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The Fairest of Them All
A mathematical exploration of fairness.
Playing a (hopefully fair) board game. Image credit: Brendan Humphreys from Pexels.

Whether it's a game on the playground or a policy proposal on Capitol Hill, debates about fairness are commonplace. But what does it mean for something to be fair? It's a word that many of use frequently, but fewer of us (myself included) would be able to rigorously define.

In this story, I'd like to think more critically about what it means to be fair. We'll explore a few different meanings of fairness, and by surveying the landscape, we can start to develop a framework for thinking about fairness more precisely. We'll see where mathematics can help in the formulation of fairness, and where our modeling falls short. We'll also examine research that suggests we are, in some sense, hard-wired to favor fairness.

In general, fairness can get complicated. But in some cases, it's relatively straightforward to define and even measure. Let's start simple and begin with some scenarios where fairness shouldn't be too difficult to pin down: games of chance.

Fairness and Probability

Let's start simple. Consider a coin flipping game. Heads, I win; tails, I lose. Is this game fair?

Well, it depends on the coin. If the coin is equally likely to land on either side, then neither player is advantaged, and the game is fair. But if the coin is more likely to land on one side, most would argue that the game is unfair.

It's unfair for a couple of reasons. First, one player is unreasonably advantaged over the other (it's not as though this is a game of skill). Second, if the coin is biased, it's not necessarily true that the disadvantaged player will know about the deception. Otherwise, why agree to play the game?

In this case, fairness feels fairly straightforward in principle: the game is fair if the coin is just as likely to land on one side as it is on the other. But in practice, how can we determine whether or not the game is fair? After all, a slight bias in the coin may be hard to detect. It's easier to agree on the ideals than on the details. Fairness may make sense in the abstract, but how can we ensure that the coin we use actually adheres to our sense of fairness?

A Frequentist Approach

If you don't know whether or not the coin is biased, there are a few ways to avoid getting roped into an unfair game. One is a frequentist approach: we flip the coin many times and use the proportion of flips that land on heads to measure likelihood that the coin is biased.

Flipping the coin once tells us nothing about whether the game is fair. Flipping it twice doesn't tell us much either: even if the coin is fair, it will land on the same side both times 50% of the time. But what if we flip the coin 100 times? or 10,000? Or --- wait for it --- one billion times?

As the number of flips grows, the proportion of flips that land on any one side begins to approximate a normal distribution, or bell curve. This means that we can estimate the likelihood of certain outcomes by calculating the area under that curve.

Here's a demonstration of what we're getting at. Below, you can adjust the true probability that a coin lands on heads, and then model how many times you'd like to flip a coin. The histogram will adjust accordingly, and show you how likely it to have a certain number of coin flips land heads up.

Note that as the number of coin flips increases, the probability distribution begins to center around the true probability of getting heads. For example, after 100 flips, there's a roughly 95% chance that heads should have appeared between 40 and 60 times. Similarly, there's a roughly 99% chance that it should have appeared between 37 and 63 times. Depending on your tolerance for unlikely events, a head tally outside of one of these two ranges should give you pause.

A Bayesian Approach

The more you flip the coin, the more confident you can be in whether it's fair or unfair. But you'll never be able to pin down the true probability of the coin landing on heads, no matter how many trials you run. For this reason, there's another approach worth examining: the Bayesian one.

The biggest difference with the Bayesian model is that we assume the true likelihood of our coin landing on heads is always unknown. The best we can do, then, is try to come up with a probability distribution for our coin, which changes every time we flip the coin and gather new information.

In order to work with this approach, we also need a prior distribution. This is a fancy way of saying that we need to make an assumption about how likely with think the coin is to be fair before we've collected any data. Here we'll consider two different priors.

In the first, we'll assume that any heads probability is equally likely: maybe the coin is fair, maybe it's much more likely to land on heads, maybe it's much more likely to land on tails. Who knows? ¯\_(ツ)_/¯

For the second prior, we'll assume that the coin is much more likely to be fair than not. If you expect that the coin hasn't been doctored in any way, this seems like a reasonable hypothesis.

The prior you choose has a significant impact on how the model evolves. If you assume any outcome is equally likely, a string of consecutive heads will strongly suggest bias. But if you start by assuming the coin is likely to be fair, it takes more evidence to significantly move the probability distribution.

In the demonstration below, you can start with either prior, and then mimic as many coin flips as you want. A nice bell shape around 50% represents a coin that's likely to be fair, while a curve that's biased towards either side suggests that the coin isn't balanced.

Both approaches---frequentist and Bayesian---have their tradeoffs. If you're interested in learning more about these approaches, there's a nice article available on the MIT open courseware website.

(As a mathematical aside, the standard Bayesian model for trying to estimate the probability of a coin landing on heads involves the Beta function. You can read more about the derivation of the above model here.)

Changing the Game

Regardless of your philosophy (frequentist or Bayesian), these examples show that there are ways to assess whether or not the game is fair. It's not perfect, but the more data you collect, the more certain you can be.

But if you're worried about the possibility of a biased coin and don't have the opportunity to collect data, there's another solution: change the rules of the game. In fact, there's a way to create a fair game even out of a biased coin.

Consider the following variation on the coin-flipping game, described in a 1951 paper by mathematician John von Neumann. We have a coin, but we have no idea whether it's fair or not. You inspect the coin and see that the two sides of it are different, so the probability that it lands on heads is not 0 or 1.

Here are the rules: flip the coin twice. If it comes up on the same side both times, the game continues. If it comes up heads and then tails, I win. If it comes up tails and then heads, you win.

So, what are each of our chances of winning? Well, in any given round there are four possible outcomes. If we let H represent the coin landing on heads, and T represent the coin landing on tails, these outcomes are HH, HT, TH, and TT. In two of these scenarios (HH and TT), the game proceeds to a new round.

But since we know the probability of landing on heads isn't 0 or 1, we also know that eventually we'll get either the HT or TH outcome. And because the coin flips are independent---that is, because the outcome of the second flip doesn't depend on the outcome of the first---the probabilities of these two events are the same!

The takeaway here is that it's not hard to take rules which seem fair and skew them so that the game is unfair. In order to ensure agreement between the rules and the implementation, we've seen two possible approaches. The first is data-driven: we flip the coin many times and use that data to make judgments about whether or not the coin is biased. The second is system-driven: we change the rules of the game to remove potential sources of bias. This can be helpful if the data we need is unavailable or prohibitively expensive to acquire.

Fair Division

There are plenty of other scenarios in which the fairness question doesn't have a straightforward answer.

Imagine, for instance, that you sign a lease for an apartment with two friends. Each of you will have your own bedroom, but in order to preserve roommate harmony, you need to decide who gets which room and how much each person pays.

Questions of fairness immediately come up. Should the rent be split evenly? What if one room is clearly better than the others? What if one roommate has a strong preference for a specific room, even if it isn't necessarily the best one?

Fairness quickly becomes fuzzy. After all, people prioritize different things. You may even have conflicting desires, making it difficult to rank your preferences in the first place. In situations like these, can we determine a fair way to assign rooms and costs so that everyone is happy?

It turns out that the we can, and the argument comes from a seemingly unrelated result in geometry. This connection was first made explicit in a paper titled Rental Harmony: Sperner's Lemma in Fair Division, by Francis Edward Su.

To understand Su's argument, it helps to work with a concrete example. Let's suppose three roommates are trying to find a fair division of a $1,600 rent for a three bedroom apartment. (The argument works with more roommates, but the geometry is easiest to visualize with 3.)

We need a handful of assumptions on each person's decision making. Here are the first two (I'll save the last one until we need it):

  1. Everyone always prefers a free room to a non-free room.
  2. No matter what the division of prices, at least one room will be acceptable to every roommate. (It's never the case that all of the rooms will be out of a specific roommate's budget.)

Given these two assumptions, here's how we can find a fair division of prices:

  1. Draw a triangle. Each corner of the triangle represents a pricing scheme where one of the rooms is the full $1,600, and the other two rooms are free.

  2. Subdivide the triangle into a mesh of points. Each point in the mesh corresponds to a division of the $1,600 rent for the three rooms. Prices are based on how close the point is to the corners. For instance, a mesh point in the middle of the triangle corresponds to each room having the same price.

  3. Assign each point in the mesh to a roommate. How you assign the points doesn't matter, but you want to be sure that for every small triangle formed by the mesh, each corner is owned by a different roommate.

  4. Go through the mesh points, and for each one, ask the owner of the point which room they would prefer at the prices for that point.

  5. Once you find a small triangle in the mesh where each roommate selects a different room at the corners, you're finished! You've found a fair division of the rent. If the differences in room prices are too large, you can refine the mesh and play again. (You're guaranteed to find another acceptable triangle within the triangle you've just found.)

The best way to appreciate this approach is to go through it. So here's an example of the algorithm in action, with three roommates deciding between three rooms: a green one, an orange one, and a purple one. (You can mouse over any point to see the room prices at that point.)

Here are some questions people typically ask after playing around with this algorithm:

How do you know you can find a triangle in the mesh where each roommate selects a different room? This requires proof! I'll let you fill in the details if you're curious, but a couple of hints: first, note that along any side of the triangle, the colors of the points are predetermined, because exactly one of the rooms is free along each side, and people will always select a free room over a non-free room.

The key idea in the proof is to think of each small triangle in the mesh as a room. Play with the demonstration and see if you can figure out how it decides which room to move to next. (This is an implementation of Su's "trap-door" argument.)

Is the fair collection of prices unique? Not necessarily! You're actually guaranteed to find an odd number of triangles in the mesh where every corner corresponds to a different room. You're also allowed to traverse through the mesh in a different way: The New York Times ran an article about this algorithm back in 2014 using a different traversal method.

If you take finer and finer meshes, how do you know you'll keep finding solutions inside of older solutions? Here's where Su's third assumption comes in handy: if a roommate prefers a given room for a set of prices which converge to some limiting price, then the roommate will still prefer that room at that limiting price.

Intuitively, this division algorithm seems fair: the process doesn't favor one roommate over another, and everyone winds up with an acceptable room at an acceptable price. In game theoretic terms, this arrangement is called envy-free, since no roommate would prefer to trade rooms with anyone else.

Beyond Mathematics: Fairness from Behind the Veil of Ignorance

We've now seen several examples of math applied to questions of fairness. However, all of them have assumed that every participant has equal say. While this may be true when deciding where to live or whether to play a game, it's clearly not always the case. In politics, for example, not every voice at the table is equally strong, nor are those voices free of bias. In these more complex cases, what does fairness mean, and how can we try to ensure it?

Rather than drawing from mathematical literature, let's address this question from the perspective of political philosophy. One classic thought experiment that attempts to address this question comes from John Rawls, a philosopher who taught at Harvard University until his death in 2002. Rawls made many contributions to the fields of moral and political philosophy during his career. The one we'll focus on is called the veil of ignorance.

Before we examine the veil of ignorance, let's connect it to the question of fairness. In his book Justice as Fairness, Rawls says, "The most fundamental idea in this conception of justice is the idea of society as a fair system of social cooperation over time from one generation to the next" (p. 5, emphasis added).

So what does fairness mean in this context? Later on, Rawls defines fair terms of cooperation as ones that "each participant may reasonably accept, and sometimes should accept, provided that everyone else likewise accepts them" (p. 6). Just as we've uncovered in our mathematical examples, fairness in Rawls' writing connotes a sense of reciprocity: if an agreement can't be accepted by all parties, this may be because it isn't fair.

But still, the question remains: what can we do to ensure that an agreement has been entered into under fair circumstances? Rawls recognized this issue, writing that the circumstances must "situate free and equal persons fairly and must not permit some to have unfair bargaining advantages over others. Further, threats of force and coercion, deception and fraud, and so on must be ruled out" (p. 15). This seems all well and good. In practice, however, ensuring that fair agreements are entered into fairly can be quite challenging.

The solution Rawls proposes is the veil of ignorance. Here's how he introduces it:

[T]he parties are not allowed to know the social positions or the particular comprehensive doctrines of the persons they represent. They also do not know persons' race and ethnic group, sex, or various native endowments such as strength and intelligence, all within the normal range. We express these limits on information figuratively by saying the parties are behind a veil of ignorance. (p. 15)

The idea here is to situate negotiators in such a way that they don't know specifics about the community they are representing. The goal is for the lack of information to minimize arguments based on self-interest.

Rawls discusses the veil of ignorance as it pertains to the creation of society, but the same ideas can be applied to scenarios that are not as grand in scope. (In fact, it may be more straightforward to do in simpler scenarios; trying to apply the veil of ignorance to complex social systems can be quite challenging in practice.) Even in the context of our coin-flipping game, the veil of ignorance can be used to help ensure the game is played fairly.

When laying out the ground rules, players who argue from the veil of ignorance don't know beforehand who will bring the coin, who will flip the coin, who will win if the coin lands on heads, and so on. This lack of information makes means that both parties should be more willing to agree to rules prohibiting biased coins. Thinking about a scenario from the veil of ignorance can help in the development of checks and balances.

Why Fairness?

As we've seen, fairness involves a certain degree of symmetry: in a fair agreement, equal parties should have equal bargaining power. This isn't to say that fair agreements must necessarily produce fair outcomes; after all, it's not unfair if a football team is bested by a more prepared opponent. But it may be unfair if the rules are applied differently to two equally-matched teams in order to produce a winner.

Through all of this discussion, though, a question lingers: does fairness even matter? After all, in practice, life often seems quite unfair. Instead of trying to increase fairness, why not focus on something like utility, or something even more concrete, like wealth?

There are at least two responses to this question, one moral, one practical. The moral argument is that increasing fairness aligns with the idea of people being created equal, which isn't necessarily true of other metrics. Note that fairness does not imply everyone should be granted the same quantity of resources. But it does align closely with equality of opportunity and in treatment by the institutions of justice.

The practical argument is that it turns out humans care about fairness. This fact has been borne out in a number of economics experiments, which are summarized in a 2006 paper titled The economics of fairness, reciprocity and altruism--experimental evidence and new theories. Two illustrative examples come in the form of games: the ultimatum game and the "dictator game.

Both games involve two people and a pot of money. In the ultimatum game, one person proposes a division of the money. The other person can then accept that proposal or reject it. If the second person accepts, the pot is split according to the agreement. But if the second person rejects, both people get nothing.

A fair division might seem like a 50-50 split. But since the second person can't counter the offer, a rational choice for the first person would be to offer the smallest amount of money possible. If the pot is $100 and you propose a split of $99.99 to $0.01, we both walk away with more money if I accept, even though the split is so lopsided.

In practice, people rarely exhibit this sort of extreme rationality. Experimental data shows that a majority of people offer between $40 and $50. Similarly, offers of less than $20 are rejected somewhere between 40% and 60% of the time. As to why these offers tend to be rejected, in the paper mentioned above, the authors write that "In general, the motive indicated for the rejection of positive, yet 'low', offers is that subjects view them as unfair" (p. 622).

For people on the receiving end of the offer, then, fairness seems to be a significant factor in their decision. But what about for the folks who are making the offer? Is fairness a factor for them as well, or are they merely offering more than what their rational self-interest would prefer in order to increase the likelihood that the offer is accepted?

One way to answer this question is to consider an alternative to the ultimatum game called the "dictator game." This game also involves splitting a sum of money between two parties, but this time the first person's offer can't be rejected. Here, the rational thing to do if you're the first person is to just take all the money. However, "In experiments, proposers typically dictate allocations that assign the Recipient on average between 10 and 25 percent of the surplus, with modal allocations at 50 percent and zero. These allocations are much less than proposers' offers in ultimatum games, although most players do offer something" (p. 622). While these offers can't be attributed directly to an ingrained sense of fairness, altruism does appear to play a role decision-making even when people have all of the power.

Conclusion: The Reasonable Ineffectiveness of Self-Interest in Modeling

But if rational self-interest isn't our primary motivator, why do so many models assume otherwise? One reason is that capitalism tends to breed competitive markets. When competition is the driving factor, incentives for fair play tend to be diminished. In other words, when it comes to competitive markets:

[R]ational individuals will not express their other-regarding preferences in these markets because the market makes the achievement of other-regarding goals impossible or infinitely costly. However, a large amount of economic activity takes place outside competitive markets – in markets with a small number of traders, in markets with informational frictions, in firms and organizations, and under contracts which are neither completely specified nor enforceable. Models based on the self-interest assumption frequently make very misleading predictions in these environments, while models of other-regarding preferences predict much better (p. 618).

In other words, the failure of fairness to rear its head in many economic models is because the systems being modeled don't concern themselves with being fair!

But running a business is not the same as nurturing a society or healing a planet. While healthy competition can be helpful, it's also important for citizens to feel they are being treated fairly, both by one another and by their institutions.