Just Fair

Preamble: I have not blogged in a while. I have quite a few things to say, and have started at least 7 blog posts but never found the steam to complete them. Last Friday, I was having a rather interesting conversation with my colleagues, and that was cut short by a prior dinner arrangement. Having left the conversation topic unended, I decided that it’s a good point to jump off and continue blogging.

I lean slightly left towards Marxism, and I made it clear what it is that appeals to me. What appeals to me about Marxism is that it is most sci-fi in nature. “From each according to his ability, to each according to his need” is probably one of the most Star Trek-esque thing you can say. Indeed, I dream about a future where society functions like this, and I am actively working towards making such a change in society.

Of course there are other bits of Marxism that are I think outdated - the concept of class warfare, and proleteriats needing to seize control of Das Kapital* by that I mean, means of production. I think this is a very good pun is in my opinion, a very 19th century view. I do however, note a similarity between today’s society and the society that Marx lived in, one on the verge of a technological revolution* Das Kapital was published just as the dust of the Industrial Revolution was settling. Its observations of course, were made by Marx DURING what we call now the Industrial Revolution . Just as I note that the philosophy of Marx’s time was that people find meaning of life through work, we are similarly in a period where the same has happened. Think of how you would introduce yourself to other people - it’s your name, followed by what you do.

And there we were, seated at the table. Me, J and P were discussing my Marxist leanings. J posited a very interesting question, which I have paraphrased to omit the amount of obscenities that are wont to come about around groups of male humans speaking:

Imagine if there were two students, A and B. A is super hardworking, and does all the work during the semester. A even does extra work to understand the subject deeply. B on the other hand is a party animal, preferring to skip classes and not study, and would rather spend his time partying. Then comes exam time. Obviously A does better than B. But here's the twist. The lecturer for whatever reason, approaches A and proposes that A averages out his grade with B. If you were A would you do that?

This is obviously a variant of the legendary socialism classroom experiment story that has been floating around the Internet for some time now.

Putting aside obvious flaws with the analogy (which I will expand on in the later part of this post), I answered with an affirmative - that I would indeed be okay to average out the grades.

After cries of indignation and disbelief, and logical reasonings why my decision is abnormal, I laid out an alternative view:

Imagine if there were two students, A and B. Both A and B are extremely interested in the subject, and are motivated enough to study hard for the knowledge. B however, has a string of rotten luck and poor health, which causes her to skip classes and study time as B has to go to the hospital quite a bit. Come exam time, and the teacher makes the same offer. Would you average out your grades if you were A?

Both scenarios are the same, mechanically speaking, even if the motivations are different. Both J and P said that they would at least think about averaging the grades. Incidentally, during the dinner I adjourned to, I posited the both situations and the guests at the dinner too had similar responses as J and P - that it would not be fair (and that they would not agree to average grades in the first scenario and that they would consider in the second scenario).

This harks to a concepts of justice, equity and fairness - concepts not commonly seen in academic economic literature* OK, I lied. There is an entire field of economics dedicated to this topic - it's just very new - about fewer than 20 years. Matthew Rabin invented a whole new way of looking at utility based on fairness as a preference in the utility curve. Fehr and Schmidt came up with ways to encode guilt and compassion when looking at games. The past 10 years have been pretty much replicating experiments across cultures, with not much new or groundbreaking .

Definitions

But to further discuss this issue, let’s first define the concepts of justice, equity and fairness in the context that I wish to discuss. The main context is that the concepts are to be discussed in terms of distribution of resources, which economics is mainly concerned about.

Justice is the idea that the distribution of resources is done in such a way that the receiver deserves the goods received. This makes it an extremely prickly subject, because justice means different things to different people, and it comprises a wide range of ethics and philosophies, as I shall try to paint in broad strokes here. To religious nutbags, the words of the deity in question is just. If the deity says to allocate resources in a certain manner, it is just. Anything else is unjust and blasphemous. To capitalists* meant in the broadest sense of the word, i.e. if you are Ayn Rand , it’s only just that people get what they work for. Any other way of gaining resources is unjust. To legal theorists like John Rawls, justice is a form of fairness* wait, what? circular reference detected .

Equity and equality refers to the idea that all participants within a transaction should be treated the same (or similarly), without a party being unfairly preferenced. Of the three concepts, this is the simplest concept. There is a subtle difference in the terms equality and equity. In recent years, equality has taken to mean equality of opportunity - it means that all people participating within a certain social situation should be afforded the same opportunity. For the sake of this article, we’ll call this equity. Equality is concerned with the distribution of resource, not the distribution of opportunity* For the purposes of this article, let's say that opportunity is distinct from being a resource . If everyone had an equal share of the goods, then equality is achieved. Note that an equal share does not mean that it is fair, and by extension, just.

Fairness is a concept that is related to both the concepts above. Usually when discussing economic concepts, fairness is not well defined. It’s quite hard indeed to define fairness. For the sake of simplicity, let’s say that fairness is defined as the prospensity to avert inequity and maximize justice. And I believe that this is innate in all humans (and chimps too). Given a game like the Dictator Game, it’s been shown in multiple experiments, across all cultures, that people playing the role of the dictator tend to split the earnings evenly (I believe a 50-50 split is the most common split). It’s still one of the most interesting game theory experiments to be run on the ideas of altruism and reciprocity and fairness.

fairness

Fairness, in my opinion, is a complex value* think of it as a vector/complex number if you will that has components in justice and equality. Obviously it has components in other values like reciprocity, compassion and debtedness, but for the sake of simplicity in this article, we’ll just say it’s comprised of justice and equality. We spend a lot of time thinking things in life are either fair or unfair, creating the red line and the dotted borders above. However, if we think depeer, there are other quadrants which can be filled, as I will explain in an example below.

Coincidentally, in the field of psychology, fairness can also be defined in three components - sameness, deservedness and need. The definitions of which sound pretty close to the three concepts I have defined above. However, since I have no expertise in psychology, I shall say no further.

Back to last Friday. What J and P and my dinner guests felt for the first situation was that it was unjust to average the grades in the first scenario above, while it’s at least just to average the grades in the second scenario. It is equal, for sure, but it’s clear that most people feel that in Situation 1, the averaging of grades would be unjust (Student A does NOT deserve it), but it is unarguable that it is equal. Likewise with Situation 2, where it’s somewhat just (student A may deserve it), and equal.

The talk of justice opens up all sorts of cans of worms. But the one I want to talk about is the arbitrator problem.

The Arbitrator Problem

The arbitrator problem can be illustrated with a third scenario:

A, B and C are students in a class. A is the aforementioned party dude in the class in Scenario 1. B is the aforementioned hardworking one in both Scenarios. And C is the aforementioned ill-lucked one. You play the role of the teacher. Do you average marks between A and B? or B and C? or C and A? Or between all three?

In this scenario, the teacher/lecturer is playing the judge - the arbitrator who will solve the problem of justice and equality. The problem is then the system is reliant upon the arbitrator to make good judgment calls. You’d be lucky if in a situation the arbiter is Superman. But even Superman can fail. In a very clear cut scenario like the one above, it’s easy to make good judgment calls. If A is the nephew of the teacher, and the teacher decides to average the grades for the benefit of A, we can clearly see that it’s unjust. We call this corruption in real life. Real life isn’t clear cut like the above. Real life is murky, with twists and turns.

Which brings me to the main objection of the whole analogy to begin with. There exists an arbitrator in all of the scenarios above. It’s clearly meant to be the analogue of the government (or dictator), where the students are the citizenry of the country. The scenarios above are very shallow analogues of central governments (the teachers) distributing resources (grades) to the people (students). Except, reality doesn’t work like that. There has been experiments on central government directly distributing resources, but none has worked well so far. The problem with this analogy is that it ignores the fact that the citizenry interact amongst themselves.

Of course there will come a day when the hypothetical optimal distributor viable - say a super computer who knows everyone’s indifference and utility curves over a basket of every possible item - I’d be willing to let such a computer distrubute resources and forego the marketplace dynamics if such a supercomputer were possible. But that’s a story for another day, probably one involving star dates.

Games

Let’s say we accept that the only form of information transfer is between the teacher and each student. How would a situation like the scenarios be represented as a game?

It turns out there is indeed a game theory experiment that is similar to the scenarios above - the Public Goods Game. The main difference between the scenarios above (let’s call it the Averaged Grades Game) and the Public Goods Game is that the Averaged Grades Game requires work as an input and some other form of utility as a reward.

This is a important distinction. Firstly, I believe the concept of justice can only be measured if the player has to work (i.e. put through some minor discomfort) for it, and some form of equality has to be factored in as well. The usual design tricks of giving participants some initializing tokens that can be exchanged for money does not fully capture the injustice that is felt if real work were done. A variant of the Public Goods Game where the initial play money is “earned” (or with different levels of “income”) approximates this. Having work unit fuzzies the connection between how much is invested and how much is returned.

Secondly, the Averaged Grades Game has no multiplier (or, has a multiplier set to 1). Again, this is a distinct difference. It’s been shown that without a multiplier, the Public Goods Game doesn’t work. I believe it will work for the Averaged Grades Game, because information is leaked to the player through the averaged grade, as opposed to told to the player.

Thirdly, the rewards the participants get is solely through the “exam results”. In the Public Goods Game, the rewards the players get is the sum of what they kept for themselves, and the the multiplied returns from the public pool.

Lastly the analogue for the Public Goods Game is a public good, which may or may not feature in a participant’s preference set. Whereas in analogue for the Averaged Grades Game is the results of an exam, of which preference for is a strictly increasing monotonic function* Think about it, name me a sane person who would prefer lower grades to higher grades .

Description

Here’s a more formal description of the Averaged Grades Game:

Players are given a task to solve puzzles, where a puzzle is a work unit. The more puzzles players can solve, the more reward (a monetary unit) the player earns. However, at the end of the game, all the players’ rewards are pooled together, and the averaged reward is returned to the player. The player is told at the end of the round, how much he/she has earned, and what he/she will be getting.

Variations can include a surprise rounds of not averaging the results, announced only after the round has ended. The amount of leaked information can also be used to test for justice. For example, at the end of every round, the proportion of students who score more than 50% is broadcast (this will make those who have done extra work feel “vengeful” and “punish” those who haven’t pulled their weight).

Equilibria

Depending on how the game is framed, there may be two Nash equilibria. The first Nash equilibrium of this game is the same as the Public Goods Game - zero work units will be expended. Not contributing any work is the only move a player can make in which other players’ decisions will not make the player worse off. However, just like real life experiments with the Public Goods Game, I expect reality to not converge towards 0. Instead, I believe it will converge on a level that is just above 0.

The second Nash equilibrium is the maximum work units being produced. This would happen if the game were expressed as a modified Public Goods Game where the players only get rewarded after contributing to the pool, and don’t get to keep their initial tokens. I don’t see this happening in real life situations either. People tend to punish, and feel guilt, especially with successive rounds.

Simulations

Given that I don’t have access to a behavioural economics lab, I decided I would replace human participants with some silicon ones, so I wrote some simulations* Yes, I am aware that I can use matrices and linalg, but I wrote it this way to be clear about what is happening, but hey, Python 3.4 features are actually useful! , simulating a few things:

  • Each student has some baseline "intelligence"* This is a prickly one too. I use "intelligence" as a term as a catchall. Think of it as confounding variables that are encapsulated into one, if you will. that does not change.
  • Each student puts in work. The amount of work put in is a function of the intrinsic motivation of the student* Again, this encodes things that are not directly measurable .
  • Results are a function of both intelligence and work
  • After each round, the student receives both a score, and an averaged score
  • The student chooses to make adjustments on the work put in for his/her next exam
  • The adjustment is subject to two other factors: motivation, and locus of identity of the student

The key metric we’d be looking at is the trend of the averaged scores over time, as well as the effort put in by each student over time. Here are the results of a hundred runs. Here’s the code in 200 lines of Python (skip to results and discussions):

Results

And here’s the result:

[caption id=“attachment_2827” align=“aligncenter” width=“1920”]100 runs of Average Results over time, overlaid with each other 100 runs of Average Results over time, overlaid with each other[/caption]

Here we note that as expected the average result does indeed drop after an initial spike. This is how it looks like when it’s smoothed and averaged across all the runs:

100 runs smoothed

We’d also have to consider the distribution of scores for each time period. To do that, I have plotted a box plot of the scores of a particular run (Run 47).

run 47 boxplot scores

Discussion

The simulation is obviously a very very very simple simulation. If you hadn’t understood the code, here’s a general gist of what it does: The study period is drawn from a normal distribution with a mean and standard deviation that is the same for everyone in the first period. When the second period starts, the motivations of the students affect the mean of which the study period is drawn from. The locus of identity of the student affects the standard deviation from which the study period is drawn.

The exam scores are a linear function of the study period and intelligence (of which is only a small component), and noise drawn from a normal distribution with constant mean and standard deviation across all time periods.

It should hence be clear why the average scores converge around the mean of the initial study period. This is a reasonable assumption. Nobody goes to university with a goal of not putting a single ounce of effort into studying.

Another part of the code that I don’t particularly like is the amount of clamp() that is used. This was necessary to clamp down on extreme values, but it may turn out that I don’t quite have a feel for drawing from normal distributions, so instead of tweaking the starting mean and standard deviations, I decided to just cheat by using clamp().

If anyone is willing to make a better program, please, do by all means fork the code and write me your results. I am keen to know.

On Socialism and Equality and Equity

I had mentioned briefly that I would say yes to the first scenario - that I would be willing to average my grades with the scumbag who doesn’t study. Furthermore, I too mentioned that the classroom example is a poor analogue for a central authority distributing resources to the public. Saying this appears to make me go from Oliver Queen to Lonnie Machin.

The issue is one of semantics. I’m not saying the government has no role in distributing resources, but rather, I think the government has a role as a player in the same game. In fact, I would say the government is an entity that it charged with the moral role of ensuring equity amongst other players by using the rules of the game.

What this means is that things like affirmative actions, specifically quota-based ones, are in my opinion, not good, while things like universal living wage is a good thing. The difference between them is universal minimum wage plays by the rules of the game - transactions, etc - while things like affirmative action justly creates equal opportunity by making providing unequal distribution.

Consider a scenario where a university has places for 100 students. Let’s say the university has a 20% quota for peoples of [your choice of disadvantaged demographic]. It just happens that there are 120 students who are equally qualified to enter. Out of that 120 students, 20 of them are from [your choice of disadvantaged demographic] families, so they’re instantly accepted. This means that only 80% of the others would be accepted. While the intention is noble and just (I fully agree with equal opportunity), it is not difficult to reframe the statistics in such a way that it’s unfair. Observe:

100% of students of [insert disadvantaged demographic] gets accepted to the university, while 80% of the students of higher socioeconomics status gets accepted. Does that mean that if I'm a [insert disadvantaged demographic] student, I can still get into this university without good grades?

I grew up in a country where racial quota plays a role in everything, from education to buying houses. I can tell you from first hand experience it is not fun. Even when you logically convince yourself that it’s for the betterment of society, one can’t help but feel that there is some level of unfairness at play. Obviously there is no just basis for having racial quotas, but even if the quotas were based on socioeconomic status (which I feel is most just), I’m quite sure that the feelings will remain.

And yes, I am aware in the the reframing above, I am also conflating entrances from quota with entrances from ability. It doesn’t mean that if a person is accepted as part of a disadvantaged group, the person does not have the ability to be accepted if they’re not part of the disadvantaged group.

Quota-based affirmative action almost always guarantees that a proportion of people of higher skills will not be accepted. That’s rather the whole point of quota-based affirmative actions: There exists a group that is disadvantaged in some way, and do not have the same opportunity as someone who is not in the disadvantaged group. Chances are high that the main opportunity gap is one of skill. That is to say the average skill of the disadvantaged group that is lower than the non-disadvantaged group, mainly due to pre-existing life conditions. If both groups are of the same average skill, then the opportunity gap must lie elsewhere. If the opportunity gap lies elsewhere, a quota based system will not help.

Another issue is that the disadvantaged demographic definition is subject to the arbitrator problem. Someone has to decide that X, Y, Z features are features of the disadvantaged demographic. There is a distinction of feature quality too. For example, socialeconomic status is a much better feature than say, race or ethnicity. One is easily defined and verified, the other is too fuzzy - everyone knows about the ridiculous one drop rules, for example.

This in part is why I am more in favour of softer affirmative action (such as targeted advertising* Of course bear in mind that my past work with advertising has coloured my view ). Or even things like universal living wage all sound like fairly good ideas to me. These are actions in which an arbitrator does not have absolute power. Yes, I am aware that the free market is flawed, and needs to be somewhat reigned in (by said arbitrator/government/state), but the power is not as absolute as in enforcing quotas. I have digressed too much though. I may actually spend another blog post writing about that.

And Yet...

And yet, if you recall earlier in the blog post, I said I would be willing to average the grades with the scumbag who doesn’t study. Considering it as a solely one-off event, I would likely say no. However, when the situation was posited to me, I instantly started thinking about other variations in which the ill-scoring student was subject to other conditions* I blame alcohol . It is from this thinking that a sense of fairness rooted in higher weightage of equality which prompted me to say yes.

Further Work

This is just a brain dump that arose from Friday Night Drinks at work. However I have a feeling that this is actually a good econs experiment that can be carried out. If anyone would like to carry it out, please do, and also keep me informed. I’d be delighted to read your papers* HAH! As if anyone would be inspired by this blog post enough to write a paper... .

TL;DR

Congratulations for making it so far through my ramblings. I’m not sure what you gained from it. My views on political systems is abysmal and pessimistic at best, so please don’t follow what I say.

I posited a situation, based on a popular story online. I then defined the concepts of justice, equality and fairness. From there, I digressed slightly to a problem with using a classroom setting as an analogy for government, and tried to define the classroom interaction as a game. I pointed out the similarities with a already known game, and then discussed the possible equilibrium states. I then continued with a simple simulation of the game, and discuss the results. I then end with a rant on socialism and equality and equity.

Also, I tried to hide multiple Justice League related puns and/or references. I think I failed in that aspect.

Reading Material

If you’re interested in this topic, here are some reading material on the topic:

  • Gary Bolton and Alex Ockenfels' A Theory of Equity, Reciprocity and Competition
  • Erns Fehr and Klaus Schmidt's A Theory of Fairness, Competition, and Cooperation
  • Matthew Rabin's Introducing Fairness into Game Theory and Economics
  • John Rawls' A Theory of Justice* It's a book. A very hard and dry book to get through. I didn't bother finishing it - just skimmed through
  • Amartya Sen's Collective Choice and Social Welfare
  • Ken Binmore's Game Theory and Social Contracts
  • Most of Al Roth's works
  • Most microeconomic literature
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