# The Calculus of Democracy

Posted by Graham Wheeler on Tuesday, July 20, 2010

Earlier this year I went to a “guys movie night” at a friends house. Before the movie we sat around drinking Croatian Schnapps and eating Chinese food, and the conversation turned to politics. There was general agreement that our various political systems were highly suspect and not very democratic, but Ray, the Texan lawyer with the Croatian wife who had provided the schnapps, explained the Australian system of instant runoffs and asserted that this was at last a fair system.

For those not familiar with this system, let me explain: voters get to rank the candidates in order of preference. The votes for everyone’s first preference are counted, and the candidate who gets the least votes is eliminated. That candidate is also struck from the voters choices and the process is repeated until there is a candidate that has more than 50% of the vote.

This seems fair on the surface - if someone obviously gets too few votes they should be out of the running, and everyone gets their preferences for the remaining candidates counted. But back in 1785, the Marquis de Condorcet pointed out paradoxes in this system. For example, if there are three candidates A, B, C and three voters who rank them A-B-C, B-C-A, and C-A-B, the system is a tie. In general, this is known as Condorcet’s paradox.

Still, this system is more fair then the plurality voting systems used in Canada, India, the UK and the USA, where in each district the winner takes all. In systems with multiple parties the winner may get way less than half the vote and still get to represent the entire district. Often two dominant parties with similar policies will split the vote allowing a less popular party to win a majority.

Sometimes a hybrid system with a second run-off election between the top two candidates is used - but due to the vagaries of plurality systems, the “top two” candidates may not really be the top two at all.

In two party systems like the USA parties mostly compete for the center, causing the difference between them to become all but irrelevant (in economics this is known as Hotelling’s law). Over time these systems tend to lose the interest of the voters, occasionally being re-energized when someone breaks the mold and retargets the extreme instead, as George W. Bush did.

Another problem with plurality systems is the way in which voting district boundaries are drawn up. A candidate may even win the election while losing the popular vote overall, as George Bush did in 2000. Economic and other factors often cause areas to have a preference for one party over another, and ruling parties often exploit this by redrawing boundaries while they are in power to favor their chances, a process known as gerrymandering after the 19th century governor of Massachusetts, Elbridge Gerry, who drew up a district map so convoluted it was compared to a salamander (see photo).

Yet another common system is proportional representation, where each district has representatives allocated to each party in proportion to the percentage of votes that party obtained. Once again this is subject to issues with how districts are divided up. It may seem to make sense to have just one district, the entire country - the system used in Israel - but the disadvantage is that voters end up with little control over which individuals represent them, even if they do have some control over which parties do. Furthermore, such systems often end up with weak coalition governments that are hamstrung and ineffective, or in which minority parties can wield excessive power as “king-makers” for their bigger rivals.

So what makes a fair system? In 1963 the economist Kenneth Arrow came up with four attributes that he felt described a fair voting system - and then went on to prove Arrow’s impossibility theorem - no voting system could satisfy all four conditions. Democracy, it seems, isn’t.