Notes on the Voting Paradox

Dhammika Dharmapala

Suppose that there are 3 voters – A, B, and C, and 3 possible levels of spending on a public good – low (L), medium (M) and high (H). The table below represents the preferences of the voters – i.e. how each voter ranks the alternatives (in the first row, “1” means that voter A ranks alternative H first, “2” means that A ranks M second, “3” means that A ranks L third etc).

	L	M	H
A	3	2	1
B	1	3	2
C	2	1	3

A prefers higher levels of spending; C prefers M, followed by L, with H being worst; B would like spending to be low (in which case her taxes will be low, and she can send her children to a private school); failing this, she would like a high level of spending (so she would send her children to the public schools, which would be of high quality); the worst outcome would be M, as taxes would be moderately high, so she could not afford private schooling, but the public schools would not be of high quality.

Suppose that there is ‘pairwise’ voting over the alternatives – i.e. the voters vote over 2 alternatives, then over the winner and the remaining alternative.

Consider the vote over L v. M: A and C would vote for M, while B would vote for L; M wins

Now, consider M v. H: H wins (A and B will vote for H)

That is, society prefers M to L and H to M. Consistency then requires that it should also prefer H to L (i.e. if H is better than M, which is itself better than L, then it should be the case that H is better than L).

However, consider L v. H: B and C will vote for L, so L wins

Thus, society’s preferences, as revealed through majority voting, are inconsistent. Choosing the level of provision of public goods through majority voting does not guarantee a consistent outcome in this sense.

Note:

- with the preferences we have assumed, the outcome depends on the agenda – i.e. the order in which proposals are voted on. Suppose we start with L v M - the outcome is H; if we start with L v H, we get M; if we start with M v H, we get L. Any outcome is possible depending on the choices we start with.

- if defeated proposals can be reintroduced at a later stage, then we can get the phenomenon of cycling: suppose we start with L v M, then M v H, leading to a win for H; but, if L can then be proposed, L will win against H, so we go back to L v M and start the cycle over again – voting can continue forever with no outcome

- this feature of majority voting is known as Condorcet’s paradox, after the Marquis de Condorcet, who was the first to discover it, in 1785.

The above paradox applies to majority voting rules. There are many other ways of aggregating individual preferences – e.g. a rule of 2/3 (a supermajority), approval voting etc. Perhaps these will lead to consistent choices of public goods?

In fact, the possible inconsistency of outcomes is not unique to majority voting. In the early 1950’s, Arrow proved a remarkable theorem that extends Condorcet’s idea to all conceivable methods of aggregating individual preferences into social decisions. Basically, the theorem (Arrow’s Impossibility Thrm) states that, if we impose a set of reasonable constraints on the outcomes of social decisionmaking, then it is impossible to find any method of social decisionmaking that will guarantee consistent social decisions.

Arrow’s theorem applies to any decision problem with 3 or more alternatives – we will consider 3 (X, Y, Z) for convenience. Arrow’s conditions are:

U: unrestricted domain – no restrictions are placed on the preferences that individuals are allowed to have.

P: Pareto-principle – if every individual prefers X to Y, then society should prefer X to Y

I: independence of irrelevant alternatives: society’s ranking of X and Y depends only on individuals’ rankings of X and Y, not their attitude to Z

D: nondictatorship – no individual is allowed to be a dictator (a ‘dictator’ is someone for whom the following is true: if he prefers X to Y, and all other individuals prefer Y to X, society nonetheless prefers X to Y)

These are apparently reasonable conditions to impose on social decisionmaking. Nonetheless, there is no rule for social choice that satisfies all 4 conditions, and guarantees consistent outcomes.

Implications of Arrow’s theorem: does not mean that societies cannot reach collective decisions through democratic processes (of course, they do so all the time). Rather, what it means is that there may be some arbitrariness about these decisions – it undermines the idea that decisions made by voting have a particular legitimacy – has had a vast influence on political science and philosophy.