No snowflake in an avalanche ever feels responsible.
-(Disputed) Voltaire or Stanislaw J. Lec
Now that Halloween is behind us, it is time to move onto really frightening matters. Tuesday November 2nd is Election Day in the United States, and democratic elections are very deeply mathematically problematic. The primacy of the democratic system has been so often repeated to twentieth and twenty-first century citizens that most people do not consider examining its structure. However, if its foundations are inspected, one does not find pillars resting on bedrock, but impossible Escherian structures that cannot exist, but do. At the base of democracy is voting, and voting is embroiled in many paradoxes.
Whoever gets the most votes wins, whoever is voted against the least wins, right? It would be an absurd for an election to conclude with the candidate that the majority voted against winning, wouldn’t it? Consider the following: Four candidates, Groucho, Harpo, Chico, and Zeppo are in a very tight political race. When November comes, and the votes are tallied these results are found:
It would seem fundamental to a democracy that the will of the majority is accounted for, and numerous contingencies have been proposed in the event of a situation like this. The most common is the requirement of a run-off election between the top candidates. Does this alleviate the problem? Consider another election, this time there are three candidates: Moe, Larry, and Curly. There are also three types of voters:
Moe Voters: Moe Voters ideally prefer Moe to Larry, and prefer Larry to Curly.
Curly Voters: Curly voters prefer Curly to Larry, and prefer Larry to Moe.
Larry Voters: Larry voters ideally prefer Larry, but are split in their preference of the other candidates.
There is an election producing these results:
From the first round, no one holds a majority- no one obtained more than fifty percent of the votes. So there is a run-off election between Moe and Curly. In this election, the Larry Party is split with 8% voting for Moe and 4% voting for Curly. Here is the rundown:
Moe wins with 53%! However, it is questionable how well this result represents the will of the majority. If the contest was between Moe and Larry instead, the Curly voters would prefer Larry to Moe and give their votes to Larry. So would all the Larry voters. Therefore, when it is Moe v Larry, Larry wins with 55%. Likewise, in a contest between Larry and Curly, the Moe voters give their votes to Larry, and Larry again wins, this time with 57%. Notice no matter the outcome, it is hard to justify the statement that the will of the majority is being served.
This paradox is due to preference being an intransitive relation. A relation is transitive if when the relation holds between the first element and the second element, and it holds for the second element and third element, it also holds for the first element and the third element. For example, if Fred is bigger than Barney who is bigger than Wilma, we can conclude that Fred is bigger than Wilma. Size is a transitive relation. However, if Rhet loves Scarlet and Scarlet loves Ashley, we cannot conclude that Rhet loves Ashley.
Love is an intransitive relation. Preference is intransitive like love, and it is because of this that the paradox emerges. This paradox, The Voting Paradox, was first recognized by the eighteenth century French mathematician and philosopher, the Marquis de Condorcet. It was latter revived by Duncan Black, a Canadian economist, and became part of Kenneth Arrow’s 1972 Nobel Prize winning work. Kenneth Arrow presented five conditions that are essential to democracy. They are listed and explained in Morton Davis’s 1980 book, Mathematically Speaking:
1.) The decision making procedure must yield a unique preference order. Whatever the preferences of society’s members, the procedure should come up with one and only one preference order for society.
2.) Society should be responsive to its members. The more individuals like an alternative, the more society should like it too.
3.) Society’s choice between two alternatives is based on its member’s choices between those two alternatives.
4.) The decision making procedure should not prejudge. For any two alternatives X and Y, there must be some possible individual preferences that allow society to prefer X to Y. Otherwise, Y is automatically preferred to X and the group preferences are unresponsive to those of its members.
5.) There is no prejudgement by an individual. Arrow assumes there is no dictator, that is, society’s choices are not identical to the choices of any single individual. If this condition didn’t have to be satisfied, it would be easy enough to find a voting mechanism, but Arrow wouldn’t consider it representative of the individuals in the whole group.
These five precepts have been almost universally hailed as reasonable conditions for a democracy. Arrow did more though, he proved that it is impossible to have a democratic system in which the will of the majority always wins and have it satisfy all five of his conditions. If Arrow’s 5 conditions are defining of democracy, then that means that for there to be a democracy the will of the majority must sometimes be overruled. These paradoxes are then inherent to our electoral system, if it is indeed democratic. I offer them here only as something to think about when you are waiting for the voting booth. That is, if I haven’t scared you away from it.
If voting could change anything, it would be illegal.