Post on 21-Dec-2015
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