1. Partition Equilibrium Always Exists in Resource
Selection Games
Elliot Anshelevich1, Bugra Caskurlu1, Ameya Hate1
© Rennselaer Polytechnic Institute
Equilibrium Concepts
Nash Equilibrium
•Selfish agents have to choose their own strategy
•Outcome and payoff of players decided by collective strategy
•A set of strategies is a Nash equilibrium if no player can do
better by unilaterally changing its strategy
∀i: ci(si,s-i) ≤ ci(si
’,s-i)
•Limitations:
•Players cannot co-operate
Rensselaer
Computer Science
1.Rensselaer Polytechnic Institute
Resource Selection Games
Properties of Nash equilibrium
•Minimum makespan : u
•Depends on number of players and latency
functions of resource
•Nash equilibrium always has minimum
makespan
•For machine i : mi = maxz {fi(z) ≤ u}
-Type-1 machine: fi(mi) = u
-Type-2 machine: fi(mi) < u
•An allocation A of jobs onto machines is a
Nash equilibrium if and only if
-each type-2 machine i is allocated
exactly mi jobs and
-each type-1 machine i is allocated either
mi or mi−1 jobs,
-with at least one type-1 machine i
allocated exactly mi jobs.
In any Nash equilibrium...
•High machines, Low machines
•q = number of high machines
•q is same for any NE: depends on number of
players and latency function of resources
•Low level li = fi(mi − 1)
Conditions for Partition
Equilibrium
Strong Equilibrium
•Any subset of agents can form a coalition
•Coalition can deviate if all its members benefit: Strong
deviation
•Strong equilibrium exists if no coalition has incentive to
strongly deviate
∀A,∃i∈A: ci(sA,s-A) ≤ ci(sA’,s-A)
•Limitations:
•Deviation requires all members to benefit
Strong Deviation
Super-Strong Equilibrium
•Coalition deviates if at least one of its member benefits
and no member is worse off: Weak deviation
•Super-Strong equilibrium exists if no coalition has
incentive to weakly deviate
∀A,∃i∈A: ci(sA,s-A) < ci(sA’,s-A)
•Limitations:
•Unrealistically constrained
Weak Deviation
Partition Equilibrium
•Assumes that the set of players is partitioned: ∏ =
{A1,…,Ak}
•Dictated by social groups that players belong to
•Only coalition formed by Ai∈∏ are allowed to weakly
deviate
•Partition equilibrium exists if no Ai∈∏ can weakly
deviate
∀A∈∏ ,∃i∈A:
ci(sA,s-A) < ci(sA’,s-A)
•We have a set of n identical players, and a
set of m resources
•Player chooses one resource from the set of
resources
•If x players select resource i: then cost for
each player = fi(x)
•Simple congestion game but leads to
interesting results: first step
•SE always exists but SSE may not
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Our Results
•Partition Equilibrium always exists for Resource
Selection Games
•Can be computed efficiently
•PE ∩ NE ≠ ∅
•Existence of Super-Strong equilibrium is
efficiently decidable
NE + PE !
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u
Type-1:
Type-2:
m1 m2
m3
m4
1 2
u
Type-1:
l1
l2
q = 2
•Theorem: Given a Nash equilibrium allocation A and a coalition Tk, let xi denote
the number of jobs of the coalition Tk allocated to machine i in A. Then coalition
Tk does not have an improving deviation if for every high machine i such that xi ≥ 2
the following conditions are satisfied:
-for every low machine j such that lj > li, we have that xj ≥ xi and
-for every low machine j such that lj ≤ li, we have that xj ≥ xi − 1.
1 2
u
q = 3
3
l1 > l2 => x1 ≥ x2
l2 ≥ l3 => x3 ≥ x2 -1
x2=3
x1=3
x3=2
Computing Partition Equilibrium
•Algorithm:
-Begin with empty allocation
-Order active machines on non-increasing li value
-For a given coalition, place jobs sequentially with rollover
-Remove machine when it has mi jobs
-When q machines are high, place remaining jobs arbitrarily to make the
rest low
Coalition 1: 6 players Coalition 2: 4 players q = 2
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u
1 2 43
u
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u
1 2 43
u
Active machines: 1, 2, 3, 4 Active machines: 1, 3, 4
Equilibrium for RSG •2 resources and 3 players
•Load on machine: fi(x) = x