Cross-posted from substack

1. What are non-deterministic elections?

Voting is at the heart of democracy—it's how we collectively decide who will lead us and shape public policy. Traditionally, most democratic elections rely on deterministic voting systems. These are the systems where the outcome is based on a clear majority, and the winner is the one who secures the most votes.
This approach has its objections, for example the problem of the "tyranny of the majority", where the minority's voice is often silenced, and the concentration of power is in the hands of just over half the voters.^[1]

In response to these concerns, recent research has started exploring non-deterministic voting systems. Unlike deterministic systems, these introduce an element of chance, beyond just using randomness to break ties.
It might be best if I first give a simple example of a non-deterministic voting system so you can envision how this would work.
 

Example: The Random Ballot Method

One of the simplest non-deterministic voting systems is called the Random Ballot. Here’s how it works: voters cast their ballots like normal, but instead of counting the votes, one ballot is randomly selected to decide the winner. If you control 51% of the vote, you don’t automatically win—you just have a 51% chance of your candidate being selected. The same applies to any minority group; if they control 30% of the vote, they have a 30% chance of winning.

This might seem unfair at first glance, but it actually offers a more proportional distribution of power over time. In traditional systems, controlling a majority of votes means controlling all the power. In non-deterministic systems, power is spread according to the actual support each group has. The random ballot method, for instance, perfectly aligns power with voting percentage.

Non-Deterministic Systems

Interestingly enough, the idea of using randomness in decision-making isn’t new. In ancient Athenian democracy, officials were often chosen by sortition—essentially, by random lot. This was based on the belief that every citizen had an equal right and ability to contribute to governance, and it was a way to prevent a small elite from holding too much power.

Today, we see echoes of this practice in the use of citizens' assemblies, where a group of citizens is randomly selected to deliberate on policy issues. Non-deterministic voting systems share a similar ethos, aiming to distribute power more fairly and prevent the dominance of the majority.

Of course, not all non-deterministic systems are as simple or as proportional as the Random Ballot. Take for example the First-to-get-two system. It works like the random ballot, but instead of picking just one ballot at random we keep randomly drawing ballots until we have two that voted for the same candidate. This creates a power dynamic somewhere between purely random and majority-dominated. We can visualize how much power each voting system gives to a voting bloc with the following graph:

So over time most deterministic voting systems give 100% of the power to a candidate who can get >50% of the votes. We can visualize this as a step function (blue line). Of course some voting systems require a supermajority in which case it is still a step function but more to the right (purple line).
Now look at the random ballot (green line), see how the distribution of power over time is perfectly proportional to the amount of votes a candidate/party receives? But not all non-deterministic voting systems are like that, the “First-to-get-two” approach I explained earlier, is somewhere in between perfectly proportional and the step function (red line).

While I think that the random ballot and first-to-get-two are too simplistic of a voting system to be any good, I do hope that they serve as a way to show the features of non-deterministic voting systems in general, and how a more sophisticated non-deterministic system may positively influence elections.^[2]

In this post, I’ll outline the reasons philosophers have given for why we should participate in democratic elections, and see whether these reasons hold up when it comes to non-deterministic voting systems. While I'll focus on elections where a single candidate is chosen, much of what I’ll discuss applies to other collective decisions, like referendums or parliamentary elections as well. I'll assume that these elections are free and fair, and that their outcomes are actually realized —since our goal is to focus on the inherent qualities of the voting system and not on outside factors like corruption.

2. Reasons to vote in non-deterministic elections

2.1. Prudentialist and Act-Consequentialist reasons to vote

Many people believe we have a duty to vote in a democracy. However, not everyone agrees, for instance, in their paper "Is there a duty to vote?", philosophers Lomasky and Brennan challenge two key arguments that support it: the prudentialist argument and the act-consequentialist argument.

1. Prudentialist Argument: This argument says that we should vote to advance our own interests. The logic goes like this:
   • You should work to promote your own interests.
   • Voting is a way to do that.
   • Therefore, you should vote.
2. Act-Consequentialist Argument: This is similar, but it shifts focus from self-interest to promoting the interests of others:
   • You should work to promote the interests of others.
   • Voting is a way to do that.
   • Therefore, you should vote.

The core idea behind both arguments is that election results affect people's well-being. However, critics like Lomasky and Brennan argue that the chance of an individual vote making a decisive impact on an election is so small that it hardly justifies the time and effort of voting. They point out that the probability of one vote swinging an election is nearly zero.

Philosopher Derek Parfit, however, presents a counter-argument. He agrees that the chances of one vote deciding an election are slim, but argues that small probabilities still matter when the stakes are high. For instance, if you were a nuclear engineer with a 1 in a million chance of causing a disaster, that tiny probability would still be worth worrying about. Similarly, voting is a small personal cost with a potential huge collective impact. According to Parfit, even though the chances of your vote being decisive are slim, the overall benefit of voting is still significant when considering the whole society.^[3]

2.2. Prudentialist and Act-Consequentialist Arguments in Non-Deterministic Elections

One major assumption behind Lomasky and Brennan's critiques is that the voting system in question is deterministic. In non-deterministic systems, where each vote has an influence on the probability of a candidate winning, rather than directly determining the outcome, these two arguments for going to the polls become more compelling.

For instance, in a Random Ballot system, the influence of any voter is proportional to 1/N (where N is the number of voters). This influence is larger compared to deterministic systems, where the likelihood of any one vote being pivotal is much smaller. So, these two arguments become stronger for non-deterministic systems, since each vote carries a more consistent influence.

2.3. Fallibility

Lomasky and Brennan also argue that voters are often fallible—they make mistakes or misjudge candidates. Political outcomes are unpredictable, and campaign promises often don’t translate into actual policy changes. This uncertainty diminishes the expected benefits of voting. They suggest that the duty to vote is only valid if you’re well-informed and if the election is close enough that your vote could matter. The fallibility of voters applies to both deterministic and non-deterministic elections. The unpredictability of politics doesn’t disappear in non-deterministic systems. We can model this mathematically, so for the mathematically inclined: here’s my research paper (the basis for this blogpost) which includes more detailed and academic versions of all the arguments, and more formalisms.
TL;DR Unlike the prudentialist and act-consequentialist argument, this argument doesn’t make your reasons to vote in a non-deterministic election automatically stronger, but neither does it them automatically weaker. Instead it depends on the believed competency of the voter. For a voter who believes they are more competent than average, it strengthens the argument, for voters who believe they are less competent than average it weakens it.

2.4. Influence of Voting on the Winner’s Mandate

Another reason to vote is to influence the size of the winning candidate’s mandate—their authority to govern based on how many votes they receive. Even if your vote doesn’t decide the winner, it could help increase your preferred candidate's legitimacy or diminish the mandate of a losing candidate. A large mandate is often seen as giving a candidate more power to enact their policies.

However, research shows that a party’s effectiveness is mostly not tied to its mandate size. Also, an individual’s contribution to the mandate is almost negligible.

Suppose however that this research turns out to be incorrect, would the mandate argument be stronger in non-deterministic elections? After all, the argument works similarly; while randomness plays a role in determining the winner, vote counts still serve as a measure of public support. However, since non-deterministic systems can lead to winners with smaller mandates, supporters of minority candidates might have a stronger reason to vote to prevent an outcome where the winner lacks sufficient legitimacy. So I guess the strength of the argument depends on whether your candidate is a minority candidate or not.

2.5. The "Saving Democracy" Argument

The saving democracy argument says that the real power of voting isn’t just about influencing specific outcomes; it’s about keeping democracy healthy and stable by ensuring enough people participate. A high voter turnout signals that citizens are invested in their government. This engagement pressures elected leaders to be more responsive, knowing they are accountable to an active electorate. In short, even if your single vote doesn’t tip the scales, it still strengthens democracy by showing that people care and are watching.

But does it really? A single person's choice not to vote won’t collapse the system. Isn’t it more plausible that democracy’s health comes from knowing citizens could vote, not necessarily that they do. Many elections with low turnout still function well, so why is turnout considered essential to democracy’s quality?

In any case, the strength of this argument doesn’t seem to change depending on whether or not the election is deterministic.

3. Non-Instrumental reasons to vote

Most of us see voting as a way to influence election outcomes. But some philosophers argue there's also a broader, non-instrumental reasons to vote. Let’s look at some of them:

3.1. The Generalization Argument: A Kantian Perspective

What would happen if everyone abstained from voting? Wouldn’t democracy collapse? If democracy is inherently valuable, shouldn’t you vote to help maintain it?

This is a Kantian perspective on voting. Philosopher Immanuel Kant’s believed that one should only act on principles they’d be comfortable universalizing. In other words, an individual should only take actions they would be okay with everyone else taking. Applied to voting, if everyone adopted the principle “only vote when it’s in your interest,” democracy might cease to function as no one would feel obliged to participate regularly.

However, this argument faces some limitations. For instance, if we generalize that everyone must perform certain actions essential to society—like farming or building homes—it doesn’t logically mean everyone should be a farmer or a builder. Similarly, while democracy requires voters, it doesn’t require that every single person vote.

Furthermore, this argument may not apply to every reason for abstention. If someone genuinely believes that none of the parties represent their views, then abstention might seem principled rather than neglectful. For example, if a person adopts the principle “don’t vote if no candidate aligns with your political beliefs,” other people would vote and democracy could still function (assuming others do have candidates that align with their political beliefs).

3.2. The Free-Riding argument

Maybe a better way to frame this issue is through the lens of free-riding. Society generally frowns upon those who benefit from public goods, such as roads or clean air, without contributing to their maintenance, such as through taxes. Democracy can be seen as a public good that requires active participation to thrive. Voting, then, could be understood as a way of "paying your share" to sustain democracy, and abstaining from voting as a form of free-riding.

However, the free-riding perspective has its limitations when applied to voting. Voting, unlike taxes, doesn’t place an extra burden on others if someone abstains. In fact, abstention increases the influence of those who do vote. The less people vote, the more your vote influences the election.

In any case, the generalization argument doesn’t strengthen or weaken depending on whether or not the election is deterministic.

3.3. Voting as Self-Expression

Another argument for voting is that it’s valuable not just because of its instrumental effects, but because it is an expressive act—a way to affirm one’s identity, values, and support for democracy. This aligns with the ‘expressive theory of voting’, which holds that people vote not only to influence government but also as a means of self-expression. For example, even if sending flowers to a hospitalized friend doesn’t directly improve their health, it can still be seen as valuable because it serves as an expression of care and support.

Voting, in this sense, is viewed as a civic gesture, much like observing a two-minute silence on Remembrance Day. Citizens may feel a moral obligation to vote as an acknowledgment of democratic values and the historical sacrifices made to secure these rights. One could argue that by choosing not to vote a person is implicitly rejecting or downplaying the value of democratic participation.

But here too, questions arise. Unlike public gestures such as standing during a remembrance event, voting is anonymous and private, making it less visible as an expression of support for democracy. Furthermore, if voting were purely expressive, it wouldn’t explain the phenomenon of strategic voting, where individuals vote tactically rather than authentically, often supporting a candidate they dislike just to prevent a worse outcome.

In non-deterministic elections, like "Random Ballot," where the chance element reduces the incentive for strategic voting, it’s easier for people to vote authentically. In such cases, the expressive argument for voting could be seen as stronger, as individuals might feel freer to vote in alignment with their true preferences.
 

Conclusion

I made a little table to summarize the results:

The only argument that weakens our reason to vote in non-deterministic elections is the “polluting the polls argument”, which we looked at and subsequently dismantled in my last post. The rest of the arguments either vary in strength (depending on the circumstances), are equally strong, or are strengthened in non-deterministic elections.

So overall, it looks like the average voter has stronger reasons to vote in non-deterministic elections than in deterministic ones.

(If you want to see more detailed/academic versions of these arguments, check out the paper by me and Jobst Heitzig)

1. ^{^}

   While this post focuses on fairness, there are also other reasons to oppose it, such as that it could drive minority groups to separatism and violence (e.g. the separatists in Sri Lanka)

2. ^{^}

   For a more sophisticated non-deterministic voting system, see e.g. MaxParC

3. ^{^}

   This argument neglects opportunity cost. Even if voting is one way to contribute to the public good, it’s not the only way. In fact, maybe alternative actions—like encouraging others to vote— have an even greater impact than voting yourself. If you convince multiple people to vote, your influence could be much larger than if you had just cast your own ballot. This perspective suggests that while voting is a good action, it’s not necessarily a duty. It could fall into the realm of “supererogation”—morally good but not required.