The Communication Complexity of Coalition Formation Among Autonomous AgentsA. D. Procaccia & J. S. Rosenschein

Lecture Outline• Coalition formation

• Cooperative games• Solution concepts

• Communication Complexity• Model• Fooling Set

• Motivation

• Results

• Conclusions

Cooperative Games

• Cooperative n-person game =def (N;v). N={1,…,n} is the set of players, v:2N→R.

• v(S) is the value of coalition S.

• Payoffs to players are x=(x1,…,xn).

• Coalition structure =(S1,…, Sr) =def partition of N.

• Payoff configuration (x; ), s.t. j=1,…,r:)()( j

Siij SvxSx

j

Coalitions

Solution Concepts• Given coalition structure, wish to find payoff

division which is stable: agents are not motivated to deviate.

• Different notions of stability:• The core.• Shapley value.• The nucleolus.• Equal excess theory. • A horde of others.

In paper

In talk

Solution Concepts: The Core

• The core: C=def{(x; ): S, x(S) ≥ v(S)}

• No coalition can improve its payoff.• The core is sometimes empty.

Communication Complexity

• Player i holds private input zi.

• Goal: compute binary-valued function f(z1,…,zn).

• Players broadcast bits according to a protocol; in the end, all players know the value of f.

• Communication complexity: worst-case number of bits sent in best protocol.

• Ignore computations.

Communication Complexity: Example

• 2 players, each player holds 2 bits. Wish to determine whether all bits are 1.

a(00)=0

a(01)=0

a(10)=0

a(11)=1

a(00)=0

a(01)=0

a(10)=0

a(11)=1

b(00)=0

b(01)=0

b(10)=0

b(11)=1

b(00)=0

b(01)=0

b(10)=0

b(11)=1

0

0 1

I

II

Fooling Set• A set H of input vectors is a fooling set

for f iff:1. (z1,…,zn) in H, f(z1,…,zn) = f0.2. For every two distinct vectors z,z’ mix of

coordinates s.t. image is 1-f0; e.g. f(z1,z2’,z3’,…)=1-f0.

• Lemma: fooling set of size m lower bound of log(m) on communication complexity.

Motivation• Significant body of work on the

computational complexity of coalition formation.

• Virtually none on the communication complexity.

• Analysis of communication complexity particularly appropriate in this case.

Bounds• Each agent has constant info O(n) upper

bound.

• Lower bounds of (n) using fooling set: what is the function f?• The core: is nonempty?• Singleton solution concepts (Shapley, nucleolus,

equal excess): is the value of player 1 greater than 0?

Lower Bound for the Core• Lemma:

Sufficient to produce fooling set of this size.• Weighted majority: [q;w1,...,wn]. Values are

0/1, v(S)=1 iff sum of weights in S is at least q.• n’=n/2+1. H = all weighted majority games

with q=n’-1 and binary weights s.t. exactly n’ are 1.

• i0=argmax{xi}, S = all players with wi=1 and ii0.

)(

12/log n

n

n

Assume grand coalition forms

n=4,n’=3,q=2

Lower Bound for the Core II

i0 S

Lower Bound for the Core III

n=7, n’=4, q=3

z1 z2 z3 z4 z5 z6 z7

z1 z2 z3 z4 z5 z6 z7

Closing remarks• Results: tight bound of (n) on communication

complexity of four solution concepts. • May be a problem when communication is

severely restricted. • Future:

• Other lower bound methods for other solution concepts.

• Perhaps lower bound can be breached with respect to specific nontrivial games or environments.