Deterministic, Strategyproof, and Fair Cake Cutting

Proceedings of the Twenty-Sixth International Joint Conference on Artiﬁcial Intelligence (IJCAI-17)

Vijay Menon and Kate Larson David R. Cheriton School of Computer Science University of Waterloo {vijay.menon, kate.larson}@uwaterloo.ca

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) modification 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 determinism 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 valuations 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 considered 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 |I| = 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] = {1, ··· , n}. Each mechanisms, CCEA and MEA, for piecewise constant valua- 1 agent i ∈ [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+ ∪ {0}, 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 I∈X x∈I 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 ∈ [n] [2013] and Branzeiˆ and Miltersen [2015]. Kurokawa et al. are , i.e., i for all . Addi- i hungry v (x) > 0 [2013] studied envy-free cake cutting and their main result tionally, we say that agent is if i for all x ∈ [0, 1] was a parameterized protocol for piecewise linear valuations. . Additionally, they also showed that there is no mechanism 2.1 Input Models of complexity bounded only by a function of the number of agents and the total number of breakpoints that is strate- The way we have defined the valuation functions above, gyproof and envy-free for piecewise constant valuations. it is not necessary that they have a discrete representa- The other most relevant paper that operates in the tion. Therefore, most of the work in cake cutting assumes Robertson-Webb model is that of Branzeiˆ and Miltersen a query model—commonly known as the Robertson-Webb [2015]. Branzeiˆ and Miltersen [2015] showed that any de- query model—that allows only the following two types of terministic and strategyproof protocol is a dictatorship when queries. there are only two hungry agents and for the case when there 1. Eval query: given an interval [x, y], eval(i, x, y) asks are more than two hungry agents they showed that there is at agent i for its value for [x, y], i.e., eval(i, x, y) = least one agent who gets an empty piece. Additionally, they Vi(x, y). 1Robust envy-freeness and robust proportionality are much 2. Cut query: given a point x and r ∈ [0, 1], cut(i, x, r) stronger notions than envy-freeness and proportionality, respec- asks agent i for the minimum (i.e., the leftmost) point y tively; see [Aziz and Ye, 2014] for more details. such that Vi(x, y) = r.

353 Proceedings of the Twenty-Sixth International Joint Conference on Artiﬁcial Intelligence (IJCAI-17)

While the Robertson-Webb model is the most widely stud- In this paper, we use (approximate) proportionality as our ied, there is recent work (e.g., [Chen et al., 2010; 2013], [Bei main fairness criterion. In the definition below, proportional- et al., 2012], [Aziz and Ye, 2014]) which restricts the agents’ ity refers to the special case when = 0. valuation functions to ones that have a concise representa- 1 Definition 1 (-proportionality). For any ∈ 0, n , a mech- tion, thus in turn giving rise to another model (referred to as anism M is said to be -proportional if it always returns an the direct-revelation model) where the entire valuation func- allocation (A , ··· ,A ) such that ∀i ∈ [n],V (A ) ≥ 1 − . tions of the agents are given as input to a mechanism. In this 1 n i i n paper, we consider one fundamental class of restricted valu- Along with (approximate) proportionality, the other ma- ations, namely, piecewise constant valuations. An agent i is jor property we are concerned with is (approximate) strate- said to have a piecewise constant valuation if [0, 1] can be par- gyproofness. Informally, for any ∈ [0, 1], a cake cutting titioned into a finite number of intervals such that vi is a con- mechanism M is said to be -strategyproof if an agent can stant over each interval (i.e., it is a step function). A special gain a utility of at most by misreporting her valuation func- case of piecewise constant valuations are the piecewise uni- tion. More formally, we have the following definition. form valuations where, for some constant c, the function can Definition 2 (-strategyproofness). For any ∈ [0, 1], a only take the values 0 or c. One of the reasons why piecewise mechanism M is said to be -strategyproof if for every constant valuations are interesting is because they are expres- 0 agent i ∈ [n], and for all vi, v−i, Vi(Mi(vi, v−i)) ≥ sive enough to be able to approximate any general valuation 0 Vi(Mi(vi, v−i)) − . function. In this paper, we make use of both input models. All Strategyproofness refers to the special case in the above = 0 our negative (impossibility) results are in the direct-revelation definition when . model, while the algorithm we present is in the Robertson- In addition to the above mentioned properties, we also Webb model. Note that the existence of a mechanism in the talk about (approximate) envy-freeness and Pareto-efficiency. ∈ [0, 1] Robertson-Webb model implies an existence in the direct- For an , a cake cutting mechanism is said to be envy-free (A , ··· ,A ) revelation model, but the converse is not necessarily true, and - if it always returns an allocation 1 n ∀i, j ∈ [n],V (A ) ≥ V (A ) − hence our negative results, in particular, are strong. Also, all such that i i i j . Envy-freeness = our negative results are true for even the restricted class of refers to the special case in the above definition when 0 Pareto-efficient piecewise constant valuations and hence naturally carry over . A cake cutting mechanism is said to be (A , ··· ,A ) to anything more general (like piecewise linear valuations). if it always returns an allocation 1 n that is not Pareto-dominated—meaning that there is no other allocation 0 0 0 2.2 Properties of Cake Cutting Mechanisms (A1, ··· ,An) such that ∀i ∈ [n],Vi(Ai) ≥ Vi(Ai), with the inequality being strict for at least one agent. Note that non- A direct-revelation cake cutting mechanism M takes the val- wastefulness, as defined above, can be considered as a weak uation functions (v1, ··· , vn) of the agents (which we refer form of efficiency where all it is doing is restricting a mecha- to as a profile) as input and it outputs an allocation A = (A1, nism from allocating a piece of zero value to an agent if there ··· ,An), where Ai is the allocation to agent i, and ∀i, j(6= i), is some other agent who has a non-zero value for it. Ai and Aj are disjoint (except at the boundaries of the intervals). While in general Ai can be any arbitrary piece of cake, sometimes we concern ourselves only with contiguous allo- 3 Non-wasteful Mechanisms cations where each Ai is a single interval. Throughout, we Chen et al. [2010; 2013] were the first to consider direct- assume that the mechanism allocates (to some agent) all the revelation mechanisms for cake cutting and they proposed pieces of cake that are valued at greater than zero by at least a polynomial-time deterministic mechanism that is strat- one of the agents. Additionally, we often use Mi(vi, v−i) to egyproof, proportional, envy-free, and Pareto-efficient for denote Ai, where v−i denotes the valuation functions of all piecewise uniform valuations. In light of such a result, one the agents except i. natural question that arises is if there exists a mechanism A mechanism is said to be non-wasteful if it allocates ev- for at least the more general class of piecewise constant val- ery piece that is desired—meaning that the piece is assigned uations that is strategyproof, Pareto-efficient, and fair (for a value greater than zero—by at least one agent to an agent some notion of fairness). We already have an answer to this who desires it. More formally, if I is a subinterval of [0, 1] question due to Schummer [1996] who considered the prede- and D(I) = {i ∈ [n] | Vi(I) > 0}, then a mechanism is fined divisible goods setting and showed that the only mech- non-wasteful if it always makes an allocation A such that anism that is strategyproof and Pareto-efficient is a dictator- ∀I,I ⊆ ∪i∈D(I)Ai. In this paper, we consider both non- ship. And since this predefined goods setting is a special case wasteful and wasteful mechanisms. Additionally, for all our of the setting with piecewise constant valuations, we know negative results we assume free disposal, meaning that the that there is no hope for achieving strategyproofness, Pareto- mechanism can throw away pieces of cake that aren’t valued efficiency, and any notion of fairness. Therefore, in this con- (at greater than zero) by any of the agents without incurring text, another notion to consider is non-wastefulness which, as a cost. Note that the existence of a mechanism without the defined in Section 2, is a weaker notion of efficiency. In fact, free disposal assumption implies the existence of a mecha- Aziz and Ye [2014] considered this notion and showed an im- nism with the free disposal assumption, but the converse is possibility result saying that there is no strategyproof, robust not necessarily true. Hence, in the context of negative results, proportional (which is a much stronger notion of proportion- such an assumption only makes them stronger. ality; see [Aziz and Ye, 2014]), and non-wasteful mechanism

354 Proceedings of the Twenty-Sixth International Joint Conference on Artiﬁcial Intelligence (IJCAI-17)

[ ] 1 Aziz and Ye, 2014, Theorem 7 . Below, we show a stronger This in turn implies that for every 0 ≤ 1, 2 < n such that impossibility which says that there is no deterministic and 1 1−n(31+2) 31 + 2 < , we can always find a 0 < δ < n n(1−31) non-wasteful mechanism that is even 1-strategyproof and 2- proportional for , such that 0 ≤ 3 + < 1 . to fit in the definition of the function w(x) such that the 1 2 1 2 n inequality—and hence our theorem—will be true. 1 Theorem 1. For any n ≥ 2 and 0 ≤ 1, 2 < n such that 1 With respect to the above theorem, we highlight the fol- 31 + 2 < n , there is no deterministic and non-wasteful mechanism for n agents with piecewise constant valuations lowing corollaries which are basically special cases when 1 and are zero, respectively. The first one rules out the pos- that is 1-strategyproof and 2-proportional. 2 sibility of guaranteeing some value (however small) to all the Proof. Let us assume that there exists a deterministic, 1- agents if the mechanism has to be non-wasteful and strate- strategyproof, 2-proportional, and non-wasteful mechanism gyproof. The second one, on the other hand, provides a lower M for n agents. First, consider the profile (u, u, y, ··· , y), bound for the natural question on whether we can come up where with mechanisms that can guarantee proportionality and non- ( ( n x ∈ [0, 2 ] 0 x ∈ [0, 2 ] wastefulness and at the same time allow only a maximum u(x) = 2 n y(x) = n gain of , where 0 ≤ < (1 − 1 ), from misreporting. Unfor- 2 n 2 n 0 x ∈ ( n , 1] n−2 x ∈ ( n , 1]. tunately, we do not have an accompanying upper bound result Let M make an allocation A1, A2 to agent 1 and agent 2, for this and so the question of an upper bound remains open. respectively. Without loss of generality, we can assume that 1 Corollary 2. For any n ≥ 2 and 0 ≤ < n , there is no de- |A1| ≥ |A2|. Now, since M is non-wasteful, |A1| + |A2| = terministic, strategyproof, -proportional, and non-wasteful 2 1 n , and so this implies that |A1| ≥ n . mechanism for n agents with piecewise constant valuations. Next, let us consider the profile (v, u, y, ··· , y), where 1 1 Corollary 3. For any n ≥ 2 and 0 ≤ < , there is no de- v(x) = when x ∈ A1 and 0 otherwise. Since M is 3n |A1| terministic, -strategyproof, proportional, and non-wasteful non-wasteful and 1-strategyproof, agent 1 has to get a util- mechanism for n agents with piecewise constant valuations. ity of at least 1 − 1 from the allocation in this profile, because if it allocates anything less, then agent 1 will deviate 0 0 4 Removing Non-wastefulness to Profile 1. Therefore, if M allocates A1 and A2 to agent 0 1 and 2, respectively, then (1 − 1) · |A1| ≤ |A1| and so While non-wastefulness is certainly a desirable property, it 0 |A2| ≤ |A2| + 1 · |A1|. seems like the negative results in the previous section were Finally, consider the profile (v, w, y, ··· , y), where largely driven by this constraint as it severely restricts the  1−δ kind of allocations that a mechanism can make. So, what  x ∈ A1  |A1| if we remove this constraint? Do similar impossibilities still w(x) = δ hold, or are there mechanisms that satisfy some of those prop- |A | x ∈ A2  2 erties? These are the questions we try to answer here.  0 otherwise Chen et al. [2010; 2013] provided a deterministic and Now, here, since M is non-wasteful and because agent 1 strategyproof mechanism for piecewise uniform valuations does not desire any part of A2, agent 2 gets at least A2. How- and one of the main open questions they had posed was on ever, agent 2 has a total utility of only δ for A2, and since we generalizing their mechanism for piecewise constant valua- assumed that M is also 2-proportional, it has to get a piece tions. Subsequently, Aziz and Ye [2014] and Branzeiˆ and of length, say, `, of A1, where Miltersen [2015] had posed the same question regarding the 1 existence of mechanisms that are strategyproof and propor- ` 1 ( n − 2 − δ) δ + (1 − δ) ≥ − 2 =⇒ ` ≥ |A1| · . tional for piecewise constant valuations. Here we address a |A1| n (1 − δ) special case of this question when the mechanism makes con- 00 00 If M allocates A1 and A2 to agent 1 and 2, respectively, in tiguous allocations and show that there exists no determinis- 1 ( −2−δ) tic direct-revelation mechanism that is strategyproof and - this profile, then |A00| ≥ |A | + ` ≥ |A | + |A | · n . 2 2 2 1 (1−δ) 0 ≤ < 1 0 proportional, for any n . Additionally, we also prove Also, from the profile (v, u, y, ··· , y) we know that |A2| ≤ a stronger result (Proposition 4) for the case of two agents. ( 1 − −δ) n n 2 Informally, at a high-level, the proofs of both of these results |A2| + 1 · |A1|. Therefore, if 2 |A2| + |A1| · (1−δ) > n follow the same theme as in the proof of Theorem 1 where we 2 (|A2| + 1 · |A1|) + 1, then agent 2 can deviate from (v, u, y, ··· , y) to (v, w, y, ··· , y) and as a result gain strictly construct valuation profiles with an aim of arriving at a con- more than . So, considering this inequality, we have, tradiction. However, unlike in Theorem 1, we now no longer 1 have the non-wasteful constraint, and so this requires some 1 n ( − 2 − δ) n additional arguments which use the fact that the allocations |A | + |A | · n > (|A | + |A |) + 2 2 1 (1 − δ) 2 2 1 1 1 are contiguous. Due to space constraints, the proofs of both of the statements below have been deferred to the full version ( 1 − − δ) 1 n 2 2 1 [ ] · − 1 > 1 (as |A1| ≥ ) Menon and Larson, 2017 . n n (1 − δ) n 1 Proposition 4. For any 0 ≤ 1, 2 < 2 such that 1 − n(31 + 2) 1 δ < . 1 + 2 < 2 , there exists no deterministic mechanism for n(1 − 31) two hungry agents with piecewise constant valuations that

355 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17) makes contiguous allocations and is 1-strategyproof and 2- Input: An agent i’s piecewise constant valuation function proportional. vi(x), > 0, and k Output: A piecewise constant function wi(x) that - As an aside, note that the 2 = 0 case in Proposition 4 essentially says that if we insist on contiguous and proportional approximates vi(x) 2k allocations, then one cannot provide any guarantee on strate- 1: N ← , x0 ← 0, xN+1 ← 1 gyproofness (i.e., such mechanisms cannot be -strategyproof 2: for j ∈ {1, ··· ,N} do for any feasible ). Later on, in Theorem 8, we show how one 3: xj ← cut i, xj−1, 2k , cj ← 2k·(x −x ) 1 j j−1 can circumvent this and obtain a 4 -strategyproof and propor- 4: wi(x) = cj, x ∈ [xj−1, xj] tional mechanism for two agents by giving each of them at 5: end for most two contiguous pieces. 6: if XN 6= 1 then 1 (1− 2k ·N) Theorem 5. For any n ≥ 2 and 0 ≤ < n , there is no de- 7: wi(x) = , x ∈ [xN , xN+1] terministic mechanism for n agents with piecewise constant xN+1−xN 8: end if valuations that makes contiguous allocations and is strate- 9: return w (x) gyproof and -proportional. i Given Theorem 5, the first natural question that arises is if Algorithm 1: -approximation of a piecewise constant function us- 2k we can extend it to the case when the mechanism can make ing cut queries. arbitrary allocations. Unfortunately, we do not have an answer to this question. Instead, below, we show a connection at most k breakpoints by another piecewise constant function between direct-revelation mechanisms and mechanisms in the w(x) such that W ([0, 1]) = 1 and for any piece of cake X, Robertson-Webb model for the case when agents have piece- Z Z Z wise constant valuations. We believe that it will be useful in v(x) dx − ≤ w(x) dx ≤ v(x) dx + . (1) either of the cases—i.e., if the answer to the above question X 2 X X 2 is either in the positive or if we want to prove that such a Proof (sketch). Algorithm 1 describes how to convert v(x) 2k mechanism does not exist. In the case that such a mechanism to w(x) by using cut queries. It is easy to see that exists, the connection basically shows that when given an up- W ([0, 1]) = 1. To prove that for any piece of cake X, per bound on the maximum number of breakpoints in any R R R X v(x) dx − 2 ≤ X w(x) dx ≤ X v(x) dx + 2 , con- agent’s valuation function, one can construct a mechanism sider some arbitrary X. If we denote the breakpoints of in the Robertson-Webb model that approximately achieves v by {b1, ··· , bk} (since there are only a maximum of k both of the properties. And in the case that one wants to breakpoints), where 0 = b0 < b1 < ··· < bk+1 = 1, prove that such a mechanism does not exist, one can use this and if we let Ci = X ∩ [xi−1, xi], then we know that connection to prove a non-existence (of mechanisms that are R w(x) dx = PN+1 R w(x) dx. Additionally, also note -strategyproof and -proportional for some > 0) in the X i=1 Ci that if [x , x ] does not contain any breakpoint b of the Robertson-Webb model and then map it back to show that i−1 i j original function v(x), then R w(x) dx = R v(x) dx since there is no (finite) direct-revelation mechanism that is strate- Ci Ci gyproof and proportional. we know that v(x) is constant over this interval (as there are no breakpoints) and so v(x) = = w(x), ∀x ∈ Proposition 6. Let P ,P ,P denote the property of strate- 2k·(xi−xi−1) 1 2 3 [x , x ] v(x) k gyproofness, proportionality, and envy-freeness, respectively, i−1 i . This, along with the fact that has at most k C and let P ⊆ {P ,P ,P }. Given k, an upper bound on breakpoints, implies that there are at most i’s for which 1 2 3 R w(x) dx 6= R v(x) dx. Now, if we assume without loss the maximum number of breakpoints in any agent’s valuation Ci Ci function, and any n ≥ 2, if there exists a (finite) determinis- of generality that these are C1 ··· ,Ck, then using the fact that R w(x) dx ≤ and R v(x) dx ≤ we have that tic direct-revelation mechanism for n agents with piecewise Ci 2k Ci 2k constant valuations that satisfies Pi, ∀Pi ∈ P , then, for any for every i ∈ {1, ··· , k}, > 0, there exists a mechanism in the Robertson-Webb model Z Z − ≤ w(x) dx − v(x) dx ≤ . (2) for n agents with piecewise constant valuations that asks at 2k 2k 2k Ci Ci most (n ) queries on each input and is -Pi, ∀Pi ∈ P . Now, since R [w(x) − v(x)] dx = Pk (R [w(x) − X i=1 Ci Proof (sketch). Since in the Robertson-Webb model we only v(x)] dx), we can now use equation 2 to prove our claim. have query access to the valuation functions, the first step is to basically “learn” these functions to a sufficiently good ap- Given the above claim, we can now build a mechanism proximation using only eval and cut queries. Subsequently, MRW in the Robertson-Webb model in the following way, once we have that, we can essentially feed these functions to where (v1, ··· , vn) are the original valuation functions and the direct-revelation mechanism in order to create a mecha- (w1, ··· , wn) are the corresponding -approximated func- nism in the Robertson-Webb model that achieves an approx- tions. imate version of all the properties. For the first step, the key RW ∀i ∈ [n], Mi (v1, ··· , vn) = Mi(w1, ··· , wn). (3) observation is following claim. RW 2k It is clear that M makes at most (n ) queries on 2k RW Claim 1. Given an > 0, we can use cut queries to each input. To prove that M is -Pi for all Pi ∈ P satis- -approximate any piecewise constant function v(x) that has fied by M, let us consider each of the properties separately.

356 Proceedings of the Twenty-Sixth International Joint Conference on Artiﬁcial Intelligence (IJCAI-17)

i) When M satisfies P1 (strategyproofness). Since M Procedure: modifiedEP([a, b],S) is strategyproof, the only way for an agent i with a valuation Input: ([a, b],S), where [a, b] ⊆ [0, 1] is the cake to be RW function vi to manipulate M is to pretend as if its func- proportionally allocated among k = |S| agents 0 tion is some vi and answer the queries accordingly. This in 1: If k == 1, then return ([a, b]) 0 turn will result in wi being the -approximated function in- 2: for each i ∈ S do k stead of wi if it reported truthfully. However, we know that b 2 c 0 0 3: ci = cut i, a, k · Vi(a, b) ∀ wi, w−i,Wi(Mi(wi, w−i)) ≥ Wi(Mi(wi, w−i)). This in turn implies that using equation 1 we have, 4: end for 5: Let {d1, ··· , dk} be the sorted order of the ci’s, SL ← Vi(Mi(wi, w−i)) + ≥ Wi(Mi(wi, w−i)) ≥ {i ∈ S | ci ≤ d k }, and SR ← S \ SL. 2 b 2 c 0 0 h i Wi(Mi(wi, w−i)) ≥ Vi(Mi(wi, w−i)) − . 6: Proportionally allocate the cake d k , d k +1 2 b 2 c b 2 c 0 among the k agents using the Even-Paz algorithm. And so, now, using equation 3, we have that for all vi, v−i, RW 0 RW h i Vi(Mi (vi, v−i)) − Vi(Mi (vi, v−i)) ≤ . 7: Recursively call modifiedEP on a, db k c and The other two cases can be proved similarly and we defer h i 2 d k , b with SL and SR, respectively. Return the the complete proof to the full version. b 2 c+1 allocations so formed along with the proportional allo- Although, as mentioned above, it remains open as to h i cation of the piece d k , d k . whether a strategyproof and proportional mechanism exists, b 2 c b 2 c+1 another goal one could still consider is on finding better pro- Main: portional mechanisms (in terms of -strategyproofness). Be- 8: output modifiedEP([0, 1],S = {1, ··· , n}). low, we begin the pursuit of this goal and as our final question in this paper ask if there are positive results at least regard- Algorithm 2: Modified Even-Paz algorithm. ing mechanisms that are proportional and -strategyproof for some < (1 − 1 ). We answer it with a “Yes” and in par- n valuations. Here we addressed a special case of this and we ticular we first prove—by showing a bound on the maximum showed (in Theorem 5) that for any n ≥ 2 agents there is no gain an agent can get by misreporting—that the well-known deterministic mechanism that makes contiguous allocations [ ] Even-Paz algorithm Even and Paz, 1984 satisfies the above and is strategyproof and even -proportional. Although the criterion for the case when there are at least four agents. And contiguous allocations constraint captures some interesting following this, we present a minor modification to the same scenarios, we believe that answering the above question with- that has better incentive compatible properties than the orig- out this (rather strong) constraint is important. In particular, inal Even-Paz algorithm for the cases when there are two or when given the fact that there exists randomized mechanisms three agents. Due to space constraints, proofs of Proposition 7 that are a) truthful in expectation, proportional, and envy- and Theorem 8 have been deferred to the full version. free for piecewise linear valuations in the direct-revelation Proposition 7. For n ≥ 2 agents, the Even-Paz algorithm is model [Chen et al., 2013] and b) truthful in expectation and - deterministic, proportional, and -strategyproof, where proportional in the Robertson-Webb model [Branzeiˆ and Mil-  1 when n = 2 or n = 4 tersen, 2015], we believe that resolving this question can give  2 us a clearer picture regarding the limits of determinism. = 2 when n = 3 or n = 5 3 Moving away from the above result, in Section 4 we  2 1 − n when n ≥ 6. showed (in Theorem 8) that there exists a proportional mech- Next, we show that the bounds on can be improved for anism that has better incentive-compatible properties than the the cases when there are two or three agents. In particular, we Even-Paz algorithm for the cases when there are two or three show that Algorithm 2 is proportional and -strategyproof, agents. With regards to this result, one of the main ques- 3 3 1 tions that arise is the following and we suggest it as an open where = (1 − 2n ) or = (1 − 2n + 2n2 ) depending on whether n is even or odd, respectively. And so, for the cases problem: for n ≥ 2 agents, what is the minimum achiev- when there are two or three agents, the gain is 1 or 5 , respec- able for which there exists a deterministic mechanism that 4 9 tively, as opposed to 1 or 2 in the Even-Paz algorithm. is proportional and -strategyproof? It is unclear how we can 2 3 provide a lower bound, or improve the proposed mechanism Theorem 8. Algorithm 2 is deterministic, proportional, and to get smaller values of . And also, at the same time there -strategyproof, where is the question of whether we can make Algorithm 2 non- ( 3 wasteful when agents have concisely representable valuation 1 − 2n when n is even = functions. Therefore, given our lower bound (Theorem 1) 1 − 3 + 1 when n is odd. 2n 2n2 in Section 3, another problem that remains open is whether 5 Discussion we can guarantee non-wastefulness along with proportionality and -strategyproofness, where 1 ≤ < (1 − 1 ). One of the main open questions raised by Chen et al. [2010; 3n n 2013] (and also by Aziz and Ye [2014] and Branzeiˆ and Mil- tersen [2015]) was on the existence of deterministic mecha- Acknowledgments nisms that are strategyproof and fair for piecewise constant We thank Simina Branzeiˆ for useful discussions.

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