Entitlement (fair division)

Where a number of persons cooperate to pay for a facility or to gain from an enterprise there is the question of how costs or gains should be divided. In general deciding such entitlements is a cooperative game as the various parties can form coalitions against others, for instance as in a union versus a company.

The Shapley value is one common method of deciding bargaining power as can be seen in the airport problem. In economics an allocation which cannot be improved upon by any coalition is said to have the core property. Welfare economics on the other hand tries to determine allocations depending on fairness criteria.

The people can also agree on their relative entitlements by a consensus process. For instance they could say what they think everyone else is entitled to and if the assessments agree then they have an agreed impartial consensus division.[1]

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 constituencies of equal size. Sometimes, however, this can prove impossible – for instance, in the European Union or United States. 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.

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 1/4.[2]
• 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.[3]
• 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 evenly 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 with an estate of 120 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.[4]