Fair Division - Assumptions

Assumptions

Fair division is a mathematical theory based on an idealization of a real life problem. The real life problem is the one of dividing goods or resources fairly between people, the 'players', who have an entitlement to them. The central tenet of fair division is that such a division should be performed by the players themselves, maybe using a mediator but certainly not an arbiter as only the players really know how they value the goods.

The theory of fair division provides explicit criteria for various different types of fairness. Its aim is to provide procedures (algorithms) to achieve a fair division, or prove their impossibility, and study the properties of such divisions both in theory and in real life.

The assumptions about the valuation of the goods or resources are:

• Each player has their own opinion of the value of each part of the goods or resources
• The value to a player of any allocation is the sum of his valuations of each part. Often just requiring the valuations be weakly additive is enough.
• In the basic theory the goods can be divided into parts with arbitrarily small value.

Indivisible parts make the theory much more complex. An example of this would be where a car and a motorcycle have to be shared. This is also an example of where the values may not add up nicely, as either can be used as transport. The use of money can make such problems much easier.

The criteria of a fair division are stated in terms of a players valuations, their level of entitlement, and the results of a fair division procedure. The valuations of the other players are not involved in the criteria. Differing entitlements can normally be represented by having a different number of proxy players for each player but sometimes the criteria specify something different.

In the real world of course people sometimes have a very accurate idea of how the other players value the goods and they may care very much about it. The case where they have complete knowledge of each other's valuations can be modeled by game theory. Partial knowledge is very hard to model. A major part of the practical side of fair division is the devising and study of procedures that work well despite such partial knowledge or small mistakes.

A fair division procedure lists actions to be performed by the players in terms of the visible data and their valuations. A valid procedure is one that guarantees a fair division for every player who acts rationally according to their valuation. Where an action depends on a player's valuation the procedure is describing the strategy a rational player will follow. A player may act as if a piece had a different value but must be consistent. For instance if a procedure says the first player cuts the cake in two equal parts then the second player chooses a piece, then the first player cannot claim that the second player got more.

What the players do is:

• Agree on their criteria for a fair division
• Select a valid procedure and follow its rules

It is assumed the aim of each player is to maximize the minimum amount they might get, or in other words, to achieve the maximin.

Procedures can be divided into finite and continuous procedures. A finite procedure would for instance only involve one person at a time cutting or marking a cake. Continuous procedures involve things like one player moving a knife and the other saying stop. Another type of continuous procedure involves a person assigning a value to every part of the cake.