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Pareto Optimality in Coalition Formation
Haris Aziz Felix Brandt Paul Harrenstein
Technische Universität München
IJCAI Workshop on Social Choice and Artificial Intelligence, July 16, 2011
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Coalition formation
“Coalition formation is of fundamental importance in a wide variety ofsocial, economic, and political problems, ranging from communicationand trade to legislative voting. As such, there is much about theformation of coalitions that deserves study.”
A. Bogomolnaia and M. O. Jackson. The stability of hedonic coalitionstructures. Games and Economic Behavior. 2002.
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Coalition formation
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Hedonic Games
A hedonic game is a pair (N,�) where N is a set of players and�= (�1, . . . ,�|N|) is a preference profile which specifies for each player i ∈ N hispreference over coalitions he is a member of.
For each player i ∈ N, �i is reflexive, complete and transitive.
A partition π is a partition of players N into disjoint coalitions.
A player’s appreciation of a coalition structure (partition) only dependson the coalition he is a member of and not on how the remainingplayers are grouped.
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Classes of Hedonic Games
Unacceptable coalition: player would rather be alone.
General hedonic games: preference of each player over acceptablecoalitions
1 :({1, 2, 3} , {1, 2} , {1, 3} | {1} ‖
)2 :({1, 2} | {1, 2, 3} , {1, 3} , {2} ‖
)3 :({2, 3} | {3} ‖ {1, 2, 3} , {1, 3}
)Partition {{1}, {2, 3}}
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Classes of Hedonic Games
General hedonic games: preference of each player over acceptablecoalitions
Preferences over players extend to preferences over coalitions
Roommate games: only coalitions of size 1 and 2 are acceptable.
W-hedonic games: preference over coalitions only depends on the worstplayers in the coalitions
B-hedonic games: preference over coalitions only depends on the bestplayers in the coalitions
Other hedonic settings: anonymous games, 3-cyclic games, room-roommategames, house allocation.
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Classes of Hedonic Games
In a W-hedonic game, each player i has preferences over otherplayers and i’s preference of a coalition S containing i depends on theworst players in S \ {i}.
Example (W-hedonic game)
1 : (3 , 2 | 1 ‖ )2 : (1 | 3 , 2 ‖ )3 : (2 | 3 ‖ 1)
1 :({1, 2, 3} , {1, 2} , {1, 3} | {1} ‖
)2 :({1, 2} | {1, 2, 3} , {1, 3} , {2} ‖
)3 :({2, 3} | {3} ‖ {1, 2, 3} , {1, 3}
)
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Individual Rationality & Pareto Optimality
“The requirement that a feasible outcome be undominated via one-person coalitions (individual rationality) and via the all-person coalition(efficiency or Pareto optimality) is thus quite compelling.”
R. J. Aumann. Game Theory. The New Palgrave Dictionary ofEconomics. 1987
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Individual Rationality
An outcome is individual rationality (IR) if each player is at least as happy as bybeing alone.
1 :({1, 2, 3} , {1, 2} , {1, 3} | {1} ‖
)2 :({1, 2} | {1, 2, 3} , {1, 3} , {2} ‖
)3 :({2, 3} | {3} ‖ {1, 2, 3} , {1, 3}
)
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Pareto Optimality
Vilfredo Pareto (1848–1923)
An outcome is Pareto optimal (PO) if there exists no outcome in which eachplayer is at least as happy and and at least one player is strictly happier.
A minimal requirement for desirable outcomes
An IR & PO partition is guaranteed to exist
Can also be seen as a notion of stability
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Contributions
Relate Pareto optimality to ‘perfection’
A general algorithm — Preference Refinement Algorithm (PRA) — tocompute a PO and IR partition
A general way to characterize the complexity of computing and verifying aPO partition
A number of specific computational results for various hedonic settings
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Is Serial Dictatorship the Panacea?
Serial Dictatorship to compute a PO outcome: An arbitrary player is chosen asthe ‘dictator’ who is then given his most favored allocation and the process isrepeated until all players have been dealt with.
1 :({1, 2, 3} | {1, 2} | {1, 3} | {1} ‖
)2 :({1, 2} | {1, 2, 3} | {1, 3} | {2} ‖
)3 :({2, 3} | {3} ‖ {1, 2, 3} , {1, 3}
)If preferences over coalitions are not strict, then serial dictatorship does notwork
Even if preferences over players are strict, preferences over coalitions mayinclude ties
Does not return every Pareto optimal partition even if preferences overcoalitions are strict
Serial dictatorship can be ‘unfair’
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Preference Refinement Algorithm (PRA)
PRA Serial Dictatorship
can simulate Serial Dictatorshipcan handle ties cannot handle ties‘complete’ cannot return every PO partition‘fairer’ ‘less fair’
Table: PRA vs. Serial Dictatorship
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Perfection
A partition is perfect if each player is in one of his most favored coalitions.
PerfectPartition is the problem of checking the existence of a perfect partition.
1 :({1, 2, 3} , {1, 2} , {1, 3} | {1} ‖
)2 :({1, 2} | {1, 2, 3} , {1, 3} , {2} ‖
)3 :({2, 3} | {3} ‖ {1, 2, 3} , {1, 3}
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Preference Refinement Algorithm (PRA)
1 : (3 , 2 , 1)2 : (1 , 3 , 2)3 : (2 , 3 ‖ 1) 3
1 : (3 , 2 | 1)2 : (1 , 3 , 2)3 : (2 , 3 ‖ 1) 3
1 : (3 , 2 , 1)2 : (1 | 3 , 2)3 : (2 , 3 ‖ 1) 3
1 : (3 , 2 , 1)2 : (1 , 3 , 2)3 : (2 | 3 ‖ 1) 3
1 : (3 , 2 | 1)2 : (1 | 3 , 2)3 : (2 , 3 ‖ 1) 3
1 : (3 , 2 | 1)2 : (1 , 3 , 2)3 : (2 | 3 ‖ 1) 7
1 : (3 , 2 , 1)2 : (1 | 3 , 2)3 : (2 | 3 ‖ 1) 7
1 : (3 , 2 | 1)2 : (1 | 3 , 2)3 : (2 | 3 ‖ 1) 7
Figure: Running PRA on a W-hedonic game where N = {1, 2, 3} and1 : (3 , 2 | 1)2 : (1 | 3 , 2)3 : (2 | 3 ‖ 1)
.
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Preference Refinement Algorithm (PRA)
Input: Hedonic game (N,�)Output: Pareto optimal and individually rational partition
1 Qi ← Coarsest acceptable coarsening of �i for all i ∈ N2 Q ← (Q1, . . . ,Qn)3 J ← N4 while J , ∅ do5 i ∈ J6 Use Divide & Conquer to find some Q ′i better than Qi s.t.
PerfectPartition(N, (Q1, . . . ,Qi−1,Q ′i ,Qi+1, . . . ,Qn)) exists.7 if such a Q ′i exists then
8 Q ← (Q1, . . . ,Qi−1,Q ′i ,Qi+1, . . . ,Qn)9 else
10 J ← J \ {i}11 end if12 end while13 return PerfectPartition(N,Q)
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General Technique To Prove Tractability
LemmaLet (N,R) be a hedonic game, for which the following conditions hold:
any coarsening of R can be computed in polynomial time, and
PerfectPartition can be solved in polynomial time for the coarsening.
Then, PRA runs in polynomial time
(even if each equivalence class has an exponential number of coalitions orthere are an exponential number of equivalence classes!)
TheoremA Pareto optimal and individually rational outcome can be computed efficiently for
W-hedonic games
Roommate games
House-allocation with existing tenants
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General Technique To Prove Tractability
LemmaLet (N,R) be a hedonic game, for which the following conditions hold:
any coarsening of R can be computed in polynomial time, and
PerfectPartition can be solved in polynomial time for the coarsening.
Then, PRA runs in polynomial time(even if each equivalence class has an exponential number of coalitions orthere are an exponential number of equivalence classes!)
TheoremA Pareto optimal and individually rational outcome can be computed efficiently for
W-hedonic games
Roommate games
House-allocation with existing tenants
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General Technique To Prove Tractability
LemmaLet (N,R) be a hedonic game, for which the following conditions hold:
any coarsening of R can be computed in polynomial time, and
PerfectPartition can be solved in polynomial time for the coarsening.
Then, PRA runs in polynomial time(even if each equivalence class has an exponential number of coalitions orthere are an exponential number of equivalence classes!)
TheoremA Pareto optimal and individually rational outcome can be computed efficiently for
W-hedonic games
Roommate games
House-allocation with existing tenants
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W-hedonic games
Core stable partition may not exist
Checking whether a core stablepartition exists isNP-hard [Cechlárová andHajduková, 2004]
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W-hedonic games
Computing a PO & IR partition isin P: utilize PRA and show thatPerfectPartition is in P
Polynomial-time reduction fromPerfectPartition to clique packingfor reduced graph
Need to check whether verticescan partitioned into cliques of size2 or more
Sufficient to check whether thevertices can be partitioned intocliques of size 2 or 3.
Hell and Kirkpatrick [1984] andCornuéjols et al. [1982] presenteda P-time algo which achieves theabove
1 : (3 , 2 | 1)2 : (1 | 3 , 2)3 : (2 , 3 ‖ 1)
1
1′
1′′
2
2′
2′′
3
3′
3′′
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General Technique To Prove Intractability
LemmaFor every class of hedonic games for which verifying a perfect partition is in P,NP-hardness of PerfectPartition implies NP-hardness of computing a Paretooptimal partition.
TheoremComputing a Pareto optimal partition is NP-hard for
general hedonic games
B-hedonic games
anonymous hedonic games
three sided matching with cyclic preferences games
room-roommate games
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Conclusions
PRA (Preference Refinement Algorithm) to compute PO outcomes.
PerfectPartition is intractable⇒ PO is intractable
PerfectPartition is solvable for different coarsenings⇒ PO can be solved.
Game Verification Computation
General coNP-C NP-hardGeneral (strict) coNP-C in PRoommate in P in PB-hedonic coNP-C (weak PO) NP-hardW-hedonic in P in PAnonymous coNP-C NP-hardRoom-roommate coNP-C (weak PO) NP-hard3-cyclic coNP-C (weak PO) NP-hardHouse allocation in P in Pw. existing tenants
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Conclusions
PRA (Preference Refinement Algorithm) to compute PO and IR outcomes.
PerfectPartition is intractable⇒ PO is intractable
PerfectPartition is solvable for different coarsenings⇒ PO can be solved.
Game Verification Computation
General coNP-C NP-hardGeneral (strict) coNP-C in PRoommate in P in PB-hedonic coNP-C (weak PO) NP-hardW-hedonic in P in PAnonymous coNP-C NP-hardRoom-roommate coNP-C (weak PO) NP-hard3-cyclic coNP-C (weak PO) NP-hardHouse allocation in P in Pw. existing tenants
THANK YOU!
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Conclusions
PRA (Preference Refinement Algorithm) to compute PO and IR outcomes.
PerfectPartition is intractable⇒ PO is intractable
PerfectPartition is solvable for different coarsenings⇒ PO can be solved.
Game Verification Computation
General coNP-C NP-hardGeneral (strict) coNP-C in PRoommate in P in PB-hedonic coNP-C (weak PO) NP-hardW-hedonic in P in PAnonymous coNP-C NP-hardRoom-roommate coNP-C (weak PO) NP-hard3-cyclic coNP-C (weak PO) NP-hardHouse allocation in P in Pw. existing tenants
THANK YOU!
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References
K. Cechlárová and J. Hajduková. Stable partitions with W-preferences. DiscreteApplied Mathematics, 138(3):333–347, 2004.
G. Cornuéjols, D. Hartvigsen, and W. Pulleyblank. Packing subgraphs in a graph.Operations Research Letters, 1(4):139–143, 1982.
P. Hell and D. G. Kirkpatrick. Packings by cliques and by finite families of graphs.Discrete Mathematics, 49(1):45–59, 1984.
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