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The three-party voting paradox

By Margaret Harris

Last month, with space in the print edition of Physics World tight, I cut a couple of sentences from the end of a short review I’d written of Len Fisher’s new book, The Perfect Swarm. This turned out to be a bad idea. How bad? Well, you can read the published review here, but this is how it originally ended:

“UK readers may be particularly interested in [Fisher’s] explanation of how, in a three-way race, voting is not transitive. This means that it would be theoretically possible for a majority to prefer the Tories to Labour, another majority to prefer Labour to the Lib Dems, and a third majority to prefer the Lib Dems to the Tories. Ouch.”

Ouch indeed. As keen observers of British politics will have noticed, the possibility of a tight three-way race got a lot less theoretical last week, after Lib Dem leader Nick Clegg’s performance in the first-ever televised debate between party leaders boosted the (usually third-place) Lib Dems’ poll ratings.

But although I’ve clearly blown my chance of being hailed as a political oracle, it’s not too late to take a closer look at the mathematics behind a three-way race – and particularly at the Condorcet paradox, the technical name of the situation I (almost) described.

Named after the Marquis de Condorcet (an 18th-century French nobleman who, Fisher writes, “had more fun with voting systems than most politicians”), the simplest Condorcet paradox occurs when there are three voters and three political parties A, B and C:

Voter 1 prefers A to B, but would rather have B than C
Voter 2 prefers B to C, but would rather have C than A
Voter 3 prefers C to A, but would rather have A than B

In these circumstances, no matter who is declared the winner, two-thirds of the electorate will have preferred someone else. There is no “Condorcet winner” – no party which, when compared to all others, is preferred by more people.

Of course, this is an extreme example. However, even in more realistic voting scenarios, there are very few voting systems that will always produce a Condorcet winner when one exists. The “first-past-the-post” system that currently operates in the UK (and that the Tories want to keep) is obviously not one of them; indeed, there have been plenty of recent election results where more people voted against the winning party than voted for it. But the “alternative vote” that Labour has suggested also fails the Condorcet test, as does the Single Transferable Vote system advocated by the Liberal Democrats (although there are more complex variants of STV that do comply).

Worse, the economist Kenneth Arrow proved in 1950 that the Condorcet paradox is just the start. Arrow showed that no voting systems simultaneously satisfy all of the “fairness criteria” that Arrow specified (you can read more about his criteria here, but two of them are transitivity and a rule against dictatorships). It’s an insight that won Arrow a Nobel Prize for Economics, and presumably sent voting-reform fans into a deep depression.

I couldn’t possibly tell you how to vote – we at are officially neutral – but if you want to drive away an annoying political canvasser from your doorstep, I’d recommend asking them how their party would solve the Condorcet paradox.

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  1. Dr. Iain Salisbury

    1) There is no such thing as the Nobel Prize for Economics.
    2) In the British voting system, we select representatives, not delegates, and party affiliations are merely one consideration among many. “Proportional representation” is an oxymoron, since handing votes to party apparatchiks, who then go down a list of increasingly inadequate hacks and time-servers, produces delegates, who have no direct link to the electorate.
    I concur with the article, insofar as it describes the consequences of various establishment gerrymanderings.

  2. dan.hatton

    Iain, I think you really know that the closed list system you describe isn’t the only available form of proportional representation. The single transferable vote, or even an open list system, gives voters control at the individual-candidate level.
    Incidentally, did Arrow actually use the phrase “fairness criteria”? It strikes me that “independence of irrelevant alternatives”, while it’s a nice idea, has very little to do with fairness.

  3. dan.hatton

    Thanks Margaret. Nice work.

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