The Moose Lodge: voting for a number, rather than for a candidate

The following story seeks to illustrate by example some the problems of vote splitting with plurality voting. To simplify and clarify some basic concepts, this example shows people voting for a number (in this case, an amount of money), rather than for a specific human candidate.

The local Moose Lodge chapter collects monthly dues from its members, in order to to pay for the annual holiday party, and to pay for repairs and improvements to the lodge. The higher the dues, the better the party and the better shape the lodge will be in (and, of course, the poorer the members will be!).

As might be expected, members have varying ideas as to how high the dues should be. Some think they should be only $5 a month, while others would prefer they be as high as $25. Most people, however, prefer something a little more middle ground, in the $12 to 15 range.

At one of their meetings, Grand Moose Master Dave proposes that the chapter hold a vote to decide how much dues should be for the coming year. Prior to voting, he asks for to nominate "candidates", that is, amounts that can be voted on. Larry says "I nominate that we have dues of $8 a month". Max nominates $20 a month dues. They then hold a vote between the two choices, and $8 wins, 55 votes to 45.

todo: bar graph, showing a bell curve of the amounts people prefer. It will also show which of them would be likely to vote for $8 (all those picking values less than $14) and those that voted for $20

After voting, a disappointed Max becomes curious, and decides to do a survey. He asks each member what exact amount they would have preferred, and makes a little graph. He is not surprised to see that people tended to vote for the option closest to their ideal amount.

Strategic nomination

The next year, the dues come up for another vote. This time, Max -- noting that the lodge roof has a leak and that they only have enough money to serve Pabst Blue Ribbon at the holiday party -- decides he is going to be smarter about it this time.

todo: graph showing people who should vote for $8 and who should vote for $18

First, he does another survey -- this time prior to the voting -- and notes the results are not so different than last year. After Larry nominates $8 again, Max looks at his graph, does a quick calculation, and decides to nominate $18....which according to his survey should beat $8, since a few of the "middle ground" people should now vote for the higher amount.

What he is done is to be strategic in his nomination -- even though he would prefer a higher value, he nominates a value he thinks can win.

The Spoiler (a.k.a. "Vote splitting")

Before the vote is held, though, Joe decides to nominate $21 as a third option. Max, of course, is pleased to see that he's got some support on the high end.

todo: graph showing people who voted for $8, $18, and $21

Unfortunately though, as the votes are being counted, Max realizes that Joe didn't do him a favor. Now, several of the people he had counted on to vote for $18 now are voting for $21 instead! Needless to say, another year of $8 dues follows, thanks to Joe's misguided nomination. Another terrible christmas party, with cheap beer and a leaky roof.

todo: graph showing people who voted for $5, 6, 9, 12 etc with 21 being the winner

The next year, there are 13 separate amounts nominated. People start nominating amounts that are not even what they prefer, but are simply intended to split the votes of the "other side". The results, instead of capturing any sort of "will of the people", seem almost random. "This is democracy?" wonders Max. "What happened?"

The formation of parties

Parties and Duverger's Law: Maurice Duverger, a French sociologist, observed that two party systems are the inevitable consequence of single-winner elections which use plurality voting. In other words, the whole reason we have a Republican and Democratic party, and all the associated partisanship, is as a by product of our electoral system. You can read more about Duverger's Law here (Wikipedia link, opens in new window).

After a few years of this chaos, Max decides to organize those who want higher dues. He calls his group "The Extravagant Party". The Extravagants get together before the meeting, and agree to decide amongst themselves what amount they want to nominate, and not to nominate any other amounts (even if the decided upon amount is not their ideal). They try to pick an amount that is high enough to satisfy their needs, but also that they think will win. In other words, they decide to be strategic about it.

Meanwhile, the people who want lower dues form The Cheapskate Party. Throughout the years, the Cheapskates win some years, and the Extravagants win other years.

Finally, a complete fix for all problems

This is a little better, but certainly not ideal. Steve notes that it seems like a car whose steering wheel had only two positions, right or left. The voting never results in a nice compromise that is "middle of the road", but tends to result in either a fairly low value or a fairly high amount. He proposes to reform the dues-voting process. He suggests that they change it so that everyone simply votes for the exact amount they prefer, and they select the average amount.

Unfortunately, this proves to be a complete disaster. The Extravagants vote for higher and higher values to swing it further their way, and the Cheapskates vote for lower and lower values. After having a vote that somehow results in dues of negative 1,452 dollars a month, Steve says, "um, I'm sorry folks, I think I must have meant median, not average". The median, also known as the 50th percentile, is the amount right in the middle, where half the people vote for higher amounts and half vote for lower amounts.

todo: graph showing showing it selecting the median of $15

So from that day on, the amount of dues is determined by the median of everyone's preferred amount. Soon, everyone realizes that there is no longer a reason for organizing into parties, and no reason to vote strategically -- they will get the best results by simply voting for the exact amount they prefer. Everything quickly stabilizes, and every year a result is selected that is a nice, reasonable compromise....somewhere in the $15 range.