I have always thought that Australia has had a perfect electoral system, at least for the House of Representatives. Compulsory ranked voting with instant run-off? What’s not to like?! The electorate is forced to show up once every four years and have their say, and no-one’s vote is wasted, or so I thought. I was recently made aware of two theorems in social decision theory, Arrow’s Theorem, and the Gibbard-Satterthwaite Theorem, that have made me more uneasy about our voting system, and voting systems in general.

Arrow’s Theorem

Arrow’s Theorem, or Arrow’s Impossibility Theorem, is stated as follows (Taken from here):

Let $latex A$ be a set of outcomes, along with a process which aggregates individual’s preferences into a single preference order on $latex A$.

Whenever $latex A$ has more than two outcomes, then at least one of the following must not hold:

• (Weak Pareto Efficiency) If alternative a is ranked higher than alternative b for all individuals, then a is aggregately ranked strictly higher than b.
• (Non-dictatorial) There is no individual such that that individual’s strict preferences always prevails.
• (Independence of irrelevant alternatives) The social preference between x and y should only depend on the pairwise individual ordering of x and y. For example, adding a third candidate to the election should not affect the winner unless the third candidate wins.

Clearly having a non-dictatorial system is extremely important, though one could critique the power our media institutions have in choosing the winner of an election, and Weak Pareto Efficiency is also important. However, this usually means that an electoral system will fulfil these two properties, which means that the third property is usually unfulfilled. This means that in Instant Runoff Voting, we have situations where it may be advantageous to us to not enter our true preferences on the ballot. What type of situations could this occur in?

An Example

Well, let’s consider a scenario where we have the following first preference polling numbers:

Greens: 33% (33% Labor)

Labor: 30% (14% Liberal, 16% Greens)

Liberal: 37%

Suppose your favoured party is The Greens, followed by Labor, and then Liberal. Suppose also that you know that the preference distributions is as above. In the run-off, Labor will be eliminated, and the preferences allocated as follows:

Greens: 49%

Liberal: 51% (Winner)

and we have ended up with our least favoured outcome.

However, if instead 3% of Greens voters put Labor as their first preference instead, we get:

Greens: 30% (30% Labor)

Labor: 33% (14% Liberal, 19% Greens)

Liberal: 37%

and now The Greens will be eliminated, and the final results will be:

Labor: 63% (Winner)

Liberal: 37%

and we have ended up with our second choice by a landslide.

So in what situation will this come about? Well, as we can see above, it will happen when we have an extremely close election between three or more parties, and the preferences of the centre party are not sufficient to elect the minor party. It may be tactical then for the Greens voter to instead put Labor as their first preference.

You can see, however, that this requires detailed knowledge of both the first preference votes and the preference run-offs for each party, which is rarely the case. In the absence of such information, I would conjecture that it is a dominant strategy to put in one’s truthful preferences.

Alternative Voting Mechanisms

The previous section had me thinking, is there a way to bypass Arrow’s Theorem? It only holds for ranked choice ballots, right? Well, there are a plethora of other ways to vote! See Cardinal Voting among others, which does not use a ranked system, and therefore sidesteps Arrow’s Theorem. However, there exists a much more general theorem, the Gibbard-Satterthwaite Theorem, which says that for any voting mechanism, one of the following must hold:

• There are only two alternatives,
• (Non-dictatorial) There is no individual such that that individual can choose the winner,
• (Existence of Tactical Voting) There is no action which best defends the individual’s desired outcome in every situation.

So it can be seen that there is no voting system which will satisfy all three of these simultaneously, and to me, that is a little bit troubling.

Voting photo taken from here.