  Cutting a cake is a metaphor for the problem of dividing a resource (cake)
among several agents. The problem becomes non-trivial when the agents have
different valuations for different parts of the cake (i.e. one agent may like
chocolate while the other may like cream). A fair division of the cake is one
that takes into account the individual valuations of agents and partitions the
cake based on some fairness criterion. Fair division may be accomplished in a
distributed or centralized way. Due to its natural and practical appeal, it has
been a subject of study in economics. To best of our knowledge the role of
partial information in fair division has not been studied so far from an
information theoretic perspective. In this paper we study two important
algorithms in fair division, namely "divide and choose" and "adjusted winner"
for the case of two agents. We quantify the benefit of negotiation in the
divide and choose algorithm, and its use in tricking the adjusted winner
algorithm. Also we analyze the role of implicit information transmission
through actions for the repeated divide and choose problem by finding a
trembling hand perfect equilibrium for an specific setup. Lastly we consider a
centralized algorithm for maximizing the overall welfare of the agents under
the Nash collective utility function (CUF). This corresponds to a clustering
problem of the type traditionally studied in data mining and machine learning.
Drawing a conceptual link between this problem and the portfolio selection
problem in stock markets, we prove an upper bound on the increase of the Nash
CUF for a clustering refinement.
