Praesidium

Saturday, October 29, 2005

Median Voter Theorem

My weekly theory of voting class has been put back to tomorrow (yes, Saturday) morning, so I'll bring you half the presentation today and half tomorrow. This bit's on the Median Voter Theorem - tomorrow May's theorem (which is a bit more interesting).

c) Proof of Median Voter Theorem

The median voter theorem state that if voters’ “curves are singled-peaked, Omed. will be able to get a simple majority over any of the other motions a1,…, am put forward”[1]. This is quite straightforward, at least if n (number of voters) is odd[2]. Any proposed solution to the right of the median voter’s ideal will be able to gain the support of all voters to its right, and those between the proposal and the median who are ideologically closer to the proposal (for whom the proposal represents something slightly too far right, as opposed to quite a bit too far left). Opposition, on the other hand, will include the median voter himself and all to the left – amounting to [(n-1)/2]+1, i.e. a majority. Similarly there will be a majority of the median voter and all to his right that would defeat any move to the left.

The original theorem assumed a single left-right spectrum in one-dimensional issue space. In fact, it can be extended to multidimensional space[3], if the ‘median’ voter has the others in opposing pairs around him. For example, imagine a cross-shape with the median voter at the intersection of the two lines, and four others at compass points (N, S, E, W). The status quo is the median voter’s ideal. A move to the NW, for example, will be opposed by three – the median voter and those at S and E. (Note though, a move directly to N will get one for, two against [median and S] and two indifferent [W and E] – it’s possible if N could offer the latter two some small incentive that they might vote for such a move).

d) Evaluation of Median Voter Theorem

The median voter theorem explains why in two party systems we end up with two parties with very similar platforms. Those on the ideological extremes are assumed their safe, core voters. It is those in the middle of the spectrum that make the difference between 40% and 60% support – and most crucially, the median voter that tips the balance from (say) 49.5% to 50.5%.

The median voter theorem introduces some predictability into political outcomes. It does not mean that the median voter is an Arrovian dictator, however. There is no one individual who will get his way however others vote[4]. Indeed, who is ‘the median’ depends on the voting pattern. One reason why economic redistribution in democracies has perhaps been less than those fearful of extending the franchise expected in the 19th century is that, while the median member of society may have a below (mean) average income, because the poorer generally seem less likely to vote, the median voter may have an above-average income.

Another consequence of the median winning position is that no one is too far away from the outcome, and this limits dissatisfaction – as opposed to someone on the far right winning, leaving the left very dissatisfied. It does not minimise aggregate dissatisfaction (the sum of the differences between each individual and the outcome) though, because those to the left of the median could be much more extreme left wing than those to the right, and thus aggregate dissatisfaction would be reduced if the outcome was moved to the left. Further what may matter on a divisive issue like abortion might be which side of the spectrum one is on, more than how far. If the median voter is pro-life, it wouldn’t matter much to pro-choices that they were only a moderate pro-lifer – still the other side would have won[5].

In any case, the model is not always applicable. Even on a single dimension, voters’ preferences need not be single-peaked. During the Vietnam War, for example, many Americans apparently favoured either a complete withdrawal of troops or sending more in – do it properly or not at all. In such a case, the middle ground need not be stable, as it could be defeated by a proposal at either extreme. When it comes to multidimensional matters, while a median voter can exist, there may not be a unique equilibrium.

Real world politics doesn’t always fit economic models/predictions perfectly. It’s quite possible for parties to drift from the median – e.g. the ‘loony left’ dominated Labour in 1983, though out of touch with the public. At the same time, the Conservatives were further to the right – but Thatcherite policies encouraging house and share ownership effectively helped move the median voter to the right. To regain power, New Labour have moved to the right in pursuit of this median. However, if a party strays too far from its core supporters, new parties may arise on the fringes of the political spectrum – even if they have no hope of winning, the can act as pressure groups – e.g. communists keep socialists ‘honest’ to left-wing ideals.

[1] D. Black (1998) The Theory of Committees and Elections (McLean et al eds.) p.23.
[2] If n is even, the median lies between two voters.
[3] D. C. Mueller (2003) Public Choice III pp.92-3. C.f. C. Plott (1967) ‘A Notion of Equilibrium and Its Possibility under Majority Rule’ American Economic Review 57 787-806.
[4] ‘The median’ is a placeholder, like Rawls’ ‘worst off group’.
[5] Of course, compromises are possible on almost all political decisions. Even on abortion, compromises might involve state funding, maximum time allowed after conception and the stringency of medical reasons required.

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