April 1, 2013

Duverger's Law
Two's a party, three's a crowd



"Do you want to stay embedded with one of the Twin Powers? Are you on board with voting for the lesser of two evils, as you see it, for ever and ever? These two choices will leave you essentially directionless, and neither speaks of your values. So leave the corporate run two party system and choose a party that has your values and direction." [1]

That's what is says in the Green Party's official mission statement. They implore citizens to imagine a world beyond the two-party system. You gotta love their passion. And the Green Party does pose an important question: why is it that Americans only have two choices at the ballot box?

The answer is that our system is structurally distinct from countries that have many political parties, like Germany or Denmark for example. The vast majority of elections in the United States, from dog catcher to Senator, use a plurality voting system (a.k.a. a "first-past-the-poll" system.)* This simply means the candidate with the most votes wins, even if multiple candidates were on the ballot, and as a result the person with the most votes out of the bunch doesn't have a full majority. [2] In this post, I'll explain Duverger's Law, the political science hypothesis that says in election systems like America's,  only two political parties can survive. In response to the Green Party's question, the reason I don't vote for a third party is because unless we had a Constitutional Amendment, third parties cannot be viable in the U.S. That's, like, not a thing. So contrary to the Green Party's premise, if you cast your ballot for a third party, then in reality you are just voting to elect the candidate that is, as you see it, the worst of two evils.

Game Outline.

Let's keep it super simple: imagine a hypothetical congressional district with exactly 100 voters. Based on recent polls, campaign strategists can rank these voters from most progressive to most conservative. We'll illustrate this as a number line from 0 to 100 where the most conservative voter is #1 at the very right of the number line and the most progressive voter is #100 at the very left (they can literally do this in the real world.) [3]

So then, for example, voter #18 is the 18th most conservative voter in the district. Let's say that voter #18 decides to run for Congress as a Republican, and is facing a tough three-way race against voter #50, a Democrat, and #84 the Green candidate. By the way, these numbers were chosen simply for the sake of concreteness, and are not intended as an evaluation of these parties' relative ideological positions in the real world.



Each of the folks in the district will vote for the candidate whose views are closest to their own. So the Republican wins voters #1 - #34 (including herself), who are closer to her than to the Democrat. The Democrat wins the votes of #35 - #67 (including herself), who are closer to her than to either of the other candidates. The Green candidate wins #68 - #100 (including herself), which is all remaining voters to the left of the other candidates.



If you do some quick math, you'll see that in the illustration above, the Green candidate and the Democrat each get 33% of the votes, leaving 34% to the Republican. In the American system, the Republican would win in this case, because even though a majority of people voted against her, she has still has more votes than any other candidate. 

Equilibrium: Two Coalition Parties.

Clearly, the Democratic candidate and Green candidate would realize that if one of them dropped out  and they formed a political party together, they could beat the Republican candidate. But how do they decide which one of them drops out and has to endorse the other's platform?

In this example, if the Green candidate dropped out and formed a political party with the Democrat based around the Democrat's platform, and the Republican just ran as her own party, then the Green candidate's voters would now vote for the Democrat, because they agree with her more than with the Republican. So the newly formed Democratic Party would win 66% to 34%.


But on the other hand, if the Democrat dropped out and joined a political party based around the Green candidate's ideas, then the district's voters would split in two at the midpoint between the Green platform and the Republican platform, which is between #51 and #52. So now the newly formed Green Party would lose to the Republican Party 49% to 51%.



Since there's no way for the Green candidate to win in this scenario, she would be better off dropping out and letting the Democrat win than staying in the race, splitting the progressive vote, and it being her fault that the Republican won. Moreover, Duverger's Law states that even if the Green candidate refused to drop out, over time Green voters would migrate to the Democrat because they see that their favored candidate has no shot at victory.

Conclusion.

I totally get why the Green Party wishes we didn't have a two party system. They are right about one thing: with only two parties, a lot of voters end up voting for a candidate they don't completely agree with. But that's just the system we have. Our system could be described as "survival of the biggest."  You can only win by forming the largest coalition. Therefore, the equilibrium is that there is not room for more than two viable US political parties.

It is important to note that the reason that progressives do better forming a coalition party around the Democratic platform than the Green platform has nothing to do with one of them starting out as the bigger progressive party in the initial three-way race. We controlled for that by assuming they start out the same size (33% each in our example.) The reason is that the Democrats are more moderate, and therefore will appeal to more voters between themselves and the Republicans.

Smaller factions can still have a big impact by joining one of the two parties with a shot at winning, and thereby pulling of center gravity leftward within that progressive coalition. But if you vote for a third party, then you are helping to elect the canidate that least shares your values.

The Green Party's official platform includes one structural reform called "Instant Runoff Voting," which they argue would make third parties viable. My next post will play out what would happen if we adopted this proposal. Read next post>>


*Footnote: the U.S. Presidential elections work slightly differently. A candidate must receive a majority of electoral college delegates, not just a plurality, in order to clinch the Presidential election. If no candidate receives a majority, the President is chosen by the House of Representatives and the Vice President is chosen by the Senate. The game theory model of this system would look way more complicated. So for simplicity I drew a model of plurality voting systems, which are most elections in this country, but it would have the same result if we looked at a voting system where you must get a majority: with regard to parties, it's survival of the biggest, even more so if you have to reach the higher threshold of 50% + 1. Maybe I'll model this more complicated game at some point.