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;:::; jNj) is a preference profile which specifies for each player i 2 N his preference over coalitions he is a member of. For each player i 2 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: f1; 2; 3g ; f1; 2g ; f1; 3gjf 1gk 2: f1; 2gjf 1; 2; 3g ; f1; 3g ; f2gk 3: f2; 3gjf 3gkf 1; 2; 3g ; f1; 3g Partition ff1g; f2; 3gg 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 n fig. Example (W-hedonic game) 1:(3 ; 2 j 1 k ) 2:(1 j 3 ; 2 k ) 3:(2 j 3 k 1) 1: f1; 2; 3g ; f1; 2g ; f1; 3gjf 1gk 2: f1; 2gjf 1; 2; 3g ; f1; 3g ; f2gk 3: f2; 3gjf 3gkf 1; 2; 3g ; f1; 3g 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: f1; 2; 3g ; f1; 2g ; f1; 3gjf 1gk 2: f1; 2gjf 1; 2; 3g ; f1; 3g ; f2gk 3: f2; 3gjf 3gkf 1; 2; 3g ; f1; 3g 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: f1; 2; 3gjf 1; 2gjf 1; 3gjf 1gk 2: f1; 2gjf 1; 2; 3gjf 1; 3gjf 2gk 3: f2; 3gjf 3gkf 1; 2; 3g ; f1; 3g 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: f1; 2; 3g ; f1; 2g ; f1; 3gjf 1gk 2: f1; 2gjf 1; 2; 3g ; f1; 3g ; f2gk 3: f2; 3gjf 3gkf 1; 2; 3g ; f1; 3g 14 / 21 Preference Refinement Algorithm (PRA) 1:(3 ; 2 j 1) 2:(1 j 3 ; 2) 3:(2 j 3 k 1) 7 1:(3 ; 2 j 1) 1:(3 ; 2 j 1) 1:(3 ; 2 ; 1) 2:(1 j 3 ; 2) 2:(1 ; 3 ; 2) 2:(1 j 3 ; 2) 3:(2 ; 3 k 1) 3 3:(2 j 3 k 1) 7 3:(2 j 3 k 1) 7 1:(3 ; 2 j 1) 1:(3 ; 2 ; 1) 1:(3 ; 2 ; 1) 2:(1 ; 3 ; 2) 2:(1 j 3 ; 2) 2:(1 ; 3 ; 2) 3:(2 ; 3 k 1) 3 3:(2 ; 3 k 1) 3 3:(2 j 3 k 1) 3 1:(3 ; 2 ; 1) 2:(1 ; 3 ; 2) 3:(2 ; 3 k 1) 3 Figure: Running PRA on a W-hedonic game where N = f1; 2; 3g and 1:(3 ; 2 j 1) 2:(1 j 3 ; 2) . 3:(2 j 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 2 N 2 Q (Q1;:::; Qn) 3 J N 4 while J , ; do 5 i 2 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 n fig 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 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 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 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 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 j 1) in P: utilize PRA and show that 2:(1 j 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.