The Talmud, Game theory and HR

Long, long ago a man owed three people 100,200 and 300 (let us call them Mr.100, Mr.200 and Mr.300), he died without paying his lenders; the lenders found that estate owned by man was insufficient to recover the amount. How do to divide the amount obtained after selling estate?

You too may have faced similar problems, ex. you promised to pay back money you owed to your friends based on performance bonus (assuming you work for company were performance link pay system is so transparent that you can actually calculate bonus in advance), and then your HR head tells you that you will get lesser amount as bonus, lesser than what you promised you friends. How do you handle the situation? For that matter any other situation where you have to pay more than what you have.

Coming back to our problem, Jewish religious text The Talmud offered solution to this problem.

Solution was…
1. If estate was worth 100, then it is given in equal proportion i.e. each gets 33.3.
2. If estate was worth 300, then it is given in proportion of 50,100 and 150
3. If estate was worth 200, then it is given in proportion of 50, 75 and 75.

While case 1 and 2 can be explained logically either equal distribution (each gets 1/3) or in proportion (each gets half of what needs to be recovered), but 3rd case was not making any sense.
Why should Mr. 100 get half of amount i.e. 50 while those Mr.200 and Mr.300 get 75 each? This remained mystery for long time, till noble prize winning mathematician Dr. Robert Aumann solved it using game theory-treating it as a cooperative/coalition game.

aumann

Solution lay in concept of “equal division of contested sum”, which was accepted principle during that era.

Solution was arrived in following fashion.

1. Determine what amount is contested by 3 lenders, lender with least amount gets half the amount contested, other two also get equal amount. So in our case, estate is worth 200, so Mr.100 will get half of what he should get i.e. 50, Mr. 200 and Mr.300 also get 50 each.

2. With this Mr.100 is now out of picture, so remaining 50 is equally divided between Mr.200 and Mr.300, so they get 25 each, so total amount is 75 each. Hence division of 50, 75, 75.

Had it been 2 person game ex. between Mr. 100 and Mr.300, then Mr. 100 would have got 50 ( half of disputed amount) and Mr.300 would also get 50 ( equal division of contested sum) and entire undisputed amount i.e. 200, so he get 50 + 200= 250. Some may find this solution strange, but that was as per social custom, and social customs are made to ensure cooperation.

Now you know how to repay your friends with whatever money you have.

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HR department always has to match declining resources (esp. during recession) with rising aspirations of employees; maybe this game can offer solutions.

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