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4. Voting and Elections: Enter Kenneth Arrow

Voting and Elections: Enter Kenneth Arrow

4. Enter Kenneth Arrow

The fact that different seemingly appealing methods can elect different winners suggests a change in perspective, from that of an election system delivering the will of the people to one in which the results are as consistent, fair, or equitable over a range of possible election patterns that the voters might provide to the system. What rules should an election system obey so that we will think it is a good system? What rules should an election system obey so that we will think it is better than some other system?

Insight using this approach was provided by Kenneth Arrow, who developed a collection of fairness, consistency or reasonability conditions (axioms) that any fair election method was to obey. What is an example of such a fairness axiom? Suppose that one has an election decision procedure based on preference ballots. Suppose a particular election where candidate A is ranked at the top of 9 schedules, as shown below:

Suppose the decision method assigns A as victor in the election. Now imagine an election in which all the ballots are the same except that instead of having 9 votes for the schedule shown above, one has 10 votes for this schedule. Would it make any sense that the decision procedure applied to these ballots now elect someone other than A? If it seems unreasonable, we might state that we require any fair election method to obey this rule. Arrow developed a variety of fairness conditions that he thought any reasonable election method should obey. He then proceeded to show that for elections where there were more than 2 candidates no election decision method obeyed all of the rules! Since Arrow's original work many investigators have developed a wide array of desirable fairness rules and showed results similar to Arrow's Theorem.

Kenneth Arrow's work is important for many of reasons, some of which go well beyond the actual theorem itself. In particular, it is very valuable to notice the axiomatic framework he was working in was in an applied area (e.g. economics, political science) rather than mathematics itself. Certainly, the most famous example within mathematics of the use of axiomatic ideas has been in geometry. Euclid's work in codifying Greek knowledge of geometry is a great landmark in intellectual history. However, it is still debated whether or not what Euclid was doing was developing a way of deducing the properties of the physical space we live in from some simple principles (axioms) or whether he realized that he was just developing an abstract mathematical system. In any case, the famous 5th postulate of Euclid seemed to many ancient writers more complex than the other assumptions he worked with. As a result many attempts were made to prove this axiom from Euclid's other axioms but without success. Eventually, this work culminated in the landmark discovery that from the mathematical point of view Euclidean geometry is but one of many geometries, and that there is a geometry where one accepts all the other axioms of Euclid except that one uses the negation of the Euclidean 5th postulate as an axiom, which is no better or worse than Euclid's geometry. This discovery has a complex history but the resulting geometry is known as Bolyai-Lobachevsky geometry or hyperbolic geometry. Although there are ways in which the work of Euclid, Bolyai, and Lobachevsky had to be improved on in modern times (notably in the work of David Hilbert), the power of the use of axiomatic thinking is a large part of this success.

When Arrow developed a series of axioms that a group decision method should obey, he was proceeding in a way that was analogous to the route that Euclid had pioneered. He had to address questions such as the independence of the axioms he had chosen in just the same way that people checked to make sure that none of the axioms of Euclid followed from some of the other axioms. Arrow chose the axioms he did because he deemed that they made sense as a list of desirable properties for a decision procedure. When Arrow showed that there was no election system that obeyed all of the axioms he proposed when there were at least 3 candidates, some people have misinterpreted the meaning of his results. Although it is true that no election method can obey all his fairness axioms, this does not mean that one can not argue that some particular method of election is not better than some other method from some point of view. As soon as one writes down an axiom, say A, one can determine which methods obey axiom A and which do not. The consequence of the mathematical analysis is in seeing that if one really cares about property A, then one should rule out those decision methods that do not obey this axiom. In some cases scholars have been able to characterize particular election decision methods. Finding a characterization of a method means finding a collection of axioms that this method obeys and no other obeys. For example, H. Peyton Young found a set of axioms that characterizes point count methods such as the Borda count.