Social Choice and Individual Values - Terminology

Terminology

The book defines a few terms and logical symbols used thereafter and their applied empirical interpretation (pp. 11–19, 23). Key among these is the "vote" ('set of orderings') of the society (more generally "collectivity") composed of individuals (“voters” here) in the following form:

• Voters, a finite set with at least two members, indexed as i = 1, 2, ... n.
• Commodities, the objects of choice (things that voters might want, goods and services), both private and public (municipal services, statecraft, etc.).
• A social state is a specification (formally, an element of a vector) of a distribution among voters of commodities, labor, and resources used in their productions.
• The set of social states, the set of all 'social states', indexed as x, y, z, . ., with at least three members.
• A (weak) ordering, a ranking by a voter of all 'social states' from more to less preferred, including possible ties.
• The set of 'orderings', the set of all n orderings, one ordering per voter.

The ordering of each voter ranks social states, including the distribution of commodities (possibly based on equity, by whatever metric, or any other consideration), not merely direct consumption by that voter. So, the ordering is an "individual value," not merely, as in earlier analysis, a purely private "taste." Arrow notes that the distinction is not sharp. Resource allocation is specified in the production of each social state in the ordering.

The comprehensive nature of commodities, the set of social states, and the set of orderings was noted by early reviewers.

The two properties that define any ordering of the set of objects in question (all social states here) are:

• connectedness (completeness): All the objects in the set are included in the ranking (no "undecideds" nor "abstentions") and

• transitivity: If, for any objects x, y, and z in the set, x is ranked at least as high as y and y is ranked at least as high as z, then x is ranked at least as high as z.

The earlier definition of an ordering implies that any given ordering entails one of three responses on the "ballot" as between any pair of social states (x, y): better than, as good as, or worse than (in preference ranking). (Here "as good as" is an "equally-ranked," not a "don't know," relation.)

An ordering of a voter is denoted by R. That ordering of voter i is denoted with a subscript as .

If voter i changes orderings, primes distinguish the first and second, say compared to ' . The same notation can apply for two different hypothetical orderings of the same voter.

The interest of the book is in amalgamating sets of orderings. This is accomplished through a 'constitution'.

• A constitution (or social welfare function) is a voting rule mapping each (of at least one) set of orderings onto a social ordering, a corresponding ordering of the set of social states that applies to each voter.

A social ordering of a constitution is denoted R. (Context or a subscript distinguishes a voter ordering R from the same symbol for a social ordering.)

For any two social states x and y of a given social ordering R:

x P y is "social preference" of x over y (x is selected over y by the rule).

x I y is "social indifference" between x and y (both are ranked the same by the rule).

x R y is either "social preference" of x over y or "social indifference" between x and y (x is ranked least as good as y by the rule).

A social ordering applies to each ordering in the set of orderings (hence the "social" part and the associated amalgamation). This is so regardless of (dis)similarity between the social ordering and any or all the orderings in the set. But Arrow places the constitution in the context of ordinalist welfare economics, which attempts to aggregate different tastes in a coherent, plausible way.

Arrow (pp. 15, 26–28) shows how to go from the social ordering R for a given set of orderings to a particular 'social choice' by specifying:

• the environment, S: the subset of social states that is (hypothetically) available (feasible as to resource quantity and productivity), not merely conceivable.

The social ordering R then selects the top-ranked social state(s) from the subset as the social choice set.

Less informally, the social choice function is the function mapping each environment S of available social states (at least two) for any given set of orderings (and corresponding social ordering R) to the social choice set, the set of social states each element of which is top-ranked (by R) for that environment and that set of orderings.

The social choice function is denoted C(S). Consider an environment that has just two social states, x and y: C(S) = C. Suppose x is the only top-ranked social state. Then C = {x}, the social choice set. If x and y are instead tied, C = {x, y}. Formally (p. 15), C(S) is the set of all x in S such that, for all y in S, x R y ("x is at least as good as y").

The next section invokes the following. Let R and R' stand for social orderings of the constitution corresponding to any 2 sets of orderings. If R and R' for the same environment S map to the same social choice(s), the relation of the identical social choices for R and R' is represented as: C(S) = C'(S).