# Coalition Formation among Farsighted Agents

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Coalition Formation

**Definition 1.**

**Definition 2.**

**Example 1.**

**Definition 3.**

**Example 2.**

## 3. Farsightedly Stable Sets of Coalition Structures

**Definition 4.**

- (
**i**) - ∀ $p\in P$, ∀ ${p}^{\prime}\notin P$ such that ${p}^{\prime}$ is obtainable from p via $S\subseteq N$, $\exists \phantom{\rule{0.166667em}{0ex}}{p}^{\u2033}\in F\left({p}^{\prime}\right)\cap P$ such that we do not have ${V}_{i}\left(p\right)\le {V}_{i}\left({p}^{\u2033}\right)$ for all $i\in S$ and ${V}_{i}\left(p\right)<{V}_{i}\left({p}^{\u2033}\right)$ for some $i\in S$.
- (
**ii**) - $\forall {p}^{\prime}\in \mathbb{P}\backslash P,$ $F\left({p}^{\prime}\right)\cap P\ne \u2300.$
- (
**iii**) - ∄ ${P}^{\prime}P$ such that ${P}^{\prime}$ satisfies Conditions (i) and (ii).

**Proposition 1.**

**Proposition 2.**

**Proposition 3.**

**Proposition 4.**

**Corollary 1.**

**Example 3.**

**Proposition 5.**

## 4. Coalition Formation with Positive Spillovers

**P.1**)

**P.2**)

**P.3**)

**Proposition 6.**

## 5. Conclusions

## Acknowledgements

## References

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## Appendix

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^{1.}Xue [3] has proposed the solution concepts of optimistic or conservative stable standards of behavior. It strengthens the farsightedness notion of the largest consistent set. A farsighted individual considers only the final outcomes that might result when making choices. But, an individual with perfect foresight considers also how final outcomes can be reached. That is, possible deviations along the way to the final outcomes should be considered. Barbera and Gerber [4] have proposed a solution concept for hedonic coalition formation games: durability. This concept assumes some form of maxmin behavior on the part of farsighted players.^{2.}For the coalitional contingent threat situation, Mariotti [5] has defined an equilibrium concept: the coalitional equilibrium. Central to his concept is the notion of coalitional strategies and the similarity with subgame perfection (except that coalitions are formally treated as players).^{3.}Konishi and Ray [8] have studied a model of dynamic coalition formation where players evaluate the desirability of a move in terms of its consequences on the entire discounted stream of payoffs.^{4.}Myopic notions of stability make assumptions about what a deviating coalition conjectures about the reaction of the non-deviating players. Hart and Kurz’s [1] notion of δ-stability assumes that non-deviating players do not move, while their notion of γ-stability supposes that former partners of the deviating players form singletons after the deviation. Hafalir [11] has investigated other rules of behavior where deviators hold the conjecture that either non-deviating players minimize the payoff of the deviating coalition, or non-deviating players merge, or non-deviating players will take the deviation as given and try to maximize their own payoff. For farsighted notions of stability such assumptions matter less to the extent that a deviating coalition considers the possibility that, once it deviates, another coalition might react, a third coalition might in turn react, and so on without limit. The behavior of reacting coalitions is endogenous.^{5.}Page and Wooders [15] have introduced the notion of path dominance core. In general, the path dominance core is contained in each farsightedly stable set. Page and Wooders [16] have analyzed the problem of club formation as a game of network formation and have identified stable club networks if players behave farsightedly in choosing their club memberships. They have found that, if there are too few clubs so that the average number of players per club is larger than the optimal club size, then the path dominance core is empty. Unlike myopic players, farsighted players may switch their club memberships to overcrowded clubs, temporarily becoming worse off, if they believe that switching might induce other members to leave those overcrowded clubs to make them ultimately better off. Thus, a non-empty path dominance core may fail to exist while a non-empty farsightedly stable set always exists.^{6.}Jackson and van den Nouweland [17] have proposed the myopic notion of strong stability which is the adaptation of δ-stability for network formation models.^{7.}Bogomolnaia and Jackson [18] have studied the partitioning of a society into coalitions in pure hedonic games, that is, in situations where the payoff to a player depends only on the composition of members of the coalition to which she belongs. They have looked for sufficient conditions for the existence of stable partitions if players are myopic. Diamantoudi and Xue [13] have analyzed the stability of partitions if players are farsighted. They have shown that, if a hedonic game satisfies the top-coalition property and preferences are strict, then the largest consistent set contains only the top-coalition partition and coincides with the unique von Neumann-Morgenstern farsightedly stable set. Hence, a singleton set consisting of the top-coalition partition is a farsightedly stable set.^{8.}Ray and Vohra [7] have provided a justification for the assumption of an equal sharing rule. In an infinite-horizon model of coalition formation among symmetric players with endogenous bargaining, they have shown that in any equilibrium without delay there is equal sharing.

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Herings, P.J.-J.; Mauleon, A.; Vannetelbosch, V. Coalition Formation among Farsighted Agents. *Games* **2010**, *1*, 286-298.
https://doi.org/10.3390/g1030286

**AMA Style**

Herings PJ-J, Mauleon A, Vannetelbosch V. Coalition Formation among Farsighted Agents. *Games*. 2010; 1(3):286-298.
https://doi.org/10.3390/g1030286

**Chicago/Turabian Style**

Herings, P. Jean-Jacques, Ana Mauleon, and Vincent Vannetelbosch. 2010. "Coalition Formation among Farsighted Agents" *Games* 1, no. 3: 286-298.
https://doi.org/10.3390/g1030286