I'm having trouble understanding the Bargaining Set, a Game Theory concept introduced by Aumann and Maschler back in 1964 and which has several variations in the literature. I see plenty of examples with $3$ Players, but not so many with $4$ and $5$ Players. Also, the few games with $4$ and $5$ Players that I found are not analysed in detail and are generally games in which the worth of each coalition depends only on its cardinality (which is not very interesting to me). I'm interested in understanding the Aumann and Maschler original definition of the Bargaining Set. Assuming a certain knowledge of cooperative Game Theory, I introduce the Bargaining Set below:

Objection: Let $x \in X(\mathcal{B};v)$ be an imputation, and let $k \neq l$ be two Players belonging to the same coalition in $\mathcal{B}$. An objection of Player $k$ against Player $l$ at $x$ is a pair $(C,y)$ such that:

(i) $C \subseteq N$ is a coalition containing $k$ but not $l$: $k \in C, l\notin C$;

(ii) $y \in \mathbb{R}^C$ s a vector of real numbers satisfying $y(C) = v(C)$, and $y_i > x_i$ for each Player $i \in C$.

Counter-objection: Let $(C,y)$ be an objection of Player $k$ against Player $l$ at $x$. Then, a counter-objection of Player $l$ against Player $k$ is a pair $(D,z)$ satisfying:

(i) $D$ is a coalition containing $l$ but not $k$: $l \in D$, $k \notin D$;

(ii) $z \in \mathbb{R}^D$, and $z(D) = v(D)$;

(iii) $z_i > x_i$ for every Player $i \in D \backslash C$;

(iv) $z_i \geqslant y_i$ for every Player $i \in D \cap C$.

Justified Objection: An objection $(C,y)$ of Player $k$ against Player $l$ is a justified objection if Player $l$ has no counter-objection $(D,z)$ to it.

Bargaining Set: Let $(N,v)$ be a coalition game, and $\mathcal{B}$ a coalitional structure. The Bargaining Set relative to the coalition structure $\mathcal{B}$ is the set $\mathcal{M}(\mathcal{B})$ of imputations in $X(\mathcal{B},v)$ at which no Player has a justified objection against any other Player in his coalition.

Therefore, my question is the following: Could anyone please provide a couple of cooperative games $(N,v)$ and their entire Aumann and Maschler Bargaining Set?

Such games $(N,v)$ should also:

(i) have $N=4$ and $N=5$;

(ii) have an empty core core for any coalitional structure;

(iii) have a characteristic function $v$ that does not exclusively depend on the cardinality of the coalitions.

Thank you all for your time.

EDIT: Partial answers and hints are of course welcome!


I am very sorry, but you are demanding too much. First of all, the bargaining set is a finite union of polytopes (bounded and closed convex sets), which makes it even from this point of view difficult to compute the entire bargaining set. Since such a set needs not to be convex! And finally more important, we have to consider a huge system of inequalities. In particular for four persons, we have to consider $150^{12}$ systems of polytopes with 41 inequalities each. To provide such an example would blow up the scope of this network. However, it is easier to figure out if a particular imputation $x$ is in the bargaining set, for the mentioned four person game, we have then just to check 197 inequalities (cf. Maschler, Solan, Zamir (2013, pp.788-794), Game Theory). To learn more about the bargaining set, I recommend to study a particular part of it, namely the kernel of a game.

  • $\begingroup$ Thank you very much for your explanation. Unless someone else comes up with an example like the one I ask for in a few hours, I'll accept your answer. I already read that part of the book you mention. But I was so surprised by it, that I tried to find the entire Bargaining Set for a "large" (4 players) game but it was impossible to me. Would you mind introducing me to the idea of the kernel, please? $\endgroup$ – Héctor Jan 11 '18 at 13:24
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    $\begingroup$ For the Kernel and Pre-Kernel have a look at the following book springer.com/gp/book/9783642395482. There you will also find some additional references. Moreover, from the computational view of the kernel, I recommend to use the Matlab toolbox MatTuGames that can be found at: mathworks.com/matlabcentral/fileexchange/…. Otherwise, grasp my e-mail from these sources, and we can start a private conversation about this topic. $\endgroup$ – Holger I. Meinhardt Jan 11 '18 at 13:40

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