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Deterministic, Strategyproof, and Fair Cake Cutting Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17) Deterministic, Strategyproof, and Fair Cake Cutting Vijay Menon and Kate Larson David R. Cheriton School of Computer Science University of Waterloo fvijay.menon, [email protected] Abstract and Webb [1998]), and more recently has attracted the atten- tion of computer scientists (see e.g. the survey by Procaccia We study the classic cake cutting problem from [2016]) partly due to the nature of the challenges involved— a mechanism design perspective, in particular fo- that often require an algorithm design or complexity point of cusing on deterministic mechanisms that are strat- view—and partly due to its relevance in the design of multia- egyproof and fair. We begin by looking at mecha- gent systems [Chevaleyre et al., 2006]. nisms that are non-wasteful and primarily show that Most work in the cake cutting literature focuses on ad- for even the restricted class of piecewise constant dressing the above described scenario and so most of it is valuations there exists no direct-revelation mecha- concerned with computing a fair allocation using propor- nism that is strategyproof and even approximately tionality or envy-freeness as the main fairness criterion (see proportional. Subsequently, we remove the non- [Aziz and Mackenzie, 2016], and also [Procaccia, 2016] for wasteful constraint and show another impossibil- a recent survey). While addressing it is indeed the core ity result stating that there is no strategyproof and part of the problem, the fact that the valuation functions approximately proportional direct-revelation mech- of the agents may be their private information entails that anism that outputs contiguous allocations, again, the designed protocols or mechanisms be strategyproof— for even the restricted class of piecewise constant meaning that there is no incentive to misreport one’s valua- valuations. In addition to the above results, we tion function—for if otherwise the agents can lie about their also present some negative results when consid- valuation functions and potentially benefit from it. There- ering an approximate notion of strategyproofness, fore, the focus of this paper, like in [Chen et al., 2010; show a connection between direct-revelation mech- 2013; Mossel and Tamuz, 2010; Maya and Nisan, 2012; anisms and mechanisms in the Robertson-Webb Aziz and Ye, 2014; Branzeiˆ and Miltersen, 2015], is to look model when agents have piecewise constant valu- at mechanisms that are not only fair, but also strategyproof. ations, and finally also present a (minor) modifica- tion to the well-known Even-Paz algorithm that has In this paper, we look at deterministic mechanisms and we better incentive-compatible properties for the cases are primarily focused on the direct-revelation model—one when there are two or three agents. where the agents reveal their entire valuation function. Our main objective is to better understand the limits of determin- ism when it comes to imposing strategyproofness and fairness 1 Introduction (or approximate notions of either of them)—i.e., to see what Imagine a scenario where there is heterogeneous divisible is or is not achievable with deterministic mechanisms—and good that is to be divided among a certain set of n agents. to this end we make the following contributions. For an appropriately chosen notion of fairness, ideally, we a) In Section 3 we begin by looking at non-wasteful mecha- would like this resource to be divided fairly among these n nisms and we show a strong impossibility which in turn agents and so a natural question that arises is on how one strengthens an impossibility result given [Aziz and Ye, would accomplish this. The cake-cutting problem metaphori- 2014, Theorem 7]. In particular, we prove (in Theo- cally refers to this very scenario and it represents a fundamen- 1 rem 1) that for any n ≥ 2 and 0 ≤ 1; 2 < such that tal problem in the theory of fair division. More formally, in n 3 + < 1 , there is no deterministic and non-wasteful the cake cutting problem, the cake is modelled as the interval 1 2 n mechanism for n agents with piecewise constant valua- [0; 1] and each of the n agents is assumed to have a valua- tions that is -strategyproof and -proportional. tion function over the cake. The goal, as described above, 1 2 is to partition the cake so that it is fair according to some b) In Section 4 we remove the non-wasteful constraint and chosen notion of fairness. Starting with the work of Stein- provide three main results. The first one (Theorem 5) [ ] 1 haus 1948 , it has been studied over the last several decades shows that for any n ≥ 2 and 0 ≤ < n , there is by mathematicians, economists, and political scientists (see no deterministic mechanism for n agents with piece- e.g. the books by Brams and Taylor [1996] and Robertson wise constant valuations that makes contiguous alloca- 352 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17) tions and is strategyproof and -proportional. The sec- also provided a randomized mechanism that is strategyproof ond result (Proposition 6) establishes a connection be- in expectation and -proportional for any > 0. tween direct-revelation mechanisms and mechanisms in Finally, we also briefly remark on the predefined divisible the Robertson-Webb model for the case when agents goods setting which has been extensively studied in fair di- have piecewise constant valuations. And finally, the vision (see, for instance, [Cheung, 2016] and the references third one (Theorem 8) is a positive result where we therein). In the predefined goods setting, the task is to fairly present a (minor) modification to the well-known Even- allocate m divisible items among n agents, where the private Paz algorithm that has better incentive-compatible prop- information of an agent is the value she has for each of the erties for the cases when there are two or three agents. items. So, in effect, the predefined goods setting can be con- sidered as a special case of the setting of cake cutting with 1.1 Related Work piecewise constant valuations—i.e., one where the valuation The cake cutting problem has been studied using two in- functions of all the agents have the same set of breakpoints— put models—Robertson-Web model and the direct-revelation and therefore all the negative results in that setting carry over model. Therefore, we can classify the work on strategic as- to the cake cutting setting. However, some of the nice positive pects of cake cutting into two classes based on the input results with regards to strategyproof mechanisms in this set- model used and look at each of them separately. ting (see, e.g., [Cole et al., 2013] and [Cheung, 2016]) do not Among related papers that operate in the direct-revelation carry over to the cake cutting setting as here, informally, the model [Chen et al., 2010; 2013; Mossel and Tamuz, 2010; difficulty in achieving strategyproofness arises from the fact Maya and Nisan, 2012; Aziz and Ye, 2014; Li et al., 2015; that a mechanism has to guard against two types of manip- Alijani et al., 2017], the ones which are most relevant to ulations: i) with regards to the breakpoints in the valuation results in this paper are the works of Chen et al. [2010; functions and ii) with regards to the values assigned to the 2013] and Aziz and Ye [2014]. Chen et al. [2010; 2013] pieces between two consecutive breakpoints. were the first to look at strategyproof cake cutting and their main result was a deterministic, strategyproof, envy-free, pro- 2 Preliminaries portional, and Pareto-optimal mechanism for the case when The cake—which is a heterogeneous divisible good—is mod- the agents have piecewise uniform valuations. Additionally, elled as the unit interval [0; 1]. A piece of cake X is a finite they also presented randomized algorithms that are truthful in union of disjoint (except at the boundaries of the intervals) expectation, proportional, and envy free for piecewise linear subintervals of [0; 1]. An interval is denoted by I and the valuations. length of an interval I = [x; y] is given by jIj = y − x. In The other most relevant work that uses the same model is any cake cutting instance, there are n agents who want a share [ ] that of Aziz and Ye 2014 . They present two deterministic of the cake and we denote them by [n] = f1; ··· ; ng. Each mechanisms, CCEA and MEA, for piecewise constant valua- 1 agent i 2 [n] has a non-negative, integrable, and private value tions, where CCEA is robust envy-free and non-wasteful, and density function (also referred to as their utility/valuation MEA is Pareto-optimal and envy-free. Although both of them function), v : [0; 1] ! R+ [ f0g, that indicates how agent et al. [ i generalize the mechanism proposed by Chen 2010; i values different parts of the cake, and given a piece of cake ] 2013 , they do not remain strategyproof for piecewise con- X, the value for X is defined by V (X) = R v (x) dx = stant valuations. Additionally, they also showed that there is i X i P R v (x) dx. (Note that the fact that the valuation no mechanism that is strategyproof, robust proportional1, and I2X x2I i non-wasteful for piecewise constant valuations. functions are integrable implies additivity—meaning for two I I0 V (I [I0) = V (I)+V (I0) Among papers that use the Robertson-Webb model, the disjoint intervals and , i i i .) With- two that are most related are the works of Kurokawa et al. out loss of generality, we assume that the valuation functions normalized V ([0; 1]) = 1 i 2 [n] [2013] and Branzeiˆ and Miltersen [2015].
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