Learning to Identify Winning Coalitions in the PAC Model

1
Learning to Identify Winning Coalitions in the
PAC Model

• A. D. Procaccia J. S. Rosenschein

2
Lecture Outline

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

3
Simple Cooperative Games

• Cooperative n-person game def (Nv). N1,,n
  is the set of players, v2N?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
4
PAC Model

• Sample space X wish to learn target concept
  cX?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(?,?).

5
VC-Dimension

• X sample space, C contains functions cX?0,1.
• Sx1,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.

6
VC Dimension Example
X R, Cf ?a,b s.t. f(x)1 iff x is in a,b

• X sample space, C contains functions cX?0,1.
• Sx1,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.

7
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.

8
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
9
VC-dim(C)

• 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

• F is antichain if ?A,B in F A?B.
• Sperners Theorem F antichain of subsets of
  1,..,n. Then

10
Restricted Simple Games

• Dictator
• Single minimum winning coalition with one player.
  
• VC-dim ?logn?.
• Junta Coalition
• Single minimum winning coalition.
• VC-dim n.

11
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.

12
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!