[caption id="" align="alignright" width="370"] Photo credit: The National[/caption]

One phrase I kept hearing over and over last week in coverage of Egypt's election results is that the two frontrunners were each "worse than the other." The two top finishers, Mohamed Morsi and and Ahmed Shafiq, received a combined total of 48.5 percent of the votes. Overall turnout was estimated at about 43 percent. Some commentators took this as bad news, arguing that a mere one-fifth of the eligible voting population had decided which candidates would make it to the run-off round. I take a more neutral view, and see the election results as an illustration of two well-known paradoxes of democracy.

The first paradox, named after the Marquis de Condorcet, is that in an election with three or more candidates, it may be impossible to select a single candidate that satisfies majority preferences.  The canonical example is with three candidates and three voters, but let's use the five candidates who received more than 10 percent of Egyptian votes (of the other eight candidates, none received more than 1.1 percent). Of these five, two will continue to the next round.

Suppose the Egyptian electorate voted according to its true preferences (i.e. not strategically). For simplicity's sake, let's round the votes of the top five candidates to 25, 25, 20, 15, and 15 percent. We know that Morsi's and Shafiq's supporters each consider their own candidate to be better than the other. Now suppose that the remaining 50 percent are evenly split between the two candidates (they each prefer their own candidate first, but half think Morsi is better than Shafiq and half think the opposite). This means that 50 percent of voters think that Shafiq is the better choice between the two, and the other half think Morsi is better than Shafiq.

We also do not know how much better the voters rank their preferred candidate. It could be that everyone who likes Shafiq best likes Morsi the least, rather than second best. There is no way to accurately deduce the overall preferences of Egyptian's from the initial results without making strong assumptions like we did here. This is why the runoff is required. But this example does explain how so many people can be unhappy with the results of a democratic election.

The second paradox of voting is known as Arrow's impossibility theorem. This theorem--one of the few great contributions of game theory to modern social science--says that no election system can accurately translate ranked preferences over three or more candidates into an overall ranking for the community while maintaining three criteria. These three are:

  1. If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
  2. If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
  3. There is no "dictator": no single voter possesses the power to always determine the group's preference.

For the first round of elections, the second criterion is most important. Changing the amount by which Shafiq or Morsi is preferred could have changed the results. In the runoff elections, the third principle will be violated. Once you are down to only two candidates, majority rule takes over. Egyptians got rid of one dictator, Mubarak, but exchanged him for the median voter. We will see how satisfied they are with that decision later this month.