The Communication Complexity of Coalition
Formation Among Autonomous Agents
A. 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.
Coalition
• v(S) is the value of coalition S.
s
• Payoffs to players are x=(x1,…,xn).
• Coalition structure =(S1,…, Sr) =def partition
of N.
• Payoff configuration (x; ), s.t. j=1,…,r:
x( S j )   xi  v( S j )
iS j
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.
In talk
Shapley value.
The nucleolus.
Equal excess theory.
A horde of others.
In paper
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.
I
a(00)=0
a(01)=0
a(10)=0
a(11)=1
b(00)=0
II
b(01)=0
0
b(10)=0
b(11)=1
0
1
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:
n


  (n)
log 
 n / 2  1
•  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.
Assume grand coalition forms
• i0=argmax{x
},
S
=
all
players
with
w
=1
and
ii
.
i
i
0
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.