How Individuals Can Choose, Even Though They Do Not Maximize Utility

1.0 Introduction
I think of this post as posing a research question. S. Abu Turab Rizvi re-interprets the primitives of social choice theory to refer to mental modules or subroutines in an individual. He then shows that the logical consequence is that individuals are not utility-maximizers. That is, in general, no preference relation exists for an individual that satisfies the conditions equivalent to the existence of an utility function. I have been reading Donald Saari on the mathematics of voting. What are the consequences for individual choice from interpreting this mathematics in Rizvi's terms?

I probably will not pursue this question, although I may draw on these literatures to present some more interesting counter-intuitive numerical examples.

2.0 Arrow's Impossibility Theorem and Work-Arounds
Consider a society of individuals. These individuals are "rational" in that each individual can rank all alternatives, and each individual ranking is transitive. Given the rankings of individuals, we seek a rule, defined for all individual rankings, to construct a complete and transitive ranking of alternatives for society. This rule should satisfy certain minimal properties:

• Non-Dictatorship: No individual exists such that the rule merely assigns his or her ranking to society.
• Independence of Irrelevant Alternatives (IIA): Consider two countries composed of the same number of individuals. Suppose the same number in each country prefer one alternative to another in a certain pair of alternatives, and the same number are likewise indifferent between these alternatives. Then the rule cannot result in societal rankings for the two countries that differ in the order in which these two alternatives are ranked.
• Pareto Principle: If one alternative is ranked higher than another for all individuals, then the ranking for society must rank the former alternative higher than the latter as well.

Arrow's impossibility theorem states that, if there are at least three alternatives, no such rule exists.

Arrow's work has generated lots of critical and interesting research. For example, Sen considers choice functions for society, instead of rankings. A choice function selects the best alternative for every subset of alternatives. That is, for any menu of alternatives, a choice function specifies a best alternative. Consider a rule mapping every set of individual preferences to a choice function. All of Arrow's conditions are consistent for such a map from individual preferences to a choice function.

Saari criticizes the IIA property as requiring a collective choice rule not to use all available information. In particular, the rule makes no use of the number of alternatives, if any, that each individual ranks between each pair. The rule does not make use of enough information to check that each individual has transitive preferences. (Apparently, the IIA condition has generated other criticisms, including by Gibbard.) Saari proposes relaxing the IIA condition to use information sufficient for checking the transitivity of each individual's preference.

Saari also describes a collective choice rule that includes each individual numbering their choices in order, with the first choice being assigned 1, the second 2, and so on. With these numerical assignments, the choices are summed over individuals, and the ranking for society is the ranking resulting from these sums. This aggregation procedure is known as the Borda count. Saari shows that Borda count satisfies the relaxed IIA condition and Arrow's remaining conditions.

3.0 Philosophy of Mathematics
Above, I have summarized aspects of the theory of social choice in fairly concrete terms, such as "individuals" and "society". The mathematics behind these theorems is formulated in set-theoretic terms. The referent for mathematical terms is not fixed by the mathematics:

"One must be able to say at all times - instead of points, straight lines, and planes - tables, chairs, and beer mugs." - David Hilbert (as quoted by Constance Reid, Hilbert, Springer-Verlag, 1970: p. 57)

"Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." -- Bertrand Russell

4.0 An Interpretation
Rizvi re-interprets the social choice formalism as applying to another set of referents. A society’s ranking, in the traditional interpretation, is now an individual’s ranking. An individual’s ranking, in the traditional interpretation, is now an influence on an individual’s ranking. Rizvi’s approach reminds me of Marvin Minsky's society of mind, in which minds are understood to be modular. Rizvi examines the implication’s of Sen’s impossibility of a Paretian liberal for individual preferences under this interpretation of the mathematics of social choice theory.

Constructing natural numbers in terms of set theory allows one to derive the Peano axioms as theorems. Similarly, interpreting social choice theory as applying to decision-making components for an individual allows one to analyze whether the conditions often imposed on individual preferences by mainstream economists can be derived from this deeper structure. And, it follows from Arrow's impossibility theorem, these conditions cannot be so derived in general. Individuals do not and need not maximize utility. On the other hand, Sen's result explains how individuals can choose a best choice from menus with which they may be presented.

References

• Kenneth J. Arrow (1963) Social Choice and Individual Values, Second edition, Cowles Foundation
• Alan G. Isaac (1998) "The Structure of Neoclassical Consumer Theory", working paper (9 July)
• Marvin Minsky (1987) The Society of Mind, Simon and Schuster
• Donald G. Saari (2001) Chaotic Elections! A Mathematician Looks at Voting, American Mathematical Society
• S. Abu Turab Rizvi (2001) "Preference Formation and the Axioms of Choice", Review of Political Economy, V. 13, N. 2 (Nov.): 141-159
• Amartya K. Sen (1969) "Quasi-Transitivity, Rational Choice and Collective Decisions", Review of Economic Studies, V. 36, N. 3 (July): 381-393 (I haven't read this.)
• Amartya K. Sen (1970) "The Impossibility of a Paretian Liberal", Journal of Political Economy, V. 78, N. 1 (Jan.-Feb.): 152-157

Militant Voting

This post illustrates a phenomenon that is a possibility in pairwise voting. Consider a constituency of 30 voters deciding among six candidates for a given vacancy. (I take this example from Donald Saari.) Table 1 describes the preferences of these voters. For example, the first row shows that ten voters prefer Anne to Barb, and Barb to Carol, and so on.) The voters are asked to choose between successive pairs of candidates, as shown in Figure 1. In the first election, Debra defeats Elaine. But Carol defeats Debra in the next choice. And so on, until Flicka is the only choice standing, after a landslide victory. It seems that clearly Flicka is the consensus choice. Strangely enough, though, every voter prefers Carol, Debra, and Elaine to Flicka. The voting system doesn’t seem to allow for a true expression of the preferences of the members of the electorate.

Table 1: Voter Preferences

Number	Preference Ranking
10	A > B > C > D > E > F
10	B > C > D > E > F > A
10	C > D > E > F > A > B

Figure 1: Pairwise Elections in Example

Militant was a Trotskyite tendency practicing entryism within the British Labour Party. They seemed to have figured out how to use party discipline to have their way in Liverpool in the early 1980s:

"All political parties on the Monday before the Council meeting on a Wednesday, have a caucus meeting to decide the line of approach at the Council. The agenda for the meeting comes out on a Friday. The ten or twelve Militant members ... meet on either the Friday or the Saturday and go through the agenda to look for important policy decisions and for important vacancies. They then have a meeting with the broad left of the Labour group on Sunday morning at Pirrie Labour Club ... those Militants turn up in their full strength. There are generally about twenty people and the ten or eleven Militants there. They carry the majority vote there that commits the broad left for the meeting of the Labour group. On the Monday night at the Labour group meeting of forty-two members the commitment that Militant have made themselves, plus other people they've taken along at the meeting on Sunday morning, gives them a majority. ... so you find that of forty-two Labour councilors, ten Militants control the policy of the Labour group." -- Eddie Roderick, as quoted by Michael Crick

Crick says, "Roderick's analysis may be a rather simplified version." At any rate, Figure 2 illustrates how the Council seems to have made their decisions. Figures 1 and 2 look quite similar. Maybe Saari’s math describes more than a theoretical possibility.

Figure 1: Pairwise Voting on the Liverpool City Council

References

• Michael Crick (1984). Militant, Faber and Faber
• Donald G. Saari (2004). "Geometry of Chaotic and Stable Discussion", American Mathematical Monthly, V. 111, N. 5: 377-393

Friedman Blinded Me With Science

Scientists aspire to develop theories that observations can potentially demonstrate to be wrong. Here I examine whether this aspiration can possibly be achieved when economics is practiced in keeping with one of two views on methodology, the deductive-nomological or the instrumental view. I get the argument below from Donald P. Green and Ian Shapiro, Pathologies of Rational Choice Theory: A Critique of Applications in Political Science (Yale University Press, 1994).

Consider the covering law model, also known as the deductive-nomological view of scientific methodology. In this view, scientists formulate universal laws, in some sense. In an application of a scientific law, the hypotheses or antecedents are asserted to be true. That is, the statement of scientific law is conjoined with initial conditions. One then checks that the consequent holds. If observation is inconsistent with the consequent and one is sure that the initial conditions are true, the law is refuted.

Milton Friedman advocates instrumentalism, in which the assumptions of a scientific theory are false. (Actually, his famous essay, "The Methodology of Positive Economics", is so incoherent, Friedman can be interpreted as advocating almost any methodology you care to name. But let's stick with a widely argued view.) In Friedman's view the antecedents are always false in a significant theory:

"Truly important and significant hypotheses will be found to have 'assumptions' that are wildly inaccurate descriptive representations of reality, and, in general, the more significant the theory, the more unrealistic the assumptions" -- Milton Friedman

Thus, if one holds that economic theories state covering laws and that economists are and should be instrumentalists, economic theories cannot be refuted by observation. The logical implications of false antecedents need not be true.

Can economics be a science if it is practiced in keeping with Friedman's strictures?