Envy-free

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In mathematical sociology and especially game theory, envy-free is a property of certain fair division algorithms for a divisible heterogeneous good over which different players may have different preferences.^[1]

A scheme is envy-free if each recipient believes that (according to their measure) no other recipient has received more than they have. A procedure for envy-free division was first published by Steven Brams and Alan D. Taylor in 1995.

The concept generalizes naturally to chore division: in this case, a division is envy-free if each player believes her share is smaller than the other players'. The crucial issue is that no player would wish to swap her share with any other player.

Two siblings dividing the last piece of cake is a simple and practical example. The first sibling divides the cake into two pieces, and the second sibling chooses which piece to take. Since both siblings wish to maximize their share of the cake, the first sibling will divide the cake evenly in his estimation and the second sibling will take the one perceived as more desirable. Even if there is icing unevenly on the cake that the siblings want, the first sibling can divide the cake to compensate for the perceived benefit of the icing in his view making them even, and then the second sibling chooses the piece he prefers.

With three siblings the process is more complicated, but similar. The first sibling this time divides the cake into three pieces, then passes the knife and the cake pieces to the second sibling. The second sibling then is allowed to make one cut and adjust the part cut off to any other piece. The third sibling then picks one piece/pile, the first sibling choose the next, and the middle sibling takes the remaining piece. The first sibling has an incentive to evenly divide the cake in the first place, and the second sibling then has a chance to make sure that at least two pieces are even. The third sibling then takes the largest piece available, and the first sibling takes the next biggest piece, leaving the middle sibling with the remaining piece.

At each step of the process, to maximize their own cake allotment each sibling cutting seeks to make all pieces even, and each sibling choosing takes the largest piece.

These methods do break down. If even divisions are not possible, there will be a result with envy; and if preferences are different it is possible that the utility received may not be even, making the division "unfair" even though no one would want to trade for another's share.

The study and debate over which methods are most fair with which situations continue.