## 3-cornered electoral fights and Arrow’s impossibility theorem

Recently Alex Au wrote a couple of posts here and here questioning if opposition parties’ attempt to avoid 3-cornered fights might be undemocratic to voters since they offer the electorate fewer choices. Now of course this might appear to most people as a stupid conclusion to reach. There are many reasons to believe that doing so would help only the PAP, but this post would not attempt to go down that line. Lucky has done so here and has E-Jay here. To begin let’s start by quoting Alex’s argument that having more opposition parties contest in the same district would result in a more democratic outcome:

But this logic is based on an assumption which, strangely, no one has yet interrogated: that there is a relatively inelastic pool of voters who would not want to vote for the PAP and that they would vote for whichever opposition party happened to be standing in their constituency. You see this assumption at work whenever someone talks about not “splitting” the vote.

…

From what I have seen,

opposition parties are not interchangeable, and even if there are days when one is frustrated with the PAP,not all of the other parties are always preferable to the PAP.…

But the trend to differentiate themselves by articulating convictions and policy directions can only mean that they will become less and less interchangeable. I may like Party K and Party L for their positions; I may not like Party M and Party N. So, come election time,

why should I be denied the opportunity to choose among K, L, M and N?

However, it is not necessary to argue as E-Jay did, that 3-cornered fights can only be considered a more democratic electoral representation if Singapore were more democratic. Something known as Arrow’s impossibility theorem proves that result does not hold *mathematically* regardless of that setting. In other words, 3-cornered fights are undemocratic and unfair purely from a *rigorous and mathematical* point of view. I will attempt to argue that offering the electorate more choices may paradoxically result in a less fair and undemocratic outcome from a strictly theoretical perspective, without taking into account Singapore’s electoral history and landscape. Similarly the argument that having one opposition party give in the does not necessarily imply that the move was done purely out of a motive to deny the PAP their votes without any consideration whether doing so might actually result in an electoral outcome which is more reflective of the electorate’s ranking preference.

Arrow’s impossibility theorem

In 1952, Kenneth Arrow, a mathematical economist proved an important mathematical result in social choice theory which would turn out to have wide reaching theoretical and practical implications. This was published in his PhD thesis and was subsequently renamed after him as Arrow’s impossibility theorem. What does this theorem say? Arrow’s theorem shows that any election for which there are three or more choices cannot satisfy all of the following conditions simultaneously (taken from Wikipedia):

- If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
- If every voter’s preference between X and Y remains unchanged, then the group’s preference between X and Y will also remain unchanged (even if voters’ preferences between other pairs like X and Z, Y and Z, or Z and W change).
- There is no “dictator”: no single voter possesses the power to always determine the group’s preference.

What Arrow’s theorem says is that even if we grant that all voters have their own valid preferences and ranking and the most democratic of all electoral systems, the election result will always be suboptimal if there are 3 or more candidates. In short, democratic and fair 3-cornered elections are *impossible *even from a strictly theoretical or mathematical standpoint.

This point may be hard to see without taking an example as follows. I shall adapt an example from Investopedia for the following.

## An electoral example

Let’s suppose there are three parties, the ruling PAP, the GOP (Good Opposition Party), and the LOP (Lousy Opposition Party) all contesting in the same district populated by 100 eligible voters. In this setting, we assume that PAP has 45 voters who would vote for them but would vote instead for GOP if PAP were not contesting. In other words, assume that they are wary of upsetting the government by taking a chance and voting in the LOP . The GOP on the other hand, has 35 voters who back them firstly but who rank the PAP as their 2nd choice because they too likewise are wary of giving their votes to the LOP who might run the country into the ground if they ever win. Lastly, the LOP has only 20 voters who are considered die-hard opposition supporters, they would vote for GOP if LOP withdraws but never for PAP.

The voters’ ranking preference may be presented as follows:

- 45 votes PAP>GOP>LOP
- 35 votes GOP>PAP>LOP
- 20 votes LOP>GOP>PAP

It can be seen from the above that of the non-PAP voters, a clear majority of 35 out of 55 (or 64% close to the 66% vote share PAP won the last GE) back the GOP because they truly believe and support what the GOP stands for and not just because they’re running against the PAP. Hence we have an overwhelming majority of non-PAP voters who are rational voters who would vote PAP if GOP decides to withdraw to preserve the existing social order. This is exactly what Alex Au would like to see right? Having an electorate where most non-PAP voters vote for an opposition party because they believe in what they stand for instead of simply being anti-PAP. However, it’ll be shown that presenting 3 or more choices actually makes the election worse off and less democratic.

What happens on election day? If all three parties contest, the PAP wins with 45 votes (Scenario 1). Alternatively, suppose the LOP withdraws. All its 20 supporters vote for their next choice, the GOP (Scenario 2). As a result, the GOP wins 55 votes to 45 votes. It’s clear that LOP’s withdrawal helped the GOP to win, but there’s something else at work here.

In the case of Scenario 1, the PAP won despite the fact that they were the 1st or 2nd choices of only 80 voters. The GOP **lost **even though they were either the 1st or 2nd choice of **all 100 voters**. In Scenario 2, the GOP wins when the LOP withdraws. Hence the resulting outcome is paradoxical in the sense that giving the electorate more ballot choices actually results in a *less* fair and equitable outcome.

This might appear paradoxical prima facie. But read through it again and you’ll see Arrow’s impossibility theorem at work here. Alex Au’s stated ideal that giving the electorate more choices in the ballot actually resulted in a less optimal outcome where a party which ranked 1st or 2nd in everyone’s list loses out to a party which has lesser support in the electorate’s 1st or 2nd place ranking

For argument’s sake, let’s consider a third possibility (Scenario 3), the one where the GOP withdraws and the LOP remains to contest. In this case, the PAP wins with 80 to 20 votes. Again we get a suboptimal result where the winning party is the choice of only 80 voters, while a non-contesting party (GOP) ranks 1st or 2nd in everyone’s list. It’s interesting to note that in this case of 2 contesting parties, the result obtained does not violate Arrow’s theorem; the PAP which is supported by 80 voters in 1st or 2nd place trumps the LOP which enjoys the 1st/2nd place support of only 20 voters.

In closing, what Scenario 3 shows is that if either opposition party in a district has to withdraw, the less popular one should give way to the more popular one in order to ensure a more equitable and democratic outcome reflective of the electorate. Of course the big problem in any such move would be to determine which party to give way to the other and this post in no way endorses one party over another. Contrary to Alex’s opinion restricting voters’ choices by having one opposition party withdraw can and may result in a *more* democratic outcome. So to conclude, unless Alex Au has suddenly managed to find a disproof of Arrow’s impossibility theorem, he’s flat out wrong that 3-corner fights are good for democracy.

As an afterthought note that the above-mentioned scenarios would not have occurred if elections were won by majority vote. Perhaps this should be considered a good reason to push for run-off majority-vote elections instead of merely one where a plurality of votes wins.

## Further reading

Arrow’s Theorem Proves No Voting System is Perfect

Arrow’s impossibility theorem (Wikipedia)

The Arrow Impossibility Theorem: Where Do We Go From Here

Su, Francis E., et al. “Arrow’s Impossibility Theorem.” *Math Fun Facts*.

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Daily SG: 23 Mar 2011 « The Singapore DailyMarch 23, 2011 at 11:15 AM

It was a delight to read this.

Just a thought…guess who among the MIWs has a First-Class Honours in Mathematics from Cambridge?

Methinks there could be some mathematical rationale behind the seemingly invincible GRCs strategy – FBD could you dig that up?

crescitMarch 24, 2011 at 9:43 AM