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.
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Even when only money is to be divided and 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.
Shared costs or gains
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.
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 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.
In the Talmud
The Talmud has a number of examples where entitlements are not decided on a proportional basis.
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.