Download - Pareto Optimality in Coalition Formationhaziz/po_slides.pdf · Pareto Optimality An outcome is Pareto optimal (PO) if there exists no outcome in which each player is at least as happy

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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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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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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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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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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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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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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