## Sunday, January 15, 2012

### Equations of States for Election (選舉狀態方程式)

(WARNING) This post has nothing to do with beading.
I still hope that some of you might be interested in the discussion.

At the beginning of the 2008 election year, the local Kuomintang (KMT) party of Taiwan had a landslide victory in parliamentary election by winning the 80% of the seat out of 58% of the total population votes. Here, I have recalculated the fractions of seats and votes based on the two-party approximation, i.e. I ignored all other votes except those voting for the two major parties, KMT and Democratic Progressive Party (DPP). The apparent disproportionation raises some puzzles on the fairness of the new electoral system just adopted for the election, in which only one representative get elected in a single district. However, this is very similar to the electoral system adopted by the US presidential election - winner of a state get all electoral votes of that state. So I guessed one might be able to see some empirical correlation from the US presidential elections of past one eighty years (1932-2004).

Figure 1 shows the relationship between the fractions of electoral votes vs. popular votes for US presidential elections since 1932 (Franklin D. Roosevelt won the election in that year.) Here x is defined as the fraction of popular votes obtained by republican's presidential candidate and y is defined as the fraction of electoral votes obtained by republican's presidential candidate. Interestingly, x and y are very similar to mole fractions commonly used in chemistry.
As an example from the US presidential election in 2000, Bush vs. Gore, we have popular votes: 50456002 (Republican) 50999897 (Democrat) and electoral votes : 271 (Republican), 266 (Democrat). Thus, in this case, x=0.497, y=271/(271+266)=0505.

The resulting data (blue circles in the figure) look like a titration curve, which can be fitted by y=tanh(ξ x) pretty well (blue curve). As a chemist, I deliberately write this curve in the form of the Henderson-Hasselbalch equation with an extra parameter ξ, denoting the cooperativity effect. There are fluctuations around the fitting curve, of course. That is the reason why G. W. Bush got elected, even though Gore had more popular votes in 2000. Additionally, this equation is symmetric, meaning that the electoral system is basically fair. But the fraction of seats of one party is a highly nonlinear function (close to a step function) of fraction of votes. The small parties are hard to get any seat in the parliament. Also, this equation does not say that we can predict the result either since we still need to know the fraction of votes, which is impossible to know before the election.

OK, all these data are already there before the 2008. What I found is that the 2008 re-election of legislative representatives of Taiwan fell pretty well on the same fitting curve or equation of states for a particular electoral system.

Later that year, in US presidential election, B. Obama got elected as the 44th president of United States by winning the 52% of popular votes and 67% electoral votes, which is again described by the election curve pretty well.

Now, we have a re-election of the legislative representative again in Taiwan yesterday (Jan. 14, 2012).
The results go like:

Fraction of total votes obtained by these two parties: (KMT: 48.18%; DPP: 43.8%)
Rescale the ratio by considering only these two parties: (KMT:52%; DPP: 48%)

Seats obtained by these two parties (KMT:44; DPP: 27)
Fraction of seats obtained by these two parties (KMT: 62%; DPP:38%)

This result satisfies the empirical equation of states for our electoral system pretty well. I consulted some local experts on election a few years ago. They told me this correlation is well known to them. However, the information I got from the media seems to contain so many details and is usually not very useful for getting a big picture. If this empirical relationship is valid, one might still need to provide a microscopic interpretation possibly based on some sort of statistical theory or ensemble theory that chemists and physicists commonly used for making connections between microscopic states and macroscopic properties.

slides of a talk I gave in Nov. 2008 right after US 2008 presidential election

2008 年1月 臺灣 立委選舉改用「單一選區兩票制」，國民黨以 53.48% （僅考慮國 ，民兩黨，則為 58% 相對比例）， 得到近八成的席次，很多人對此產生”不公”，或是「票票不等值」的疑慮。

2008年底美國總統大選漂亮地落在這條曲線上。

（選票分率 國：48.18%；民43.8%）