Entitlement (fair division)

Entitlement in fair division describes that proportion of the resources or goods to be divided that a player can expect to receive. The idea is based on the normal idea of entitlement. Entitlements can in the main be determined by agreeing on a cooperative game and using its value as the entitlement.

Even when only money is to be divided and a some fixed amount has been specified for each recipient the problem can be complex. The amounts specified may be more or less than the amount of money and the profit or loss will then need to be shared out. A proportional adjustment is normally used in law nowadays and is the default assumption in the theory of fair division. Other rules however are often used and this article describes the basis underlying the common variants.

Entitlement in the Talmud

The Talmud has a number of examples where entitlements are not decided on a proportional basis.

* The disputed garment problem. If one person claims the whole of a cloth and another half then it is divided 3/4 and /4. [ [http://www.come-and-hear.com/babamezia/babamezia_2.html#chapter_i Babylonian Talmud: Tractate Baba Mezi'a 2a] The disputed garment]

* The estate division problem. Three wives have claims to 100, 200 and 300 zuz. Three cases are considered, if the estate is 100 zuz then they get 33 and a third each, if 200 then 50, 75, 75, and if 300 then 50, 100 and 150. [ [http://www.come-and-hear.com/kethuboth/kethuboth_93.html#PARTa Tractate Kethuboth Folio 93a] The estate division problem]

* Profits from a joint fund. If two people put 200 and 100 into a fund and buy an ox for ploughing and use it for that purpose they divide the profit between them. But if they instead then slaughter the ox they divide the profit in proportion. This is discussed in the Babylonian Talmud just after the estate division problem.

* Ibn Ezra's problem. This is a later problem of estate division that was solved in a different way. A man dies bequeathing 120, 60, 40 and 30 to his four sons. The recommendation was to award (120-60)/1+(60-40)/2+(40-30)/3+(30-0)/4 to the first and sums with leading terms removed for the rest ending with 30/4 for the last. This allocation is different from the previous estate division

These solutions can all be modeled by cooperative games. The estate division problem has a large literature and was first given a theoretical basis in game theory by Robert J. Aumann and Michael Maschler in 1985. [ [http://www.elsevier.com/framework_aboutus/Nobel/Nobel2005/nobel2005pdfs/aum16.pdf Game Theoretic Analysis of a bankruptcy Problem from the Talmud] Robert J. Aumann and Michael Maschler. Journal of Economic Theory 36, 195-213 (1985)]

Voting

Voting can be a very nonlinear process. The allocation of seats by size of population can leave small constituencies with no voice at all. The easiest solution is to have equal size constituencies. Sometimes however as for instance in the European Union or United States this can prove impossible. Ensuring the 'voting power' is proportional to the size of constituencies is a problem of entitlement.

There are a number of methods which compute a voting power for different sized or weighted constituencies. The main ones are the Shapley-Shubik power index, the Banzhaf power index. These power indexes assume the constituencies can join up in any random way and approximate to the square root of the weighting as given by the Penrose method. This assumption does not correspond to actual practice and it is arguable that larger constituencies are unfairly treated by them.

hared costs or gains

Where a number of persons cooperate to pay for a facility or to gain from an enerprise there is the question of how costs or gains should be divided. The Shapley value is one common method as can be seen in the Airport problem.

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