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I hate deadlock!
I hate deadlock!

I hate deadlock!

Well... I am doing a experiment on automatic coalition formation. It works fine until the deadlock comes...

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.

A Review on Multilateral Bargaining - Part 1

From this post, I will start to write a series of reviews on multilateral bargaining and coalition formation mechanisms. I would like to do this to resort some of my understanding on related topics.

This series will review some approaches on both economic (game theory approaches) and multi-agent systems.

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I think many people who are interested in game theory may have heard the name of Nash and Rubinstein, who have both made outstanding contribution to the game theory. The former one proposed the famous Nash bargaining solution and the later one designed the well-known Rubinstein Model. However, those two approaches are mainly developed in the field of the bilateral bargaining problem (even some parts can be extended to multilateral bargaining problem). For the multilateral bargaining problem, I noticed that there are also some great researchers doing some interesting and unique researches.

The most different between bilateral bargaining and multilateral bargaining is that the later one is effected by coalition matters and therefore some theories such as Nash bargaining solution will be not suitable any more. Instead, it is very necessary to take the coalition formation issue into consideration. 

The earliest paper I have found is [3] by Robert J. Aumann and Michael Maschler, which discussed how the coalition can be "stable" under n-player bargaining games. For a simple example, 

Example 1.1  In a bargaining game there are three players, labeled as 1, 2, 3. Any two of them can work together as a coalition. The coalition {1,2} and {1,3} can obtain 100 to work together, whilst coalition {2,3} can gain only 50. Otherwise, they can obtain nothing, such that

v(1)=v(2)=v(3)=0; v({1,2})=v({1,3})=100, v({2,3})=50; v({1,2,3})=0

In example 1.1, if player 1 proposes an utility partition (80,20,0) for coalition structure {{1,2},3},then player 2 can object by pointing out that (0,21,29) for {1,{2,3}}; on the other hand,  (75,25,0) {{1,2}.3} is stable, because an objection of player 2, e.g. (0,26,24) {1,{2,3}} can be met by a counter objection (75,0,25) {{1,3},2}.

Therefore, Aumann proposed a definition such that a stable coalition satisfies that for each "objection" of a subset of the coalition, there is a "counter objection" against the "objection".

In this paper[3], Aumman have enumerated such a lot examples for n-player games, n=2, 3, 4, 5, and in pretty good simpleness and clearness. This work has build a foundation for many later works.

The one who proposed a rigorous analysis is in Elaine Bennett's paper[4] (also a later version [5]). In his work, the interrelationships among coalitions became a key issue in multilateral bargaining games. Also the outside option has been discussed in depth. (Aumann has noticed there are some factors as an "alternatives"). I will write more on this in my next update.

Also I plan to write reviews on following topics in my future updates:

3C/3P games (one man out)

Nash stable

Bargaining solutions

Coalition formation mechanisms (in multi-agent systems)

Negotiation mechanisms (in multi-agent systems)

......

[1] Rubinstein, Ariel. "Perfect equilibrium in a bargaining model." Econometrica: Journal of the Econometric Society (1982): 97-109.

[2] Nash Jr, John F. "The bargaining problem." Econometrica: Journal of the Econometric Society (1950): 155-162.

[3] Aumann, Robert J. The bargaining set for cooperative games. Defense Technical Information Center, 1961.

[4] Bennett, Elaine. "Multilateral bargaining problems." Economic Department, University of California at Los Angeles WP594 (1986).

[5] Bennett, Elaine. "Multilateral bargaining problems." Games and Economic Behavior 19.2 (1997): 151-179.