1. The Communication Complexity of Coalition Formation Among Autonomous Agents A. D. Procaccia & J. S. Rosenschein

2. Lecture Outline Coalition formation Cooperative games Solution concepts Communication Complexity Model Fooling Set Motivation Results Conclusions

3. Cooperative Games Cooperative n-person game = def (N;v). N={1,…,n} is the set of players, v:2 N →R. v(S) is the value of coalition S. Payoffs to players are x=(x 1,…,x n ). Coalition structure  =(S 1,…, S r ) = def partition of N. Payoff configuration (x;  ), s.t.  j=1,…,r: Coalition s

4. 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

5. 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.

6. Communication Complexity Player i holds private input z i. Goal: compute binary-valued function f(z 1,…,z n ). 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.

7. 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 b(00)=0 b(01)=0 b(10)=0 b(11)=1 0 01 I II

8. Fooling Set A set H of input vectors is a fooling set for f iff: 1.  (z 1,…,z n ) in H, f(z 1,…,z n ) = f 0. 2.For every two distinct vectors z,z’  mix of coordinates s.t. image is 1-f 0 ; e.g. f(z 1,z 2 ’,z 3 ’,…)=1-f 0. Lemma: fooling set of size m  lower bound of log(m) on communication complexity.

9. 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.

10. 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?

11. Lower Bound for the Core Lemma:  Sufficient to produce fooling set of this size. Weighted majority: [q;w 1,...,w n ]. 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. i 0 =argmax{x i }, S = all players with w i =1 and i  i 0. Assume grand coalition forms n=4,n’=3,q=2

12. Lower Bound for the Core II i 0 S

13. Lower Bound for the Core III n=7, n’=4, q=3 z1z1 z2z2 z3z3 z4z4 z5z5 z6z6 z7z7 z1z1 z2z2 z3z3 z4z4 z5z5 z6z6 z7z7

14. 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.