Ori Livson

Is there a right way to vote?

Arrow's Impossibility Theorem and the 2025 Australian Federal Election

• Mathematics
• Social Choice-Theory

There are many ways to hold elections; the Wikipedia page on Electoral Systems is full of colour-coded world maps of different ways to do so, including 24 types of electoral systems for lower house / unicameral elections alone (see Figure 1).

The one thing all electoral systems have in common is that they all have trade-offs.

By trade-off, I mean some seemingly desirable guarantee about the behaviour of elections doesn’t necessarily hold. For example, in some electoral systems, one cannot guarantee that a candidate always benefits from receiving more votes. The study of these trade-offs in social decision-making processes such as elections, trial by jury, budget allocation, etc. is a major theme of the field of Social Choice Theory.

The purpose of this post is to introduce key topics of Social Choice Theory using examples from the 2025 Australian Federal Election. A particular phenomenon of interest in this election is how many lower-house candidates won or lost their seats based on who came 3^rd in their race. The story of the electoral system trade-off that phenomenon corresponds to is a consequence of Social Choice Theory’s most famous result: Arrow’s Impossibility Theorem. Moreover, this story provokes an interesting question of whether that trade-off is a bug or a feature of Australia’s electoral system.

Note: the election figures of this post are accurate as of 12.05.2025 (9 Days after the election) but are subject to change as counting continues.

The 2025 Australian Federal Election

Australia uses an electoral system called Instant Run-Off Voting, whereby voters rank candidates 1^st, 2^nd, 3^rd, etc. in whatever order they wish, and a winner is decided using the following procedure:

1. To begin, each candidate is assigned a ballot total of those that rank them 1^st.
2. If any candidate’s ballot total is greater than 50%, they are declared the winner.
3. Otherwise, the candidate with the lowest ballot total is eliminated and each of their ballots are redistributed to next highest-ranked candidate - updating the ballot totals (see Figure 2).
4. Repeat 2-3 until a winner is declared.

This system has many interesting dynamics, for example, a party’s first preferences share can rise by just 0.46% from 33.34% to 34.8%, and it means the difference between losing with 68/151 seats (Labor, 2019) and winning with 93/151 seats (Labor, 2025).

Since 1946, the party that holds government in Australia’s has been a contest between two major parties: the Labor Party (a left-wing party) and the Liberals (a right-wing party, typically in coalition with the National party). That said, the performance of various minor parties and independent candidates often influences which major party wins. In this post, I want to focus on one minor party in particular: the Greens party (a far-left party) and for simplicity, will pretend no other candidates exist.

In this election, ballots typically ranked candidates from the three parties according to the left-right alignment per voter ¹, i.e.:

• If a voter ranked the Liberals 1^st or 2^nd, their next preference was typically Labor rather than Greens. This is because the Greens are further left-wing than Labor, so, a right-wing Liberal voter typically preferred Labor next.
• If a voter ranked Labor 1^st or 2^nd, their next preference was typically Greens.
• If a voter ranked the Greens 1^st or 2^nd, their next preference was typically Labor.

In other words, a first-preference Liberal ballot typically had a most-to-least preferred ordering: Liberal < Labor < Greens, a first-preference Labor ballot had ordering: Labor < Greens < Liberal, and a first-preference Greens ballots had ordering: Greens < Labor < Liberal.

How 3^rd-Place Flips Electorates

Consider electorates where the final three candidates were Liberal, Labor and Greens, with the Greens in a narrow lead. As previously discussed: in 2025, preferences primarily flowed in one of two ways:

1. If the Liberals were ranked 2^nd and Labor 3^rd, Labor ballots by in large flowed to the Greens, pushing the Greens vote past 50% to a victory.

2. Conversely, if Labor was ranked 2^nd and the Liberals 3^rd, the Liberal voter’s next preference by in large flowed to Labor, pushing the Labor vote past 50% to a victory.

Case 2 materialised several times; see Figure 3 for an example in the electorate of Melbourne. As discussed in Footnote 1, these two cases aren’t the norm in every election. However, in 2025, the fact that the Greens vs Labor outcome depended on the Labor vs Liberal outcome demonstrates that Instant Run-Off Voting has a trade-off known as the violation of the Independence of Irrelevant Alternatives axiom. This axiom is the main subject of Arrow’s Impossibility Theorem.

The Indenpendence of Irrelevant Alternatives Axiom

A social decision-making process satisfies the axiom of “Independence of Irrelevant Alternatives” if:

The decision of alternatives A vs. B depends only on A and B.

In other words, the addition or removal of any other alternative C shouldn’t affect the decision of A vs. B. In hindsight, this may seem obviously unattainable for many electoral systems. However, to understand the intuition for desiring the Independence of Irrelevant Alternatives axiom, consider the following anecdote about philosopher Sidney Morgenbesser.

Morgenbesser is at a diner, ordering dessert and is told by a waiter that he can choose between blueberry or apple pie. So, he orders apple pie. Soon after, the waiter comes back and explains that cherry pie is also an option. Morgenbesser then replies “In that case, I’ll have blueberry pie!”.

Not only is this a strange response by an individual, it would still be a strange outcome of a group of customers voting on what pie to serve. Perhaps strange because of the sense in which the cherry pie option is an afterthought.

Arrow's Impossibility Theorem

Kenneth Arrow’s 1951 Nobel Prize winning work “Social Choice and Individual Values” questioned whether the axiom of Independence of Irrelevant Alternatives is ever feasible in electoral systems ². He argued that it is not by showing it is incompatible with the following other three non-negotiable axioms. In the context of voting, those axioms are as follows:

• Unrestricted Domain: each voter should be allowed to cast any valid ballot they wish.
• Unanimity: if every voter prefers candidate A over candidate B, then A should win against B.
• Non-Dictatorship: no single voter should always get their way in every possible election ³.

Unanimity failing, or a voter being a dictator are absurd on face. However, the argument for Unrestricted Domain being non-negotiable is more subtle. Indeed, imagine an electoral system for which Unrestricted Domain does not hold, i.e., there is a situation where the populace casts their votes, and their government rejects the election outright. Arguably, the electoral system in question is impractical or undemocratic. Impractical due to the rejection of a likely time-consuming and expensive election, and undemocratic due to individuals not being allowed to vote how they wish all the time.

Arrow’s Impossibility Theorem states that there is no electoral system that satisfies all of: Unrestricted Domain, Unanimity, Non-Dictatorship and the Independence of Irrelevant Alternatives axioms

In other words, every electoral system has the trade-off of violating at least one of those four axioms ⁴. It doesn’t matter if we use Instant Run-Off voting as Australians do, first-past the post as the English do, two-round voting as the French do, assign points to votes as the Nauruans do, thumbs-up and thumbs-down votes as the Latvians do, or use the clamouring of spears on shields like the Ancient Spartans did ⁵.

The Why: Condorcet Paradoxes

The suspicion that a theorem like Arrow’s Impossibility Theorem holds, largely grew out of the heated debates preceding the French Revolution on how voting ought to be done. A pivotal momenet came when the French polymath Nicolas de Condorcet observed a problem with a voting-method known as Pairwise Majority Voting ⁶.

The voting-method is simple: use the ballots to assess the candidates head-to-head, and rank winners and losers transitively. Transitivity means, for example, that if the Greens win against the Liberals, and the Liberals win against Labor, then the Greens win against Labor.

Pairwise majority voting satisfies the Independence of Irrelevant Alternatives axiom by definition, as well as Unanimity and Non-Dictatorship. However, it fails Unrestricted domain due to presence of Condorcet Paradoxes.

The simplest example of a Condorcet Paradox occurs for the election of 3 candidates by 3 voters as per Figure 4. The calculation for Greens win vs Liberals is shown, and one can likewise calculate that Liberals win vs. Labor, and Labor wins vs. Greens in the same way. However, by transitivity (i.e., the Greens win vs Liberals, which wins vs Labor) it must also be the case that Greens win vs Labor, a contradiction!

In fact, this isn’t just a toy problem: in the 2014 Victorian state elections, had Victoria used pairwise majority voting rather than instant run-off voting, a Condorcet Paradox would have occurred in the electorate of Prahran ⁷.

Although many proofs of Arrow’s Impossibility Theorem have been found since 1951, Condorcet Paradoxes were typically merely seen as intuition for why Unrestricted Domain fails when the other axioms hold. However, in 2024, Prof. Massimo D’Antoni devised an ingenious proof that in the strict case (i.e., when ties between candidates aren’t allowed): Unrestricted Domain fails because one can always find an election that produces a Condorcet Paradox. A proof that this holds in the full case of Arrow’s Impossibility was shown in a 2025 paper I wrote with Mihkail Prokopenko ⁸.

Arrow's Impossibility Theorem and Australia: a Bug or a Feature?

Using the term “trade-off” to describe the failure of the Independence of Irrelevant Alternatives axiom in the Australian electoral system is not an argument that the Australian system is unfair or undemocratic. To the contrary, the trade-off arguably represents a sophistication in the Australian system, wherein:

The Australian Electoral System does not simply average the left-right alignment of voters.

Moreover, labelling Labor as the centre-candidate for simplicity, one can argue:

1. Melbourne’s 2010-2025 Greens victories were possible given a weak enough centre (i.e., Labor 1^st preference votes).
2. Otherwise, the system meets conservative voters in the middle, given a strong enough middle. For example, despite Melbourne being further left-leaning than Labor on average (due to its 40.1% Greens and 31.4% Labor 1^st preference votes), the Greens’ -9.5% and Liberals’ +4.1% swings sufficed to flip that seat to the centre ⁹ ).

Is Voting Doomed?

I don’t believe as some do, that Arrow’s Impossibility Theorem spoils the project of fair voting itself. The way I see it is that:

Arrow’s Impossibility Theorem is a testament to voting being a highly expressive process.

By “expressive”, I mean an election outcome consist of preferences that no individual necessarily initially held, i.e., the whole is different to its parts. However, expressivity means trade-offs are inevitable. In other words, if one desires voting without trade-offs (e.g., one where the Independence of Irrelevant Alternatives axiom holds), only a very dull system is feasible (e.g., one with a dictator).

This principle can be found in other domains. For example, if one wanted English to be unambiguous, or free from Self-Referential Paradoxes (e.g., the sentence “This sentence is false”), one would need to remove so many words and grammatical forms that English would also have to become very dull.

In fact, my aforementioned 2025 paper demonstrates formal links of this sort by relating the trade-offs of Arrow’s Impossibility Theorem to the trade-off of incompleteness in logic (specifically, Gödel’s First Incompleteness Theorem). A post outlining that paper is coming soon.

Addendum

Here, I’d like to discuss some other interesting, related topics in Social Choice Theory that manifest in the Australian System. This will hopefully emphasise the relevance of Social Choice Theory and also drive home the more philosophical argument about expressiveness vs trade-offs.

Firstly, I want to discuss the assumption of voter’s preferences respecting left-right alignment. Not only is that not true for every voter, but some voters stand to gain by voting against their honest preferences. For example, in Australia, voters who most prefer one major party may preference the other major party last (i.e., below fringe candidates). This is done in order to prevent the less desired major party from securing a majority too early in the instant run-off voting process. In Social Choice Theory, this is called Strategic Voting, and has been common in Australia at various times (see Footnote 1). An well-known Arrovian result for strategic voting is the Gibbard–Satterthwaite Theorem.

Secondly, on a more positive note, when Strategic Voting is low, the Australian System typically satisfies Monotonicity, meaning “a candidate always benefits from receiving more votes”. This is because a candidate receiving more votes typically further increases the ballots they receive from preference flows via closely aligned candidates, i.e., ballots that likely preferences them soon after. In other words, a candidate receiving more votes tends towards eliminating closely aligned candidates earlier, which causes their ballots to flow sooner.

Monotonicity can’t be taken for granted in the best of times. For example, it does not hold in the French national system, which holds up to two “choose-one” elections, where the second is held between the final two in the event of a majority not being reached in the first round. Monotonicity does not hold in this system because if a candidate cannot attain a majority in the first round, they best ensure their second round opponent is one they can actually defeat head-to-head. Thus, if a candidate receives too many votes, that may prevent all defeatable candidates from appearing in the second round.

Finally, a fascinating result from Social Choice Theory that reiterates the value of all these chaotic edge-cases and trade-offs is May’s Theorem, which states that the only monotone voting method where all voters and all candidates are interchangeable is two-party majority voting. Thus, to have the expressiveness of multi-candidate and preferential voting, one must learn to embrace the trade-offs.