There is a heterogeneous resource that contains both good parts and bad parts, for example, a cake with some parts burnt, a land-estate with some parts heavily taxed, or a chore with some parts fun to do. The resource has to be divided fairly among n agents, each of whom has a personal value-density function on the resource. The value-density functions can accept any real value positive or nega...

A cake is a metaphor for a heterogeneous, divisible good, such as land. A perfect division of cake is efficient (also called Pareto-optimal), envyfree, and equitable. We give an example of a cake in which it is impossible to divide it among three players such that these three properties are satisfied, however many cuts are made. It turns out that two of the three properties can be satisfied by ...

Cake cutting, a subfield of fair division, refers to the allocation of a heterogeneous, divisible, and continuous good among n players who have heterogeneous preferences. The challenge is to allocate the cake in such a way that is fair by using a procedure that encourages the players to be truthful about their preferences. There are several different notions of fairness studied in the literatur...

Cake cutting is a playful name for the problem of fairly dividing a heterogeneous divisible good among a set of agents. The agent valuations for different pieces of cake are typically assumed to be additive. However, in certain practical settings this assumption is invalid because agents may not have positive value for arbitrarily small “crumbs” of cake. In this paper, we propose a new, more ex...

We consider the classic problem of envy-free division of a heterogeneous good (aka the cake) among multiple agents. It is well known that if each agent must receive a contiguous piece then there is no finite protocol for the problem, whenever there are 3 or more agents. This impossibility result, however, assumes that the entire cake must be allocated. In this paper we study the problem in a se...

We show that no finite protocol (even if unbounded) can guarantee an envy-free division of a cake among three or more players, if each player is to receive a single connected piece.

We present a moving-knife procedure, requiring only 11 cuts, that produces an envy-free allocation of a cake among four players and discuss possible extensions to five players.

We analyze a class of proportional cake-cutting algorithms that use a minimal number of cuts (n − 1 if there are n players) to divide a cake that the players value along one dimension. While these algorithms may not produce an envy-free or efficient allocation—as these terms are used in the fair-division literature—one, divide-and-conquer (D&C), minimizes the maximum number of players that any ...

Protein hydrolysate (PH) was prepared from the defatted cottonseed cake by using immobilized or native trypsin on polyamide (nylon 6,6) matrix using glutaraldehyde coupling under the controlled conditions of pH and temperature. Immobilization of trypsin on nylon beads retained 30% of the free enzyme activity and was stable against leaching for more than 30 days. Foaming and emulsification prope...