A bargaining approach to the Owen value and the Nash solution with coalition structure

The mechanism by Hart and Mas-Colell (1996) for non-transferable utility (NTU) games is generalized so that a coalition structure among players is taken into account. The new mechanism yields the Owen value for transferable utility (TU) games with coalition structure as well as the consistent value (Maschler and Owen 1989, 1992) for NTU games with trivial coalition structure. Furthermore, we obtain a solution for pure bargaining problems with coalition structure which generalizes the Nash (1950) bargaining solution.     

On the Least Core and the Mas-Colell Bargaining Set

We show that the least core of a TU coalitional game with a finite set of players is contained in the Mas-Colell bargaining set. This result is extended to games with a measurable space of players in which the worth of the grand coalition is at least that of any other coalition in the game. As a consequence, we obtain an existence theorem for the Mas-Colell bargaining set in TU games with a measurable space of players. Journal of Economic Literature Classification Number: C71.     

A Theory of Gradual Coalition Formation

We study noncooperative multilateral bargaining games, based on underlying TU games, in which coalitions can renegotiate their agreements. We distinguish between models in which players continue to bargain after implementing agreements ("reversible actions") and models in which players who implement agreements must leave the game ("irreversible actions"). We show that renegotiation always results in formation of the grand coalition if actions are reversible, but that the process may otherwise end with smaller coalitions. On the other hand, we show that the grand coalition cannot form in one step if the core of the game is empty, irrespective of the reversibility of actions.     

On core stability, vital coalitions, and extendability

If a TU game is extendable, then its core is a stable set. However, there are many TU games with a stable core that are not extendable. A coalition is vital if there exists some core element x such that none of the proper subcoalitions is effective for x. It is exact if it is effective for some core element. If all coalitions that are vital and exact are extendable, then the game has a stable core. It is shown that the contrary is also valid for matching games, for simple flow games, and for minimum coloring games.     

On core stability,vital coalitions,and extendability

If a TU game is extendable, then its core is a stable set. However, there are many TU games with a stable core that are not extendable. A coalition is vital if there exists some core element x such that none of the proper subcoalitions is effective for x. It is exact if it is effective for some core element. If all coalitions that are vital and exact are extendable, then the game has a stable core. It is shown that the contrary is also valid for matching games, for simple flow games, and for minimum coloring games.     

Excluded coalitions and the distribution of power in parliaments

In this article, we introduce a new value for cooperative games. This value is based on the Shapley (1953) value and takes into account that players exclude coalitions with other players. One example of such exclusions are the coalition statements of parliamentary parties. A case study demonstrates the application of the new value for these situations.     

Games of Status

A status game is a cooperative game in which the outcomes are rank orderings of the players. They are a good model for certain situations in which players care about how their "status" compares with that of other players.
We present several formal models within this class. Included are authoritarian status games (where coalitions may assign positions in the rank ordering to nonmembers) and oligarchic status games (where they are unableto do so). We consider the issues of a value concept for authoritarian games and that of core existence for oligarchic games. We then add a transferable resource to the models, obtaining "games of wealth and status."
Finally, we consider an interesting variant, called a "secession game," where coalitions have the right to secede from the grand coalition and form their own smaller "subsocieties," each with its own hierarchy.     

Stability when mobility is restricted by the existing coalition structure

In many social and economic situations the optimal solution requires the formation of coalitions that partition the set of players. When the individual player is small relative to the size of the existing coalitions, it seems realistic to assume that the prevailing coalition structure dictates the set of possible blocking coalitions. Specifically, it is assumed that an individual does not consider forming any coalition, but rather joining an already existing one. Two solution concepts for these games are investigated: structural equilibrium and stable payoffs, which are derived from the application of ψ-stability to the core and to the bargaining set, respectively. To this end an extension of the bargaining set to games without side payments is offered. Both solution concepts are shown to exist for some coalition structure. However, while structural equilibrium may fail to exist for any non trivial coalition structure, for every coalition structure there exists a stable payoff.     

The vector lattice structure of the n-person TU games

We show that any cooperative TU game is the maximum of a finite collection of a specific class of the convex games: the almost positive games. These games have non-negative dividends for all coalitions of at least two players. As a consequence of the above result we show that the class of modular games is a set of generators of the distributive lattice of all cooperative TU games. Finally, we characterize zero-monotonic games using a strong max-convex decomposition.     

Stability and Largeness of the Core

In general, there are examples of TU games where the core is stable but is not large. In this paper, we show that the extendability condition introduced by Kikuta and Shapley (1986, “Core Stability in n-Person Games,” Mimeo) is sufficient for the core to be stable as well as large, for TU games with five or fewer players. We provide a counter example when the number of players is six. We then introduce a stronger extendability condition and show that it is necessary and sufficient for the core to be large. Our proof makes use of a well-known result from the theory of convex sets. Journal of Economic Literature Classification Number: C71.     

On the stability of coalition structures

We resolve a seeming conflict between a non-existence result on solutions to coalition formation in hedonic games [Barberà, S., Gerber, A., 2007. A note on the impossibility of a satisfactory concept of stability for coalition formation games. Economics Letters 95, 85–90] and the universal existence of stable coalition structures in TU games under the χ-value [Casajus, A., 2008. Outside options, component efficiency, and stability, Games and Economic Behavior (forthcoming). doi: 10.1016/j.geb.2007.04.003].     

Cooperative games in permutational structure

Summary. By a cooperative game in coalitional structure or shortly coalitional game we mean the standard cooperative non-transferable utility game described by a set of payoffs for each coalition being a nonempty subset of the grand coalition of all players. It is well-known that balancedness is a sufficient condition for the nonemptiness of the core of such a cooperative non-transferable utility game. In this paper we consider non-transferable utility games in which for any coalition the set of payoffs depends on a permutation or ordering upon any partition of the coalition into subcoalitions. We call such a game a cooperative game in permutational structure or shortly permutational game. Doing so we extend the scope of the standard cooperative game theory in dealing with economic or political problems. Next we define the concept of core for such games. By introducing balancedness for ordered partitions of coalitions, we prove the nonemptiness of the core of a balanced non-transferable utility permutational game. Moreover we show that the core of a permutational game coincides with the core of an induced game in coalitional structure, but that balancedness of the permutational game need not imply balancedness of the corresponding coalitional game. This leads to a weakening of the conditions for the existence of a nonempty core of a game in coalitional structure, induced by a game in permutational structure. Furthermore, we refine the concept of core for the class of permutational games. We call this refinement the balanced-core of the game and show that the balanced-core of a balanced permutational game is a nonempty subset of the core. The proof of the nonemptiness of the core of a permutational game is based on a new intersection theorem on the unit simplex, which generalizes the well-known intersection theorem of Shapley. Received: October 31, 1995; revised version: February 5, 1997     

A Risk‐Dominant Allocation: Maximizing Coalition Stability

This paper presents a rule to allocate a coalition’s worth for superadditive games with positive externalities. The allocation rule awards each member their outside payoff, plus an equal share of the surplus. The resulting allocation maximizes coalition stability. Stable coalitions are Strong Nash equilibria since no subset of members has an incentive to leave. Similarly, no subset of non‐members has an incentive to join a stable coalition if the game is concave in this region. The allocation is risk‐dominant. All stable coalitions are robust to the maximum probability of 50% that players’ deviate from their individual best‐responses. The paper compares the allocation to the Shapley value and the Nash bargaining solution, and illustrates why these traditional rules result in small coalitions when applied to issues such as international environmental agreements.     

Theories of coalitional rationality

This paper generalizes the concept of best response to coalitions of players and offers epistemic definitions of coalitional rationalizability in normal form games. The (best) response of a coalition is defined to be an operator from sets of conjectures to sets of strategies. A strategy is epistemic coalitionally rationalizable if it is consistent with rationality and common certainty that every coalition is rational. A characterization of this solution set is provided for operators satisfying four basic properties. Special attention is devoted to an operator that leads to a solution concept that is generically equivalent to the iteratively defined concept of coalitional rationalizability.     

On weighted Kalai–Samet solutions for non-transferable utility coalitional form games

We investigate the implications of the axiom of coalitional concavity for non-transferable utility coalitional form games. This axiom says that if the feasible set of some coalition is uncertain whereas the feasible sets of other coalitions are known, then all players in the coalition with the uncertain feasible set should (weakly) benefit from reaching a compromise before the uncertainty is resolved. By imposing this axiom, in addition to other minor axioms, we characterize the weighted Kalai–Samet [Econometrica 53 (1985) 307] solutions: these solutions coincide with the weighted egalitarian solutions on the domain of bargaining problems, and with the weighted Shapley values on the domain of transferable utility coalitional form games.     

The bargaining set of a large game

We study the equivalence between the Mas-Colell bargaining set and the core in the general context of TU games with a measurable space of players. In the first part of the paper, we study the problem without imposing any restriction on the class of games we consider. In the second part, we first introduce a new class of exact games, which we call thin games. For these games, we show not only that the Mas-Colell bargaining set is equal to the core, but also that it is the unique stable set in the sense of von Neumann and Morgenstern. We then study the relation between thin games, exact non-atomic market games and non-atomic convex games. Finally, by further developing “thinness” related ideas, we prove new equivalence results for a class of non-exact market games as well as a class of non-exact, non-market games.     

Lexicographic composition of simple games

Certain voting bodies can be modeled as a simple game where a coalition's winning depends on whether it wins, blocks or loses in two smaller simple games. There are essentially five such ways to combine two proper games into a proper game. The most decisive is the lexicographic rule, where a coalition must either win in G₁, or block in G₁ and win in G₂. When two isomorphic games are combined lexicographically, a given role for a player confers equal or more power when held in the first game than the second, if power is assessed by any semi-value. A game is lexicographically separable when the players of the two components partition the whole set. Games with veto players are not separable, and games of two or more players with identical roles are separable only if decisive. Some separable games are egalitarian in that they give players identical roles.     

Coalition Formation and Potential Games

In this paper we study the formation of coalition structures in situations described by a cooperative game. Players choose independently which coalition they want to join. The payoffs to the players are determined by an allocation rule on the underlying game and the coalition structure that results from the strategies of the players according to some formation rule. We study two well-known coalition structure formation rules and show that for both formation rules there exists a unique component-efficient allocation rule that results in a potential game. Journal of Economic Literature Classification Numbers: C71, C72.     

Global public goods and coalition formation under matching mechanisms

This paper links coalition theory with matching mechanisms in the presence of global public goods among heterogeneous players. This matching coalition may achieve Pareto‐improving outcomes while avoiding side payments. The paper characterizes conditions of coalition profitability and stability at both interior and corner equilibria. It is generally much harder to satisfy stability conditions than profitability conditions. A matching coalition is more profitable but less stable with a larger matching rate. Empirically there is no stable coalition but this can be overcome by introducing reputation mechanisms. There always exists a stable grand matching coalition if players value their reputation. The matching coalition faces a trade‐off between matching depth and breadth.     

Demand bargaining and proportional payoffs in majority games

We study a majoritarian bargaining model in which players make payoff demands in decreasing order of voting weight. The unique subgame perfect equilibrium outcome is such that the minimal winning coalition of the players that move first forms with payoffs proportional to the voting weights. This result advances previous analysis in terms of one or more of the following: a) the simplicity of the extensive form (finite horizon with a predetermined order of moves); b) the range of the majority games covered; c) the equilibrium concept (subgame perfect equilibrium is sufficient for a unique prediction).