Fairness
Fairness is a subject on which there is seldom any agreement. People like to talk about fairness: this is fair, this is unfair. It is almost impossible to pin down what exactly is meant in each particular case. The root problem might be that our intuition of fairness is informed by the brain that is capable of but a few arithmetic operations without conscious effort. Consider a well-defined problem of fair division. Suppose there is an estate E and n claimants for this estate with debts d(1)<=....<=d(n) with the total of D. One has to divide the estate E into x(1),...x(n) payments with the sum E<=D. General fairness requires that equal claims correspond to equal compensation and the compensation is order preserving.
It is our intuition that when E->0, fair division should be equal (rationing of scarce resources during the war), whereas if D->E (plentiful resource), the division should be proportional. Both of these divisions are easy to work out and intuitively graspable even without calculations. However, in real life it is neither one of these two extremes. One has to devise a general method of fair division that produces intuitively correct extremal behavior and then explain in what way is it fair. That can be done; the problem is that this fair division (i) is not intuitively clear, and (ii) computationally demanding. The result is the brain reverting to its default modes of equal vs. proportional division and the conflict of these two intuitions.
It seems that this difficulty has been recognized ages ago. The Talmud gives several examples of bankruptcy problem; in 1985, "the Talmud approach" has been generalized by Aumann and Maschler, winning the first author the Nobel Prize for game theory in economics.
http://www.elsevier.com/framework_aboutus/Nobel/Nobel2005/nobel2005pdfs/aum16.pdf
The Talmud recommends the CG ("contested garment") division: x(1)=.5*[E-max(E-d(2),0)+max(E-d(1),0)].
Suppose A claims the whole garment and B claims 1/2 of the garment. Since only half of the garment is claimed by B, one half should go to A. The remainder is split evenly between A and B, so the resulting division is 3:1. You can test yourself whether you percieve this CG division as fair. I do not. On the other hand, there are excellent game-theoretical, geometrical, and least squares arguments indicating that the CG division is indeed most fair. For example, it minimizes (x(1)-d(1))^2 + (x(2)-d(2))^2 subject to constraint x(1)+x(2)=E.
http://www.u-bourgogne.fr/leg/documents-de-travail/e2008-07.pdf
http://www.ams.org/samplings/feature-column/fcarc-bankruptcy
http://www-users.math.umd.edu/~jmr/MathTalmud.html
For a larger group, the procedure is to make the CG division between the first and the rest of the claimants (forming a coalition ranked according to their claim), reduce the estate by x(1), and continue this procedure for the second and the rest of the claimants. If the division is not order preserving, the remaining asset is divided equally within the coalition. This method gives the correct extremes, but yields rather complex divisions in the intermediate cases. So counter-intuitive are these divisions that the sages themselves questioned whether the examples given in the Mishna (e.g., E=200 for d={100,200,300} is x={50,75,75}) might be a scribal error. Aumann showed that this scheme is uniquely self-consistent: for any i and j, {x(i),x(j)} is the CG division of x(i)+x(j) for claims {d(i),d(j)}. It is also a self-dual (one can focus on the losses rather than the rewards and obtain the same division).
So here we have fair division that is as fair as fair can be from the purely mathematical standpoint and declared so by the Oral Torah itself. However, working this division out for, say, five claimants requires quite a bit of calculations that I cannot do in my head in seconds -- and when I do the numbers, these do not feel right. Worse, if in the middle of the estate division another claimant shows up, the whole group needs to be re-ranked, the procedure should begin from the start and the resulting payments may change very substantially even if the added claim is small. Once again, this goes against our deeply held intuitions of fairness (as this does not occur in equal and proportional divisions), and perhaps this is the true reason why the Talmud method has never been put in practice, even in the Talmudic time.
This seems to be the general case. People have come with all kinds of axioms that appear to promote fairness, but their consistent implementation results in counter-intuitive outcomes:
...There are many solution concepts for fair division: Nash, Kalai-Smorodinski, Shapley, Talmud, consistency, proportional fairness, max-min fairness, etc. Any one solution concept will usually violate the axioms associated with some other solution concept. If axioms are meant to represent intuition, then counter-intuitive examples are inevitable. A ‘perfect’ solution to a bargaining, arbitration or voting problem is unattainable. One must choose a solution concept on the basis of what properties one likes and what counter-intuitive examples one wishes to avoid. http://www.statslab.cam.ac.uk/~rrw1/talks/qtalk2.pdf
So (even in the most transparent case) fairness based on the intuitive notions results in counter-intuitive outcomes that are as disputed as anything in real life. What are we supposed to do then? Pick and choose our axiomatic systems, as suggested above? But then everyone can fit fairness to set prejudice. Fair division consistent with our built-in notions of fairness turns out to be impossible.
This is so... unfair..
It is our intuition that when E->0, fair division should be equal (rationing of scarce resources during the war), whereas if D->E (plentiful resource), the division should be proportional. Both of these divisions are easy to work out and intuitively graspable even without calculations. However, in real life it is neither one of these two extremes. One has to devise a general method of fair division that produces intuitively correct extremal behavior and then explain in what way is it fair. That can be done; the problem is that this fair division (i) is not intuitively clear, and (ii) computationally demanding. The result is the brain reverting to its default modes of equal vs. proportional division and the conflict of these two intuitions.
It seems that this difficulty has been recognized ages ago. The Talmud gives several examples of bankruptcy problem; in 1985, "the Talmud approach" has been generalized by Aumann and Maschler, winning the first author the Nobel Prize for game theory in economics.
http://www.elsevier.com/framework_aboutus/Nobel/Nobel2005/nobel2005pdfs/aum16.pdf
The Talmud recommends the CG ("contested garment") division: x(1)=.5*[E-max(E-d(2),0)+max(E-d(1),0)].
Suppose A claims the whole garment and B claims 1/2 of the garment. Since only half of the garment is claimed by B, one half should go to A. The remainder is split evenly between A and B, so the resulting division is 3:1. You can test yourself whether you percieve this CG division as fair. I do not. On the other hand, there are excellent game-theoretical, geometrical, and least squares arguments indicating that the CG division is indeed most fair. For example, it minimizes (x(1)-d(1))^2 + (x(2)-d(2))^2 subject to constraint x(1)+x(2)=E.
http://www.u-bourgogne.fr/leg/documents-de-travail/e2008-07.pdf
http://www.ams.org/samplings/feature-column/fcarc-bankruptcy
http://www-users.math.umd.edu/~jmr/MathTalmud.html
For a larger group, the procedure is to make the CG division between the first and the rest of the claimants (forming a coalition ranked according to their claim), reduce the estate by x(1), and continue this procedure for the second and the rest of the claimants. If the division is not order preserving, the remaining asset is divided equally within the coalition. This method gives the correct extremes, but yields rather complex divisions in the intermediate cases. So counter-intuitive are these divisions that the sages themselves questioned whether the examples given in the Mishna (e.g., E=200 for d={100,200,300} is x={50,75,75}) might be a scribal error. Aumann showed that this scheme is uniquely self-consistent: for any i and j, {x(i),x(j)} is the CG division of x(i)+x(j) for claims {d(i),d(j)}. It is also a self-dual (one can focus on the losses rather than the rewards and obtain the same division).
So here we have fair division that is as fair as fair can be from the purely mathematical standpoint and declared so by the Oral Torah itself. However, working this division out for, say, five claimants requires quite a bit of calculations that I cannot do in my head in seconds -- and when I do the numbers, these do not feel right. Worse, if in the middle of the estate division another claimant shows up, the whole group needs to be re-ranked, the procedure should begin from the start and the resulting payments may change very substantially even if the added claim is small. Once again, this goes against our deeply held intuitions of fairness (as this does not occur in equal and proportional divisions), and perhaps this is the true reason why the Talmud method has never been put in practice, even in the Talmudic time.
This seems to be the general case. People have come with all kinds of axioms that appear to promote fairness, but their consistent implementation results in counter-intuitive outcomes:
...There are many solution concepts for fair division: Nash, Kalai-Smorodinski, Shapley, Talmud, consistency, proportional fairness, max-min fairness, etc. Any one solution concept will usually violate the axioms associated with some other solution concept. If axioms are meant to represent intuition, then counter-intuitive examples are inevitable. A ‘perfect’ solution to a bargaining, arbitration or voting problem is unattainable. One must choose a solution concept on the basis of what properties one likes and what counter-intuitive examples one wishes to avoid. http://www.statslab.cam.ac.uk/~rrw1/talks/qtalk2.pdf
So (even in the most transparent case) fairness based on the intuitive notions results in counter-intuitive outcomes that are as disputed as anything in real life. What are we supposed to do then? Pick and choose our axiomatic systems, as suggested above? But then everyone can fit fairness to set prejudice. Fair division consistent with our built-in notions of fairness turns out to be impossible.
This is so... unfair..