Entitlement (fair division)
Encyclopedia
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.
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.
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.
has a number of examples where entitlements are not decided on a proportional basis.
These solutions can all be modeled by cooperative game
s. 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.
Fair division
Fair division, also known as the cake-cutting problem, is the problem of dividing a resource in such a way that all recipients believe that they have received a fair amount...
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
Entitlement
An entitlement is a guarantee of access to benefits based on established rights or by legislation. A "right" is itself an entitlement associated with a moral or social principle, such that an "entitlement" is a provision made in accordance with legal framework of a society...
. Entitlements can in the main be determined by agreeing on a cooperative game
Cooperative game
In game theory, a cooperative game is a game where groups of players may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players...
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.
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 gameCooperative game
In game theory, a cooperative game is a game where groups of players may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players...
as the various parties can form coalitions against others, for instance as in a union versus a company.
The Shapley value
Shapley value
In game theory, the Shapley value, named in honour of Lloyd Shapley, who introduced it in 1953, is a solution concept in cooperative game theory. To each cooperative game it assigns a unique distribution of a total surplus generated by the coalition of all players...
is one common method of deciding bargaining power
Bargaining power
Bargaining power is a concept related to the relative abilities of parties in a situation to exert influence over each other. If both parties are on an equal footing in a debate, then they will have equal bargaining power, such as in a perfectly competitive market, or between an evenly matched...
as can be seen in the Airport problem
Airport problem
In mathematics and especially game theory, the airport problem is a type of fair division problem in which it is decided how to distribute the cost of an airport runway among different players who need runways of different lengths. The problem was introduced by S. C. Littlechild and G. Owen in 1973...
. In economics an allocation which cannot be improved upon by any coalition is said to have the core
Core (economics)
The core is the set of feasible allocations that cannot be improved upon by a subset of the economy's consumers. A coalition is said to improve upon or block a feasible allocation if the members of that coalition are better off under another feasible allocation that is identical to the first...
property. Welfare economics
Welfare economics
Welfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it...
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
VotingVoting
Voting is a method for a group such as a meeting or an electorate to make a decision or express an opinion—often following discussions, debates, or election campaigns. It is often found in democracies and republics.- Reasons for 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
European Union
The European Union is an economic and political union of 27 independent member states which are located primarily in Europe. The EU traces its origins from the European Coal and Steel Community and the European Economic Community , formed by six countries in 1958...
or United States
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
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
Banzhaf power index
The Banzhaf power index, named after John F. Banzhaf III , is a power index defined by the probability of changing an outcome of a vote where voting rights are not necessarily equally divided among the voters or shareholders.To calculate the power of a voter using the Banzhaf index, list all the...
. 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
Penrose method
The Penrose method is a method devised in 1946 by Professor Lionel Penrose for allocating the voting weights of delegations in decision-making bodies proportional to the square root of the population represented by this delegation...
. This assumption does not correspond to actual practice and it is arguable that larger constituencies are unfairly treated by them.
Entitlement in the Talmud
The TalmudTalmud
The Talmud is a central text of mainstream Judaism. It takes the form of a record of rabbinic discussions pertaining to Jewish law, ethics, philosophy, customs and history....
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.
- 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.
- 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 game
Cooperative game
In game theory, a cooperative game is a game where groups of players may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players...
s. 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
Michael Maschler
Michael Bahir Maschler was an Israeli mathematician well known for his contributions to the field of game theory...
in 1985.