: No voter in the society is a dictator. That is, there is no voter i in the society such that for every set of orderings in the domain of the constitution and every pair of distinct social states x and y, if voter i strictly prefers x over y, x is socially selected over y.
| Condition D: There is no voter i in {1, ..., n} such that for every set of orderings in the domain of the constitution and every pair of social states x and y, x y implies x P y. |
- Arrow's Theorem : The constitution is impossible, that is, the 4 conditions of a constitution imply a contradiction.
| Each voter has an ordering (by attribution). Yet a set of orderings used as an argument of the voting rule does not carry over to a social ordering, with a corresponding loss of social adaptivity and constitutional generality, whether descriptive or prescriptive. |
# Pareto is stronger than necessary in the proof of the theorem that follows above. But it is invoked in Arrow (1963, ch. VIII) for a simpler proof than in Arrow (1951). In the latter, Arrow uses 2 other conditions, that with (2) above imply Pareto (1963, p. 97; 1987, p. 127):
- 3a. Monotonicity M (Positive Association of Individual and Social Values), as in Arrow (1987, p. 125): For a given set of orderings with social ordering R, such that state x is socially preferred to state y, if the preference for x rises in some individual ordering(s) and falls in none, x is also socially preferred to y in the social ordering for the new set of orderings.
Arrow (1951, p. 26) describes social welfare here as at least not negatively related to individual preferences.
- 3b. As defined by Arrow (1951, pp. 28–29), an Imposed Constitution is a constitution such that for some alternative social states x and y and for any set of orderings , ..., in the domain and corresponding social ordering R, the social ranking is x R y.
- Non-imposition N (Citizens' Sovereignty): A constitution is not to be imposed.
Under imposition, for every set of orderings in the domain, the social ranking for at least one x and y is only x R y. The vote makes no difference to the outcome.
| For the special case of all preferring y over x, the vote still makes no difference. If the invariant social ranking applies to only one pair of distinct social states, the constitution would violate N. In this respect, as a representation of excluding convention, N is thorough. |
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