Political Paradoxes

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:

Clearly, Zeppo has triumphed- he wins! However, the vast majority, 70%, did not vote for Zeppo! Sometimes in democratic elections, the will of the majority can be overlooked.

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 Intransitive

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.



About Webster

Transitionally, I’m a math student and tutor en route to becoming a math professor. Permanently, I’m a mathematics enthusiast. I study mathematics professionally, and as a leisure activity. At the time of writing this, I’m a generalist. I have let to reach the depth of understanding that requires specialization. Though I eagerly await that time, I do enjoy the ‘now’ and find there is bountiful food for thought at any level.
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5 Responses to Political Paradoxes

  1. James Rossi says:

    I know this is lame but I would buy a shirt that says, “Love is Intransitive.”

    Also for some additional stats to consider is voter fraud which is 100% a certainty in any election, but it appears there is a bit of a lax attitude towards as both major parties commit it. However, that is likely less an issue with 2 dominant parties. In Europe where there can be upwards of tens of parties all with not insignificant minorities it must make pretty strong waves that I’m not sure if can be mathematically worked out. Could be too social an issue to be accurately described mathematically.

    • Webster says:

      I’m considering making shirt designs out of some of the things I’ve written about here. I’ll try to mock something up for “Love is Intransitive.”

      I would have discussed other aspects of voting, but I was pressed for time and decided that instead I would post about one of the most classical problems. Concerning your statement that it is too social an issue, one of the amazing things about social issues is that as larger and larger populations are analyzed the deviations between individuals become less and less and the population becomes more predictable. Populations are predictable where individuals are not, and so with modest simplification even the most complex social behavior is usually very mathematically modelable. However, that is another topic, perhaps for another time.

  2. Annabelle says:

    Having taken 3 political science classes, 2 of them American Politics, all I can say is “Know ye now, Bulkington?”

    I have been to the other side and seen shit that no man should ever see.

    Aka don’t vote, and if you vote, know that you are voting for the opponent candidate as much as you are voting for your own candidate because the whole foundation of our country is that SHIT IS IMPOSSIBLE TO CHANGE so no matter who is in office doing what, everything will always be the same as it always has, with a few random deviations that cancel out when taken as a whole.

    Also incumbency advantage. The reelection rate is 88 percent for the Senate and 96 percent for the House. Since WWII, 90% of incumbents who ran for reelection were successful (thank you Wikipedia, as always).

    ALSO think of the times when people did vote. Specifically, for Hitler lol (I know I just lost my argument but I am just playin). HE WON 84.6% of the electorate. THAT’S INSANE. Everyone voted for him. Look what happened. Also ANDREW JACKSON. Jacksonian democracy was the best time American Democracy has ever experienced. 80% of people (white dudes) voted!!!!! Typical white dudes, paving trails of tears. Nobody likes Jackson anymore.

    Idk man. When you vote just know that you are voting for a corrupt system (built on beautiful, beautiful ideals omg ❤ i love america seriously) where the legislature isn't even an Aristotelian legislature unless it insures no change, the executive officer is a puppet to the ~*~*~*~special interests~*~~* aka the only good part of politics, and the supreme court is "POINTLESS AND HARMFUL" according to my American Politics professor. How heartbreaking is that shit? Are your dreams crushed yet? Clearly I am still not over this. Brb seeking therapy for Professor Sherer crushing my deep adoration for the American political system. Also everyone who hates candidates for being bought out by ~special interests~ can bite me because those candidates wouldn't accept money from those orgs/business unless they already wanted supported their platforms/wanted to enact those policies. Special interests/lobbyists are the only cool people in the government. Everyone else sucks.

    • Webster says:

      Love the line “I have been to the other side and seen shit that no man should ever see.”

      I’ve always thought the implementation of the Supreme Court was a stroke of genius, why did your professor call it: “POINTLESS AND HARMFUL?”

      As for the balls-to-the wall cynicism, wait for the follow up post.

      Also points for Moby Dick quote, kudos- that is some cold-blooded stuff!

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