The Boltzmann fair division for distributive justice

You have a cake, and there are three people. You need to divide it fair. ⅓ and 120° are the good choices. This is an easy one, which is 1/n in n people. Our society is more complicated. The demand of the cake is required by your ability and contribution. This is often greedy and competitive. The Boltzmann distribution is based on entropy maximization and provides the most probable, natural, and unbiased distribution of a physical system.

Ej is the division potential, and j is players.

β is a division constant. (β≧0)

When β is Zero, all players receive an equal amount of cake. When β increases to a large value, only a few players having made the highest cake contributions receive most of the cake.

The player’s need for the cake as the need values Dj satisfy: uj(0) = 0, uj(Dj) = tanh(1)≅ 0.762. This means that if a player receives what they need (Dj), they satisfy 76.2%.

Homogeneous cake cutting is the total number of cake units, Ej is the division potential of player j, and Pj is the Boltzmann probability that a cake unit is allocated to player j.

Heterogeneous cake cutting is the total number of cake units with flavor i which is the weight factor expressing player j’s preference for flavor i, and the Boltzmann probability that a cake unit of flavor i is allocated to player j.

In β≧0.029,equality starts decreasing.

Discrepancy theory

There is a finite set of elements {1,2,・・n}.

S1,...,Sm ⊆ {1,...,n}

There are two colors.

S={{い,ろ,は},{い,に},{に,ほ},{は,ほ}}

∴

This is the discrepancy.

Then you see the binary distribution.

You see -1={{い,ろ,は}{い,に}{に,ほ}} and 1={は,ほ}. This is at most 50%.