Aumann & Game Theory in the Talmud.

Suppose there are three creditors of a bankrupt business. The first is owed 100. The second is owed 200 and the third is owed 300. But  the recoverable assets are only 500. How much should each receive? Aristotle thought that fair division should be proportionate to what is owed. Thus each creditor sustains the same percentage loss. There is no discrimination between the small guy and the big guy. 

Back then, Aristotle had been the tutor of Alexander. Greece was a Great Power which could conquer Egypt and Iran and everywhere in between. Thus, the Greeks thought they could enforce the law even in a manner adverse to any coalition of the strong such that overall allocative efficiency was higher and thus ad valorem tax yield would be maximized.

 

What of weaker polities? How would they resolve problems of contested division? Presumably, they'd want to avoid too many such problems coming to Court because of the risk that a coalition of the powerful would defy judgments and just take what they felt entitled to. This might push the polity into a death spiral. Better to have a 'common knowledge' bias in favor of the strong from the get go. 

Aumann & Maschler, in 1985, looked at a 'concede and divide rule' for bankruptcy problems such that, in the above case, 'constrained equal loss' was imposed. Then the pay out would be 66.66 to the guy owed a 100 (as opposed to 83.33 under Aristotle's rule) 166.67 to the guy owed 200 (which is what he'd get under the Aristotle rule) and 266.67 to the guy owed 300 (which is about 17 dollars more than he'd get under equi-proportionate sacrifice). What is happening here is that there is a bias whereby the small guy pays the big guy in order to get something rather than nothing. That's a fair description of how Justice works in a Society with a weak government or where Judges can't enforce their rulings. Notice that the guy in the middle has no incentive to side with the little guy. Either way he gets the same result. But, clearly, this is a regressive social order where the middling side with the rich against the poor. 

 Suppose we lived in a world where the size of one's claim is not objectively or endogenously defined. Contracts in such a world would then seek to maximize entitlement iff a dispute arises. Thus, if I lend ten dollars I stipulate that either I should get ten dollars or else the entire estate. Everybody else would do the same. Large lenders would only lend against property or insist on guarantors. In practice this is what obtains in the real world. The Bank does 'secured lending' getting the first mortgage while the Loan Company picks up riskier second mortgages at higher interest rates.  

Behind it is a Finance Industry which seeks 'residuary control rights'.  The middling factorize- i.e. sell on- their debt for a small discount. In the event of a bankruptcy, it is the small trade creditor who takes the hit. Thus, even today, we see that the rich can protect themselves by 'securing' their loans or selling on the risk whereas the poor must scramble for scraps. 

In Talmudic law, once the option of enslaving the debtor was lost, obligations gave rise to an automatic lien  (shi’bud) over his debtor’s property. This was a real right (in rem) attaching to the obligation, which for a very long time was regarded as stronger than the personal right (in personam) afforded by the obligation. However, since the ability to assert control over land was confined to more powerful individuals and families, the Law in this respect would have limited applicability. It appears the Talmud also used other decision rules to drive home the point that the weaker party must be content with a smaller payout under bankruptcy or other such proceedings. This explains the game theoretic notion- specifically, the 'Shapley value' or relative power based distribution of the gain from cooperation-  that Robert Aumann found in Talmud. 

The following is from Wikipedia-

Consider the 'contested garment (CG) rule,[1] also called concede-and-divide,[2] is a division rule for solving problems of conflicting claims (also called "bankruptcy problems"). The idea is that, if one claimant's claim is less than 100% of the estate to divide, then he effectively concedes the unclaimed estate to the other claimant. Therefore, we first give to each claimant, the amount conceded to him/her by the other claimant.

Which is why one should concede nothing and claim everything. 

 The remaining amount is then divided equally among the two claimants.

So, the rule to follow when lending is to either settle for specific payment or, if that does not happen, gain a claim to the entire estate. But the latter course was only open to those able to assert control rights in realty.  I suppose, as Jews lost control of land and became a diasporic trading community, post-Talmudic law moved towards sophisticated notions of personal obligation rather than liens on land. I imagine that the spiritual and moral qualities of Judges/Rabbis promoted righteousness in a manner which increased mutual trust and allocative efficiency. Still, the fact remains, the Babylonian Talmud was on the side of those with money and power. This becomes apparent when we look at the case Aumann cites- 

Where there are three people all of whom are wealthy enough to have a loan outstanding worth the entire estate then it is merely a question of chance as to which was the bigger creditor. So they get the same payout. Here the prudence of the smallest debtor is rewarded. But then, it is likely, that the richer guys would have smaller exposure w.r.t unsecured debt. In other words, even among the rich, the rule favors the richest. 

The case where the estate is 200 does initially appear mysterious. The smallest debtor should get 33, the next 66, and the biggest 100 under Aristotle's rule. However, if the two smaller debtors combine their debt, they jointly get 100 so, only if they both have an incentive not to do so, would this rule have any application. It penalizes the biggest lender but the fellow should have taken a lien on real property. His lack of prudence in this regard is what is being punished.  I suppose, if the cost of taking out a lien is the same no matter what the size of the debt, then it makes sense to give bigger lenders an incentive to do so. Still, we are speaking of bad lending practices save in the final case where nobody was lending more than the value of the estate. In the third case, the Talmudic solution converges to the Aristotelian. 

I suppose, if we were to look at what actually happens in bankruptcy proceedings, we'd find that 'mechanism design' in our own societies are more Talmudic than Aristotelian- they discriminate against the little guy because it is preferable that legal and financial scope and scale economies are realized. 

Interestingly, as in modern times, money isn't all that mattered. 'Voice' too was important. Shouting louder and claiming more was rewarded- presumably because it was a proxy for some other sort of countervailing power- 

The CG rule first appeared in the Mishnah, exemplified by a case of conflict over a garment, 

one guy claims half a garment. The other claims it all. The latter gets the lion's share. 

hence the name. In the Mishnah, it was described only for two-people problems. But in 1985, Robert Aumann and Michael Maschler have proved that, in every bankruptcy problem, there is a unique division that is consistent with the CG rule for each pair of claimants.[1] They call the rule, that selects this unique division, the CG-consistent rule (it is also called the Talmud rule).[2]ment rule'-

This is because the rule won't be applied save where irrationality or negligence obtained. In other words, the rule has a deterrent effect against seeking for its own application. 

It makes sense for a judge with little power to split things down the middle rather than listen to lengthy arguments or seek to pursue an equitable or more allocatively efficient solution. Those capable of taking control of land would always prefer to get a lien over the whole realty of the debtor. Maybe, that was a good thing. Perhaps it improved security over all. What is certain is the priority given to written agreements together with the collective obligation to educate orphans, created a Law minded community which, however, was not bound by rules or strategic considerations, but wholly by the Holy Spirit. 

Aumann's paper came out at precisely the time when many 'intellectuals' were waking from Marxist slumber and rediscovering the glories of Scripture. It is certainly possible that learned Rabbis can guide us towards great spiritual truths by discussing cases like the following about a 'Safek'- i.e. the first son of a levirate marriage who is accounted the heir of the deceased brother. The 'Yavam', (i.e. paternal Uncle who married the widowed mother) however, may be his biological father. 

This departs from what we would expect- viz. the Safek gets 50 and the other two 25 each- but the difference is not by very much. Essentially, fraternity is maintained by the Safek taking an 18 percent haircut and compensating his brothers with that money. However, what Aumann is not telling us is that this is a rather artificial case which depends on the rule that if the firstborn predeceases his father, his heirs take the double share, which he would otherwise have inherited from his father’s estate. In this case, the 'Yavam' must have been the older brother which is why the Safek accepts one third (assuming the grandfather had only two sons).  What this means is that the system is biased against the younger brothers- particularly the Safek's younger half-brother. That's why the latter can't ally with the Safek but must form a unit with his full brother. Notice that their initial claim is more than it should be, unless their dad was the eldest- but then a claim is just a claim. Judges in a weak polity have to go with the flow. In any case, if one coalition satisfies the other claimants and takes the remainder- a big headache for the judge is avoided. Here too those with deep pockets or local influence will prevail. 

Justice is merely a service industry. However, if State Capacity increases, it can enforce Aristotelian rather than Talmudic fair division which gets us to a more allocatively efficient pooling equilibrium. Indeed, it can go further. It can seek equi-marginal sacrifice with the poor taking a smaller haircut because they would feel that hardship more intensely. This shifts risk onto broader shoulders. But Governance too is a service industry. Broader shoulders may shrug off fiscal shackles if they don't like the public good mix the administration is providing. 

Does Aumann's paper really show 'game theory in the Talmud'? No. It shows that all 'Law & Econ' embodies 'concrete models' of the reverse game theory that is mechanism design. But none genuinely 'optimize the price of anarchy' or any such nonsense. We are not speaking of magic or mysticism but merely of mechanisms which, however, the rich and smart could always find better alternatives to. God, however, was with Israel, because the Jews, aiming to be a nation of scriptural savants, entered into the mysteries of Faith which, however, are non-rival even if sublimely utilitarian. 

Aumann, unlike any Indian Game theorist I know of, has shown, with both mathematical brilliance and Scriptural scholarship, that the Talmud had one type of mechanism which mathematical economists only began to understand very recently, which, however, was nothing very special and which was superseded as times changed and Jews had to adapt. What is marvellous about those Babylonian Jews is that their descendants in distant Poland and Lithuania and elsewhere cultivated both piety and intellectual and artistic excellence under adverse conditions. It was not the letter of the law, but its spirit that the chosen people continue to glorify.