Important Note: It is crucial to understand that "share value" is subjective, and that each player may even have a different notion of how much the set to be divided is worth.

For instance, if Jane values an estate to be divided fairly among 3 people at $300,000, then Jane considers her fair share to be at least $300,000 / 3 = $100,000. But if Paul values the same estate at $450,000, then Paul considers his fair share of the estate to be at least $450,000 / 3 = $150,000. And, if Rose values the estate at $375,000, then Rose considers her fair share of the estate to be at least $375,000 / 3 = $125,000.

Fair Division Scheme
A fair division scheme is any systematic procedure for solving a fair division problem.

Working definition:
A fair division scheme is a set of rules that, when properly applied, produces a fair division of the objects to be divided.

Continuing the example above (with Jane, Paul, and Rose), a fair division scheme, when properly applied, will guarantee Jane a share of the estate that she agrees is worth at least $100,000 (and thus is fair from Jane's perspective); will guarantee Paul a share of the estate that he agrees is worth at least $150,000 (and thus is fair from Paul's perspective); and will guarantee Rose a share of the estate that she agrees is worth at least $125,000 (and thus is fair from Rose's perspective). Looking at it this way, there is no 'objective' VALUE of the estate against which one can judge the 'true' fairness of the division; everything is subjective and if Jane, Paul, and Rose get what each agreed beforehand was a fair share then we must agree that the division is fair.

Even if each player agrees on the value of the set to be fairly divided, they may still not agree on the value of individual pieces of the the set.

For instance, even if all three (Jane, Paul, and Rose) agree that the above estate is worth a certain amount, they still may assign different values to certain items; Jane may value the antique furniture at $75,000 and the cars at $30,000 while Paul may value the cars at $75,000 and the anitique furniture at $30,000.

We expect any fair division scheme to satisfy the following conditions:

• The procedure is decisive.
  If the rules are followed, a fair division is assured.
• The procedure is internal to the players.
  No outside intervention is required to achieve a fair division.
• Players have no knowledge of each others’ value systems
  They are not aware of each others likes, dislikes, etc.
• Players are rational.
  They make logical, not emotional decisions.

Special Note
A Fair Division Scheme does NOT guarantee each player a fair share. A player may misplay the game and, by so doing, receive less than his/her fair share. A Fair Division Scheme DOES, however, guarantee that no other players or bad luck can deny a player his/her fair share.

Some Share Value Exercises

The Cheesecake Problem

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