Fair by Design: Multidimensional Envy-Free
Mechanisms
Ahuva Mu’alem∗
December 25, 2013
Abstract
The model we present addresses the following common scenario: a group of agents
wants to divide a set of items fairly, and at the same time seeks to optimize a global
goal. In this paper we focus on a natural task-scheduling scenario, in which each item
is a task and we want to find an allocation that minimizes the completion time of
the last task in an envy-free manner, where no individual agent prefers anyone else’s
allocated task bundle over its own. In the scheduling literature, this optimization goal
is called makespan minimization, and the agents are treated as machines. We give
tight deterministic bounds on the approximation factors achievable for the following
standard scenarios: (1) two unrelated machines and (2) m ≥ 2 related machines.
A natural question to ask is whether envy-free pricing techniques can improve the
current approximability and inapproximability bounds for truthful mechanisms for the
task-scheduling problem studied in the seminal paper of Nisan and Ronen [26]. Here,
we present several bounds for envy-freedom in-expectation, and give a partial answer to
this question. We find that in multidimensional settings for two unrelated machines,
truthful in-expectation is a far stronger constraint (i.e. more restrictive) than envy-free
in-expectation.
1
Introduction
Traditionally fairness analysis focuses on the individual performance each participant receives, while allocation algorithms consider optimizing overall criteria. This paper formulates
the fair by design approach to combine these two point of views.
Consider a project with different tasks to be assigned to a heterogeneous group of employees. As a motivating scenario assume that Alice is a technician who specializes in repairing
antennas and Bob specializes in repairing battery charges. Suppose that Carol the customer
∗
Software Engineering Dept. Ort Braude College of Engineering, Karmiel, Israel. This research was done
while the author was a Post-Doctoral Fellow at the Social and Information Sciences Laboratory, California
Institute of Technology. Email: [email protected], Web: sites.google.com/site/ahuvamualem
1
has two antennas and two battery chargers for repair. Consider two possible allocations: (1)
the allocation in which Alice receives two antennas and Bob receives two battery charges;
(2) the allocation in which each technician receives an antenna and a battery charger. If the
goal of the manager is to complete the repair as soon as possible to please Carol, then the
first allocation is preferable. However, if the manager’s goal is to expand the expertise of
Alice and Bob then the second allocation is preferable. Observe that both allocations are
fair from the point of view of Alice and Bob. This is not the case in general.
A natural challenge is determining a fair allocation such that the last task completes as
soon as possible. In general, no such allocation may exist if the tasks are indivisible. For
instance, in a project with a single task, the fastest employee should be assigned the task.
However, this allocation would not be considered fair from the perspective of the fastest
employee. This suggests that some (additional) reward should be allowed to guarantee a
fair division of the tasks. It is convenient to assume that rewards are granted in the form
of monetary payments. In the scheduling literature, the above optimization goal is called
makespan minimization of unrelated machines. In the economics literature, an allocation
algorithm coupled with a price function is called a mechanism.
We treat each machine as a distinct economic agent and say that an allocation is envyfree if no agent would prefer to exchange its assigned set of tasks with those of any other
agent. Informally, a heterogeneous group of machines is called unrelated in the scheduling
literature and multidimensional in the economics literature, while a homogeneous group is
called related and single-dimensional, respectively.
The design of mechanisms for our general problem can be regarded as solving an optimization problem with binding envy-freedom constraints. However, even if we allow monetary
payments, it could be that no feasible solution exists: we are faced with an inherent clash
between global optimization goals and envy-free pricing constraints (regardless of any computational considerations). Thus the need to consider approximations motivates the following
definition: we say that an allocation a is a ρ-approximation if the makespan of allocation a is
no more than a factor of ρ times the makespan of an optimal allocation, where ρ ≥ 1.1 Our
main question is how well can fundamental global goals be approximated in an envy-free
manner?
Relation to Truthful Mechanisms
The design of envy-free mechanisms is intimately connected to the well-studied class of
truthful mechanisms [8, 10, 30, 15]. A mechanism is truthful if no agent can ever improve
its utility by misreporting his valuation or cost. In many interesting settings, truthful mechanisms are essentially equivalent to mechanisms that select envy-free allocations with the
1
Additionally, if we consider a maximization problem (such as profit maximization) we say that an
allocation a is a ρ-approximation if the value of allocation a is at least a factor of ρ1 times the value of an
optimal allocation, where ρ ≥ 1.
A fair-design problem exhibits an upper bound of ρU and a lower bound of ρL if there exists an envy-free
ρU -approximation mechanism and if an envy-free (ρL − ²)-approximation mechanism is impossible for every
² > 0, respectively. If ρU = ρL we say that the bounds are tight.
2
smallest supporting price vectors [8].
The seminal paper of Nisan and Ronen [26] initiates the study of truthful mechanisms in
computerized settings, examining the problem of how well the makespan goal on unrelated
machines can be approximated. They showed that truthful mechanisms for two machines
exhibit a tight bound of 2 [26], and they circumvent this impossibility result by employing
randomization.
In a truthful in-expectation mechanism, each bidder prefers to truthfully report his value
to the mechanism since this gives him higher expected utility. Nisan and Ronen presented a
truthful in-expectation mechanism for two machines that exhibits an upper bound of 74 (their
upper bound was later improved to 1.5963 by [22]). The lower bound known for truthfulness
in-expectation is 32 [24].
As tightening the bounds for unrelated machines is a central problem of algorithmic
mechanism design, this raises the question of whether envy-free techniques can tighten the
current upper and lower bounds (1.5963 vs. 32 ) known for truthful in-expectation mechanisms
for two unrelated machines.
In this paper we study envy-free in-expectation mechanisms to minimize the makespan.
We present a 43 -approximation envy-free in-expectation mechanism for two unrelated machines. By the simple fact that the envy-free in-expectation upper bound of 43 is smaller than
the truthful in-expectation lower bound of 32 [24], we get that our 34 -approximation envy-free
in-expectation mechanism is not truthful in-expectation. Intuitively, this suggests that envyfree bounding techniques cannot be applied straightforwardly to tighten the current truthful
in-expectation bounds for minimizing the makespan on two unrelated machines. We conclude
that in multidimensional settings, truthful in-expectation is a far more restrictive constraint
than envy-free in-expectation.
Overview and Results
In this paper we study two canonical objectives over multidimensional domains: profitmaximizing combinatorial auctions for general bidders and makespan-minimizing scheduling
for unrelated machines.
We start by formally defining the notion of an envy-free allocation mechanism. In Section
2, we briefly state a known characterization of envy-free allocation mechanisms in terms of
locally-efficient bundle assignments [13]. Importantly, this characterization does not involve
price functions.
In Section 3, we study envy-free combinatorial auctions for general bidders. In this
scenario, a profit-maximizing auctioneer has a collection of items for sale, and bidders
compete for subsets of items. Envy-free prices can be interpreted as anonymous nondiscriminatory prices. We describe √
an envy-free mechanism that requires polynomial communication and achieves (min{n, O( k log(min{k, n}))})-approximation with respect to the
maximum profit, where k is the number of items and n is the number of bidders. On the
negative side, we show that any envy-free profit-maximizing mechanism with approximation
ratio strictly better than n requires exponential communication.
In Section 4, we study envy-free scheduling mechanisms. There are k tasks that are to
3
be scheduled on m 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. This canonical optimization
problem was extensively studied by [20].
We consider minimizing the makespan in the context of envy-free design. Specifically,
using the characterization reviewed in Section 2, we derive general bounds on the approximability of deterministic envy-free mechanisms that seek to minimize the makespan on
unrelated machines. We exhibit a lower bound of 2 − m1 of the best approximation ratio
achievable by any deterministic envy-free mechanism for m unrelated machines. We present
a deterministic ( m+1
+ ²)-approximation envy-free mechanism. This mechanism is polyno2
mial time computable when m is a constant. Observe that for m = 2 we obtain a tight
result of ( 32 + ²). Similar deterministic upper and lower bounds for unrelated machines were
achieved independently in a recent work by Hartline et al. [14].
In section 5, we focus on the case of two unrelated machines and show that envy-free
in-expectation mechanisms are more powerful than randomized envy-free mechanisms. Informally, a randomized envy-free mechanism specifies a probability distribution over deterministic envy-free mechanisms. While an envy-free in-expectation mechanism essentially specifies
a probability distribution over allocations (so that each agent cares about his expected load)
assuming that the agents are risk neutral. We first show that randomized envy-free mechanisms are not more powerful than their deterministic counterparts. That is, no randomized
envy-free mechanism for two unrelated machines can achieve an approximation ratio better
than the deterministic lower bound of 23 . To circumvent this impossibility we show a polynomial time computable ( 43 + ²)-approximation envy-free in-expectation mechanism for two
unrelated machines. We show that this mechanism is optimal under the assumption that
the mechanism is symmetric.
Section 6 focuses on related machines [16] which is a special interesting case of the
unrelated model. As opposed to our inapproximability results overviewed so far for the
multidimensional settings, we show that the envy-freedom constraint does not impose any
further burden if the machines are related. Specifically, there exists a polynomial-time
computable deterministic envy-free mechanism that achieves the approximation ratio of 1+²
with respect to the optimal makespan. Additionally, we characterize envy-free pricing in
single-dimensional environments.2
Related Work
The notion of envy-free allocations was introduced by Foley [11]. A recent survey on cakecutting and related models of divisible goods appears in [29].
Envy-free allocations without money of indivisible goods were studied from a computational point of view by Lipton et al. [21]. In this setting, envy-free allocations might not
2
The current paper supersedes "On Multidimensional Envy-Free Mechanisms" that appeared as an extended abstract in [23]. The current version includes a new section about envy-free in-expectation and
randomized mechanisms.
4
exist, and thus they consider approximations for the minimum possible envy.
Profit Maximization Envy-free profit maximization approximations for combinatorial
auctions were first studied by Guruswami et al. [12]. For unit-demand bidders with limited supply they showed an O(log n)-approximation algorithm, and an O(log n + log k)approximation for single-minded bidders with unlimited supply, where n is the number of
bidders and k is the number of distinct types of items.3 The latter result was extended to
an O(log n + log k)-approximation for general bidders with unlimited-supply, √
by Balcan et
al. [2]. Cheung and Swamy [5] used an LP-based technique to obtain an O( k log umax )approximation for single-minded bidders with limited-supply, where umax is the maximum
number of item supply. None of these papers studies profit-maximizing envy-free pricing of
combinatorial auctions for general bidders when supply is limited.
Unrelated Machines Makespan minimization is NP-hard even on identical machines [4,
27]. This fundamental scheduling problem was extensively studied by Lenstra, Shmoys and
Tardos [20]. They presented a 2-approximation polynomial-time algorithm for minimizing
the makespan of unrelated machines. They also showed that the problem cannot be approximated in polynomial-time within a factor of less than 32 . Horowitz and Sahni presented an
FPTAS4 for any fixed number of unrelated machines [17].
Recently, Cohen et al. [7] improved our deterministic results for envy-free makespan
minimization for m ≥ 3 unrelated machines. They presented a polynomial time computable
deterministic envy-free O(log m)-approximation mechanism. Additionally, they showed that
no envy-free mechanism can achieve a better bound than O( logloglogmm ).
A paper by Nisan and Ronen [26] defines the notion of algorithmic mechanism design [27].
In this paper each machine is treated as a strategic agent. They consider the unrelated
machine setting. Their paper proves that not only is it impossible to minimize the makespan
in a truthful manner, but that any approximation ratio strictly better than 2 cannot be
achieved by a truthful deterministic mechanism (their lower bound was later improved from
2 to 2.61 for m ≥ 3 by Koutsoupias and Vidali [18]). They also showed that there is a
computationally efficient truthful mechanism that achieves an approximation ratio of m. A
lower bound of 2 − m1 for truthful in-expectation mechanisms for unrelated machines was
given by Mu’alem and Schapira [24] while a randomized truthful upper bound of m+5
was
2
given by Lu and Yu [22], who also provide a randomized truthful upper bound of 1.5963 for
the two machine case [22].
Related Machines In this setting the type of each related machine can be described easily
by a single positive number (associated with its speed). Hochbaum and Shmoys describe a
3
A unit demand-bidder is essentially interested in every item, but would like to buy at most one item. A
single-minded bidder would like to buy a specific subset of items. In the unlimited-supply setting, the number
of copies of each item is as large as the number of bidders.
4
An FPTAS algorithm is an (1 + ²)-approximation algorithm that runs in polynomial time in the size of
the input and 1² .
5
PTAS for minimizing the makespan of related machines [16].5 This canonical problem was
first studied from an algorithmic mechanism design perspective in [1]. Archer and Tardos
designed a 3-approximation mechanism based on a randomized rounding of the optimal
fractional solution [1]. Recently, [9, 6] closed this gap and showed truthful mechanisms that
achieve an approximation ratio of 1 + ².
2
Characterizing Envy-Free Multidimensional Mechanisms
This section characterizes envy-free mechanisms. The characterization is stated in terms of
the local efficiency of the allocation rule. We begin with the general framework.
2.1
The Setting
We consider a finite set K of k indivisible items and a set N of n agents. We assume that
agents value combinations of items. Formally, each agent i ∈ N has a valuation function
vi () (or vi , for short) that describes its nonnegative valuation for each subset S of items, i.e.
vi (S) is the maximum finite amount of money agent i is willing to pay for S. A subset S of
items is sometimes called a bundle.
Every valuation vi ∈ Vi satisfies the following three conditions: (1) No externalities meaning that the valuation of agent i depends only on its allocated bundle. (2) 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 )). (3) Normalization - meaning that the value of the
empty bundle is always zero. The domain of all possible valuations of agent i is denoted by
Vi , where V = V1 × V2 × · · · × Vn .
An allocation a = (a1 , . . . , an ) is a partition of items among the agents, where ai denotes
the bundle allocated to agent i, a1 ∪ a2 ∪ · · · ∪ an ⊆ K (observe that not all items need to
be allocated), and ai ∩ aj = ∅, whenever i 6= j. The set of all possible allowed allocations is
denoted by A.
An allocation rule f : V → A maps an n-tuple of valuations v = (v1 , v2 , . . . , vn ) ∈ V
to an allocation a ∈ A. A mechanism specifies an allocation and a set of prices for every
possible valuation of the agents. Formally, a mechanism is a tuple M = (f, p1 , p2 , . . . , pn )
(or M (f, p), for short) where f is an allocation rule and the pricing function pi : V → R
assigns a price to each agent i ∈ N .
A mechanism M (f, p) is individually rational if agents always receive a nonnegative
utility. Formally, if for every v = (vP
1 , v2 , . . . , vn ) we have that vi (f (v)) − pi (v) ≥ 0. The
profit collected by the mechanism is ni=1 pi (v).
Definition 1 (Envy-Free Mechanism) Let M = (f, p) be a mechanism. Let i, j ∈ N
and let v = (v1 , . . . , vn ) ∈ V be any n-tuple of valuations. Denote by a ∈ A the allocation
5
A PTAS is an (1 + ²)-approximation algorithm that runs in polynomial time in the size of the input,
assuming ² is a fixed constant.
6
that f outputs for v. A mechanism M is said to be envy-free if for every pair of agents i, j
it holds that:
vi (ai ) − pi (v) ≥ vi (aj ) − pj (v).
We say that an allocation rule f : V → A is envy-free achievable if there exists a price
function p such that the mechanism M = (f, p) is envy-free.6
Example 1 Consider a mechanism for a single item with two agents, where the agent with
the highest value wins the item and pays the average of both values. The other agent pays
zero. By symmetry, it is enough to consider the case where v1 ≥ v2 . By the fact that
2
2
v1 − v1 +v
≥ 0 ≥ v2 − v1 +v
, we immediately obtain that agent 1 does not envy agent 2, and
2
2
vice versa. Therefore the mechanism is envy-free.
Definition 2 For arbitrary allocation a ∈ A and valuation v ∈ V, let
Ψ(v, a) = Σni=1 vi (ai ).
We call Ψ(v, a) the social welfare of the allocation a with respect to v.
2.2
Locally Efficient Allocations
In order to be able to state the characterization theorem we start with some definitions.
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 agent i in aβ is allocated all bundles ak with
β(k) = i.
We also consider the special case where the function β : N → N is a permutation.
Definition 4 (Locally-Efficient Bundle Assignment) An allocation a = (a1 , . . . , an ) is
said to be a 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 ).
Clearly, the allocation rule f ∗ (v) ∈ argmaxa∈A Ψ(v, a) which maximizes the social welfare, produces locally-efficient bundle assignments. Specifically, locally-efficient allocations
always exist.
6
The agents in our setting are non-strategic, they always report their true valuations. Observe also that
we use the notion of "achievable" rather than the notion of "dominant-strategy implementable".
7
2.3
Characterizing Envy-Freedom
We now state a known characterization of envy-free bundle pricing mechanisms in terms
of local efficiency by Haake et al. [13]. We provide the proof (based on a shortest path
argument) in the appendix.
Theorem 1 (Haake et al. [13]) A deterministic allocation rule f : V → A is envy-free
achievable if and only if the allocation f (v) is a locally-efficient bundle assignment with
respect to v, for every v ∈ V .
Intuitively, the above characterization allows us to focus on the allocation rule alone
in order to prove or disprove envy-free achievability rather than considering the interplay
between the allocation rule and the price function, Based on this characterization, the supporting prices can be calculated in polynomial time. More formally:
Claim 1 Given any allocation a ∈ A and values vi (aj ), i, j = 1..n, it can be decided in polynomial time whether supporting envy-free prices exist. Furthermore, if individually-rational
nonnegative envy-free prices exist they can be computed in polynomial time.
2.4
Cost Minimization Problems
The above characterization theorem is stated for value maximization problems (such as
profit maximization). It also applies to cost minimization problems (such as makespan
minimization).
The Setting. As before, we will still have a finite set K of k indivisible items and a set N of
n agents. Each agent i ∈ N now has a nonnegative cost function ci () ∈ Ci (or ci , for short)
that describes its cost incurred by each subset S of items.
A mechanism M (f, p) for a cost minimization setting is individually rational if agents
always receive a nonnegative utility. That is, if for every c = (c1 , c2 , . . . , cn ) we have that
−ci (f (c)) − pi (c) ≥ 0.
For notational simplicity, we define pbi (c) = −pi (c). In particular, pbi is called a reward
and it specifies the monetary transfer that agent i receives from the mechanism, while pi
refers to the price that agent i gives to the mechanism. Therefore, the nonnegative utility
requirement is equivalent to
pbi (c) − ci (f (c)) ≥ 0.
Similarly, the inequality in Definition 1 is equivalent to
pbi (c) − ci (ai ) ≥ pbj (c) − ci (aj ).
An allocation a = (a1 , . . . , an ) is said to be a locally-efficient bundle assignment with
respect to c = (c1 , . . . , cn ), if for every permutation π : N → N it holds that:
Ψ(c, a) ≤ Ψ(c, aπ ),
8
where Ψ(c, a) = Σni=1 ci (ai ) and Ψ(c, aπ ) = Σni=1 ci (aπi ).
Proposition 1 A deterministic allocation rule f : C → A is envy-free achievable if and
only if the allocation f (c) is a locally-efficient bundle assignment with respect to c, for every
c ∈ C. Furthermore, given any allocation a ∈ A and costs ci (aj ), i, j = 1..n, it can be decided
in polynomial time whether a supporting envy-free reward function pb exists. Additionally,
if individually-rational nonnegative envy-free reward function exists it can be computed in
polynomial time.
3
Profit-Maximizing Combinatorial Auctions
In this section we consider the problem of maximizing the auctioneer’s profit in a combinatorial auction. Here, we relate to the agents as bidders; each bidder can have a different
value for each bundle of items and thus it is a multidimensional setting. Envy-free prices
can be interpreted as anonymous non-discriminatory prices. We quantify the number of bits
required to be communicated to determine a profitable allocation.7
We show
√ an envy-free mechanism that requires polynomial communication and achieves
(min{n, O( k·log(min{n, k}))})-approximation with respect to the maximal envy-free profit.
However, we show that an envy-free mechanism with approximation ratio strictly better than
2 (with respect to the optimal profit) requires exponential communication in the worst case.
We then show that a similar impossibility result applies to approximation ratios strictly better than n, where n is the number of bidders. Based on the work of Guruswami et al. [12],
Lehmann et al. [19] and Nisan and Segal [28] we can state the main results of this section.
Proposition 2 Any envy-free profit-maximizing mechanism for Combinatorial Auctions that
achieves an approximation ratio better than 2 requires exponential communication.
Proposition 3 Any envy-free profit-maximizing mechanism for Combinatorial Auctions that
achieves an approximation ratio better than n requires exponential communication.
Proposition 4 There exists an envy-free
mechanism for combinatorial auctions for general
√
bidders which achieves a (min{n, O( k ·log(min{n, k}))})-approximation for maximizing the
profit and which requires polynomial communication.
4
Deterministic Envy-Free Approximability of Unrelated Machines
This section focuses on minimizing the makespan for unrelated machines. For this model we
prove that not only is it impossible to minimize the makespan in an envy-free manner, but
7
An introduction to communication complexity for combinatorial auctions can be found in [27, Chapter
11].
9
that any approximation ratio better than 2− m1 cannot
¡ m+1 be achieved
¢ by an envy-free deterministic mechanism. We then present an envy-free
· (1 + ²) -approximation mechanism.
2
Specifically, for m = 2 we get the tight deterministic result of 23 + ². Similar deterministic
upper and lower bounds for unrelated machines were achieved independently in a recent
work by Hartline et al. [14].
4.1
The Setting
The unrelated machine scheduling setting (R||Cmax ) is a special case of the combinatorial
auction setting: There are k tasks to be scheduled on m machines.8 Every machine i is an
agent with a nonnegative cost function ci (). Formally, ci ({j}) (or simply ci (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. 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. Formally, ci (S) = Σj∈S ci (j) for every S.
In the unrelated machines setting these costs can be arbitrary (every (k · m)-tuple of
nonnegative costs is feasible), and thus it is a multidimensional scheduling problem.
Let a ∈ A be an arbitrary allocation of tasks to the machines ("scheduling"), where
all tasks must be assigned, and each task is assigned to exactly one machine. The load of
machine i is its cost ci (ai ) = Σj∈ai ci (j).
The makespan denoted by r(a, c) is the maximum load of all machines, that is
r(a, c) = max{c1 (a1 ), c2 (a2 ), . . . , cm (am )}.
Given c and m, the makespan minimization problem involves finding an allocation a that
minimizes the term r(a, c). We shall use the notation r(a) instead of r(a, c), when c is clear
from the context.
4.2
A Deterministic Lower Bound
The optimal allocation with respect to makespan may not be envy-free achievable. In this
subsection, we will use the characterization theorem to prove a lower bound on the approximability of envy-free mechanisms.
Theorem 2 No envy-free mechanism for m unrelated machines can achieve an approximation ratio better than 2 − m1 with respect to the optimal makespan.
Proof: Suppose not. Let M (f, pb) be a deterministic envy-free mechanism that achieves an
approximation factor of 2− m1 −². Let ²0 < ². We shall consider two cases. For m = 2, consider
the following instance with two tasks: c1 (1) = 1, c1 (2) = 0.5, c2 (1) = 1.5 − ²0 , c2 (2) = 1,
(see the matrix below):
8
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 agents, whereas here the agents are the machines.
10
µ
1
0.5
0
1.5 − ² 1
¶
The first column represents the costs of running the first task on the first and the second
machines, respectively. Similarly, the second column represents the costs of the second task
on each of the machines.
Clearly, there exist exactly 4 possible allocations. The makespan of allocating the first
task to the first machine and the second task to the second machine is 1. However, this
allocation is not a locally-efficient bundle assignment and thus by Proposition 1 is not envyfree achievable, as one can easily verify that: 2 = c1 (1) + c2 (2) > c1 (2) + c2 (1) = 2 − ²0 (recall
that this is a cost minimization setting). Furthermore, any other allocation has a makespan
≥ 1.5 − ²0 > 1.5 − ²; this contradicts the assumption that M (f, pb) is a deterministic envy-free
mechanism that achieves an approximation factor of 1.5 − ².
For m ≥ 3, consider the following matrix


∞
∞ ···
∞
1
1 − m1




1


∞
1
1
−
∞
·
·
·
∞


m






..
..
..
.
.


.
.
.
.






1


∞
∞
·
·
·
∞
1
1
−
m 





 2 − m1 − ²0

∞
···
∞ ∞
1
First, consider the allocation b = ({1}, {2}, . . . , , {m}), where task i is assigned to machine
i. The makespan of this allocation is 1. Observe that the optimal makespan cannot be strictly
smaller than 1, since the cost of the first task on each machine is at least 1. Therefore, b is
an optimal allocation with respect to the makespan.
However, b is not a locally-efficient bundle assignment: To see this, let
bb = ({2}, {3}, {4}, . . . , {m}, {1})
be the assignment marked with italics in the above matrix. Now, Ψ(c, b) = m > (m − 1)(1 −
1
) + 2 − m1 − ²0 = m − ²0 = Ψ(c, bb).
m
Let a be an allocation with an approximation ratio of at most 2− m1 −². First, no machine
in a is assigned two or more tasks (since the sum of any two elements at each row is at least
2 − m1 ). Additionally, the first task must be assigned to the first machine (otherwise, the
makespan would be ≥ 2 − m1 − ²0 > 2 − m1 − ²).
Now, the second task must be assigned to the second machine (since assigning it to any
other machine would results in a makespan ≥ 2 − m1 ). Repeating the last argument, task i
must be assigned to the i0 th machine for every i = 3..m. This implies that a = b and that
11
every allocation 6= b has an approximation ratio strictly larger than 2 − m1 − ² with respect
to the optimal makespan. This contradicts the assumption that M (f, pb) is a deterministic
envy-free mechanism that achieves an approximation factor of 2 − m1 − ².
4.3
A Deterministic Upper Bound
In this subsection we show how to convert any ρ-approximation algorithm into an envy-free
( m+1
· ρ)-approximation mechanism. Applying the conversion technique to the deterministic
2
FPTAS for any fixed number of machines presented by Horowitz and Sahni [17] yields a
polynomial-time-computable
¡
¢deterministic envy-free mechanism that achieves an approximation ratio of m+1
·
(1
+
²)
with respect to the optimal makespan. This result is nearly
2
optimal for a small number of machines. Specifically, for m = 2 we get a tight result of 32 + ².
We first consider several procedures to reconfigure a given allocation into a locally-efficient
bundle assignment. Let [m] = {1, 2, . . . , m} denote the set of machines.
Definition 5 (The Function β ∗ ) Let a = (a1 , . . . , am ) ∈ A be an arbitrary allocation.
Define the function β ∗ : [m] → [m] as follows: β ∗ (j) ∈ argmin i=1,...,m ci (aj ), j ∈ [m]. That
is, β ∗ (j) is the machine with the minimal cost for the bundle aj (breaking ties arbitrarily).
∗
The allocation aβ can be constructed by the following "history-independent" procedure:
at step j = 1..m, re-assign bundle aj to a machine with the minimal cost for this bundle.
Note that this construction may result in a single machine receiving more than one bundle
- or even all bundles.
Definition 6 (The Permutation π ∗ ) Let a = (a1 , . . . , am ) ∈ A be an arbitrary allocation.
∗
Define π ∗ to be a permutation such that aπ is a locally-efficient bundle assignment. If there
is more than one such permutation, then arbitrarily choose one.
∗
∗
Fact 1 If the cost of each machine is additive, then b = aβ and aπ are locally-efficient
∗
bundle assignments. In particular, Ψ(c, aπ ) ≤ Ψ(c, a), and Ψ(c, b) ≤ Ψ(c, bπ ) for every
permutation π.
We are now ready to state the algorithm.
Algorithm 1 (Bundle-Local-Search) Input: c = c1 , c2 , . . . , cm and an allocation a.
∗
• If the makespan of aπ is at most
m+1
2
∗
times the makespan of a, then output aπ .
∗
• Otherwise, output aβ .
∗
Informally, if the makespan of aπ is not too far from the makespan of a, we output
π∗
a . Otherwise, we re-assign each bundle ai independently to the fastest machine for that
particular bundle.
12
Theorem 3 Algorithm 1 always
¡ m+1 outputs
¢ a locally-efficient bundle assignment and guarantees
an approximation ratio of 2 · r(a) with respect to the makespan in polynomial time.
Proof: Algorithm 1 always outputs a locally-efficient bundle assignment and thus by Proposition 1 is envy-free achievable.
∗
Assume without loss of generality, that the highest loaded machine in allocation aπ is
∗
∗
machine 1. If r(aπ ) ≤ m+1
· r(a) then Algorithm 1 outputs the allocation aπ . Otherwise,
2
we get for the highest loaded machine:
∗
∗
c1 (aπ ) = r(aπ ) >
Additionally,
∗
m+1
· r(a).
2
∗
Ψ(c, aβ ) ≤ Ψ(c, aπ ) ≤ Ψ(c, a) ≤ m · r(a).
Putting these together,
∗
Ψ(c, aβ ) ≤ r(a) + m · r(a) −
m+1
m+1
· r(a) =
· r(a),
2
2
where r(a) upper-bounds the value of the machine with the minimum cost for the bundle
∗
∗
aπ1 , and m · r(a) − m+1
· r(a) upper-bounds the total minimum cost of the bundles aπi , i ≥ 2.
2
In the worst case, all bundles will be allocated to the same machine. We thus have the
following:
∗
∗
r(aβ ) ≤ Ψ(c, aβ ) ≤
m+1
· r(a).
2
Theorem 4 (Horowitz and Sahni [17]) For any fixed number of machines and ² > 0
there exists an (1 + ²)-approximation algorithm for minimizing the makespan on unrelated
machines. The running time of this algorithm is polynomial in the number of tasks k and in
1
.
²
Theorem
5 ¢For any fixed number of machines and ² > 0 there exists an envy-free
¡ m+1
· (1 + ²) -approximation mechanism for minimizing the makespan on unrelated ma2
chines. The running time of this mechanism is polynomial in the number of tasks k and in
1
.
²
Proof: For every input and ² > 0 we can simulate the (1 + ²)-approximation algorithm in
Theorem 4 to get a nearly optimal allocation a with respect to the makespan, and we then
can simulate ¡Algorithm 1 on
¢ allocation a. By Theorems 3 and 4 and Proposition 1 this yields
m+1
an envy-free 2 · (1 + ²) -approximation mechanism whose running time is polynomial in
k and 1² .
13
5
Envy-Free in-Expectation Mechanisms for Unrelated
Machines
In the previous section we showed a lower bound of 32 for two machines. Here we circumvent
this impossibility by employing randomization.
Agents in an envy-free in-expectation setting care about their expected load (rather than
the actual assigned load). The main result in this section is the existence of an envy-free inexpectation mechanism with an approximation ratio of 43 + ² for two unrelated machines. We
then show that this bound is tight under the assumption that the mechanism is essentially
symmetric from the perspective of agents.
5.1
The Setting
Any envy-free in-expectation mechanism can be regarded as a mechanism that produces a
probability distribution over possible allocations for every instance of a given problem. In
turn, for each agent this implies agent-specific probability distribution over tasks.
The cost function of each of the agents in such mechanisms can therefore be viewed as
assigning costs to probability distributions over tasks, assuming risk neutrality as defined
below.
Definition 7 (Risk Neutrality) Let ci be a cost function. For every probability distribution D over the tasks 1..k we define the extended cost function Eci as follows:
Eci (D) = Σt=1..k P rD [t] · ci (t),
where P rD [t] is the probability of task t in D.
Definition 8 (Envy-Free in-Expectation Mechanism) Let ∆(A) be the set of all possible allowed probability distributions over allocations. Let f : C → ∆(A) be an allocation
rule that maps a tuple of costs c = (c1 , . . . , cn ) ∈ C to a probability distribution over possible
allocations ∈ ∆(A).
A mechanism M (f, pb) is said to be envy-free in-expectation if for every pair of agents i, j
and every c it holds that:
pbi (c) − Eci (Di ) ≥ pbj (c) − Eci (Dj ),
where pbi , pbj are the the reward that agents i, j receive, and Di , Dj are the associated agentspecific probability distributions over tasks produced by f for c.
5.2
An Upper Bound of
4
3
for Two Unrelated Machines
We now describe our allocation algorithm for the envy-free in-expectation of two machines.
Algorithm 2 (Random-Bundle-Local-Search) Input: c = c1 , c2 .
14
1. Simulate the (1 + ²)-approximation algorithm in Theorem 4 to get a nearly optimal
allocation a with respect to the makespan.
2. If a is a locally-efficient bundle assignment, then output the allocation a and the rewards
pb as in Proposition 1.
3. Let b0 be the allocation that allocates all the tasks to machine 1. Let b00 be the allocation
that allocates all the tasks to machine 2. If min(r(b0 ), r(b00 )) ≤ 34 r(a) then output the
allocation argmin(r(b0 ), r(b00 )) and the rewards pb as in Proposition 1.
4. Otherwise, let aR be the reverse allocation that allocates the bundle a2 to machine 1
and a1 to machine 2. Output the allocation a with probability 12 and the allocation aR
with probability 12 . The reward pb to each machine will be max(r(b0 ), r(b00 )).
Theorem 6 Algorithm 2 yields an envy-free in-expectation ( 34 +²)-approximation mechanism
for minimizing the makespan on two unrelated machines whose running time is polynomial
in the number of tasks k and 1² .
Proof: In steps 2 and 3 Algorithm 2 deterministically outputs locally-efficient bundle assignments, and thus by Proposition 1 is envy-free achievable.
Let a∗ be the optimal allocation with respect to the makespan. If Algorithm 2 terminates
at the third step then clearly min(r(b0 ), r(b00 )) ≤ 43 r(a) ≤ 43 · r(a∗ ) · (1 + ²).
In the last step, each machine gets each task with probability 21 and therefore envyfreedom holds by using exactly the same reward for each machine. It remains to show that the
makespan of the last step is within 34 (1 + ²) of the optimal makespan. Clearly, if Algorithm 2
terminates at the last step, then a 6= b0 , b00 . Additionally, a1 , a2 6= ∅, min(r(b0 ), r(b00 )) >
4
r(a), and aR is a locally-efficient bundle assignment.
3
We first claim that
5
ci (aR
· r(a), i = 1, 2.
i ) <
3
By using the facts that min(r(b0 ), r(b00 )) > 43 · r(a) and ci (ai ) ≤ r(a), i = 1, 2, we get that
1
R
R
R
ci (aR
i ) > 3 · r(a), i = 1, 2. Now, the local efficiency of a translates to c1 (a1 ) + c2 (a2 ) <
5
c1 (a1 ) + c2 (a2 ) ≤ 2 · r(a). Putting these together we have that ci (aR
i ) < 3 · r(a), i = 1, 2.
Finally, we get the desired approximability bound on the expected makespan:
1
1
1
5
4
4
· r(a) + · r(aR ) ≤ (1 + ) · r(a) = · r(a) ≤ · r(a∗ ) · (1 + ²).
2
2
2
3
3
3
5.3
A Tight Lower Bound for Balanced Mechanisms
We now show that our envy-free in-expectation mechanism is optimal (regardless of computational considerations) under the assumption that the mechanism is balanced. Formally:
Definition 9 (Balanced Mechanism) Let Di (c) denote the associated agent-specific probability distribution over tasks for agent i produced by a mechanism M (f, pb) for the input
15
c = (c1 , . . . , cn ). A distribution Di (c) is called truly random if there exists a task t such that
0 < P rDi (c) [t] < 1.
A mechanism M (f, pb) is balanced if a truly random Di (c) for c = (c1 , . . . , cn ) and agent
i implies that P rDi (c) [t0 ] = P rDj (c) [t0 ] for every agent j and task t0 = 1..k.
Intuitively, the assumption of balance implies that any non-deterministic allocation must
be symmetrical from the perspective of the agents: all agents’ truly random probability
distributions over tasks must be identical. However, observe that it still allows a rather large
degree of freedom in choosing the probability distribution over possible allocations.
Theorem 7 No balanced envy-free in-expectation mechanism for two unrelated machines
can achieve an approximation ratio strictly better than 43 with respect to the makespan.
Proof: To show this bound we solve a linear program based on the the following instance
with two machines and two tasks:
1
5
c1 (1) = 1, c1 (2) = , c2 (1) = − ², c2 (2) = 1.
3
3
The optimal makespan of this instance is 1, however the optimal allocation with respect
to the makespan is not locally-efficient. Additionally, it is easy to check that the makespan
of any other deterministic allocation is at least 43 .
The optimal approximation ratio achievable by any balanced envy-free in-expectation
mechanism for the above instance can be computed by the following linear program:
Minimize
x1 + ( 53 − ²)x2 + 43 x3 + ( 83 − ²)x4
s.t.
x1 + x2 + x3 + x4 = 1
−x1 + x2 − 2x3 + 2x4 + 1.5x5 − 1.5x6 ≥ 0
( 32 − ²)x1 − ( 23 − ²)x2 + ( 83 − ²)x3 − ( 83 − ²)x4 − x5 + x6 ≥ 0
x5 − x6 = 0
xi ≥ 0, i = 1, 2, 3, 4
The variables x1 , x2 , x3 , x4 represent the probabilities of allocations a1 , a2 , a3 , a4 in the
optimal envy-free in-expectation mechanism, where
a1 = ({1}, {2}), a2 = ({2}, {1}), a3 = ({1, 2}, ∅), and a4 = (∅, {1, 2}).
The variables x5 , x6 represent the reward to the first and the second machines, respectively. The objective function being minimized is the expected makespan.
The first and the last constraints imply that x1 , . . . , x4 is a probability distribution over
the set of allocations. Now, if the first machine does not envy the second machine, then:
16
x5 − x1 − 13 x2 − 43 x3 ≥ x6 − 13 x1 − x2 − 43 x4 . By rearranging we get the second constraint.
Similarly, if the second machine does not envy the first machine, then x6 − x1 − ( 53 − ²)x2 −
( 83 − ²)x4 ≥ x5 − ( 53 − ²)x1 − x2 − ( 83 − ²)x3 ; by rearranging we get the third constraint.
Finally, the fourth constraint x5 = x6 is implied by the assumption that the mechanism is
balanced.
The optimal solution is (x1 , x2 , x3 , x4 , x5 , x6 ) = (0.5, 0.5, 0, 0, 0, 0), with objective value 34 .
This probability distribution induces a balanced envy-free in-expectation mechanism, where
each machine gets exactly one task uniformly at random. Clearly, the solution is feasible.
To verify the optimality of this solution observe that:
4
3
− ² ≤ ( 43 − ²)(x1 + x2 + x3 + x4 ) + 12 (x5 − x6 ) + 13 (−x1 + x2 − 2x3 + 2x4 + 1.5x5 − 1.5x6 )
≤ x1 + ( 35 − ²)x2 + 43 x3 + ( 83 − ²)x4 .
5.4
Inapplicability of Randomized Envy-Free Mechanisms
A randomized mechanism is a probability distribution over deterministic mechanisms. Practically, for every c the mechanism M (f, pb) produces a distribution DM (c) over deterministic
mechanisms and outputs a deterministic mechanism drawn from this distribution (observe
that Definition 7 is redundant here). We next show that no randomized envy-free mechanism can provide a strictly better lower bound than any deterministic envy-free mechanism.
Formally,
Proposition 5 No randomized envy-free mechanism for two machines can achieve an approximation ratio strictly better than 32 (in expectation).
Proof: Consider the following instance for two machines c1 (1) = 1, c1 (2) = 0.5, c2 (1) =
1.5 − ²0 , c2 (2) = 1, taken from the proof of Theorem 2. Now, any realization of a randomized
envy-free mechanism must output a deterministic envy-free mechanism.
Recall that a deterministic envy-free mechanism must output a locally-efficient bundle
assignment. However, as we have seen in the proof of Theorem 2, the optimal allocation
with respect to the makespan for this instance is not locally-efficient. Furthermore, any other
allocation has a makespan of at least 32 times the optimal.
Therefore, no convex combination of locally-efficient bundle assignments can produce an
approximation ratio strictly better than 32 .
6
Near-Optimality of Related Machines
This section focuses on related machines so as to minimize the makespan, and shows that
the envy-freedom constraint does not impose any further burden. Specifically, we show
how to convert any ρ-approximation algorithm into an envy-free ρ-approximation mechanism in polynomial time. Applying the conversion technique to the deterministic PTAS
17
by Hochbaum and Shmoys [16] yields a polynomial time computable deterministic envyfree mechanism that achieves the approximation ratio of 1 + ² with respect to the optimal
makespan.
6.1
The Setting
The related-machines setting is a special case of the unrelated model. This problem is
denoted Q||Cmax in the scheduling literature, and is NP-hard [4], although a PTAS [16]
exists. In this model, each task j has a load lj > 0, and every machine i has a type ti > 0.
The running time of task j on machine i is ti ·lj . Additionally, machine i’s cost for performing
the task j is ci ({j}) = ti ·lj . Since the costs here are derived from the single parameter ti ∈ R,
it is considered to be a single-dimensional scheduling problem [1].
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. For convenience we use the notation l(S) = Σj∈S lj , to
denote the total load of a subset of tasks S.
6.2
Characterizing Envy-Free Single-Dimensional Mechanisms
The following simple condition characterizes envy-freedom in single-dimensional environments. As noted by [15], this characterization is nearly identical to that of the analogous
characterization of truthful mechanisms for single-dimensional environments by Myerson [25].
Definition 10 An allocation a = (a1 , . . . , am ) is called aligned if l(ai ) < l(aj ) implies that
ti ≥ tj for every pair of machines i, j.
Proposition 6 An allocation rule f is envy-free achievable if and only if f outputs an
aligned allocation for every c.
6.2.1
Proof of Proposition 6 and Frugality
We first show the "if" part of Proposition 6.
Lemma 1 If f is envy-free achievable then f outputs an aligned allocation for every c.
Proof: Suppose that f is envy-free achievable but f (c) = a, l(ai ) < l(aj ) and ti < tj for some
c, i and j. Now, (l(aj ) − l(ai )) · (tj − ti ) > 0 is equivalent to l(ai )ti + l(aj )tj > l(ai )tj + l(aj )ti .
But then exchanging the loads between machines i and j strictly decreases the overall social
cost, contradicting Proposition 1.
To show the "only-if" part of Proposition 6 it is convenient to directly use the following
reward function (rather than to use Proposition 1 to show the local efficiency of aligned
allocations).
18
Definition 11 (Allocation-Specific Frugal Reward) Suppose that f (c) = a, where allocation a is aligned. Assume that the machines are numbered from 1 to m in order of decreasing type, breaking ties in favor of the smallest load with respect to a, so that t1 ≥ t2 ≥ · · · ≥ tm
and l(a1 ) ≤ l(a2 ) ≤ · · · ≤ l(am ).
The allocation-specific frugal reward function is defined recursively as follows:
pb1 (c) = t1 · l(a1 )
pbi (c) = pbi−1 (c) + ti · (l(ai ) − l(ai−1 )) i = 2, . . . , m.
Lemma 2 If f outputs an aligned allocation for every c then f is envy-free achievable.
Proof: We will show that f , coupled with the above allocation-specific frugal reward, yields
an envy-free mechanism. Suppose that f (c) = a. Using Definition 11 we show that a slow
machine cannot envy a fast machine, and vice versa.
We have that pbi (c) − ti · l(ai ) ≥ pbd (c) − ti · l(ad ) for i < d by rearranging the following:
ti · l(ad ) − ti · l(ai ) = ti · [(l(ad ) − l(ad−1 )) + (l(ad−1 ) − l(ad−2 )) + · · · + (l(ai+1 ) − l(ai ))]
≥ td · (l(ad ) − l(ad−1 )) + · · · + ti+1 · (l(ai+1 ) − l(ai ))
= pbd (c) − pbi (c).
Similarly, we have that pbi (c) − ti · l(ai ) ≥ pbd0 (c) − ti · l(ad0 ) for d0 < i by rearranging the
following:
pbi (c) − pbd0 (c) = ti · (l(ai ) − l(ai−1 )) + · · · + td0 +1 · (l(ad0 +1 ) − l(ad0 ))
≥ ti · [(l(ai ) − l(ai−1 )) + · · · + (l(ad0 +1 ) − l(ad0 ))]
= ti · li (ai ) − ti · li (ad0 ).
The next proposition shows that the allocation-specific frugal reward function produces
the cheapest total reward with respect to a given allocation.
Proposition 7 The allocation-specific frugal reward provides the cheapest individually rational envy-free total reward supporting a given allocation.
Proof: Suppose that f (c) = a and a is aligned. Clearly by definition the allocation-specific
frugal reward is individually rational (that is, pbi (c) − ti · l(ai ) ≥ 0).
Recall that the allocation-specific frugal reward for the slowest machine is pb1 (c) = l(a1 )·t1 .
This covers exactly the cost of the first machine, and thus it is the cheapest for that machine
among all individually rational rewards.
19
Now, let pb0i−1 (c) be an arbitrary reward for machine i − 1. The envy-free reward to the
i’th machine must be larger than pb0i−1 (c) + (l(ai ) − l(ai−1 )) · ti . Otherwise machine i would
envy machine i − 1.
Observe that the allocation-specific frugal reward for machine i satisfies this constraint
with equality if pb0j (c) = pbj (c), where j = 1, 2, . . . , i − 1. This completes the proof.
6.3
A Tight Envy-Free Mechanism for Related Machines
Lemma 3 Let f be a deterministic ρ-approximation algorithm. There exists a ρ-approximation
algorithm f 0 that outputs aligned allocations. Furthermore, if f is polynomial time computable then so is f 0 .
Proof: For every input, algorithm f 0 will simulate f and then reassign the loads in a
"sorted" manner to the machines. More formally, let f (c) = a. We can gradually shift from
a to an aligned allocation a0 by the following procedure: Exchange loads between any two
machines that violate the alignment condition in Definition 10.
Similar to the classic Bubble-Sort algorithm, the shifting procedure can be implemented
in polynomial-time. It remains to show that the resulting allocation a0 is a ρ-approximation.
To do this, number the machines with respect to the aligned allocation a0 and Definition 11. For the two machine case m = 2: if a 6= a0 then clearly r(a) = t1 · l(a1 ) and after
the exchange r(a0 ) = max{t1 · l(a2 ), t2 · l(a1 )} ≤ r(a).
For the general case, observe that similarly to the previous case the makespan weakly
improves in each exchange in the above procedure.
Proposition 8 Any polynomial-time computable ρ-approximation deterministic algorithm
with respect to the optimal makespan for the related machines model can be converted to
a polynomial time computable ρ-approximation envy-free mechanism (using the allocationspecific frugal reward) in polynomial time.
Proof: Let f be a polynomial-time-computable deterministic ρ-approximation algorithm.
By Lemma 3 there exists a polynomial-time-computable deterministic ρ-approximation algorithm f 0 that outputs aligned allocations. By Lemma 2, f 0 coupled with the allocationspecific frugal reward yields an envy-free mechanism. Since the allocation-specific frugal
reward can be computed in polynomial time this concludes the proof.
Theorem 8 (Hochbaum and Shmoys [16]) There exists a polynomial-time computable
(1 + ²)-approximation algorithm for minimizing the makespan on related machines for every
fixed ² > 0.
Putting everything together, we can state the main result of this section:
Theorem 9 There exists a polynomial-time computable envy-free (1 + ²)-approximation
mechanism for minimizing the makespan on related machines for every fixed ² > 0.
Proof: The theorem is an immediate consequence of Theorem 8 and Proposition 8.
20
7
Research Directions
This paper formulates the fair by design approach and shows several tight bounds on envyfree mechanisms for makespan minimization. We now briefly consider few future research
directions.
Can we use randomization to remove the need of rewards? Specifically, is there an
envy-free mechanism without money with a reasonable approximation ratio for makespan
minimization?
Are the determinstic and in-expectation truthful lower-bounds of any reasonable multidimensional problem always bigger than its corresponding envy-free upper bounds?
Acknowledgements
I would like to thank Liad Blumrosen, Federico Echenique, Jason D. Hartline, David Kempe,
John Ledyard, Debasis Mishra, Mohamed Mostagir, Mahyar Salek, Michael Schapira and
anonymous referees for helpful discussions and suggestions.
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A
A.1
Appendix
Multidimensional Characterization: Proof of Theorem 1
Let v be an arbitrary valuation tuple. Denote by a ∈ A the allocation that f outputs for
v. For simplicity, we add a special null agent, with the null valuation function v0 (S) = 0
for every bundle S. Moreover, we also assume that f allocates the empty bundle to the null
agent, that is a0 = ∅.
In order to construct envy-free prices we define the following finite directed graph Gf,v ,
−
→
with the vertices V (Gf,v ) = {0, 1, 2, . . . , n} and the edges E (Gf,v ) = {(i, j) | i, j ∈ V (Gf,v ), i 6=
j}. Intuitively, each agent has a corresponding vertex in the graph, and each pair of distinct
agents {i, j} has a directed edge from i to j and a directed edge from j to i in the graph. Note
that the set of vertices V (Gf,v ) includes a vertex corresponding to the null agent. Finally,
the length of the directed edge (i, j) is defined as l(i, j) = vi (ai ) − vi (aj ).
23
Definition 12 (Canonical Prices) The canonical price for agent i is pi (v) = ∆i,0 , where
∆i,0 is the length of the shortest path from vertex i ∈ N to vertex 0 in the directed graph
Gf,v .
Lemma 4 If f (v) is a locally-efficient bundle assignment with respect to v for every v ∈ V ,
then f : V → A is an envy-free achievable allocation rule.
Proof: We shall verify that the canonical price induces the envy-free achievability of f .
Clearly ∆i,0 ≤ l(i, 0) < ∞. We need to show that p is well defined (that is, ∆i,0 > −∞) and
that M (f, p) is an envy-free mechanism.
It is a well known fact that if a graph has no negative cycles then ∆i,0 > −∞, for every
i ∈ N (e.g. by the correctness of the Bellman-Ford Algorithm). We first claim that every
directed cycle through {1, 2, . . . n} in the graph Gf,v has a nonnegative length. Suppose
not. Clearly every negative length cycle 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. Without loss of generality assume that (1, 2, . . . , d) is a simple
negative directed cycle. That is l(1, 2) + l(2, 3) + · · · + l(d, 1) < 0. Define the permutation
π(1) = 2, π(2) = 3, . . . , π(d) = 1 and π(i) = i for i > d. Now:
Ψ(v, a) − Ψ(v, aπ ) = l(1, 2) + l(2, 3) + · · · + l(d, 1) < 0,
contradicting the local efficiency of f .
Now we show that there are no negative cycles through vertex 0: Suppose that there
is such a cycle. Without loss of generality assume that (0, 1, 2, . . . , d) is a simple negative
directed cycle. That is l(0, 1) + l(1, 2) + l(2, 3) + · · · + l(d, 0) < 0. Then we know by the
above that l(1, 2) + l(2, 3) + · · · + l(d, 1) ≥ 0. Thus, l(0, 1) + l(1, 2) + l(2, 3) + · · · + l(d, 0) <
l(1, 2) + l(2, 3) + · · · + l(d, 1). Equivalently, l(0, 1) + l(d, 0) < l(d, 1). By definition, l(d, 1) =
vd (ad ) − vd (a1 ), l(d, 0) = vd (ad ) ≥ 0 and l(0, 1) = 0. Putting together and rearranging
we have: vd (a1 ) < 0 contradicting the the assumption that the valuation function vd is
nonnegative.
We show now that agent i > 0 cannot envy agent j 6= i. Suppose it does, then vi (ai ) −
∆i,0 < vi (aj ) − ∆j,0 . By rearranging we get l(i, j) + ∆j,0 = vi (ai ) − vi (aj ) + ∆j,0 < ∆i,0 . The
left-hand side represents a length of a directed path from i to 0 (through j) which is strictly
smaller than ∆i,0 , a contradiction to the minimality of ∆i,0 .
Lemma 5 If the allocation rule f : V → A is envy-free achievable then the allocation f (v)
is a locally-efficient bundle assignment with respect to v.
Proof: Suppose that f is envy-free achievable. Denote by a ∈ A the allocation that f
outputs for v. Let π : N → N be an arbitrary permutation. By the achievability of f , there
exists a price 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 agents. 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.
24
Theorem 1 follows from combining Lemma 4 and Lemma 5. We now show that the
canonical prices satisfy individual rationality.
Claim 2 Let f : V → A be an envy-free achievable allocation rule. Let p1 , p2 , . . . , pn be the
canonical prices. Then the mechanism M (f, p) is individually rational. Moreover, pi (v) ≥ 0,
for every i.
Proof: By Definition 12 we have vi (ai ) = vi (ai ) − vi (a0 ) = l(i, 0) ≥ ∆i,0 = pi (v). That is,
vi (ai ) ≥ pi (v) for every i ≥ 1. This shows the individual rationality.
Recall that l(0, i) = 0, for every i ≥ 1. Now, pi = ∆i,0 = ∆i,0 + l(0, i) ≥ 0, since by the
proof of Lemma 4 all cycles in Gf,v are nonnegative.
A.2
Proof of Claim 1
Claim 1 Given any allocation a ∈ A and values vi (aj ), i, j = 1..n, it can be decided in
polynomial time whether supporting envy-free prices exist. Furthermore, if individuallyrational nonnegative envy-free prices exist, they can be computed in polynomial time.
Proof: Consider the following decision algorithm: For the input pair (a, v) construct the
complete bipartite graph Hv,a with vertices {1, 2, . . . , n} and {a1 , a2 , . . . , an }, where each nondirected edge (i, aj ) has weight vi (aj ). Compute a maximum weighted bipartite matching a∗
for Hv,a . Output ’yes’ if and only if Ψ(a, v) = Ψ(a∗ , v).
By definition Ψ(a, v) = Ψ(a∗ , v) if and only if a is locally-efficient bundle assignment.
By Theorem 1, allocation a is locally-efficient bundle assignment if and only if it can be
supported with envy-free prices. This shows the correctness of the decision algorithm.
Now, by Claim 2 the canonical prices pi (v) = ∆i,0 , i = 1..n, if they exist, are individually
rational nonnegative envy-free prices. Let GR
f,v be the reverse of the graph Gf,v , that is a
complete directed graph on the same set of vertices as Gf,v , where the length of the edge
R
(j, i) in the graph GR
f,v is l (j, i) = vi (ai ) − vi (aj ) = l(i, j). The canonical prices can
be computed by solving the single source shortest paths problem, specifically, by running
the Bellman-Ford algorithm on the input pair (GR
f,v , 0) where the single source vertex is the
vertex 0.
A.3
Proof of Proposition 1
We now briefly outline how to adjust the proof of Theorem 1 and Claims 1 and 2. Let
c be an arbitrary valuation tuple. Denote by a ∈ A the allocation that f outputs for c.
We define a new finite directed graph Gf,c , with vertices V (Gf,c ) = {1, 2, . . . , n} and edges
−
→
E (Gf,c ) = {(i, j) | i, j ∈ V (Gf,c ), i 6= j}. The length of the directed edge (i, j) is defined as
l(i, j) = ci (aj ) − ci (ai ).
The canonical reward for agent i is pbi (c) = ∆1,i + α, where ∆1,i is the length of the
shortest path from vertex 1 to vertex i ∈ N in Gf,c and α ≥ 0 is a large enough constant to
be defined later.
25
(⇒) Suppose that f (c) is a locally-efficient bundle assignment with respect to c. We
shall verify that the canonical reward induces the envy-free achievability of f . Clearly
∆1,i ≤ l(1, i) < ∞. We first claim that every directed cycle through {1, 2, . . . n} in the
graph Gf,c has a nonnegative length. Suppose not. Without loss of generality assume that
(1, 2, . . . , d) is a simple negative directed cycle. That is l(1, 2) + l(2, 3) + · · · + l(d, 1) < 0.
Define the permutation π(1) = 2, π(2) = 3, . . . , π(d) = 1 and π(i) = i for i > d. Now:
Ψ(c, aπ ) − Ψ(c, a) = l(1, 2) + l(2, 3) + · · · + l(d, 1) < 0,
contradicting the local efficiency of f .
We show now that agent i cannot envy agent j 6= i. Suppose it does, then ∆1,i + α −
ci (ai ) < ∆1,j + α − ci (aj ). By rearranging we get ∆1,i + l(i, j) = ∆1,i + ci (aj ) − ci (ai ) < ∆1,j .
The left-hand side represents a length of a directed path from 1 to j (through i) which is
strictly smaller than ∆1,j , a contradiction to the minimality of ∆1,j .
We can choose a large enough α ≥ 0 such that ∆1,i + α ≥ ci (ai ) for every i ≥ 1. Clearly,
since ci (ai ) ≥ 0 we have ∆1,i + α ≥ 0. This shows that the canonical rewards are nonnegative
and satisfy individual rationality.
(⇐) Suppose that f is envy-free achievable. Denote by a ∈ A the allocation that f
outputs for c. Let π : N → N be an arbitrary permutation. By the achievability of f , there
exists a reward function pb such that: pbi (c) − ci (ai ) ≥ pbπ(i) (c) − ci (aπ(i) ), for every i ∈ N . By
rearranging we get ci (aπ(i) ) − ci (ai ) ≥ pbπ(i) (c) − pbi (c). The local efficiency then follows from
summing the inequalities over all agents. Formally:
Ψ(c, aπ ) − Ψ(c, a) = Σi∈N ci (aπi ) − Σi∈N ci (ai ) ≥ Σi∈N pbπ(i) (c) − Σi∈N pbi (c) = 0.
Now, the envy-free achievability can be decided in polynomial time by solving minimum
weighted bipartite matching as follows: For the input pair (a, c) construct the complete
bipartite graph Hc,a with vertices {1, 2, . . . , n} and {a1 , a2 , . . . , an }, where each non-directed
edge (i, aj ) has weight ci (aj ). Compute a minimum weighted bipartite matching a∗ for Hc,a .
Output ’yes’ if and only if Ψ(a, c) = Ψ(a∗ , c).
The canonical rewards can be computed in polynomial time by solving the single source
shortest paths problem. Specifically, by running the Bellman-Ford algorithm on the input pair (Gf,c , 1) where the single source vertex is the vertex 1, and by choosing α =
max{0, c1 (a1 ) − ∆1,1 , . . . , cn (an ) − ∆1,n }.
A.4
A.4.1
Communication Bounds for Profit Maximization
Lower Bounds: Proof of Propositions 1 and 2
To show the lower bounds we reduce the social welfare maximizing problem studied by [28] to
the profit maximizing problem. Recall that the problem of maximizing the socialP
welfare is
to find an allocation that maximizes the sum of valuation (formally, argmax{a∈A} n1 vi (a)).
Theorem 10 (Nisan and Segal [28]) Any algorithm for maximizing the social welfare in
combinatorial auctions that achieves an approximation ratio better than 2 requires exponential
communication.
26
Theorem 11 (Nisan and Segal [28]) Any algorithm for maximizing the social welfare in
combinatorial auctions that achieves an approximation ratio better than n requires exponential communication.
Proposition 2 Any envy-free profit-maximizing mechanism for combinatorial auctions that
achieves an approximation ratio better than 2 requires exponential communication.
The idea is to reduce the social welfare maximizing problem in Theorem 10 to the profitmaximizing problem.
Let v = (v1 , v2 , . . . , vn ) ∈ V1 ×V2 ×· · ·×Vn = V . We introduce the set D = {d1 , d2 , . . . , dn }
of n additional items and define the following valuation:

 vi (S \ D) di ∈ S
vbi (S) =

0
Otherwise.
Observe that vbi is a valid valuation satisfying No externalities, Free disposal and Normalization.
Let W ∗ (v, K) be the maximum social welfare with respect to v and the original set of
items K. Let P ∗ (b
v , K ∪ D) be the maximum envy-free profit with respect to vb and the
extended set of items K ∪ D.
Claim 3 W ∗ (v, K) ≤ P ∗ (b
v , K ∪ D) for every v.
P
Proof: Let b be an allocation of the items in K such that n1 vi (bi ) = W ∗ (v, K). Let
bd = (b1 ∪ {d1 }, . . . , bn ∪ {dn }) be an allocation based on the allocation b and the new
items.
It is enough to show that the maximum envy-free profit of vb using bd is at least
Pn
bi (S), to see the envy-freedom observe that vbi (bi ∪ {di }) − pi (bi ∪
1 vi (bi ). Let pi (S) = v
{d
bi (bj ∪ {dj }) − pj (bj ∪ {dj }). Additionally,
i ) − vi (bi ) = 0 ≥ 0 − vj (bj ) = v
Pin}) = vi (bP
n
1 vi (bi ) =
1 pi (bi ∪ {di }).
Let W (v, a, K) be the social welfare of allocation a with respect to v and K. Let
P (b
v, b
a, K ∪ D) be the maximum envy-free individually-rational profit obtainable from a
locally-efficient allocation b
a with respect to vb and K ∪ D.
Claim 4 There exists an allocation a0 such that P (b
v, b
a, K ∪ D) ≤ W (v, a0 , K).
Proof: To see this, let a0i = b
ai \ D, observe that pi (b
ai ) ≤ vbi (b
ai ) ≤ vi (b
ai \ D) = vi (a0i ).
Proof (of Proposition): Assume, for the sake of contradiction, that there exists an envyfree profit-maximizing (2 − ²)-approximation mechanism that requires sub-exponential communication. We show that this implies a (2−²)-approximation algorithm for maximizing the
social welfare in combinatorial auctions whose communication complexity is sub-exponential.
For the input v and K our algorithm will simulate the envy-free profit mechanism on the
input vb and K ∪ D. Let b
a be the allocation selected by the envy-free profit mechanism. The
output of our algorithm will be the allocation (b
a1 \ D, . . . , b
an \ D).
27
Clearly, the above algorithm requires sub-exponential communication. By Claims 3 and 4
we have that the above algorithm is a (2 − ²)-approximation algorithm with respect to the
social welfare. Specifically:
W ∗ (v, K)
P ∗ (b
v , K ∪ D)
≤
≤ P (b
v, b
a, K ∪ D) ≤ W (v, a0 , K),
2−²
2−²
where a0 = (b
a1 \ D, . . . , b
an \ D), this contradicts Theorem 10.
Proposition 3 Any envy-free profit-maximizing mechanism for combinatorial auctions that
achieves an approximation ratio better than n requires exponential communication.
Proof: We can apply the proof idea of Proposition 2 to get a contradiction to Theorem 11.
Observe that the number of bidders in the input v and K is identical to the number of
bidders in the input vb and K ∪ D, and therefore:
W ∗ (v, K)
P ∗ (b
v , K ∪ D)
≤
≤ P (b
v, b
a, K ∪ D) ≤ W (v, a0 , K).
n−²
n−²
A.4.2
Upper Bound: Proof of Proposition 3
We observe that a result by Guruswami et al. [12] can be applied to convert an approximation
algorithm for maximizing the social welfare for general bidders into an envy-free approximation mechanism for profit maximization. Guruswami et al. [12] studied envy-free profit
maximization mechanisms for unit-demand bidders.9 Applying the conversion technique to
the algorithm by Lehmann et al. [19] yields an envy-free profit-maximizing mechanism for
combinatorial auctions for general bidders that requires polynomial communication.
Theorem 12 (Guruswami et al. [12]) There exists a polynomial time computable
(2 ln(min{n, k}))-approximation algorithm for maximizing the envy-free profit of n unitdemand bidders with k items.
√
Theorem 13 (Lehmann et al. [19, 3]) There exists a (min{n, c · k})-approximation
algorithm for maximizing the social welfare in combinatorial auctions that requires polynomial
communication, where c > 0 is some fixed constant.
Proposition 4 There exists an envy-free
√ mechanism for combinatorial auctions for general
0
bidders which achieves a (min{n, c · k · ln(min{n, k})})-approximation for maximizing the
profit and which requires polynomial communication, where c0 > 0 is some fixed constant.
Proof: Consider the following mechanism that computes two allocations and supporting
prices and then selects the allocation with the higher profit:
9
We say that agent i is a unit-demand bidder if vi (S) = max{s∈S} vi ({s}), that is the agent would like to
buy at most one item.
28
Step 1: allocate the whole bundle K of k items to the bidder with the highest value. The
price for this bundle is maxi vi (K), while the price for an empty bundle is zero.
Step 2.1: simulate the algorithm in Theorem 13. Denote by a ∈ A the allocation that
the algorithm outputs for v.
∗
Step 2.2: convert a into a locally-efficient bundle assignment aπ , where π ∗ is the permu∗
tation such that Ψ(v, aπ ) ≥ Ψ(v, aπ ) for every π.
Step 2.3: apply the envy-free pricing algorithm in Theorem 12 on the input v and the n
∗
∗
indivisible bundles aπ1 , . . . , aπn .
Step 3: output the allocation with the higher profit.
Let P ∗ (v) be the maximum envy-free profit, and let P (v) be the profit of the mechanism.
The first step provides an n-approximation for maximizing the envy-free profit. Specifically:
P ∗ (v) ≤ n · max vi (K) = n · P (v).
i
The pricing method of [12] requires that the input allocation has a maximum social
∗
welfare. Since aπ is locally-efficient, it is easy to see that this allocation maximizes the
∗
social welfare when each bundle aπi is treated as indivisible, and the bidders are treated as
unit-demand bidders with respect to the new bundled-items. Furthermore, it can be verified
∗
Ψ(v,aπ )
that P (v) ≥ 2 ln(min{n,k})
(see the proof of Theorem 3.5 in [12]).
∗
let W (v) be the maximum social welfare. Clearly, P ∗ (v) is at most W ∗ (v), and therefore:
P ∗ (v)
W ∗ (v)
∗
√ ≤
√ ≤ Ψ(v, a) ≤ Ψ(v, aπ ) ≤ 2 ln(min{n, k}) · P (v).
c· k
c· k
√
Thus, P ∗ (v) ≤ min{n, 2c · k · ln(min{n, k})} · P (v).
A.5
Item Prices
Item pricing is a special case of bundle pricing, where the price of a bundle is the total of
the individual prices of the items in the bundle. Thus an agent can envy a bundle or any
sub-bundle allocated to some other agent. Despite their appeal, item prices severely restrict
the approximability of mechanisms. Formally:
Claim 5 No individually-rational envy-free mechanism for profit maximization with supporting item prices can achieve approximation ratio better than k.
Proof: Consider two bidders and k ≥ 2 identical items. Bidder 1 is a single minded bidder
with v1 (S) = k − ² if |S| = k, otherwise v1 (S) = 0 for any |S| < k. Bidder 2 is a unit-demand
bidder with v2 (S) = 1 if |S| ≥ 1. If bundle prices are allowed, we can allocate the k items to
the first bidder. This is a locally-efficient bundle assignment, and the profit of k − ² can be
extracted in an envy-free manner. However, if we require item prices the maximum profit is
at most 1.
29
Claim 6 No individually-rational envy-free mechanism for minimizing the makespan with
supporting item prices (on unrelated or related machines) can achieve approximation ratio
better than m.
Proof: To show the envy-free inapproximability, consider m identical tasks and m machines.
Suppose that machine 1’s cost for each task is 1 − ², and machine i’s cost for each task is
1, where i = 2..m. Clearly, a reward of < 1 − ² for any task violates individual rationality.
Additionally, a reward of > 1 for any task violates envy-freedom. The only possible envy-free
allocation is to assign all the tasks to machine 1 for a reward ∈ [1 − ², 1] for each task. This
gives a schedule with a makespan of m · (1 − ²), while the optimal schedule with respect to
the makespan is to allocate exactly one task to each machine.
30