Learning to Identify Winning Coalitions in the PAC Model - PowerPoint PPT Presentation

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Learning to Identify Winning Coalitions in the PAC Model

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Learning to Identify Winning Coalitions in the PAC Model. A. D. Procaccia ... It holds that: Elimination of dummies: i C s.t. C is winning but C{i} is losing. ... – PowerPoint PPT presentation

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Title: 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!
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