Learning to Identify Winning Coalitions in the PAC Model

Learning to Identify Winning Coalitions in the PAC Model A. D. Procaccia & J. S. Rosenschein

Lecture Outline • Cooperative Games • Learning: • PAC model • VC dimension • Motivation • Results • Closing Remarks

Simple Cooperative Games • Cooperative n-person game =def (N;v). N={1,…,n} is the set of players, v:2N→R. • v(C) is the value of coalition C. • Simple games: v is binary-valued. C is winning if v(C)=1, losing if v(C)=0. • 2N is partitioned into W and L, s.t. •  in L. • N in W. • Superset of winning coalition is winning. Coalitions

PAC Model • Sample space X; wish to learn target concept c:X{0,1} in concept class C. • Pairs (xi,c(xi)) given, according to a fixed distribution on X. • Produce concept but allow mistakes: • Probability  that learning algorithm fails. • -approximation of target concept. • How many samples are needed? Sample Complexity mC(,).

VC-Dimension • X = sample space, C contains functions c:X{0,1}. • S={x1,…xm}, C(S) =def {(c(x1),...,c(xm)): c in C} • S is shattered by C iff |C(S)|=2m. • VC-dim(C) =def size of largest set shattered by C. • VC dimension yields upper and lower bounds on sample complexity of concept class.

VC Dimension: Example X = sample space, C contains functions c:X{0,1}. S={x1,…xm}, C(S)={c(x1),...,c(xm): c in C} S is shattered by C if |C(S)|=2m. VC-dim(C) = size of largest set shattered by C. X = R, C={f: a,b s.t. f(x)=1 iff x is in [a,b]}

Motivation • Multiagent community shows interest in learning, but almost all work is reinforcement learning. • Cooperative games are interesting in multiagent context. • Real world simple cooperative games settings: • Parliament. • Advisers.

Minimum Winning Coalitions • Simple cooperative games defined by sets of minimum winning coalitions. • X = coalitions, C* = sets of minimum winning coalitions. {} {1} {2} {3} {4} {1,2} {1,3} {1,4} {2,3} {2,4} {3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2,3,4}

VC-dim(C*) • F is antichain if A,B in F: AB. • Sperner’s Theorem: F = antichain of subsets of {1,..,n}. Then • Theorem: {} {1} {2} {3} {4} {1,2} {1,3} {1,4} {2,3} {2,4} {3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2,3,4}

Restricted Simple Games • Dictator: • Single minimum winning coalition with one player. • VC-dim = logn. • Junta Coalition: • Single minimum winning coalition. • VC-dim = n.

Restricted Simple Games II • Proper games: • C is winning  N\C is losing. • It holds that: • Elimination of dummies: • i C s.t. C is winning but C\{i} is losing. • Same lower bound.

Closing Remarks • Easy to learn simple games with dictator or junta coalition; general games are much harder. • Monotone DNF formulae are equivalent to minimum winning coalitions. • Need to find implementation. Algorithms included!

Learning to Identify Winning Coalitions in the PAC Model