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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 of social, economic, and political problems, ranging from communication and trade to legislative voting. As such, there is much about the formation of coalitions that deserves study.”

A. Bogomolnaia and M. O. Jackson. The stability of hedonic coalition structures. 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 his preference 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 depends on the coalition he is a member of and not on how the remaining players 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 acceptable coalitions

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 acceptable coalitions

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 worst players in the coalitions

B-hedonic games: preference over coalitions only depends on the best players in the coalitions

Other hedonic settings: anonymous games, 3-cyclic games, room-roommate games, house allocation.

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Classes of Hedonic Games

In a W-hedonic game, each player i has preferences over other players and i’s preference of a coalition S containing i depends on the worst 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 of Economics. 1987

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Individual Rationality

An outcome is individual rationality (IR) if each player is at least as happy as by being 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 each player 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) — to compute a PO and IR partition

A general way to characterize the complexity of computing and verifying a PO 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 as the ‘dictator’ who is then given his most favored allocation and the process is repeated 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 not work

Even if preferences over players are strict, preferences over coalitions may include ties

Does not return every Pareto optimal partition even if preferences over coalitions are strict

Serial dictatorship can be ‘unfair’

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Preference Refinement Algorithm (PRA)

PRA Serial Dictatorship

can simulate Serial Dictatorship can 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}

) 14 / 21

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} and 1 : (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 ∈ N 2 Q ← (Q1, . . . ,Qn) 3 J ← N 4 while J , ∅ do 5 i ∈ J 6 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 if 12 end while 13 return PerfectPartition(N,Q)

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General Technique To Prove Tractability

Lemma Let (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 or there are an exponential number of equivalence classes!)

Theorem A 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

Lemma Let (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 or there are an exponential number of equivalence classes!)

Theorem A 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

Lemma Let (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 or there are an exponential number of equivalence classes!)

Theorem A 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 stable partition exists is NP-hard [Cechlárová and Hajduková, 2004]

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W-hedonic games

Computing a PO & IR partition is in P: utilize PRA and show that PerfectPartition is in P

Polynomial-time reduction from PerfectPartition to clique packing for reduced graph

Need to check whether vertices can partitioned into cliques of size 2 or more

Sufficient to check whether the vertices can be partitioned into cliques of size 2 or 3.

Hell and Kirkpatrick [1984] and Cornuéjols et al. [1982] presented a P-time algo which achieves the above