A Review on Multilateral Bargaining - Part 2

In my last post I reviewed Aumann's approaches on Multilateral Bargaining Problems[1]. He presents that a multilateral bargaining problem has a bargaining set as a kind of solution. However, the computation on finding the bargaining set is complicated and Aumann did not give an method to find such a solution in a general n-player game.

Elaine Bennett had made a great improvement on this problem[2]. Bennett proposed a solution and interpret it as a set of imputations for every potential coalition. In the imputation, the partition of utilities is related to a bargaining function f^S and agents' outside options d^S. I will explain this in detail through the example below.

Example 2.1(the example 2 in [1]) v(1)=v(2)=v(3)=v(123)=0, v(12)=20,v(23)=30,v(13)=100
Aumann gave the bargaining set for this problem such that

{(0,0,0;1,2,3), 
(20≤x1≤70,0,100-x1;13,2), 
(20,0,0;12,3)
(0,0,30;1,23)}

The outside option is the utility a player i would obtain if he broke off negotiation in a coalition S and took the initiative to form his best alternative coalition. Therefore, the player 1's outside option in coalition {13} is 20 (if he can form a coalition with player 2) and the outside option in {12} is x1. 

Therefore "the agreement in coalition S depends on the agreements in other coalitions which depend in turn on the outside options in these other coalitions, which depend in turn on the agreement in the coalition S." (like a tongue twister...) And this caused the interrelationships between bargaining problems of the severals coalitions. 

Now for instants let the bargaining function f works in a way that divides the share equally, therefore x1=45. and we then have a Bennett's solution for example 2.1. We notice that player 1's outside option for coalition {12} is 45, which is infeasible for the coalition {12}, then {12} will not form. 

Bennett had proved that there is at least one of the solution agreements is feasible in a multilateral bargaining problem.

However, a solution is not an outcome. Imaging that in example 2.1 if player 1 and 2 form a coalition, the coalition structure is broken off and player 3 will not form a coalition any more. 

Moreover, the actual outcome may be mattered if the coalitions cannot simultaneously form. 
Example 2.2(the example 3 in [2]) v(12)=40,v(23)=34,v(34)=20,v(14)=34,v(S)=0 for other S.
One solution to example 2.2 is (22,22,12,12;14,23), however, if player 4 wait for player 2 and 3 to form a coalition and leave, then he can bargain with player 1 without a viable outside option. Then player 4 will yield a payoff of 17. 

(It too late now and I have to stop here and go to sleep... I will continue a bit more on this paper in my next update:p)


[1] Aumann, Robert J. The bargaining set for cooperative games. Defense Technical Information Center, 1961.
[2] Bennett, Elaine. "Multilateral bargaining problems." Games and Economic Behavior 19.2 (1997): 151-179.