Approximate Strong Equilibrium in Job Scheduling Games.
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Conference Proceeding: On the Road to -Completeness: 8 Agents in a Singleton Congestion Game.[show abstract] [hide abstract]
ABSTRACT: In this paper, we investigate the complexity of computing locally optimal solutions for Singleton Congestion Games (SCG), in the framework ofPLS, as defined in Johnson et al. (34). Here, in an instance weighted agents choose links from a set of identical links, such that no agent has an incentive to unilateraly decrease its cost by switching to a different link. The cost of an agent is the load (the sum of the weights of the agents) on the link it chooses. The agents are selfish and try to minimize their individual cost. Agents may form arbitrary, non-fixed coalitions. The cost of a coalition is defined to be the maximum cost of its members. The potential function is defined as the lexicographical order of the agents' cost. In each selfish step of a coalition, the potential function decreases. Thus, a local minimum is a Nash Equilibrium. The neighborhood of a feasible assignment (every agent chooses a link) are all assignments, where the cost of some arbitrary non-fixed coalition of at most k reallocating agents decreases. We call this problem SCG-(k) and show that SCG-(k) isPLS-complete for k 8. On the other hand, for k = 1, it is well known that the solution computed by Graham's LPT-algorithm (22, 29) is locally optimal for SCG-(k), (20, 32). We show our result by tight reduction from the MAXCONSTRAINTASSIGNMENT problem (p;q;r)-MCA, which is an extension of GENERALIZED SATISFIABILITY to higher valued variables. Here, p is the maximum number of variables occuring in a constraint, q is the maximum number of appearances of a variable, and r is the valuedness of the variables. To the best of our knowledge, SCG-(k) is the first problem, which is known to be solvable in polynomial time for a small neighborhood andPLS-complete for a larger, but still constant neighborhood.Internet and Network Economics, 4th International Workshop, WINE 2008, Shanghai, China, December 17-20, 2008. Proceedings; 01/2008
Journal of Artificial Intelligence Research 36 (2009) 387-414Submitted 07/09; published 11/09
Approximate Strong Equilibrium in Job Scheduling Games
School of Business Administration
and Center for the Study of Rationality,
Hebrew University of Jerusalem, Israel.
School of Computer Science,
The Interdisciplinary Center, Herzliya, Israel.
A Nash Equilibrium (NE) is a strategy profile resilient to unilateral deviations, and is
predominantly used in the analysis of multiagent systems. A downside of NE is that it is
not necessarily stable against deviations by coalitions. Yet, as we show in this paper, in
some cases, NE does exhibit stability against coalitional deviations, in that the benefits
from a joint deviation are bounded. In this sense, NE approximates strong equilibrium.
Coalition formation is a key issue in multiagent systems. We provide a framework for
quantifying the stability and the performance of various assignment policies and solution
concepts in the face of coalitional deviations. Within this framework we evaluate a given
configuration according to three measures: (i) IRmin: the maximal number α, such that
there exists a coalition in which the minimal improvement ratio among the coalition mem-
bers is α, (ii) IRmax: the maximal number α, such that there exists a coalition in which
the maximal improvement ratio among the coalition members is α, and (iii) DRmax: the
maximal possible damage ratio of an agent outside the coalition.
We analyze these measures in job scheduling games on identical machines. In particular,
we provide upper and lower bounds for the above three measures for both NE and the well-
known assignment rule Longest Processing Time (LPT).
Our results indicate that LPT performs better than a general NE. However, LPT is not
the best possible approximation. In particular, we present a polynomial time approximation
scheme (PTAS) for the makespan minimization problem which provides a schedule with
IRminof 1 + ε for any given ?. With respect to computational complexity, we show that
given an NE on m ≥ 3 identical machines or m ≥ 2 unrelated machines, it is NP-hard to
determine whether a given coalition can deviate such that every member decreases its cost.
We consider job scheduling systems, in which n jobs are assigned to m identical machines
and incur a cost which is equal to the total load on the machine they are assigned to.1These
problems have been widely studied in recent years from a game theoretic perspective (Kout-
soupias & Papadimitriou, 1999; Andelman, Feldman, & Mansour, 2007; Christodoulou,
Koutsoupias, & Nanavati, 2004; Czumaj & V¨ ocking, 2002; A. Fiat & Olonetsky., 2007).
In contrast to the traditional setting, where a central designer determines the allocation of
jobs to machines and all the participating entities are assumed to obey the protocol, mul-
1. This cost function characterizes systems in which jobs are processed in parallel, or when all jobs on a
particular machine have the same single pick-up time, or need to share some resource simultaneously.
c ?2009 AI Access Foundation. All rights reserved.
Feldman & Tamir
tiagent systems are populated by heterogeneous, autonomous agents, which often display
selfish behavior. Different machines and jobs may be owned by different strategic entities,
who will typically attempt to optimize their own objective rather than the global objective.
Game theoretic analysis provides us with the mathematical tools to study such situations,
and indeed has been extensively used recently to analyze multiagent systems. This trend
is motivated in part by the emergence of the Internet, which is composed of distributed
computer networks managed by multiple administrative authorities and shared by users
with competing interests (Papadimitriou, 2001).
Most game theoretic models applied to job scheduling problems, as well as other net-
work games (e.g., Fabrikant, Luthra, Maneva, Papadimitriou, & Shenker, 2003; Albers,
Elits, Even-Dar, Mansour, & Roditty, 2006; Roughgarden & Tardos, 2002; Anshelevich,
Dasgupta, Kleinberg, Tardos, Wexler, & Roughgarden, 2004), use the solution concept
of Nash equilibrium (NE), in which the strategy of each agent is a best response to the
strategies of all other agents. While NE is a powerful tool for analyzing outcomes in com-
petitive environments, its notion of stability applies only to unilateral deviations.
numerous multiagent settings, selfish agents stand to benefit from cooperating by forming
coalitions (Procaccia & Rosenschein, 2006). Therefore, even when no single agent can profit
by a unilateral deviation, NE might still not be stable against a group of agents coordinat-
ing a joint deviation, which is profitable to all the members of the group. This stronger
notion of stability is exemplified in the strong equilibrium (SE) solution concept, coined by
Aumann (1959). In a strong equilibrium, no coalition can deviate and improve the utility
of every member of the coalition.
Figure 1: An example of a configuration (a) that is a Nash equilibrium but is not resilient against
coordinated deviations, since the jobs of load 5 and 2 all profit from the deviation demon-
strated in (b).
As an example, consider the configuration depicted in Figure 1(a). In our figures, a
job is represented by a rectangle whose width corresponds to the job’s length. The jobs
scheduled on a specific machine form a vertical concatenation of rectangles. For example,
in Figure 1(a) there are three machines, and M1processes two jobs of length 5. Note that
the internal order of jobs has no effect, since the cost of each job is defined to be the load on
the machine it is assigned to. This configuration is an NE since no job can reduce its cost
through a unilateral deviation. One might think that any NE on identical machines must
also be sustainable against joint deviations. Yet, as was already observed in (Andelman
Approximate Strong Equilibrium
et al., 2007), this may not be true.2For example, the configuration above is not resilient
against a coordinated deviation of the coalition Γ consisting of the four jobs of load 5 and
2 deviating to configuration (b), where the jobs of load 5 decrease their costs from 10 to 8,
and the jobs of load 2 improve from 5 to 4. Note that the cost of the two jobs of load 3
(which are not members of the coalition) increases.
In the example above, every member of the coalition improves its cost by a (multiplica-
tive) factor of5
improvement ratio? As it will turn out, this example is in fact the most extreme one in a
sense that will be clarified below. Thus, while NE is not completely stable against coor-
dinated deviations, in some settings, it does provide us with some notion of approximate
stability to coalitional deviations (or approximate strong equilibrium).
We also consider a subclass of NE schedules, produced by the Longest Processing Time
(LPT) rule (Graham, 1969). The LPT rule sorts the jobs in a non-increasing order of
their loads and greedily assigns each job to the least loaded machine. It is easy to verify
that every configuration produced by the LPT rule is an NE (Fotakis, S. Kontogiannis, &
Spiraklis, 2002). Is it also an SE? Note that for the instance depicted in Figure 1, LPT
would have produced an SE. However, as we show, this is not always the case.
In this paper we provide a framework for studying the notion of approximate stability
to coalitional deviations. In our analysis, we consider three different measures. The first
two measure the stability of a configuration, and uses the notion of an improvement ratio
of a job, which is defined as the ratio between the job’s costs before and after the deviation.
The third measures the worst possible effect on the non-deviating jobs, as will be explained
1.Minimum Improvement Ratio: By definition, all members of a coalition must
reduce their cost. That is, the improvement ratio of every member of the coalition is larger
than 1. Clearly, the coalition members might not share the same improvement ratio. The
minimum improvement ratio of a particular deviation is the minimal improvement ratio of
a coalition member. The minimum improvement ratio of a schedule s, denoted IRmin(s),
is the maximum over all possible deviations originated from s of the minimal improvement
ratio of the deviation. In other words, there is no coalitional deviation originating from s
such that every member of the coalition reduces its cost by a factor greater than IRmin(s).
A closely related notion has been suggested by Albers (2009), who defined a strategy
profile to be an α-SE if there is no coalition in which each agent can improve by a factor
of more than α. In our notation, a schedule s is an α-SE if IRmin(s) is at most α. Albers
studied this notion in the context of SE existence in cost-sharing games, and showed that
for a sufficiently large α, an α-SE always exists. The justification behind this concept is
that agents may be willing to deviate only if they improve by a sufficiently high factor (due
to, for example, some overhead associated with the deviation).
4. By how much more can a coalition improve? Is there a bound on the
2. This statement holds for m ≥ 3. For 2 identical machines, every NE is also an SE (Andelman et al.,
3. Throughout this paper, we define approximation by a multiplicative factor. Since the improvement and
damage ratios for all the three measures presented below are constants greater than one (as will be shown
below), the additive ratios are unbounded. Formally, for any value a it is possible to construct instances
(by scaling the instances we provide for the multiplicative ratio) in which the cost of all jobs is reduced,
or the cost of some jobs is increased, by at least an additive factor of a.
Feldman & Tamir
For three machines, we show that every NE is a5
that can deviate such that every member improves by a factor larger than
case, we also provide a matching lower bound (recall Figure 1 above), that holds for any
m ≥ 3. For arbitrary m, we show that every NE is a (2 −
draws a connection between makespan approximation and approximate stability, where
the makespan of a configuration is defined as the maximum load on any machine in the
We next consider schedules obtained from the LPT rule. We show that for m = 3,
every LPT configuration is a (1
bound, which holds for any m ≥ 3. For arbitrary m, we show an upper bound of4
The above results indicate that LPT is more stable than NE with respect to coalitional
deviations. Yet, LPT is not the best possible approximation of SE. Similar to this notion
in approximation algorithms, we define an SE-PTAS to be an assignment algorithm which
gets as input an additional parameter ε, specifying how close to an SE the schedule should
be and produces a (1 + ε)-SE in time polynomial in n,1/ε. In this paper we devise an
SE-PTAS for any fixed number of machines, which also approximates the makespan within
a factor of 1 + ?.
2. Maximum Improvement Ratio: The maximum improvement ratio of a particular
deviation is the maximal improvement ratio experienced by some coalition member. The
maximum improvement ratio of a schedule s, denoted IRmax(s), is the maximum over all
possible deviations originated from s of the maximal improvement ratio of the deviation.
In other words, there is no coalition deviation originating from s such that there exists a
member of the coalition that reduces its cost by a factor greater than IRmax(s).
This notion establishes the bounds on how much an agent would gain in a deviating
coalition for which all agents gain something from the deviation. Also, this notion is similar
in spirit to stability against a large total improvement. It also suits environments in which
individuals are willing to obey a specific player as long as they are not hurt. Interestingly,
we find that given an NE configuration, the improvement ratio of a single agent may be
arbitrarily large, for any m ≥ 3. In contrast, for LPT configurations on three machines, no
agent can improve by a factor of5
IRmax, the relative stability of LPT compared to NE is more significant than with respect
to IRmin. For arbitrary m, we provide a lower bound of 2−1
3. Maximum Damage Ratio: As is the case for the jobs of load 3 in Figure 1, some jobs
might get hurt as a result of a coalitional deviation. The third measure that we consider
is the worst possible effect of a deviation on jobs that are not members of the deviating
coalition. Formally, the maximum damage ratio is the maximal ratio between the costs of
a non-coalition member before and after the deviation. Variants of this measure have been
considered in distributed systems, e.g., the Byzantine Generals problem (Lamport, Shostak,
& Pease, 1982), and in rational secret sharing (Halpern & Teague, 2004).4In Section 5,
we prove that the maximum damage ratio is less than 2 for any NE configuration, and less
4-SE. That is, there is no coalition
4. For this
m+1)-SE. Our proof technique
4)-SE (≈ 1.1123), and we also provide a matching lower
3or more and this bound is tight. Thus, with respect to
m, which we believe to be tight.
4. In a rational secret sharing protocol, a set of players, each holding a share of a secret, aims to jointly
reconstruct it. Viewing the protocol as a game, the players’ utilities are typically assumed to satisfy the
following two basic constraints: (i) each player prefers learning the secret over not learning it, and (ii)
conditioned on having learned the secret, each player prefers as few as possible other players to learn it.
Approximate Strong Equilibrium
provide matching lower bounds.
In summary, our results in Sections 3-5 (see Table 1) indicate that NE configurations
are approximately stable with respect to the IRminmeasure. Moreover, the performance of
jobs outside the coalition would not be hurt by much as a result of a coalitional deviation.
As for IRmax, our results provide an additional strength of the LPT rule, which is already
known to possess attractive properties (with respect to, e.g., makespan approximation and
stability against unilateral deviations).
2for any LPT configuration. Both bounds hold for any m ≥ 3, and for both cases we
m ≥ 3
m = 3
Table 1: Our results for the three measures. Unless specified otherwise, the results hold for
arbitrary number of machines m.
In Section 7, we study computational complexity aspects of coalitional deviations. We
find that it is NP-hard to determine whether an NE configuration on m ≥ 3 identical
machines is an SE. Moreover, given a particular configuration and a set of jobs, it is NP-
hard to determine whether this set of jobs can engage in a coalitional deviation.
unrelated machines (i.e., where each job incurs a different load on each machine), the above
hardness results hold already for m = 2 machines. These results might have implications
on coalitional deviations with computationally restricted agents.
Related work: NE is shown in this paper to provide approximate stability against coali-
tional deviations. A related body of work studies how well NE approximates the optimal
outcome of competitive games. The Price of Anarchy was defined as the ratio between the
worst-case NE and the optimum solution (Papadimitriou, 2001; Koutsoupias & Papadim-
itriou, 1999), and has been extensively studied in various settings, including job scheduling
(Koutsoupias & Papadimitriou, 1999; Christodoulou et al., 2004; Czumaj & V¨ ocking, 2002),
network design (Albers et al., 2006; Anshelevich et al., 2004; Anshelevich, Dasgupta, Tar-
dos, Wexler, & Roughgarden, 2003; Fabrikant et al., 2003), network routing (Roughgarden
& Tardos, 2002; Awerbuch, Azar, Richter, & Tsur, 2003; Christodoulou & Koutsoupias,
2005), and more.
The notion of strong equilibrium (SE) (Aumann, 1959) expresses stability against co-
ordinated deviations. The downside of SE is that most games do not admit any SE, even
amongst those admitting a Nash equilibrium. Various recent works have studied the exis-
tence of SE in particular families of games. For example, it has been shown that in every job
scheduling game and (almost) every network creation game, an SE exists (Andelman et al.,
2007). In addition, several papers (Epstein, Feldman, & Mansour, 2007; Holzman & Law-
Yone, 1997, 2003; Rozenfeld & Tennenholtz, 2006) provided a topological characterization
for the existence of SE in different congestion games, including routing and cost-sharing