What’s Fair is Fair

Suppose you’ve got 1000 students to assign to two schools, each with 500 slots available. Everyone prefers the Good School to the Bad School. Which of the following is a fair way to decide who goes where?

Method A: Give each student a coin to flip and count on the Law of Large Numbers to insure that just about exactly 500 will flip heads. Those students go to the Good School.

Method B: Randomly assign each student to one of two groups. Then flip a single coin to determine which group goes to the Good School.

Method C: After taking note of the fact that, coincidentally, exactly half the students are white and half are black, flip a single coin to determine which race goes to the Good School.

Method D: Assign all the white students to the Good School.

(There’s also of course Method D-prime, where you assign all the black students to the Good School, but I don’t think we need to consider this one separately.)

I ask this question because economists have been very involved with the design of school-allocation mechanisms, particularly in Boston, and one of the things they worry about is fairness. So it seems important to stop and think about what fairness means in this context.

To that end, I’m going to abstract from a whole lot of other issues. Most importantly I’ll ignore the fact that some students might be willing to trade their first school choice for any number of other things, and we might want to facilitate such trades. In other words, it’s crazy to tackle the real-world version of this problem without using markets. But to focus on the “What is fairness?” issue, I’m going to ignore all that and assume that school assignments are the only things anybody cares about.

Economists often interpret fairness to mean that the mechanism should be envy-free, meaning that at some stage in the process, no student wishes he could trade positions with another. Certainly that’s true of Method A, where everyone gets a fair coin and there’s no reason to prefer your neighbor’s coin to your own. And certainly it’s true of Method B, where we’re assigned to our random groups and all await the outcome of the same coin flip. And certainly it’s true of Method C, where the groups are race-based but once again, we’re all awaiting the outcome of the same coin flip. On the other hand, it seems to be quite untrue of Method D, where all of the black students believe (correctly) that they’d be treated better if they were white.

So the orthodox view, as I understand it, is that Methods A, B and C are appropriately fair and Method D is not. But here’s why that bothers me: No matter what method we use, exactly 500 kids are going to end up in lousy schools. Under Method A, we say “Sorry kid, you got unlucky on the coin flip”. Under Method D, we say “Sorry kid, you got unlucky on your skin color”. Either way, you’ve got 500 kids who are stuck in bad schools through no fault of their own. How can one of these outcomes be in any sense fairer than the other?

If you think there’s some obvious answer to that question, try a couple more variations:

Method E: Sometime before the current generation is born, each family receives a fair coin to flip. You go to the good school if your parents (or your great-great-great-great-grandparents) flipped heads.

Compare this to Method A, where the students themselves flip the coins, say at age four. It is my understanding that while Method A counts as envy-free (because each four-year is as happy with his own coin as anyone else’s), Method E does not count as envy-free, because the unlucky kids know they are unlucky right from the start. On the other hand, I think it’s self-evidently bonkers to believe that Method A is superior to Method E on those grounds. No four-year-old is going to care whether he or his parents flipped the coin that determined his school choice. (Except, of course, in the same way that all four-year-olds want to press the elevator button by themselves — a transient preference that’s forgotten by the time you reach the ground floor.)

The comparison of Methods A and E suggests to me that envy-freeness is very much the wrong criterion here in the first place. Now let’s press on:

Method F: Before the current generation of students is born, randomly assign each family to one of two groups. Then flip a single coin to determine which group sends its kids to the good school.

Method F is to Method B as Method E is to Method A: The only difference is that the coin is flipped when you’re age minus-two instead of when you’re age four. Once again, then, it seems bonkers to prefer one to the other.

Method G: Sometime before the current generation is born, a coin is flipped to determine which race goes to the Good School.

Like Methods E and F, this is not envy-free, but it seems just as good as (in this case) Method C, which is envy-free. (Just as E and F seem just as good as A and B.)

Method H (same as Method D): Assign all the white students to the Good School

Now, Method H might appear fundamentally different from Methods E, F and G, but only in ways that are invisible to the students themselves. A four-year-old is assigned to a classroom, either because of a coin he flipped himself, or because of a coin his parents flipped, or because of a coin flipped by a referee, or because of his skin color. To the kid, any one of those looks just like a random assignment.

In fact, I want to claim that Method H is a random assignment in every relevant sense. At some time in the distant past, a bunch of historical events occurred that culminated in white people having the power to claim all the good schools for their own kids. (I do hope it’s unnecessary to mention here that this is meant to be a highly stylized story, not an accurate and detailed description of the world.) Had things gone differently back then, perhaps black people would have had that power. From the point of view of a kid being assigned to a classroom, that’s a historical event with a random outcome — just like a coin flipped by his parents before he was born.

So in brief:

  1. Everyone seems to agree that it’s fair to let each kid flip a coin.
  2. If it’s fair to let each kid flip a coin, then it’s fair to let the parents flip coins for them.
  3. If it’s fair to let the parents flip after the kids are born, then it’s fair to let the parents flip before the kids were born.
  4. If it’s fair to let the let the parents flip before the kids are born, then it’s fair to let the allocation depend on any other historical event with a random outcome.
  5. If it’s fair to let the allocation depend on any historical event with a random outcome, then it’s fair to base the allocation on race.
  6. Nobody seems to think it’s fair to base the allocation on race.
  7. What’s the deal here?

Again — if you’re going to send some kids to good schools and others to bad schools (particularly without any sort of market mechanism to mitigate misallocations), then, to me, it seems completely crazy to pretend that one method can be fairer than another. But I’ve thought about this for a total of an hour or so, so God knows I might have missed something. Tell me what it is.

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53 Responses to “What’s Fair is Fair”


  1. 1 1 Andy

    I think I can follow you through Step 3, provided at some point in time the parents agree on an upcoming random event to determine whose children go where. In that case, it does not seem to matter whether or not the outcome is observed before the children are born.

    But imagine now a strange society where each day this year, the 1000 parent-pairs gather and draw from an urn 500 of their family names, and the outcome is recorded, just for fun. So there is a list of 500 family names for January 1, 2015, and another list of 500 family names for March 7, 2015, and so on.

    Imagine now that in a few years, when all the children are born, some coalition of parents proposes that the children attending the good school should be those whose family name was drawn on September 4, 2015. Suppose your child is not on that list, but your child is on plenty of other lists. Wouldn’t the selection of September 4, 2015 strike you as unfair?

  2. 2 2 Dave

    You forgot an option: send the best 500 students to the bad school, and watch it magically become the good school…

  3. 3 3 Floccina

    I have been making arguments like that for years.

  4. 4 4 mchar

    School is only one of multiple possible areas where white children could experience an advantage in life. Regarding point 4, it’s true that letting the allocation depend on another historical event doesn’t change fairness, if fairness is defined in terms of expected utility. However, it does increase the variance. It’s reasonable to want to reduce variance.

  5. 5 5 Steve Landsburg

    mchar: If the goal is to reduce variance, and if the black kids are getting the short end of the stick in other areas, then it would seem that the solution is to reserve the good schools for the black kids. Yet it seems that economists at least, and quite possibly the general public as well, considers coin-flipping fairer than this solution. So I think there’s still something to be explained.

  6. 6 6 Cricketer

    I think the allocation based on race feels so abhorrent in part because of the nature of events that led to the whites’ domination. If, sometime in the past, a god-like figure had flipped a coin to decide who gets to be the masters and who the slaves, then most of us would agree that Method H is no worse than the others.

    However, in reality, the division was achieved through inhumane means by conscious humans. And our ethics suggests this is far worse than a coin flip. We tend to see Method H as an extension of that injustice – a white student’s predicament depends not on his/her ancestors’ luck, but on their conscious, immoral acts. To test this hypothesis, consider how many people would prefer sending black kids to Good school over sending white kids to Good school. So I say, #5 isn’t quite right. The subjugation of blacks wasn’t a ‘random event’ and that’s why Method H feels unjust.

  7. 7 7 Ari

    Steve, you make a valuable point and I don’t think you’re “wrong”, but society does treat method H as distinct from the other examples of randomness, and most of us instinctively distinguish as well. When we talk about “fairness” in the context of society, we almost always take as a given that some people are born luckier than others. A test is not “unfair” just because it rewards people born genetically smarter or stronger than others; on the contrary, that’s core to our conception of meritocracy.

    To use an even simpler example: is it “fair” that almost all US students will receive a better education than almost all students of Niger? And if not, why don’t Americans advocate redistributing a large percentage of the US education budget to Nigerian students? I think the answer is that almost always when we talk about “fairness”, we mean within the context of one lifetime of one person within one society.

  8. 8 8 Windypundit

    I think there are also issues of trust and verification that creep into our feelings about the fairness of the decision method. The system has to appear to be fair, and it has to be easy to verify that the system is as fair as it is described to be.

    Suppose that the two of us are walking across campus when we find a $100 bill on the ground, and I propose we flip a coin to see who gets to keep it.

    Method A: I’ll toss the coin, you call it heads or tails in the air, and you keep the $100 if you call it correctly.

    Method D: I pull out a small box in which I tell you I’ve placed a coin that I flipped earlier but have not looked at. You keep the $100 if the coin is heads up when I open the box.

    If you trust me completely, method A and D are equally fair. If you don’t trust me, method A seems more fair because it would be much harder for me to cheat you than method D. With method D, especially if you lose, you’re going to wonder why I wanted you to agree to win on heads rather than tails. Was it really fair? Or did I trick you?

    Andy captures some of this in his answer: You’d might trust that your child has an equal chance of being on any day’s list and yet still suspect someone is trying to game the decision by choosing the date according to which children are on it.

    Alternatively, no one thinks it’s unfair when the assignment by racial or ethnic group is clearly not the result of someone gaming the system, such as with ethnically-linked illnesses such as sickle cell anemia or Tay-Sachs disease.

    If you’re the parent of a black child, and they announce that students will be assigned by method D, aren’t you going to wonder why they picked D instead of D’? Aren’t you going to question the fairness of it?

  9. 9 9 Jonathan Kariv

    I tend to agree with #4 here. The reason method D seems unfair has a lot to do with blacks getting the short end of the stick in a lot of other areas. Method D and D’ seem different (or similar) to an extent depending on how much you focus (or don’t) on other types of white privilege.

    Here are 2 other methods.
    Method I: Richest kids go to the good school, poorest to the bad one.
    Method J: Poorest kids go to the good school, richest to the good.

    Do these strike you as “random in every relevant sense”? why/why not? Given race based income distributions it might be that I=D and J=D’.

    #2. Assuming you can identify the best kids. I suspect that an actual attempt at this will have a fair bit to do with how much help the kids get from parents early on, which correlates with parents resources (I guess you could maybe correct for that somehow but I’m not convinced it’s trivial).

  10. 10 10 nobody.really

    A card trick culminates with four seemingly random cards face down on the table. I ask if people would clap for me if, when I turn them over, they proved to be the Ace of Spades, Hearts, Diamonds, and Clubs. People say yes. I ask why. Eventually, people arrive at the answer that this would be impressive because it would be an unlikely result to arise by chance.

    Then I ask if they’d clap if when I turned over the cards, they proved to be the 2 of Clubs, the 6 of Diamonds, the Jack of Spades, and the 9 of Diamonds. Not so much. I point out that the odds of this outcome would be equal to the odds of the outcome with the four aces. People grudgingly acknowledge this.

    Then I point out that, in fact, the odds of any specific combination of four cards is equal to the odds of the four aces arising. People grudgingly acknowledge this.

    So I point out that if people would clap for me when I produce an outcome are rare as the four aces, they should clap for me if I produce any specific combination of cards at all.

    Then I propose that we skip to the chase, and people should simply clap for me now and be done with it. I bow and prepare to put away the cards.

    Generally someone will insist that I turn over the four cards anyway. I ask why they think it matters; what difference could it make in their opinion of me or this trick? They can’t really articulate why they care, but they do.

    So I point to the first card and, in a dramatic voice, cry “2 of Clubs!” I flip it over and it’s the Ace of Spades. “Wait, wait – I’ve still got three more tries. Behold the …. 6 of Diamonds!” It’s the Ace of Hearts. “Jack of Spades!” produces the Ace of Diamonds. “9 of Diamonds!” yields the Ace of Clubs.

    As people marvel at my ability to produce these four aces, I slam my fist down, kick the table in frustration, and mutter under my breath “…. stupid trick never WORKS!”

    Because, within the context of my narrative, it hadn’t. Yet people insist on being impressed when four aces arise – even ask they acknowledge that this outcome was no more or less likely than any other combination of four cards, and that I had in fact promised to produce a different combination of cards. No matter; they remain impressed with the aces.

    Of course, people are responding to the cultural background in which the ability to produce four aces seemingly at random is the goal of many a card trick.

    Similarly, I have to wonder that whenever you propose a hypothetical involving allocation of school attendance by race, you tap into a deep cultural background regarding that issue. You can protest that you’re merely discussing a stylized example all you like – you will have difficulty causing people to extract this discussion from what they know about the real-world dynamics involving education and race.

  11. 11 11 nobody.really

    Yup, people generally hate the idea that (past) “determinism” dictates the circumstances of their lives, but are more accepting of (present) “chance” – perhaps even when the probability of any given outcome is the same. Why? Four theories:

    1. Windypundit @ 8 offers a logical explanation for this: a present-day act of “chance” can be scrutinized to ensure fairness, whereas a past act, or the choice to make current circumstances depend on a given past act, may not lend itself to this kind of scrutiny.

    2. People value a sense of agency, even if illusory. People may feel safer driving a car than flying in a plane, all statistical evidence notwithstanding, because they feel they have control of their car whereas they have no control of the plane. Similarly, people want the opportunity to flip their own coin, dammit.

    3. Many people want the opportunity to mentally prepare for a momentous outcome – especially if the outcome may be bad. In truth, they may not actually care about this preparation time in retrospect; Daniel Gilbert has demonstrated that people are predictably bad about anticipating how they will feel in the future. But, anticipating how they might feel in the future, they care now about having the opportunity to prepare for bad news. And thus they value the sensation that the bad news is not a foregone conclusion.

    4. Rationale 2 and 3 combine in this final theory: People want the opportunity to influence even “random” events though prayer/faith that the universe will be benevolent to them individually. “Determinism” deprives them of the comfort/satisfaction that this kind of faith would otherwise provide (at least temporarily).

  12. 12 12 Harold

    I think #4 has the reason why we feel uncomfortable with assigning all whites to the good school, but not why we feel uncomfortable assigning all blacks to the good school, so yes, there is something else to explain. We have tried reserving good schools for the black kids -called positive discrimination or quotas. It may be effective, but it not universally popular. In fact, this is probably the fairest way to assign in the absence of other information, since we know blacks are statistically disadvantaged. if we have an interest in equality, then we should assign the blacks to the good school.

    Why don’t we send all the boys to the good school? That seems pretty random. Do we have the same problem here as with race?

    There is something about education that is different from finding money etc. The school may improve the more children with educated parents and more resources attend. If we assign based on race, which may be a good proxy for parental education and resources, then as Dave says (#2) the bad school becomes the good school.

    We also generally feel racial stereotyping leads to bad outcomes as it unreasonably distorts expectations. If we classify according to race we are reinforcing racial stereotyping, and exacerbating the problem of perceived differences between races being much larger than they are. We increase racial tension, which is already much greater than it should be based on reason and facts.

    #10. Nice presentation of the trick. Although you had apparently promised a different set of cards, really you hadn’t – you were really promising 4 aces all along. The allocation based on race may be random, but is only one of many possible random allocations -boy / girl being another example, maybe tall / short could be another. The reason why we choose the black / white one over any of the others is the real question. It may be random that we are in either one group or the other, but it is not random that we choose *that* quality to divide into two groups.

  13. 13 13 FC

    Your reasoning is correct. If you substituted labels X for race, Y for grandparents etc and told no one (including yourself) no one would complain about your logic.

    Unfortunately what is going on here is people see this as the first step in analyzing the larger game of life, where the possibility of iteration and multiple assignments may result in quite different conclusions or requirements (you have multiple coin flips, some coins are better than others, should the coins be reassigned between rounds of flipping; if they flip a head the kids can study coin-flipping and get better, should they then give some of their coin-flips to kids who got crappy coins and hence could not study? etc etc)

    If you state clearly the simple models you are building is for insight, and if you find its solution may have no bearing on the solution to the larger problem that would be ok with you (although no one will believe you!).

    Unfortunately in physics there has been a lot of success by solving a simpler problem, gaining insight, then solving the larger problem. While sympathetic to this approach personally, ut is sometimes easy to forget that this may not be true in policy domains, or that even if the approach has validity, that it will be followed properly instead of used for some biased gain.

  14. 14 14 RPLong

    It’s just signaling. As long as it looks like we care about fairness, we can get away with any allocation mechanism we want to, no matter how unfair it actually is; and if anyone second-guesses us, well, that’s just a difference of political opinion.

  15. 15 15 Harold

    “5.If it’s fair to let the allocation depend on any historical event with a random outcome, then it’s fair to base the allocation on race.”

    Maybe it is this. What is the event with a random outcome? In any coin toss scenario we would only have to go back to before the coin toss. When do we go back to in the race scenario?

  16. 16 16 Alan Wexelblat

    I think Cricketer has hit the nail on the head in that you cannot compare an impartial and provably fair (or unfair) object – in this case the coin – to a series of conscious and deliberate acts.

    Relatedly we believe ourselves (generally) to be more knowledgeable and fairer than in the past. You can certainly find past publications of “scientific” articles that purported to show the superiority of caucasian persons over “negro” persons. We now repudiate these papers as nonsense and bad science. If, however, we construct a present system that uses this repudiated nonsense it presents a contradiction.

  17. 17 17 khodge

    More as an aside, except for #1, the system can be gamed: Once the parents know the outcome they can: increase/decrease the number of children, pass as a different race, move, home-school, etc. There, at least, has to be two flips to keep the fairness neutral, a flip for the student and a flip for the school (much like what developed for the military draft in the 60s), at a point in time where few parents will be able to alter their plans.

  18. 18 18 Pete

    Every day I visit your site at it brightens my day when you’ve posted. I have zero objections to your stylized logic, but I have a different reason to object to your race based methods.

    Schools are only part of the equation. Let’s say that the top twenty students at either school come out geniuses, but as you work your way down the rankings within each school, the good school kids start to be better than the bad school kids, and when we get to the bottom half or two thirds of the bad school rankings, they’re below all but the delinquents and mentally retarded from the good school.

    If these schools are color coded, then people will be right to be prejudiced against the bad school race. What you then need to ask yourself is, do I want to live in that world, or in a world where people can’t use race to give them a statistically accurate picture? If you’re a hardworking student at the bad school that is able to overcome your school’s limits, you’re going to have to put up with prejudice anyway. Does this prejudice incentivize the bad school students to not bother? How about the color that goes to the good school? A very lazy student will still have the positive association with his skin color. Maybe there is less incentive for people at each school to try as hard as they otherwise would.

    One more thing to note is that this means whole families, perhaps even across generations, will be condemned to the bad school. If it was random, year to year, or student by student, then a family might be able to pay for tutoring or give more attention to their bad scoop children. If they have to make up for children’s bad schooling for all of the children that they have, they might not be able to overcome it. Across generations could be even worse still, because you miss out on the chance for good school parents to make up for bad school children. All that being said, a family can still get very unlucky with all children going to bad schools and all generations going to bad schools.

    Stylization aside, I want to live in a world where black and white children are rewarded for their efforts in school. We aren’t there yet. There are a lot of good reasons for black children to give school everything that they can. There are still a lot of good excuses for them to slack off too. I’d generally support things that get us closer to equal opportunity and give us less reasons to assume the white person is properly educated and the black person isn’t.

  19. 19 19 nivedita

    I think methods A and B are “fair”, by which I probably mean more desirable, because they are independent of history and other factors. i.e. on the one hand, we say no matter what happened before this day, today we reset (statistically) all of you to the same position.

    There’s also the argument that perhaps we want the student body of each school to reflect society at large, and a random assignment likely best achieves that, even if you don’t know what factors to consider. i.e. if you were instead targeting a particular racial mix say 15% black 85% white, and you decided a non-random way to achieve this, say by taking the white kids alphabetically by last name and splitting them into two groups, there’s theoretically the possibility that last name correlates with something else, so you ended up biasing the groups along some other factor that you weren’t measuring, even though racial mix is ok.

  20. 20 20 Henri Hein

    This is somewhat orthogonal to your point, but your example has a personal poignancy for me. When I started high-school, I went to one of the best schools in Denmark. We then moved and I had to go to another school for a spell, and I was convinced it must have been one of the worst. I can at least say the former school was decidedly ‘good’ and the latter decidedly ‘bad.’ At the time, I did not think so much in terms of fairness, as I thought of myself as victim or beneficiary of circumstance. (I am a youngest sibling and thus learned early in life fairness is at best an ephemeral experience).

    I do think Cricketer is right, but I want to add another point. There is also something about decisions that are based on who we are, as opposed to what we produce. Say, the boss promoting their son-in-law over other, more qualified personnel. To answer Harold’s question, I think we would find it unfair to base the decision on gender as well (though maybe not equally unfair). Or height, or eye color, or alphabetical by name. The latter is as random as the coin-toss, but I think most people would call that unfair.

  21. 21 21 Henri Hein

    @10: Love it. If I saw that trick, I would become an instant fan.

  22. 22 22 Bob Murphy

    Steve,

    Let’s start with something much simpler to isolate what you are trying to say/ask with this post…

    You and I have to figure out which one of us gets something valuable (like the window seat in an airplane). You say, “Let’s flip a coin for it.”

    I come back and say, “Nah, how about instead, we declare the winner to be the guy whose last name starts with ‘M’? Oh wow, it’s me, how about that… Anyway, my procedure is just as fair as yours. After all, they both produce one winner, and that’s what we need to figure out how to allocate this single window seat.”

    Are you telling us in this blog post:

    (A) You wouldn’t object to my proposal,

    or

    (B) You of course agree that the coin flip is fair, whereas my proposal is grossly unfair, but you are having trouble explaining precisely WHY?

  23. 23 23 David Johnson

    “But here’s why that bothers me: No matter what method we use, exactly 500 kids are going to end up in lousy schools.”

    We are going through a restructure at work at the moment, and some people will lose their jobs. It seems to me that a lot of work could have been avoided in the HR department if they had borne in mind the statement above. The system that has been developed for job allocation in the name of fairness is unbelievably complex.

  24. 24 24 Harold

    Say we have one allocation to make – schools. We can decide on a coin toss and pretty much everyone seems to think that is about the best solution (without markets).

    What if we have a huge series of allocations? We can start with schools, then who gets the window seat, who gets the job, who gets arrested, etc etc.

    We could make all the allocations based on a single coin toss – whoever wins the toss gets to decide all future allocations. Before the toss we are in a position of fairness, so there should be no envy.

    However, we could also decide each allocation on a series of coin tosses. The person who gets the best school might not get the job or the window seat.

    Presumably, if we treat each allocation as an isolated event and add up all the outcomes, the expected “utility” (for want of an alternative phraseology) from each method is the same -a series of 50% vs one big 50%. So can we say the two methods are equivalent?

    Certainly not. I personally would opt for the series of coin tosses – I am risk averse enough to not risk getting the shitty end of the stick for everything. It *seems* fairer to do it the case by case way.

    Also each allocation is not necessarily an independent event. The cumulative effect of every allocation not going your way may be worse overall than 50% of each going your way.

    Allocating the school to white people is like deciding every allocation based on the single coin toss. It seems like an unnecessarily risky way to do it, and it may have a worse outcome overall because events are not isolated.

  25. 25 25 nobody.really

    We could make all the allocations based on a single coin toss – whoever wins the toss gets to decide all future allocations. Before the toss we are in a position of fairness, so there should be no envy.

    However, we could also decide each allocation on a series of coin tosses. The person who gets the best school might not get the job or the window seat.

    Presumably, if we treat each allocation as an isolated event and add up all the outcomes, the expected “utility” (for want of an alternative phraseology) from each method is the same -a series of 50% vs one big 50%. So can we say the two methods are equivalent?

    Certainly not. I personally would opt for the series of coin tosses – I am risk averse enough to not risk getting the shitty end of the stick for everything. It *seems* fairer to do it the case by case way.

    Harold is not the first person to suggest that people tend to be risk-averse, and a risk-averse person would structure social rules to mitigate the risk of being on the losing end of chance. This was the premise of John Rawls’ Theory of Justice.

    Conversely, and at the risk of some political incorrectness, Harold is also not the first person to suggest the possibility of letting the winner of a single coin toll decide all future allocations.

  26. 26 26 Andrew Macdonald

    I agree with Cricketers’s analysis – #5 is false.

    Whatever random events were involved in method D, it also significantly depended on a non-random process involving human agency. It is this latter process that raises the question of fairness.

    To make a biological analogy, it’s like the roles played by mutation and natural selection. The random aspect is only part of the story.

  27. 27 27 Steve Landsburg

    Bob Murphy:

    Are you telling us in this blog post:

    (A) You wouldn’t object to my proposal,

    I’m certainly not telling you this. My blog post is about what’s fair, not about what I’d object to. I object to stuff all the time without regard to fairness issues.

    or

    (B) You of course agree that the coin flip is fair, whereas my proposal is grossly unfair, but you are having trouble explaining precisely WHY?

    Something closer to this, though I don’t think this is exactly it either. If we’re deciding whether your kid or my kid gets the window seat, then I’m not sure that going by last name *is* any less fair than us flipping a coin. As far as the kids are concerned, your kid got the window seat and mine didn’t, and I’m not sure they’d care very much whether it’s because we flipped a coin or because you bullied me. What’s fair for us and what’s fair for the kids might be two different issues.

  28. 28 28 Scott H.

    nobody.really @ 10:

    You really don’t understand the humor in your own card trick?

  29. 29 29 Scott H.

    Steve Landsberg @ 27:

    “My blog post is about what’s fair, not about what I’d object to. I object to stuff all the time without regard to fairness issues.”

    OK. How about this approach then?

    No outcome is fair. However, there are methods we object to, and methods we don’t.

  30. 30 30 Richard R

    I think that if you let historical events determine today’s school allocation that might be fair unless historical events also affect the number and type of school children in the first place.

  31. 31 31 Ken B

    A method will be seen as unfair if the outcome can depend on the choice of an interested party. This is why D is seen as unfair. This is true even if there should be choice. Hence some of the misguided opposition to markets here.

  32. 32 32 RPLong

    “As far as the kids are concerned, your kid got the window seat and mine didn’t, and I’m not sure they’d care very much whether it’s because we flipped a coin or because you bullied me. What’s fair for us and what’s fair for the kids might be two different issues.”

    Great point!

  33. 33 33 nobody.really

    nobody.really @ 10:

    You really don’t understand the humor in your own card trick?

    Funny you would ask. I recently laughed along with Landsburg’s “Innumeracy Watch” thread, only to learn that the joke was on me. (In my defense, I would distinguish between issues of innumeracy and vocabulary; after all, I expect even excellent mathematicians might struggle to pass a math test written in unfamiliar terms.)

    But no matter; rest assured, the Innumerati will deal with him later….

    Anyway, let me respond by saying that I’m gratified if you find the trick amusing – I do, too – but I can’t say that what you’re laughing at and what I’m laughing at are the same things.

  34. 34 34 Scott H.

    Explaining the humor of a particular joke is tedious. Telling another joke… well, that’s more my style.

    Here is a kind of analogous trick that is done more at the expense of the audience than the magician. I ask a member of the audience to pick a random card out of the deck. I have them put the card back into the deck and shuffle. (Spoiler: I figure out what their exact card was using trickery!) Anyway, I take the entire deck and explain that I am now going to show them their card (which they think I don’t know.)

    I begin to turn over cards, one by one, and place them face up on the table. I do this slowly like I’m looking for something. Eventually I finally get to their card and place their card face up on the table like all the others. However, I say nothing at the time, I don’t miss a beat, and I keep on turning over cards and placing them face up on the table. Finally, after turning over 6 to 10 cards more, I suddenly stop and announce “How much do you want to bet that the next card I turn over is your card?”

    Well, by that time the audience member is REAL sure the next card on the deck is not their card, so they are typically willing to bet a lot of money — and quickly. One hundred dollars is often proffered without pause.

    I then accept the bet, put down the deck, grab their card from the mess of cards on the table, and turn it over again so that it’s now design side up. “You got that $100?”

  35. 35 35 Roger

    I define a method as fair if it is possible that all parties can reasonably consent to it.

    Then methods A and B are fair, because people consent to coin tosses to resolve disputes all the time. Method D is certainly not fair, as the blacks would never consent to it.

    Method C is probably not fair, because it assumes race-segregated schools, and many people are opposed to that.

    A possibly fair variant of method C might be to send all the blacks to the Good school, and all the whites to the Bad school. The blacks might agree, because they would be getting the Good school. The whites might also agree, because of the expectation that soon the Bad school will be the Good school, and the Good school will be the Bad. But again, some people may object to the segregation.

  36. 36 36 Bob Murphy

    I once said, “Nobody is a better troll of Landsburg than me!”

    I am still right.

  37. 37 37 Bob Murphy

    Steve, I got it! The black kids have to walk up a staircase, while the white kids get an escalator. (Inside joke everyone; #NotAllEconomists.)

    Steve, for real, I think something is screwy here right in the beginning. I think the entire paradox (or whatever you want to call it) arises from pretending we don’t know what “random” means in everyday usage.

    Look, how would a mathematician have defined a “random event” back when people thought the universe was a giant clock, following Newton’s laws? In a certain sense, nothing was random, but people still did probability theory right?

    I think that’s what solves all of these “paradoxes” here. A fair coin toss is fair by definition–it even has “fair” in the label. But more deeply, it means we don’t know the answer beforehand, even though if we knew a lot about the atoms etc. right before the toss, maybe we could predict how the coin would land.

    I think that is the issue here, and solves everything. It’s what drives my airplane seat story and what drives the typical reactions to the various allocation methods you described in your OP.

  38. 38 38 Bremen

    You may as well add Method I: give all the kids an IQ test and send the top 500 to the good school. This is basically the same as Method D (intelligence is mostly heritable, and it’s certainly not controlled by children). But most people would see it quite differently.

  39. 39 39 Vivek

    ‘the mechanism should be envy-free, meaning that at some stage in the process, no student wishes he could trade positions with another’
    That’s not what envy-free means. The stages don’t matter. Only the final allocation. Thus so long as one School is better than the other, no allocation is envy-free. Resources need to be concentrated on equalizing the Schools for envy-freedom to obtain. Alternatively, the cost- whether monetary or in terms of acquiring relevant entry qualifications or in terms of staying the course- has to be differentiated.

  40. 40 40 Steve Landsburg

    Bob Murphy (#37): You’re telling me why one method seems fair and another doesn’t, which of course is exactly the question I raised in the post.

    But maybe I framed the post wrong. A better form of the question might have been: “Why should a policymaker care whether the allocation mechanism is fair?”. Fair or not, any of these mechanisms results in 500 kids going to a bad school through no fault of their own. Is there reason (other than self-interest if you care about some kids more than others) to prefer the fair methods to the unfair ones?

  41. 41 41 Steve Landsburg

    @Vivek (#39): I think you’ll find that if you look at the paper I linked to, and related literature, the phrase “envy-free” is used as I described.

  42. 42 42 Bob Murphy

    Steve (answering your #40):

    You wrote, “You’re telling me why one method seems fair and another doesn’t, which of course is exactly the question I raised in the post.”

    I can’t tell if you are telling me I’m just repeating back what you already gave us initially, or if you are admitting, “OK right, that *is* what I asked you guys to explain, but on second thought that’s not really what’s bothering me here…”

    In case it’s the former, in my defense I’ll just repeat your early line: “Which of the following is a fair way to decide who goes where?”

    …and I haven’t seen anyone make my point about why we call it a “fair coin.”

    Anyway, as far as the broader issue: I really do think the point about randomness ties perfectly into your (clarified) question. In a situation where there are X slots and we have to come up with some way of allocating them among people, and if we have a prior view that no one kid is more deserving than another, then a random allocation is perfect. No one can argue with it.

    Let me try it this way. It seems that you are simultaneously saying:

    (1) Let’s try to figure out how to allocate slots if we don’t have a prior reason to think some of the kids are more deserving than others. What matters is simply that *some* kid get each available slot.

    (2) I am baffled as to why people think a random allocation, instead of giving it to the kids who are tallest, is fair.

    So I don’t see what the problem is here. Given (1), (2) is obvious.

    If you want to rule out (1), OK that’s fine (e.g. we wouldn’t randomly allocate children to parents), but then nobody’s moral intuition would say (2) is right, so the mystery disappears.

  43. 43 43 nobody.really

    Ok, I think I’m getting a firmer grasp on this hypothetical.

    Economists often interpret fairness to mean that the mechanism should be envy-free, meaning that at some stage in the process, no student wishes he could trade positions with another….
    Compare this to Method A, where the students themselves flip the coins, say at age four. It is my understanding that while Method A counts as envy-free (because each four-year is as happy with his own coin as anyone else’s), Method E [in which a coin flip occurs before the child is born] does not count as envy-free, because the unlucky kids know they are unlucky right from the start.

    I think we fall off the rails here. The question is “Was there some stage in the process in which no student would wish to trade positions with another?” And the answer is ”Yes: immediately before the coin flip.” True, the students were not yet born, but that’s irrelevant for purposes of judging fairness under the envy-free test.

    In short, I now understand this hypothetical as a variant of John Rawls’s Theory of Justice hypothetical, in which we are asked to design a world before we know what station we will fill in that world.

    With this understanding, I now disagree with Landsburg’s assessments of Method E, F, and G.

    And this illustrates the challenge of introducing race as a relevant variable. As other commentors have remarked, to make any scenario “envy-free” where race is involved, you’d have to find a stage in the process where members of one race would not wish to trade places with members of another. And that would seem to imply looking VERY far back – basically to the point of complete conjecture. Applying the envy-free test in this fashion would seem to eliminate the text entirely.

    Why should a policymaker care whether the allocation mechanism is fair?

    To avoid riots?

    I mean that seriously. I sense we often attribute high-minded rationales for policies that really amount to accommodating the mobs. We make a virtue of necessity. Why does the Constitution acknowledge freedom of religion, but not freedom of conscience per se? Well, religions tend to be organized. Individual contentious objectors? Not so much. So government grants tax exemptions to religious organizations, but when Thoreau conscientiously refuses to pay taxes that finance slavery, he’s imprisoned.

  44. 44 44 Ken B

    Murphy wrote:
    “So I don’t see what the problem is here. Given (1), (2) is obvious.”

    I think Steve’s questions is, why is it obvious? If the height ranking is random, why is it not perceived as a fair random selection? It’s random, right?

    I think the answer is that since it is known ex ante it excites our distrust detection mechanism. It is susceptible to manipulation. Imagine that we put 500 red and 500 black balls in a bag and announced a blind drawing, with the red ball winning. Now imagine that we do not announce until after all the balls are drawn whether red or balck is the winner. This seems unfair for, I suggest, the same reason.

  45. 45 45 Alex

    white power (the way it’s used here) is not a random outcome. There was agency involved.

    I thought ken b and Bob Murphy were onto something with the allocation being decided before the flip/race birth (before we know the outcome) vs. After (we already know it). Because maybe Then you can’t guarantee that the choice is random?

    But that can’t be right. If you said “from now on, all future black kids will go to the good schoo”, that wouldn’t seem fair.

  46. 46 46 Ken

    Cricketer,

    I think the allocation based on race feels so abhorrent in part because of the nature of events that led to the whites’ domination.

    It wasn’t due to slavery, as slavery existed in pretty much every part of the globe for most of history. There isn’t anything special about the slavery that took place in the United States between 1789 and 1863 that would led to “whites’ domination”. In fact, that white domination lead to a dramatic world wide decline in slavery and the (near) elimination of slavery in all white countries (some criminal elements in white countries still traffic in humans of all races).

    in reality, the division was achieved through inhumane means by conscious humans

    To what “inhumane means” are you referring to that resulted in “whites’ domination”? As pointed out, it slavery certainly is not what led to “whites’ domination”.

    The subjugation of blacks wasn’t a ‘random event’ and that’s why Method H feels unjust.

    What about the subjugation of other races by blacks? What about the subjugation of any group of people based on any reason? Are those random events? Isn’t your selection of black subjugation, over which you frett, which conerns a relatively short time frame and pretty small geographic area, while ignoring all others throughout history and global geography, a random selection?

  47. 47 47 Bob Murphy

    Steve, if Ken B. and I agree on this, I think we can rule out us being wrong with 99% confidence.

    Let me rephrase what is (to me) the obvious solution to all of this. When we’re trying to allocate something scarce and we a priori don’t think any one recipient is more deserving than another, then a random allocation settles all disputes. It’s not because a random allocation is the *only possible* way to do it, but it obviously satisfies the criterion we desire.

    In contrast, other possible criteria lead to disputes, because of the underlying suspicion that the proposer did it deliberately in order to steer the prize to the recipients he secretly favored.

    To give an example of what I mean: Suppose I say, “I have a quarter on my desk in the other room. Let’s go check to see if it’s showing heads or tails.” Here people would be uncomfortable, because they couldn’t be sure if I was either lying or subconsciously knew what the quarter looked like. An active coin toss destroys that (unless we’re worried about a magician).

    One last whack at this whole thing: Steve, I think maybe you are confusing things by bringing in all of your permutations. I think the real issue you are raising is to say, as an economist, that as long as we’re not wasting available school slots, then how can we object to any particular allocation? And so, in particular, if somebody says all the slots should be reserved to white hetero boys from rich families whose dads are senators, then why isn’t that just as fair as a coin toss?

    But this obviously strikes most people as unfair. The thing is, no matter *what* you propose, some group of people will object. But that’s why a lot of people (though not all) would be OK with a coin toss, because it knocks out all of the other stuff and at least guarantees everyone had an equal shot, ex ante. I don’t know if it’s so much that everyone thinks, “A coin toss is fair,” so much as everyone thinks, “There’s no non-subjective way for me to complain about this, because the only thing we can all agree on is that we shouldn’t waste slots.”

  48. 48 48 Daniel

    Bob Murphy and Ken B have this pretty well nailed.

    Regarding 46,
    Ken now I know exactly what kind of person you are. You’re the kind of person that thinks immediately as slavery was ending black people and white people were then afforded equal chances at happiness. This makes so much sense given the rest of your worldview and now I know I should never waste time arguing with you again.

  49. 49 49 Amr

    It is best to have parents flip s coin and *then* decide whether to have a child. Responsible parents might think twice about bringing up a child in a world where they can’t guarantee the opportunities that come with better education. Socially, a generation of 500 well-educated individuals is probably better than one double the size but with far lower average education.

    The method actually implemented in practice is likely the one favored by parents who are more likely to vote in elections.

  50. 50 50 iceman

    Late to the party again…of course one might start with ‘why can’t we fix the bad schools?’ Sorry that’s ducking the question.

    I agree notions of fairness often involve a sense that outcomes are based on factors within one’s control – which might make none of the above scenarios fully satisfactory, but maybe helps explain why people ‘feel’ like they at least prefer the chance to flip the coin themselves (certainly if there’s a trust issue as some have suggested). Allocation based on race understandably scores low in this regard. Of course so should alphabetical etc., but as others have said, race taps into a notion of ‘legacy’, e.g. if some have already lost a few coin flips it seems more fair to even things out. (This sounds dangerously close to endorsing a *process* principle, e.g. including knowledge of our histories which we’re not supposed to have behind the veil?) Most everyone seems to give equality of opportunity high marks for fairness.

    Interestingly though, Rawls (since he has been invoked here) might well endorse something quite different — the seemingly sensible approach (at least to economists?) of allocating based on who may be able to get the most value out of a scarce resource, e.g. based on innate ability (IQ), to the extent this benefits society as a whole. This may seem unfair from the control angle, but not necessarily unjust. (Is it better if we could predict who would work harder to maximize their innate abilities?) However Rawls’ use of fairness was muddled, as he tried to ‘veil’ his framework in rational calculation by self-interested individuals, but in the end had to rig the game in favor of his notion of distributive fairness-as-justice; in part by ruling out any pre-knowledge of our histories, abilities and choices — i.e. ‘legacies’?? — so that his “process” could not in fact endorse *process* principles, only end-state principles.

    In short, you might have more luck asking which approaches seem most *just*, as opposed to fair. Including considerations of process.

  51. 51 51 Ben

    OK Oldish thread but I hope the Professor will see this.

    Of course it is irrational for people to distinguish between two methods of allocation (differently organised lotteries) based on a counterfactual “I had a chance to win” when they did not in fact win. Can the preference for “fair” forms of allocation (immediate lotteries over generational or race based lotteries) be justified on rational grounds? Yes. With one additional assumption (which happens to be true).

    People gain utility from the success of their friends and relations, including distant relations (race here being understood as a distant kinship). Any parent knows the latter and anyone with a real friend knows the former. With a lottery based allocation most people will have friends and kin who were allocated to the good school. Though it is not possible to fairly distribute the first order good, the good school (since there will always be exactly 500 people who don’t get to go), the second order good, seeing the success of ones friends, can be more evenly distributed.

    That said, I am sure it is not the whole story. I am certain that there are many people who are irrationally attached to the “at least I had a chance/I never had a chance story”. Can that be perhaps understood as a rational irrationality? Yes. Firstly it is easier to understand (though incorrect) and can therefore act as a Schelling point around which people can rally to support the rationally justified pure lottery.

  52. 52 52 Ben

    (There is no second half – editorial mistake)

  53. 53 53 Thomas

    You’ve greatly overcomplicated this.

    People think the first three are fair because, as you say, prior to the coin flip no-one wants to be in the other group. Of course, once the flip is made it becomes ex post unequal, but the process was not unequal and was not, in that sense, unfair.

    In the last example, though, the outcome is determined up front by an agent who might have an agenda. Up front, half of the students want to be in the other group before the process even begins. It’s both ex post unequal and a priori unfair.

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