For cake cutting, cuts are parallel to an axis or segment and yield rectangular pieces. As such, cutting a cake is viewed as dividing a line segment. For pie cutting, cuts are radial from the center of a disc to the circumference and yield sectors or wedge-shaped pieces. As such, cutting a pie is viewed as dividing a circle. This difference in topology is apparent in the minimal number of cuts ...

We study the monotonicity properties of solutions in the classic problem of fair cake-cutting — dividing a heterogeneous resource among agents with different preferences. Resourceand population-monotonicity relate to scenarios where the cake, or the number of participants who divide the cake, changes. It is required that the utility of all participants change in the same direction: either all o...

Properties of discrete cake-cutting procedures that use a minimal number of cuts (n – 1 if there are n players) are analyzed. None is always envy-free or efficient, but divideand-conquer (D&C) minimizes the maximum number of players that any single player may envy. It works by asking n ≥ 2 players successively to place marks on a cake that divide it into equal or approximately equal halves, the...

We consider the problem of fairly dividing a two dimensional heterogeneous good among multiple players. Applications include division of land as well as ad space in print and electronic media. Classical cake cutting protocols primarily consider a one-dimensional resource, or allocate each player multiple infinitesimally small “pieces”. In practice, however, the two dimensional shape of the allo...

Currently more and more agents are involved in human routine activities and a mixed human-agent society is thus formed by the humans and the agents involved. The relationship between humans and agents in this mixed society is subtle and it is vital to explore the influence that agents bring into the society. In this paper, we try to investigate a few questions: will humans act differently towar...

Let X1,X2, . . . , be independent random variables with EXi = 0 and write Sn = ∑ n i=1 Xi and V 2 n = ∑ n i=1 X i . This paper provides an overview of current developments on the functional central limit theorems (invariance principles), absolute and relative errors in the central limit theorems, moderate and large deviation theorems and saddle-point approximations for the self-normalized sum S...

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 ...