The Possibility of Social Choice for 3 Alternatives

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Key Words: social choice, Arrow, Condorcet, voting, paradox of voting, algorithm, social welfare function, impossibility theorem, general possibility theorem

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In this paper, our goal is not to make an incremental contribution to the field of social choice, but to make a revolutionary one. This paper is not about offering a workaround for Arrow’s Impossibility Theorem, but is about overturning it. In so doing we accept all of Arrow’s framework except that part which is counterintuitive and flawed. We show that it is possible to have a Social Welfare Function (SWF) which complies with Arrow’s criteria. The flawed part of Arrow’s theory has to do with his treatment of ties. Arrow only considers ties among alternatives, but the range of the SWF consists of orderings. In a real sense, instead of voting for individual candidates or alternatives, the voters are voting for orderings. Therefore, ties among orderings must be considered. When they are, SWFs are possible and demonstrable (Lawrence, 1998).

We assume alternatives of the form a, b, c ... x, y, z; a preference relationship, P, an indifference relationship, I and a “preference or indifference” relationship, R. We assume a set of voter/consumers each of whom has an ordering over a given set of m alternatives characterized by a list of preferences; or preferences and indifferences. For example, aPib Pic would characterize a preference ordering over an alternative set of three alternatives by the ith voter/consumer. aIib Pic would characterize a preference and indifference ordering over the set.
We assume a SWF which is a mapping from the set of individual orderings to a social ordering. P, I and R without subscripts represent social orderings. Therefore, a social ordering would be of the form aPbPc or aPbIc or a set of consistent binary orderings of the form {aRb, bRa, aRc, cRa, bRc, cRa}, for example. An expresssion of the form aRbRc is meaningless since we need to know both aRb and bRa, for example, to maintain a 1-1 relationship between P and I information and R information. We assume a universal SWF which means that there is a mapping from every possible combination of individual orderings (the domain) to one of the set of every possible social ordering (the range). Each element of the range represents a potential social ordering. Therefore, a social ordering is assigned to each domain point by the SWF.
Arrow’s (1951, p. 13) Axiom 1 states: “For all x and y, either xRy or yRx.” A relation R which satisfies Axiom 1 is said to be complete and reflexive since xRx. The “or” in the definition is the inclusive or so that “the word ‘or’ in the statement of Axiom 1 does not exclude the possibility of both xRy and yRx. ” We write “both xRy and yRx” as {xRy, yRx}.
The choice function, C(S), is defined by Arrow (p. 15) as follows: “C(S) is the set of all alternatives in S such that, for every y in S, sRy. ” As such it can be used to specify ties among alternatives if the set, C(S), contains more than one element. Sen (1970, p. 48) says, “Arrow's impossibility theorem is precisely a result of demanding social orderings as opposed to choice functions.” In other words, if the solutions required were simply alternatives, Arrow’s Impossibility Theorem would not apply. Since Arrow only uses it in his specification of the Condition of the Independence of Irrelevant Alternatives, it would have been more natural (and certainly stronger) to define C(S) as an ordering over a subset instead of the highest ranking alternative or set of alternatives in a subset. We define an ordering function herein which strengthens the Condition of Independence of Irrelevant Alternatives and allows for ties among orderings as well as ties among alternatives.
The three possible range assignments according to Arrow’s Axiom 1 are xRy, yRx, {xRy, yRx}. Arrow goes on to define xRy AND yRx as xIy and not yRx as xPy. Arrow assumes a knowledge of P and I. Therefore, he should have defined R in terms of P and I instead of the other way round. From the point of view of this paper, we will demonstrate our results in terms of P and I rather than R.
In the P and I world Axiom 1 can be restated as “For all x and y, either xPy, yPx, xIy, {xPy,yPx}, {xPy, xIy}, {yPx, xIy} or {xPy, yPx, xIy}” {xPy, yPx} might (but does not necessarily have to) be the social ordering if half the voter/consumers prefer x to y and half prefer y to x. Similarly, {xPy, xIy} might be the social ordering if half the voter/consumers prefer x to y, and half are indifferent between x and y. Finally, {xPy, yPx, xIy} might be the social ordering if a third prefer x to y, a third prefer y to x and a third are indifferent between y and x. If P and I are primary and R defined in terms of them, then Axiom 1 can be restated as the following: “For all x and y, either not yRx, not xRy, xRy AND yRx, {not yRx,not xRy}, {not yRx, xRy AND yRx }, {not xRy, xRy AND yRx } or {not yRx, not xRy, xRy AND yRx }.” Sen (1970, p. 41-46) manages to reconstruct essentially the same proof using P and I and without using R at all.

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