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 following hold: 1) there are only two agents; 2) agents' valuations are piecewise-constant; 3) agents are hungry. The impossibility result extends to the case where the mechanism is allowed to leave some part of the cake unallocated. We also present a truthful and envy-free mechanism when each agent's valuation is piecewise-constant and monotone. However, if we require Pareto-optimality, we show that truthful is incompatible with approximate proportionality for any positive approximation ratio under this setting. To circumvent this impossibility result, motivated by the kind of truthfulness possessed by the I-cut-you-choose protocol, we propose a weaker notion of truthfulness: the proportional risk-averse truthfulness. We show that several well-known algorithms do not have this truthful property. We propose a mechanism that is proportionally risk-averse truthful and envy-free, and a mechanism that is proportionally risk-averse truthful that always outputs allocations with connected pieces.

翻译：我们研究了具有策略性代理的可分割异质资源公平分配问题（蛋糕切割问题），其中每个代理可通过操纵其私人估值以获取更优分配。（直接显示）机制以代理报告的估值为输入，输出满足给定公平性要求的分配方案。一个自然且基础性的开放问题（由[Chen等人，2010]首次提出，随后[Procaccia，2013]、[Aziz和Ye，2014]、[Branzei和Miltersen，2015]、[Menon和Larson，2017]、[Bei等人，2017]、[Bei等人，2020]等学者再次提出）是：是否存在确定性的、诚实的且无嫉妒（甚至比例公平）的蛋糕切割机制。本文通过证明不存在确定性的、诚实的且比例公平的蛋糕切割机制解决了该开放问题，即使在下述所有特殊条件同时成立时亦如此：1）仅有两个代理；2）代理的估值为分段常数函数；3）代理处于饥饿状态。该不可能性结论可推广至允许部分蛋糕未被分配的机制情形。我们同时提出了一种在代理估值为分段常数且单调情况下的诚实无嫉妒机制。然而，若要求帕累托最优性，我们证明在该设定下对于任意正逼近比，诚实性与近似比例公平性不可兼容。为规避这一不可能性结论，受“我切你选”协议所具诚实性的启发，我们提出了一种更弱的诚实性概念：比例风险规避诚实性。我们证明了若干知名算法不满足该诚实属性，并提出了一个满足比例风险规避诚实性且无嫉妒的机制，以及一个总能输出连通片分配的比例风险规避诚实性机制。