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

1 / 21 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.

2 / 21 Coalition formation

3 / 21 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.

4 / 21 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}k 2: {1, 2}|{ 1, 2, 3} , {1, 3} , {2}k 3: {2, 3}|{ 3}k{ 1, 2, 3} , {1, 3} Partition {{1}, {2, 3}}

5 / 21 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.

6 / 21 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 k ) 2:(1 | 3 , 2 k ) 3:(2 | 3 k 1) 1: {1, 2, 3} , {1, 2} , {1, 3}|{ 1}k 2: {1, 2}|{ 1, 2, 3} , {1, 3} , {2}k 3: {2, 3}|{ 3}k{ 1, 2, 3} , {1, 3}

7 / 21 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

8 / 21 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}k 2: {1, 2}|{ 1, 2, 3} , {1, 3} , {2}k 3: {2, 3}|{ 3}k{ 1, 2, 3} , {1, 3}

9 / 21 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

10 / 21 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

11 / 21 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}k 2: {1, 2}|{ 1, 2, 3}|{ 1, 3}|{ 2}k 3: {2, 3}|{ 3}k{ 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’

12 / 21 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

13 / 21 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}k 2: {1, 2}|{ 1, 2, 3} , {1, 3} , {2}k 3: {2, 3}|{ 3}k{ 1, 2, 3} , {1, 3}

14 / 21 Preference Reﬁnement Algorithm (PRA)

1:(3 , 2 | 1) 2:(1 | 3 , 2) 3:(2 | 3 k 1)

1:(3 , 2 | 1) 1:(3 , 2 | 1) 1:(3 , 2 , 1) 2:(1 | 3 , 2) 2:(1 , 3 , 2) 2:(1 | 3 , 2) 3:(2 , 3 k 1) 3:(2 | 3 k 1) 3:(2 | 3 k 1)

1:(3 , 2 | 1) 1:(3 , 2 , 1) 1:(3 , 2 , 1) 2:(1 , 3 , 2) 2:(1 | 3 , 2) 2:(1 , 3 , 2) 3:(2 , 3 k 1) 3:(2 , 3 k 1) 3:(2 | 3 k 1)

1:(3 , 2 , 1) 2:(1 , 3 , 2) 3:(2 , 3 k 1)

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 k 1)

15 / 21 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 0 6 Use Divide & Conquer to ﬁnd some Qi better than Qi s.t. 0 PerfectPartition(N, (Q1,..., Qi−1, Qi , Qi+1,..., Qn)) exists. 0 7 if such a Qi exists then 0 8 Q ← (Q1,..., Qi−1, Qi , Qi+1,..., Qn) 9 else 10 J ← J \{i} 11 end if 12 end while 13 return PerfectPartition(N, Q)

16 / 21 (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

General Technique To Prove Tractability

17 / 21 Theorem A Pareto optimal and individually rational outcome can be computed efﬁciently for W-hedonic games Roommate games House-allocation with existing tenants

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!)

17 / 21 General Technique To Prove Tractability

Theorem A Pareto optimal and individually rational outcome can be computed efﬁciently for W-hedonic games Roommate games House-allocation with existing tenants

17 / 21 W-hedonic games

Core stable partition may not exist Checking whether a core stable partition exists is NP-hard [Cechlárová and Hajduková, 2004]

18 / 21 W-hedonic games

Computing a PO & IR partition is 1:(3 , 2 | 1) in P: utilize PRA and show that 2:(1 | 3 , 2) PerfectPartition is in P 3:(2 , 3 k 1) Polynomial-time reduction from 0 PerfectPartition to clique packing 2 for reduced graph Need to check whether vertices 200 can partitioned into cliques of size 2 2 or more 10 Sufﬁcient to check whether the vertices can be partitioned into 1 cliques of size 2 or 3.

Hell and Kirkpatrick [1984] and 100 Cornuéjols et al. [1982] presented 3 a P-time algo which achieves the 30 above

300

19 / 21 General Technique To Prove Intractability

Lemma For every class of hedonic games for which verifying a perfect partition is in P, NP-hardness of PerfectPartition implies NP-hardness of computing a Pareto optimal partition. Theorem Computing 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

20 / 21 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-hard General (strict) coNP-C in P Roommate in P in P B-hedonic coNP-C (weak PO) NP-hard W-hedonic in P in P Anonymous coNP-C NP-hard Room-roommate coNP-C (weak PO) NP-hard 3-cyclic coNP-C (weak PO) NP-hard House allocation in P in P w. existing tenants

21 / 21 THANK YOU!

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.

22 / 21 Conclusions

THANK YOU!

22 / 21 References

K. Cechlárová and J. Hajduková. Stable partitions with W-preferences. Discrete Applied 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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