  We study the fair division problem on divisible heterogeneous resources (the
cake cutting problem) with strategic agents, where each agent can manipulate
his/her private valuation in order to receive a better allocation. A
(direct-revelation) mechanism takes agents' reported valuations as input, and
outputs an allocation that satisfies a given fairness requirement. A natural
and fundamental open problem, first raised by [Chen et al., 2010] and
subsequently raised by [Procaccia, 2013] [Aziz and Ye, 2014] [Branzei and
Miltersen, 2015] [Menon and Larson, 2017] [Bei et al., 2017] [Bei et al.,
2020], etc., is whether there exists a deterministic, truthful and envy-free
(or even proportional) cake cutting mechanism. In this paper, we resolve this
open problem by proving that there does not exist a deterministic, truthful and
proportional cake cutting mechanism, even in the special case where all of the
followings hold: 1. there are only two agents; 2. each agent's valuation is a
piecewise-constant function; 3. each agent is hungry: each agent has a strictly
positive value on any part of the cake. The impossibility result extends to the
case where the mechanism is allowed to leave some part of the cake unallocated.
  To circumvent this impossibility result, we aim to design mechanisms that
possess certain degree of truthfulness. Motivated by the kind of truthfulness
possessed by the classical I-cut-you-choose protocol, we define a weaker notion
of truthfulness: the risk-averse truthfulness. We show that the well-known
moving-knife procedure and Even-Paz algorithm do not have this truthful
property. We propose a mechanism that is risk-averse truthful and envy-free,
and a mechanism that is risk-averse truthful and proportional that always
outputs allocations with connected pieces.
