Multi-Dimensional Envy-Free Scheduling ... - Semantic Scholar

Multi-Dimensional Envy-Free Scheduling Mechanisms Jason Hartline

Sam Ieong

Ahuva Mualem

Michael Schapira

Aviv Zohar

Abstract We study fairness design scenarios in which each bidder follows the global goal of the mechanism designer only if the resulted allocation would be fair from his own point of view. More formally, we focus on approximation algorithms for indivisible items with supporting envy-free bundle prices. We first provide an exact characterization of envy-free mechanisms for indivisible goods in terms of local-efficient bundle-assignments. Interestingly, an envy-free mechanism always selects local-optimum allocations with respect to the social welfare. We then study the paradigmatic non-utilitarian scheduling problem of Algorithmic Mechanism Design. Specifically, we use our characterization to study the approximability of envy-free scheduling mechanisms (over multidimensional domains). We derive general bounds on the approximability of deterministic envyfree mechanisms that seek to minimize the makespan in the unrelated machines model with 1 and an upper bound of m+1 for the best m machines. We exhibit a lower bound of 2 − m 2 approximation ratio achievable by any envy-free mechanism. For m = 2 our bounds are tight. However, our upper bound is not known to be computationally efficient. Finally, we consider the important case of related parallel machines and show the envy-freeness constraint do not impose an additional computational burden.

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1

Introduction

In this paper we consider the cake-cutting problem where each bidder would like to get a fair share of the cake from her point of view. At the same time, it might be the case that the divider of the cake has certain global goal in mind. To encourage the agents to follow the global goal, the divider needs to devise a fair partition of the cake. More formally, the divider seeks a fair partition that is as close as possible to his global goal. As a typical scenario we consider a resource allocation problem in a society or a company. Suppose a new project with several tasks is just arrived. The challenge is to find a fair allocation of the tasks among the resources such that the last task of the project finishes as soon as possible. It is not hard to verify that this non-trivial global goal is different from the social-welfare maximizing goal. Naturally, this raises the question: what global goals can be achieved in a fair manner? Fair division of goods has been a central problem in economic theory. Several concepts of fairness were studied over the years. Envy-free allocations introduced by Foley [8]. An allocation is called envy-free if every bidder likes his own bundle at least as well as that of anyone else. In this paper we study envy-free allocations for indivisible items with supporting anonymous bundle prices.

Results In Section 3, we identify a characterization of envy-free mechanisms for multi-dimensional domains in terms of local-efficient bundle-assignments. Interestingly, an envy-free mechanism always selects a local-optimum allocation with respect to social welfare (the exact definition is given in the section itself). To the best of our knowledge, this is the first characterization of envy-free mechanisms with supporting bundle prices. In Section 4, we study envy-free scheduling mechanisms. We focus on the non-utilitarian NPhard multi-dimensional scheduling problem presented by Lenstra, Shmoys, and Tardos [13]. This optimization problem was formulated as a mechanism design problem by Nisan and Ronen in their seminal paper on Algorithmic Mechanism Design [17]: There are k tasks that are to be scheduled on m non-identical machines (”unrelated machines”). The total cost of a subset of tasks on machine i is the additive sum of the costs of the individual tasks on that machine. The global goal is minimizing the makespan of the chosen schedule. I.e., assigning the tasks to the machines in a way that minimizes the finishing time of the last task. Nisan and Ronen considered this global goal in the context of truthfulness (assuming players are selfish and thus should be incentivized to act in a ”truthful” manner). We consider this global goal in the context of envy-free design. Specifically, using our characterization, we derive general bounds on the approximability of deterministic envyfree mechanisms that seek to minimize the makespan on unrelated machines. We exhibit a lower 1 bound of 2 − m and an upper bound of m+1 for the best approximation ratio achievable by any 2 envy-free mechanism. For m = 2 the result is tight. However, our upper bound is not known to be computationally efficient for any m ≥ 2. This leaves several interesting open problems. In Section 5, we consider the problem of minimizing the makespan of related parallel machines in envy-free manner (a non-utilitarian NP-hard uni-dimensional setting). We show that the envy-freeness constraint does not impose a computational burden, and in particular there exists a polynomial-time computable envy-free mechanism that achieves the approximation ratio of 1 + ! w.r.t. makespan.

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Related Work Recently, the papers [7, 21] suggest the notion of locally envy-free equilibrium, to study a possible strategic behavior of the GSP ad-word position auction used by Google and Yahoo. Both papers independently identify an interesting connection between the revenue of the locally envy-free equilibrium and the VCG auction. The paper [14] studies envy-free allocations with zero prices. In this setting envy-free allocations might not exist, and thus they consider approximations for the minimum envyness. Approximations of revenue maximization of envy-free item pricing for single-minded bidders and unit-demand bidders are considered in [10, 3] and references therein. The fundamental scheduling problem for minimizing the makespan of unrelated machines is studied in [13]. This paper presents a non-trivial 2-approximation polynomial-time algorithm. They also showed that the problem cannot be approximated in polynomial-time within a factor less than 32 . A seminal paper by Nisan and Ronen [17] defines the notion of algorithmic mechanism design [18]. Assuming that the machines are the strategic agents, the paper proves that not only is it impossible to minimize the makespan in a truthful manner, but that any approximation ratio better than 2 cannot be achieved by a truthful deterministic mechanism. They also showed that there is computationally efficient truthful mechanism that achieves an approximation ratio of m. Their result is tight for m = 2. The lower bound is recently improved for m = 3 (from 2 to 2.41 [4]). Further important recent results for this multi-dimensional scheduling problem from the algorithmic mechanisms design perspective are discussed in [5]. Hocbaum and Shmoys describe a PTAS for minimizing the makespan of related parallel machines [11].1 In this NP-hard problem the type of each machine can be described easily by a single positive number (single-parameter type). This problem from algorithmic mechanism design perspective first studied in [1]. Archer and Tardos give a 3-approximation mechanism based on randomized rounding of the optimal fractional solution [1]. A truthful randomized mechanism that achieves an approximation ratio of 1 + ! is presented in [6].

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Characterizing Envy-Free Mechanisms

This section characterizes envy-free mechanisms. The characterization is stated in terms of the local efficiency of the social choice function and applies to every domain of valuations.

2.1

The Setting

We consider a set of indivisible items U and a set N of n bidders.2 We assume that bidders value combinations of items. Formally, each bidder i ∈ N has a valuation function vi () that describes his valuation for each subset S of items, i.e. vi (S) is the maximum finite amount of money i is willing to pay for S. An allocation a = a1 , ..., an is a partition of items among the bidders. Formally, ai denotes the subset of items allocated to bidder i, a1 ∪ a2 ∪ · · · ∪ an ⊆ U , and ai ∩ aj = ∅, whenever i (= j. The set of all possible allowed allocations is denoted by A. Every valuation vi ∈ Vi satisfies the following three conditions: No externalities meaning that the valuation of bidder i depends only on his allocated bundle. Free disposal meaning that the valuation is nondecreasing with the set of allocated items (for every S and T , S ⊆ T implies vi (S) ≤ vi (T )). Normalization meaning that the value of the empty bundle is always zero. Vi denotes the domain of all possible valuations 1 2

A PTAS is an (1 + !)-approximation algorithm that runs in polynomial time, assuming ! is a fixed constant. U might be an infinite set of items.

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of bidder i. Bidders have quasi-linear utilities, and so vi (ai ) − x is the overall utility of i for getting the subset ai and paying the monetary price x. A social choice function f : V → A maps an n-tuple of valuations v = (v1 , v2 , . . . , vn ) ∈ V1 × V2 × ···× Vn = V to an outcome a ∈ A. A mechanism defines an allocation and a set of prices for every possible valuation of the bidders. A mechanism is a tuple M = (f, p), where f is a social choice function and the real function pi : Vi → R assigns a finite payment to each bidder i ∈ N . Intuitively, the social choice function f represents the global goal of the mechanism designer, and the payment p determines the fairness of f . Definition 1 (Envy-Free Mechanism) Let M = (f, p) be a mechanism. Let i, k ∈ N and let v = (v1 , ..., vn ) be an n-tuple of valuations. Denote by a ∈ A the allocation f outputs for v. The mechanism M is said to be envy-free if for every bidders i, k and valuation v it holds that: vi (ai ) − pi (v) ≥ vi (ak ) − pk (v). We say that a social choice function f : V → A is envy-free achievable if there exists a payment p such that the mechanism M = (f, p) is envy-free.3 Definition 2 (Ψ(v, a)) For arbitrary allocation a ∈ A and valuation v ∈ V , let Ψ(v, a) = Σni=1 vi (a) denotes the social-welfare of the allocation a with respect to v.

2.2

Locally-Efficient Bundle Assignments and Allocations

In order to be able to state our main theorem we proceed with some definitions. Let a ∈ A be an arbitrary feasible allocation. We shall consider the following associated allocations based on a: Definition 3 (Bundle Allocation based on a and β) Let β : N → N be an arbitrary function. Let a = (a1 , ..., an ) ∈ A be an arbitrary allocation. We say that the allocation aβ is the bundle-allocation based on a and β, if bidder i in aβ is allocated all bundles ak with β(k) = i. We also consider a special interesting case, in which each bidder gets exactly one bundle of a ∈ A, based on a given permutation: Definition 4 (Bundle Assignment based on a and π) Let π : N → N be an arbitrary permutation. Let a = (a1 , ..., an ) ∈ A be an arbitrary allocation. We say that the allocation aπ is the bundle-assignment based on a and π, if the bundle allocated to bidder i in aπ is exactly the bundle ak , where π(k) = i. Definition 5 (Locally-Efficient Bundle Assignment) An allocation a = (a1 , ..., an ) is said to be locally-efficient bundle assignment with respect to v = (v1 , ..., vn ), if for every permutation π : N → N it holds that: Ψ(v, a) ≥ Ψ(v, aπ ). 3 We consider direct revelation mechanisms. However, the bidders in our setting are not strategic, they always report their true valuations. Observe also that we use the notion of ”achievability” rather than the notion of ”implementability”.

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For instance, f ∗ which maximizes the social-welfare f ∗ (v) ∈ argmaxa∈A Ψ(v, a), and f ∅ that always allocates empty bundles, clearly produce locally-efficient bundle assignments. Consider the following 2 bidders and 2 identical items setting with: v1 (1) = v1 (2) = 1.5, v2 (1) = v2 (2) = 2. The allocation a in which bidder 2 gets both items and bidder 1 gets the empty bundle is a locallyefficient bundle assignment. Similarly, the allocation a$ in which every bidder gets exactly one item is also a locally-efficient bundle assignment. Additionally, since bundles cannot be splited, a$ (= aβ ! for every β. However, there is β $ such that a = a$β .

2.3

Characterizing Envy-Freeness

We now state our characterization of envy-free bundle pricing mechanisms. More formally, we shall show that a social choice function is envy-free achievable if and only if it always produces locallyefficient bundle assignments. Our proof relies on a convex analysis proof technique [20, 15, 9] (see also [2]). However, despite this similarity, the notion of envy-free achievability is different from the notion dominant-strategy implementability (see subsection A.1). Interestingly, it is rather simple to show that envy-free achievability implies local efficiency. In contrast, it is not straightforward to see how Rochet’s cycle-monotonicity condition that characterizes dominant-strategy implementability implies global efficiency [12]. Theorem 1 A deterministic social choice function f : V → A is envy-free achievable if and only if the allocation f (v) is a locally-efficient bundle assignment w.r.t. v, for every v ∈ V . Lemma 1 If f is envy-free achievable then f (v) is a locally-efficient bundle assignment w.r.t. v. proof: Suppose f is envy-free achievable. Denote by a ∈ A the allocation f outputs for v. Let π : N → N be an arbitrary permutation. By achievability of f , there exists a payment function p such that: vi (ai ) − pi (v) ≥ vi (aπ(i) ) − pπ(i) (v), for every i ∈ N . By rearranging we get: vi (ai ) − vi (aπ(i) ) ≥ pi (v) − pπ(i) (v). The local-efficiency then follows from summing the inequalities over all bidders. Formally: Ψ(v, a) − Ψ(v, aπ ) = Σi∈N vi (ai ) − Σi∈N vi (aπ(i) ) ≥ Σi∈N pi (v) − Σi∈N pπ(i) (v) = 0. Lemma 2 If f (v) is a locally-efficient bundle assignment w.r.t. v, for every v ∈ V then f is envy-free achievable. Our proof is constructive. Let v be an arbitrary valuation tuple. Denote by a ∈ A the allocation f outputs for v. In order to describe the envy-free payment scheme we define the following finite → − directed graph Gf,v , with the vertices V (Gf,v ) = {1, 2, ..., n} and the edges E (Gf,v ) = {(i, k) | i, k ∈ V (Gf,v ), i (= k}. That is, each vertex of the graph corresponds to a bidder and the directed graph is complete (there are directed edges from i to k and from k to i, whenever i and k are arbitrary distinct vertices of the graph). Finally, the length of the directed edge (i, k) is defined as l(i, k) = vi (ai ) − vi (ak ). Definition 6 (Canonical Payment) Fix an arbitrary bidder j ∈ N . The Canonical Payment (w.r.t. bidder j) for bidder i is pi (v) = ∆i,j , where ∆i,j is the length of the shortest path from vertex i ∈ N to vertex j in the directed graph Gf,v . 4

Observe that not every envy-free payment scheme is canonical. Suppose we have two players and two items ia and ib and v1 (ia ) = 1, v1 (ib ) = 0, v2 (ia ) = 0, v2 (ib ) = 2. If bidder 1 gets item ia and bidder 2 gets ib , then the payment of zero for each item is envy-free. The canonical prices are p(ia ) = 0, p(ib ) = 2. (w.r.t. bidder 1) and p(ia ) = 1, p(ib ) = 0 (w.r.t. bidder 2). proof: (of lemma) We shall verify that the canonical payment induces the envy-free achievability of f . Fix an arbitrary bidder, without loss of generality bidder n. Clearly ∆i,n ≤ l(i, n) < ∞. We have to show that p is well defined (∆i,n > −∞) and that M (f, p) is an envy-free mechanism. By local-efficiency of f we get that there’s no negative length directed cycle in the graph Gf,v . Suppose not. If the negative length cycle is not a simple one then it can be decomposed into simple cycles (in the sense that each vertex appears at most once in each of the simple cycles) such that at least one of them is negative. Assume without loss of generality that (1, 2, 3) is a simple negative directed cycle. That is l(1, 2) + l(2, 3) + l(3, 1) < 0. Define the permutation π(1) = 2, π(2) = 3, π(3) = 1 and π(i) = i for i > 3. Now: l(1, 2) + l(2, 3) + l(3, 1) = Ψ(v, a) − Ψ(v, aπ ) < 0, contradicting the local-efficiency of f . It is a well known fact that if the graph has no simple negative cycles then ∆i,n > −∞, for every i ∈ N . We show now that bidder i cannot envy bidder k (= i. Suppose not, then vi (ai ) − ∆i,n < vi (ak ) − ∆k,n . By rearranging we get that: l(i, k) + ∆k,n = vi (ai ) − vi (ak ) + ∆k,n < ∆i,n . The left-hand side represents a length of a direct path from i to n (through k) which is smaller than ∆i,n , the length of the shortest path from i to n, a contradiction to the minimality of ∆i,n .

2.4

More About Canonical Payments

We first see that the canonical payments ensure that bidders who declare their true valuation will always get non-negative utilities (individual rationality). We then consider a poly-time computable decision problem involving canonical payments. Finally we show that any normalized revenuemaximal envy-free payment (see definition below) must be canonical. All together this showcases the applicability of anonymous bundle pricing in general and canonical payments in particular: from both the point of view of fairness design and computational efficiency. Proposition 1 Suppose f is envy-free achievable. Let p be canonical payment. Then for every v ∈ V and f (v) = a it holds that: vi (ai ) ≥ pi (v) (Individually Rationality). Furthermore, if the number of items is strictly less than the number of bidders, then there is a canonical payment (w.r.t. some j ∈ N ) such that pi (v) ≥ 0 for every i ∈ N . proof: By using canonical payment w.r.t. j, we get that: vi (ai ) ≥ vi (ai ) − vi (aj ) = l(i, j) ≥ ∆i,j = pi (v). This shows the individual rationality. Now, if |U | < n, then there always exists a bidder that gets the empty bundle. Without loss of generality assume it is bidder j. Since ∆j,j = 0, then clearly pj (v) ≥ 0. For any i (= j: we get that 0 = vj (∅) = vj (aj ) ≤ vj (ai ). This implies that l(j, i) ≤ 0. Now, pi = ∆i,j ≥ ∆i,j + l(j, i) ≥ 0, since all cycles in Gf,v are nonnegative. Proposition 2 Given any valuation v ∈ V and any allocation a ∈ A it can be determined in polynomial-time whether supporting envy-free bundle prices exist (assuming an oracle access to v with value queries). Furthermore, such prices if exist can be computed in polynomial time. 5

¯ v,a , with the set of proof: (sketch) For every given (a, v), consider the complete bipartite graph G vertices {1, 2, ..., n} and {a1 , a2 , ..., an }, while the edge (i, ak ) has the weight vi (ak ). Computing a ¯ v,a can be done in polynomial-time (e.g., [19]). The maximum weighted bipartite matching a∗ for G decision algorithm outputs ’yes’ if and only if Ψ(a, v) = Ψ(a∗ , v). Clearly, if Ψ(a, v) = Ψ(a∗ , v), then a is locally-efficient bundle assignment. Now, a canonical payment can be derived by a polynomial-time reduction to a shortest path problem and thus it is computationally efficient. Definition 7 The payments p(v) = p1 (v), p2 (v), ..., pn (v) are normalized if for every v and every i pi (v) ≥ 0, and there at least one agent k ∈ N such that pk (v) = 0. We say that p is maximal if p is normalized and for every normalized p$ it holds that Σi∈N pi ≥ Σi∈N p$i . Clearly, every envy-free payment can be mapped to an envy-free normalized payment (by adding a constant). Proposition 3 Let M (f, p∗ ) be an every free mechanism such that p∗ is a maximal payment. Then for every v there exists j ∗ = j ∗ (v) such that p∗ is the canonical payment with respect to j ∗ . Lemma 3 Let M (f, p) be an envy free mechanism, then for every i, j, v: pi (v) − pj (v) ≤ ∆i,j . proof: Assume without loss of generality that the shortest path from i to j in the graph Gf,v is i, i + 1, i + 2, i + 3, ..., j − 1, j. Now, by the envy-freeness we get that: vi (ai ) − pi (v) ≥ vi (ai+1 ) − pi+1 (v), vi+1 (ai+1 ) − pi+1 (v) ≥ vi+1 (ai+2 ) − pi+2 (v), ..., vj−1 (aj−1 ) − pj−1 (v) ≥ vj−1 (aj ) − pj (v), where f (v) = a. By adding all these inequalities and rearranging, we get that ∆i,j = vi (ai ) − vi (ai+1 ) + vi+1 (ai+1 ) − vi+1 (ai+2 ) + · · · + vj−1 (aj−1 ) − vj−1 (aj ) ≥ pi (v) − pj (v).

proof: (of proposition) p∗ is maximal, and in particular is normalized. Fix an arbitrary v ∈ V . Let k = k(v) be a bidder such that p∗k (v) = 0. We argue that p∗ must be a canonical payment w.r.t. k. By lemma 3, p∗i (v) − p∗k (v) ≤ ∆i,k . Or equivalently, p∗i (v) ≤ ∆i,k . By the fact that p∗ is normalized we get that 0 ≤ p∗i (v) ≤ ∆i,k , for every i ∈ N . This shows that the canonical payment w.r.t k is normalized (recall that ∆k,k = 0). Suppose p∗ is not the canonical payment w.r.t. k. It must be the case that there is I ∈ N such that p∗i (= ∆i,k for every i ∈ I (= ∅. In particular, we get that 0 ≤ p∗i (v) < ∆i,k , for every i ∈ I. All together we get that Σi∈N p∗i < Σi∈N ∆i,k , contradicting the maximality of p∗ .

2.5

Cost Minimization Problems

Our characterization is stated for the case each bidder is facing a maximization problem and wishes to maximize his value. Our results apply also for cost minimization problems, for which the bidders would like to minimize their costs. Technically, the inequality is reversed to the other direction in the definition 5 when considering cost minimization settings.

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The Envy-Free Approximability of Unrelated machines

We start with the formal setting [13, 17]. We then prove that not only is it impossible to minimize 1 the makespan in an envy-free manner, but that any approximation ratio better than 2− m cannot be achieved by any envy-free deterministic mechanism. We then present an envy-free mechanism that achieves m+1 2 -approximation. Our mechanism make use on the optimal allocation w.r.t. makespan, and thus is not computationally efficient. For m = 2 our result is tight. 6

3.1

The Setting

The unrelated machine scheduling setting (R||Cmax ) is a special case of the combinatorial auction setting. There are k tasks 1, ..., k that are to be scheduled on m machines 1, ..., m.4 Every machine i is a bidder with a nonnegative valuation function vi (). Formally, vi ({j}) (or simply vi (j)) specifies the cost of task j on machine i. One can think of the cost of task j on machine i as the time it takes i to complete j. For every S ⊆ [k], vi (S) = Σj∈S vi (j). That is, the total cost of a set of tasks S on machine i is the additive sum of the costs of the individual tasks on that machine. In the unrelated meachines setting these costs can be arbitrary (every (k · m)-tuple of non-negative costs is feasible), and thus it is a multi-dimensional scheduling problem. Let a ∈ A be an arbitrary allocation of tasks to the machines (”scheduling”). Let r(a, v) = max{v1 (a1 ), v2 (a2 ), ..., vm (am )}, be the load of the most loaded machine. To simplify notation we shall use the notation r(a) instead of r(a, v), when v is clear from the context. The global goal is minimizing the makespan. I.e., it is a minmax goal: assign the tasks to machines so that the last task finishes as soon as possible (each task is assigned to exactly one machine). Formally, fix an arbitrary v ∈ V . The allocation ! a is optimal with respect to the makespan if r(! a) ≤ r(a) for every a ∈ A. Let f be a social choice function. If r(f (v)) ≤ c · r(! a) for every v ∈ V then we say that f is a c-approximation algorithm (c ≥ 1 might be a constant or any function of v). A mechanism M (f, p) is called a c-approximation mechanism if f is a c-approximation.

3.2

Lower Bound

Theorem 2 Any envy-free mechanism cannot achieve an approximation ratio better than 2 − with respect to the makespan.

1 m

proof: Suppose not. Let M (f, p) be a deterministic envy-free mechanism that achieves an approx1 imation factor 2 − m − !. Let !$ < !. We shall consider two cases. For the case m = 2, consider the following valuation instance with two tasks: v1 (1) = 1, v1 (2) = 0.5, v2 (1) = 1.5 − !$ , v2 (2) = 1 (see the matrix below): " # 1 0.5 1.5 − !$ 1

The first line represents the costs of running the first task on the first and the second machine, respectively. Similarly, the second line represents the possible costs of the second task on each of the machines. Clearly, there are exactly 4 possible allocations. The optimal allocation w.r.t. the makespan assigns the first task to the first machine, and the second task to the second machine. All together the makespan is 1. However, this allocation is not a locally-efficient bundle assignment. One can easily verify that: 2 = v1 (1) + v2 (2) > v1 (2) + v2 (1) = 2 − !$ (recall that this is a cost minimization setting). Any other allocation has a makespan ≥ 1.5 − !$ > 1.5 − !, a contradiction. 4

We chose m to be the number of machines to be consistent with the formulation of Nisan and Ronen. Recall that we used n previously to denote the number of bidders, whereas here the bidders are the machines.

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For m ≥ 3, consider the following matrix (for the ease of exposition we assume m = 4):   1 1 1−m ∞ ∞     1  ∞ 1 1−m ∞        1   ∞ ∞ 1 1−m         2 − 1 − !$ ∞ ∞ 1 m

First, consider the allocation b in which task i is assigned to machine i. The makespan of this allocation is 1. Thus the optimal makespan must be ≤ 1. However, it cannot be strictly smaller than 1, since the cost of the last task on each machine is at least 1. We get that b is the optimal allocation w.r.t. the makespan. However, b is not a locally-optimal bundle assignment (compare to 1 1 the assignment marked with italics in the matrix above) as m > (m−1)(1− m )+2− m −!$ = m−!$ . 1 It is easy to verify that every other allocation has approximation ratio strictly larger than 2− m −! w.r.t. to the makespan, a contradiction.

3.3

Upper Bound

In general the optimal allocation with respect to the makespan is not locally efficient. We will thus carefully modify it. In order to describe the algorithm, we need the following: Definition 8 (The Function β ∗ ) Let a = (a1 , ..., am ) ∈ A be an arbitrary allocation. Define the function β ∗ : [m] → [m] as follows. Let β ∗ (j) ∈ argmin i=1,...,m vi (aj ), j ∈ [m]. That is, β ∗ (j) is the machine with the minimal cost for the bundle aj (breaking ties arbitrarily). ∗

Intuitively, aβ is defined by independent ”bundle-auctions”: in each step j = 1..m we allocate the bundle aj to the lowest cost machine for this bundle (independently of the history of former steps). ∗

Fact 1 If the valuation of each machine is additive, then b = aβ is a locally-efficient bundle assignment. That is, for every permutation π it holds that: Ψ(v, b) ≤ Ψ(v, bπ ). Definition 9 (The Permutation π ∗ ) Let a = (a1 , ..., am ) ∈ A be an arbitrary allocation. Define ∗ π ∗ to be a permutation such that aπ is locally-efficient bundle assignment. If there is more than one permutation, then arbitrarily choose one. ∗

∗

Clearly, aπ (= aβ in general. We now describe our main algorithm. Algorithm 1 (Bundle-Local-Search) Input: v = v1 , v2 , ..., vm . Let ! a be the optimal allocation with respect to the makespan of v. ∗

• If the makespan of ! a π is at most ∗

• Otherwise, output ! aβ .

m+1 2

∗

times the makespan of ! a, then output ! aπ . 8

Theorem 3 Algorithm bundle-local-search guarantees an approximation ratio of m+1 with respect 2 to the makespan. Moreover, algorithm bundle-local-search is envy-free achievable. All together we get that there exists an envy-free m+1 2 -approximation mechanism for minimizing the makespan on unrelated machines. proof: Algorithm bundle-local-search always outputs a locally-efficient bundle assignment and thus by theorem 1 is envy-free achievable. ∗ ∗ If the algorithm outputs the allocation ! a π , then clearly r(! a π ) ≤ m+1 a). Otherwise, 2 · r(! there exists a high loaded machine (without loss of generality machine 1) such that: ∗

∗

v1 (! a π ) = r(! aπ ) >

m+1 · r(! a). 2

That is, machine 1 determines the makespan, and based on the fact that the output of algorithm ∗ ∗ is ! a β we get that the makespan of ! a π is quite large. Additionally, ∗

All together:

∗

a π ) ≤ Ψ(v, ! a) ≤ m · r(! a). Ψ(v, ! a β ) ≤ Ψ(v, ! ∗

Ψ(v, ! a β ) ≤ m · r(! a) −

m+1 m+1 · r(! a) + r(! a) = · r(! a). 2 2

∗

∗

Intuitively, to get ! a β we first auctioned the bundle of machine 1 in the allocation ! a π among the bidders. By our construction we know that there exists a bidder with cost at most r(! a) for this bundle. In the worst case, all bundles will be allocated to the same machine. Hence we have the following: ∗

∗

r(! a β ) ≤ Ψ(v, ! aβ ) ≤

m+1 · r(! a). 2

Remark 1 Theorem 3 describes an envy-free m+1 2 -approximation mechanism. It is not a polynomialtime computable mechanism since algorithm bundle-local-search is based on the optimal allocation w.r.t. makespan. Observe that this is the only reason. Suppose that we are given an oracle access to the optimal allocation w.r.t. makespan for every v. Then all other steps in algorithm bundlelocal-search and in the computation of the associates envy-free canonical payments (proposition 2) can be done in polynomial-time.

4

Near-Optimality of Related Parallel machines

In this section we consider an interesting case in which the envy-freeness constraint do not impose a computational burden. Specifically, we consider minimizing the makespan of related parallel machines in an envy-free manner. We show that there exists a polynomial-time procedure that takes an allocation and (1) converts it to an ”envy-free” allocation without any loss in the approximation ratio, and (2) specifies supporting envy-free bundle prices. All together, based on the deterministic PTAS by Hochbaum and Shmoys [11] we can show that there exists a polynomial-time envy-free (1 + !)-approximation mechanism for every fixed !. We shall start by the formal setting. The related parallel machine scheduling setting (Q||Cmax ) is a special case of the unrelated model (R||Cmax ). In this model each task j has a load lj > 0. Additionally, every machine i is 9

a bidder with type ti . It takes ti · lj time units to perform task j on machine i. More formally, vi ({j}) = ti · lj . The total cost of a set of tasks on machine i is the additive sum of the costs of the individual tasks on that machine. In this model we can compare between meachines: if machine i1 is faster than machine i2 on task j, then machine i1 is always faster than machine i2 . For convenience we use the notation l(S) = Σj∈S lj , to denote the total load of a subset of tasks S. We also assume without loss of generality that t1 ≥ t2 ≥ · · · ≥ tm . Claim 1 Let f be a deterministic c-approximation algorithm. We can assume without loss of generality that f always allocates more load to the fastest machines. proof: We can reassign the loads in a sorted manner. Formally, let f (v) = a. We will give the fastest machine m the subset of task ai with the highest load argmaxi=1,...,m l(ai ), and machine m − 1 the second highest loaded subset of tasks, and so on. First, this reassignment can be done in polynomial-time. Second, we need to show that the resulted allocation a$ (after switching the assigned loads) is a c-approximation. For the two machine case m = 2: if a (= a$ then clearly r(a) = t1 · l(a1 ). However if we exchange the loads between the machines then: r(a$ ) = max{t1 · l(a2 ), t2 · l(a1 )} ≤ r(a). For m > 2, we can sort a as in the bubble-sort algorithm (”2 machines at each step”), and in each step the makespan can only improve. The next theorem describes directly the bundle pricing that induce the envy-freeness. Theorem 4 Any c-approximation deterministic algorithm w.r.t. makespan for related parallel machines model can be converted in polynomial-time to a c-approximation envy-free mechanism using the following payments paid by the mechanism to each of the machines: p1 (v) = l(a1 ) · t1 , and for i = 2, ..., m: pi (v) = pi−1 (v) + (l(ai ) − l(ai−1 )) · ti , where f (v) = a. proof: We first show that machine i cannot envy a faster machine d. Indeed, pi (v) − li (ai ) · ti is greater than pd (v) − li (ad ) · ti by the following: li (ad ) · ti − li (ai ) · ti ≥ (l(ai+1 ) − l(ai )) · ti+1 + · · · + (l(ad ) − l(ad−1 )) · td ≥ pd (v) − pi (v). Similarly, machine i cannot envy a slower machine d$ . Indeed, pi (v) − li (ai )ti is greater than pd! (v) − li (ad! )ti , by the fact that: pi (v) − pd! (v) ≥ (l(ad! +1 ) − l(ad! )) · td! +1 + · · · + (l(ai ) − l(ai−1 )) · ti ≥ li (ai ) · ti − li (ad! ) · ti . Based on the deterministic PTAS by Hochbaum and Shmoys [11] and theorem 4 we can show the following: Theorem 5 There exists a polynomial-time computable envy-free (1+!)-approximation mechanism for minimizing the makespan on related parallel machines for every fixed !.

Acknowledgement I would like to thank Liad Blumrosen, Federico Echenique, David Kempe, John Ledyard, Mohamed Mostagir and Mahyar Salek for helpful discussions. 10

References [1] Aaron Archer and Eva Tardos. Truthful mechanisms for one-parameter agents. In FOCS, pages 482–491, 2001. [2] Christopher P. Chambers and Federico Echenique. Profit maximization and supermodular technology. Economic Theory. Forthcoming. [3] Ning Chen, Arpita Ghosh, and Sergei Vassilvitskii. Optimal envy-free pricing with metric substitutability. In EC, 2008. [4] G. Christodoulou, E. Koutsoupias, and A. Vidali. A lower bound for scheduling mechanisms. In SODA, 2007. [5] George Christodoulou, Elias Koutsoupias, and Angelina Vidali. A characterization of 2-player mechanisms for scheduling. In ESA, 2008. [6] Peerapong Dhangwatnotai, Shahar Dobzinski, Shaddin Dughmi, and Tim Roughgarden. Truthful approximation schemes for single-parameter agents. In FOCS, 2008. [7] Benjamin Edelman, Michael Ostrovsky, and Michael Schwarz. Internet advertising and the generalized second-price auction: Selling billions of dollars worth of keywords. American Economic Review, 97(1):242–259, March 2007. [8] D. Foley. Resource allocation and the public sector. Yale Economics Essays, 7:45–98, 1967. [9] Hongwei Gui, Rudolf Muller, and Rakesh Vohra. Characterizing dominant strategy mechanisms with multi-dimensional types, 2004. Working paper. [10] Venkatesan Guruswami, Jason D. Hartline, Anna R. Karlin, David Kempe, Claire Kenyon, and Frank McSherry. On profit-maximizing envy-free pricing. In SODA, pages 1164–1173, 2005. [11] Dorit S. Hochbaum and David B. Shmoys. A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach. SIAM J. Comput., 17(3):539– 551, 1988. [12] Ron Lavi, Ahuva Mu’alem, and Noam Nisan. Two simplified proofs for roberts’ theorem, 2004. Submitted. [13] Jan Karel Lenstra, David B. Shmoys, and Eva Tardos:. Approximation algorithms for scheduling unrelated parallel machines. In FOCS, 1987. [14] Richard J. Lipton, Evangelos Markakis, Elchanan Mossel, and Amin Saberi. On approximately fair allocations of indivisible goods. In EC, 2004. [15] Dov Monderer. Monotonicity and implementability. In EC, 2008. [16] R. B. Myerson. Optimal auction design. Mathematics of Operation Research, 6:58–73, 1981. [17] Noam Nisan and Amir Ronen. Algorithmic mechanism design. Games and Economic Behavior, 35:166–196, 2001. 11

[18] Noam Nisan, Tim Roughgarden, Eva Tardos, and Vijay V. Vazirani (eds.). Algorithmic Game Theory. Cambridge University Press, expected 2007. [19] Christos H. Papadimitriou and Kenneth Steiglitz. Combinatorial Optimization: Algorithms and Complexity. Dover Publications, 1998. [20] J.C. Rochet. A necessary and sufficient condition for rationalizability in a quasi-linear context. Journal of Mathematical Economics, 16:191–200, 1987. [21] Hal R. Varian. Position auctions. International Journal of Industrial Organization, 25(6):1163– 1178, December 2007.

A

Appendix

A.1

Envy Free vs. Incentive Compatibility

It is well known that envy-freeness and truthfulness are not straightforwardly connected.5 Specifically, truthfulness do not imply envy freeness in general, and vice versa. For the sake of completeness we give two simple examples to demonstrate this. We consider a setting with 2 bidders and 2 identical items. Let f t be the following social choice function: bidder 1 wins exactly one item if v1 (1) ≥ 1, and bidder 2 wins exactly one item if v2 (1) ≥ 4. Clearly, f t is truthful achievable with the non-anonymous item prices p1 = 1, p2 = 4. However, f t is not locally-efficient bundle assignment for every v: e.g. if v1 (1) = 2, v2 (1) = 3 then only bidder 1 gets an item. However, the allocation in which bidder 2 gets the item is more efficient (and so f t is not envy-free achievable). Now, let f e be the following social choice function: If v1 = v2 then both item allocated to bidder 2. Otherwise, bidder i wins exactly one item if vi (1) ≥ 4, i = 1, 2. Clearly, f e is envy-free achievable. However, f e is not value monotone [16] and thus it is not truthful.

A.2

Open Problems

The first open problem is to close the gap between the upper bound of m+1 and the lower bound 2 1 of 2 − m for the best approximation ratio achievable by any envy-free mechanism for the unrelated machine model, m > 2. Furthermore, does the envy-freeness constraint put a further computational burden? What is the best possible polynomial-time computable upper bound? Is it true that (R||Cmax ) cannot be approximated in polynomial-time within a factor less than c in an envy-free manner, for some c >> 32 [13]?

5

Essentially a mechanism M (f, p) is truthful if declaring the true valuation is a dominant strategy of every player (see [18] for exact definition).

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