Group Choice and Majority Rule

Cyclical Majorities

Condorcet's Paradox: An individual's preferences in a group, though consistent and transitive, need not be true of the group's preferences. A majority will prefer A to B, or B to C, but another majority will prefer C to A, rendering the arrangement nontransitory. What a group ends up doing will be cyclical, but it is of great importance as legislatures from town halls to Congresses operate under this assumption.

In a group of three, an individual has 13 ways of ranking his preferences. Rankings one through six, i.e., he can go ABC, ACB, BAC, BCA, CAB, CBA, are known as strong preferences. He can put two preferences together, i.e. A, BC; B, AC; C, AB; AB, C; AC, B; BC, A; and create a series of weak preferences. The final ordering represents complete and total indifference. 2197 possible preferences between three people might be made. For the sake of simplicy, the textbook focuses on 6*6*6.

Out of 216 possible outcomes, how many are affected by Condorcet's Paradox, i.e. how many possible scenarios will develop where no majority can be met? In most situations, the odds are high that majority rule will run smoothly, meaning majority rule works most of the time.

But life isn't as simple as a Condorcet Paradox. The number of alternatives and people increase in society. We should try to derive a probablity that given the number of alternatives and the number of individuals in a group, a majority outcome can be reached, in other words, the ratio between the preferences of the group, and the alternatives.

Probablity of intransitivity (Number of preferences*number of people) = the number of problem configurations/the number of potential solutions)

As the number of group members increases, the chances of intransivity and preference cycles increases to the limit. As the number of preferences increases, the chances for intransivity also increase. In politics, we must tolerate either group incoherence, a highly compressed franchise, or a highly restricted agenda.

It is not always the case, however, that one set of strong preferences is just as likely as another to characterize the preferences of an individual. Society is interdependent and not likely to create problem scenarios as often as these equilibrium scenarios would like. However, as long as the number of preferences remains large in any sized society, the chances of majority cycling remain dangerous.

Cyclical Majorities and "Divide the Dollars"

Political fights often revolve around how to share revenue. How do we divide the deficit is also difficult: from where does one raise revenue/cut funding?

Suppose a board of three men representing different districts of the town must divide a windfall of $1,000. In each case, the more money a politican lands, the better his chances for re-election. In all instances, each politican will recieve a share of at least 0 and no more than 1000. Each politician prefers any outcome where his share is larger than the other two. The so-called "fair distribution" of each side getting 333.33 will get beat by a majority because more prefer a 500, 500, 0 to 333, 333, 333. But to prevent two from ganging up on one, X prefers 700, 0, 300 to any combination of 500, 500, 0. The final outcome of this match will be decided on other institutional features of group decision-making.

Only anti-majoritarian restrictions that allow someone to exercise agenda power, procedural rules etc., would allow anything to get done.

Tax Politics

Various social groups that want to avoid paying taxes will form unstable coalitions which leads to preference cycles.

In the Civil War, one group favored taxing wealth, another land, another no tax at all. In the ensuing fight, someone decided to tax "income," an ambiguous and undefined term that in no way guaranteed that any specific group would be damaged. Congressmen preferred a lottery to no tax at all or a specific tax on land or wealth.

One way to prevent preference cycles is to impose limits on anyone to amend legislation, therefore allowing only the status quo or the proposal. In the event of a tie, the status quo usually wins. Preference cycles emerged in the 1930s when legislators were allowed to constantly amend bills, increasing the number of preferences possible and thereby making it more likely for preference cycle gridlock. Usually limits are imposed beforehand on any legislation.

A bill designed to do-away with tax breaks for special interests was supposed to attract a strong opposition from special interests and their supporters in Congress. In other words, a bill protecting the special interests would defeat the reform bill, but as Congressmen preferred no act at all to be seen as endorsing special interests, no bill was passed. But as each of the special interest groups preferred protecting their own special interests to collective action beneficial to the whole, they could not cooperate and the reform bill passed.

Arrow's Theorem

The problem of group incoherence is a peculiarity of round-robin tournmanets or features of majority rule, but not of voting generally. If one structured the institutional arrangements of group choice differenlty, we could arrive at a system with less incoherence.

In addition to assuming a rational actor capable of defining his preference (or indifference) and staying logical about it, any group acts four different ways:

1. Universal Admissibility: Anyone in the group may adopt any strong or weak and transitive ordering over the alternatives
2. Unanimty: If every member of the group prefers j to k yet we end up with a scheme that has k ahead of j then the group preference must represent that wish.
3. Independence from Irrelevant Alternatives: If j is ranked ahead of k, in the final decision, the later elevation of l does not change j ranking ahead of k.
4. Nondictatorship: If only one person prefers k to j, then the group's decision cannot be k.

There exists no mechanism for translating the preferences of rational individuals into a coherant group preference that simultaneously satisfies all four conditions. Any scheme that satisfies all four criteria is either dictatorial or possesses intransitive solutions. Thus, there is a tradeoff between social rationality and the concentration of power. Social organizations that concentrate power provide for the prospect of social coherence. Majority rule can settle things most of the time, but not always.

Legislative Intent

Liberals prefer interpreting a silence on a particular issue as not restricting the court's or government's power. Conservatives prefer deference to the intent of the legislature. Arrow's therom states that because a group may not have a transitive set of preferences, trying to discover the intent of a legislature is a fool's errand.

Arrow's Theorem and Majority Rule

Majority rule is defined as for any pair of alternatives, if j is preferred to k by more people than it wins.

Reasonable Conditions on Preference Aggregation Methods:

Anonymity: Social preferences not influenced by who has what preference.

Neutratlity: Interchaning the ranks of alternatives in each member's preferencing has the effect of interchaning the group's preference ordering. No matter what we label them, they remain the same.

Monotonicity: The method of group choice cannot respond to changes in individual preferences. If j is strictly preferred at first, and someone changes their preferences to make J higher, J is still strictly preferred. If people are indifferent to j and k at first, but then one person prefers j to k, then j is preferred.

May's Theorem: If a group uses a handcount to satisfy deciding between any pair of alternatives, then it necessarly satisfies Universal Domain, Anonymity, Neutrality, and Monotonicity. Not all preferences are of this nature, of course, and nor should they be.


Black's Single-Peakedness Theorem and Sen's Value Restriction Theorem :

Arrow's and May's conditions are mild and innocuous conditions of fairness. Condition unanimous domain is different. The more it is applied, the greater the chance for a trade off fairness for consistency. Is it possible to restrict domain and obtain both fairness and consistency?

If in every set of alternatives, there is some alternative that is not the worst alternative, then majority rule deems it transitive. In other words, majority rule can work pretty well so long as a minimal degree of consensus exists on some alternative.