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:2 N →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. 2 N is partitioned into W and L, s.t. 1.  in L. 2.N in W. 3.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 (x i,c(x i )) 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 m C ( ,  ).

VC-Dimension X = sample space, C contains functions c:X  {0,1}. S={x 1,…x m },  C (S) = def {(c(x 1 ),...,c(x m )): c in C} S is shattered by C iff |  C (S)|=2 m. 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} {1,2,3}{1,2,4}{1,3,4}{2,3,4} {1,2,3,4} {3,4}

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

Restricted Simple Games Dictator: Single minimum winning coalition with one player. VC-dim =  logn . Junta: 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; general games are much harder. Monotone DNF formulae are equivalent to minimum winning coalitions. Need to find implementation. Algorithms included!