Overlapping coalitions, bargaining and networks

I am grateful to an anonymous referee who provided me with this information.

There exist several models of non-cooperative implementation of some specific cooperative game solution concepts as the core or the Shapley value (see for instance Kalai et al. 1979; Chatterjee et al. 1993; Lagunoff 1994; Perry and Reny 1994; Serrano 1995; Hart and Mas-Colell 1996; Serrano and Vohra 1997). However, all these studies are limited to disjoint coalitions while our setup allows coalitions to overlap.

D. Ray (2009, Chapter 14, Sect. 14.4, pp 294–296) discussed extensively the complexity of this issue.

The expression embedded coalition is also used by Myerson (1977), Macho-Stadler et al. (2007), Diamantoudi et al. (2015), among others.

I am grateful to an anonymous referee who suggested this version of the definition of a cover function.

In a setting where coalitions can overlap, it is natural to allow each player to propose as much coalitions as desired. Although the game proposed in this study can accommodate multiple proposals, it is not obvious how to model the corresponding responses, especially in the case of rejections. One way to deal with that is to make assumptions to avoid rejections on an equilibrium path, with the counterpart that the existence of equilibrium is not guaranteed. Thus, it is not easy, at least in our setting, to model multiple proposals.

Allowing for externalities adds a complexity in the formulation of the division of the conditional worth of a coalition as a player has to propose a sharing corresponding to each possible compatible cover. This is always the case in games with externalities, as the worth of a coalition is conditioned on a whole coalition structure (see for instance, Bloch 1996; Yi 1997; Ray and Vohra 1999; Hafalir 2007; Bloch and Van den Nouweland 2014). This complexity can be avoided by getting rid of externalities. However, we cannot afford that here, as the setting of overlapping coalitions looses all interest without externalities. A good discussion on the role of externalities for overlapping coalitions can be found in the study by Ray (2007), Page 294, Paragraph 5. After all, the number of compatible covers decreases sharply as the bargaining continues and the additional complexity due to externalities almost vanishes.

In this transferable utility game, the worth of a coalition can be viewed as money and the gain of a player as a cash payoff. Thus, the natural way to think of the total gain of a player who belongs to, for example, three distinct coalitions is the sum of the cash payoff that he earns from each of the coalitions.

Note that, in the paper, the sharing of the coalitional worth is endogenous. However, Binmore et al. (1985) showed that, in real life, people choose equal division as an obvious compromise.

For a complete survey on network formation, see Jackson (2005).

Other approaches have been used in the literature to bridge networks and cooperative game theory. For example, in situations where links in networks generate value, cooperative game theory solution concepts like the core, nucleolus, kernel, and others are used to predict and construct the optimal sets of market cooperation by rational actors (see for instance Bienenstock and Bonacich (1997); Bateni et al. (2010)). While those papers intended to generate the optimal networks, we construct a bargaining game of coalition formation (cover function game) to rationalize a given network as an optimal solution. Thus, our approach is different.

By degenerated game, we mean the game in which for each embedded coalition \((S,\gamma )\in {\varSigma }'\) (if the cover function is defined on \({\varSigma }'\)), we have \(v(S,\gamma )=0\).

Note that the concept of equilibrium described in Proposition 1 allows to choose a coalition to propose randomly.

Without any loss of generality, it suffices that player i makes an unacceptable proposal to only a single player j. This is because the rejection of a proposal is a consequence of the action of the first rejector.

Notice that \(x^{l}_{S}\) depends on i but for simplicity we suppress the indexation.