Learning to Identify Winning Coalitions in the PAC ModelA. 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.

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 (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}

{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: AB. • Sperner’s Theorem: F = antichain of subsets of {1,..,n}. Then

2/||

n

nF

{}

{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:

2/)dim( *

n

nCVC

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.

2/)1(

1dim

n

nVC

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!