Learning to Identify Winning Coalitions in the PAC Model
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This lecture examines the PAC model's application to cooperative games, focusing on identifying winning coalitions. The discussion includes the motivation for studying cooperative games within multiagent communities, the concept of minimal winning coalitions, and the theoretical foundations, such as VC-dimension and sample complexity. Key topics include simple cooperative games, sample space definitions, and the implications of Sperner’s Theorem on coalition formation. The talk concludes with remarks on the practical challenges of learning these games, highlighting the need for effective algorithms and implementations.
Learning to Identify Winning Coalitions in the PAC Model
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Presentation Transcript
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: AB. • 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!
