Farsightedly stable networks

A set of networks G is pairwise farsightedly stable (i) if all possible farsighted pairwise deviations from any network g G to a network outside G are deterred by the threat of ending worse off or equally well off, (ii) if there exists a farsighted improving path from any network outside the set leading to some network in the set, and (iii) if there is no proper subset of G satisfying conditions (i) and (ii). A non-empty pairwise farsightedly stable set always exists. We provide a full characterization of unique pairwise farsightedly stable sets of networks. Contrary to other pairwise concepts, pairwise farsighted stability yields a Pareto dominant network, if it exists, as the unique outcome. Finally, we study the relationship between pairwise farsighted stability and other concepts such as the largest pairwise consistent set and the von Neumann–Morgenstern pairwise farsightedly stable set.     

Farsightedly stable networks

A set of networks G is pairwise farsightedly stable (i) if all possible farsighted pairwise deviations from any network g∈ G to a network outside G are deterred by the threat of ending worse off or equally well off, (ii) if there exists a farsighted improving path from any network outside the set leading to some network in the set, and (iii) if there is no proper subset of G satisfying conditions (i) and (ii). A non-empty pairwise farsightedly stable set always exists. We provide a full characterization of unique pairwise farsightedly stable sets of networks. Contrary to other pairwise concepts, pairwise farsighted stability yields a Pareto dominant network, if it exists, as the unique outcome. Finally, we study the relationship between pairwise farsighted stability and other concepts such as the largest pairwise consistent set and the von Neumann–Morgenstern pairwise farsightedly stable set.     

Farsightedness in a Coalitional Great Fish War

We explore the implications of the farsightedness assumption on the conjectures of players in a coalitional Great Fish War model with symmetric players, derived from the seminal model of Levhari and Mirman (Bell J Econ 11:649–661, 1980). The farsightedness assumption for players in a coalitional game acknowledges the fact that a deviation from a single player will lead to the formation of a new coalition structure as the result of possibly successive moves of her rivals in order to improve their payoffs. It departs from mainstream game theory in that it relies on the so-called rational conjectures, as opposed to the traditional Nash conjectures formed by players on the behavior of their rivals. For values of the biological parameter and the discount factor more plausible than the ones used in the current literature, the farsightedness assumption predicts a wide scope for cooperation in non-trivial coalitions, sustained by credible threats of successive deviations that defeat the shortsighted payoff of any prospective deviator. Compliance or deterrence of deviations may also be addressed by acknowledging that information on the fish stock or on the catch policies actually implemented may be available only with a delay (dynamic farsightedness). In that case, the requirements are stronger and the sizes and number of possible farsighted stable coalitions are different. In the sequential move version, which could mimic some characteristics of fishery models, the results are not less appealing, even if the dominant player or dominant coalition with first move advantage assumption provides a case for cooperation with the traditional Nash conjectures.     

Coalitional stability under perfect foresight

Summary. We analyze strategic social environments where coalitions can form through binding or nonbinding agreements and actions of a coalition may impose externalities upon the welfare of the rest of the players. We define a solution concept that (1) captures the perfect foresight of the players that has been overlooked in the literature (e.g., Harsanyi [10] and Chwe [6]) and (2) identifies the coalitions that are likely to form and the “stable” outcomes that will not be replaced by any coalition of rational (and hence farsighted) players. The proposed solution concept thereby offers a notion of agreements and coalition formation in complex social environments. Received: February 12, 1996; revised version: March 3, 1997     

Networks and clubs

We formulate a club model where players’ have identical single-peaked preferences over club sizes as a network formation game. For situations with “many” clubs, we provide necessary and sufficient for non-emptiness of the farsighted core and the direct (or myopic) core. With “too few” clubs, if players are farsighted then the farsighted core is empty. In this same case, if players are myopic then the direct core is always nonempty and, for any club network in the direct core, clubs are of nearly equal size (i.e., clubs differ in size by at most one member).     

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.     

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.     

Coalition formation in general apex games under monotonic power indices

An apex game consists of one apex player and a set of minor players. We identify two key properties of apex games and use them to introduce the class of general apex games. We derive players' preferences over winning coalitions by applying strongly monotonic power indices on such a game and all its subgames and investigate whether there are core stable coalitions in the induced hedonic coalition formation game. Besides several general results, in particular, we develop conditions on the game for the Shapley–Shubik index, the Banzhaf index, and the normalized Banzhaf index.     

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.     

International Environmental Agreements—The Role of Foresight

The present paper attempts to bridge the gap between the cooperative and the non-cooperative approach employed to examine the size of stable coalitions, formed to address global environmental problems. We do so by endowing countries with foresightedness, that is, by endogenizing the reaction of the coalition’s members to a deviation by one member. We assume that when a country contemplates withdrawing or joining an agreement, it takes into account the reactions of other countries ignited by its own actions. We identify conditions under which there always exists a unique set of farsighted stable IEAs. The new farsighted IEAs can be much larger than those some of the previous models supported but are not always Pareto efficient.     

Voluntary Contributions to Multiple Public Projects

The problem of financing a set of discrete public goods (facilities, projects) by private contributions is studied. The corresponding cooperative game, the realization game , is shown to be convex. For the noncooperative setting we study a realization scheme that induces a strategic game. This contribution game is shown to be a generalized ordinal potential game ; a best–response in the contribution game implies a best response in a coordination game in which the payoff to all players is the utilitarian collective welfare function, i.e., the sum of the utility functions of the players. Strategy profiles maximizing utilitarian welfare are strong Nash equilibria of the contribution game. Each strong Nash equilibrium corresponds in a natural way with a core element of the realization game, and vice versa. Moreover, each strong Nash equilibrium is coalitional proof.     

Government versus Opposition: Who Should be Who in the 16th German Bundestag?

We model the process of coalition formation in the 16th German Bundestag as a hedonic coalition formation game. In order to induce players' preferences in the game we apply the Shapley value of the simple game describing all winning coalitions in the Bundestag. Using different stability notions for hedonic games we prove that the ``most' stable government is formed by the Union Parties together with the Social Democratic Party.     

Matching soulmates

We study iterated matching of soulmates (IMS), a recursive process of forming coalitions that are mutually preferred by members to any other coalition containing individuals as yet unmatched by this process. If all players can be matched this way, preferences are IMS-complete. A mechanism is a soulmate mechanism if it allows the formation of all soulmate coalitions. Our model follows Banerjee, Konishi, and Sönmez, except reported preferences are strategic variables. We investigate the incentive and stability properties of soulmate mechanisms. In contrast to prior literature, we do not impose conditions that ensure IMS-completeness. A fundamental result is that, (1) any group of players who could change their reported preferences and mutually benefit does not contain any players who were matched as soulmates and reported their preferences truthfully. As corollaries, (2) for any IMS-complete profile, soulmate mechanisms have a truthful strong Nash equilibrium, and (3) as long as all players matched as soulmates report their preferences truthfully, there is no incentive for any to deviate. Moreover, (4) soulmate coalitions are invariant core coalitions—that is, any soulmate coalition will be a coalition in every outcome in the core. To accompany our theoretical results, we present real-world data analysis and simulations that highlight the prevalence of situations in which many, but not all, players can be matched as soulmates. In the Appendix we relate IMS to other well-known coalition formation processes.     

Cleaner Technologies and the Stability of International Environmental Agreements

This paper shows that if countries are farsighted when deciding whether to defect from a coalition, then the implementation of cleaner technologies, as embodied by a reduction in the emission per output ratio, may either improve or jeopardize the chances of reaching an international environmental agreement. A small change in the emission per output ratio can result in a discrete jump in the stable size of a coalition and global welfare evaluated under the stable coalition size. In the case of three countries, the grand coalition may be destabilized by the implementation of cleaner technologies, ultimately resulting in higher global emissions and lower global welfare. In the case of more than three countries, implementing cleaner technologies may result in a discrete jump, either upward or downward, of the largest stable coalition size and welfare. We examine both, the case of a flow and that of a stock pollutant. In the latter case, we show that the higher the stock of pollution at the instant when the cleaner technology is implemented, the more likely that a grand coalition of three countries is destabilized. Measures that enhance the natural rate of decay of stock pollutants are shown to have similar effects on the size of stable coalitions to reductions in the emission per output ratio.     

FARSIGHTED COALITIONAL STABILITY OF A PRICE LEADERSHIP CARTEL

This paper analyzes the farsighted behaviour of firms that form a dominant price leadership cartel. We consider stability concepts such as the farsighted core, the farsighted stable sets, and the largest consistent set. We show that: (i) the farsighted core is either an empty set or a singleton set of the grand cartel; (ii) any Pareto efficient cartel is itself a farsighted stable set; and (iii) the set of cartels in which fringe firms enjoy higher profits than the firms in the minimal Pareto efficient cartel is the largest consistent set.     

The Stability of Hedonic Coalition Structures

We consider the partitioning of a society into coalitions in purely hedonic settings, i.e., where each player's payoff is completely determined by the identity of other members of her coalition. We first discuss how hedonic and nonhedonic settings differ and some sufficient conditions for the existence of core stable coalition partitions in hedonic settings. We then focus on a weaker stability condition: individual stability, where no player can benefit from moving to another coalition while not hurting the members of that new coalition. We show that if coalitions can be ordered according to some characteristic over which players have single-peaked preferences, or where players have symmetric and additively separable preferences, then there exists an individually stable coalition partition. Examples show that without these conditions, individually stable coalition partitions may not exist. We also discuss some other stability concepts, and the incompatibility of stability with other normative properties. Journal of Economic Literature Classification Numbers: C71, A14, D20.     

Large games: Fair and stable outcomes

We establish that for a broad class of large games with sidepayments, fair out-comes are nearly stable. More precisely, the Shapley value of a large game is in the ε-core and ε is very small if the game is very large. The proof uses two other results of independent interest: for large games the power of improvement is concentrated in small coalitions, and the Shapley value of a small syndicate acting together is nearly the sum of the Shapley values which accrue to the members acting alone.     

Sophisticated Experience-Weighted Attraction Learning and Strategic Teaching in Repeated Games

Most learning models assume players are adaptive (i.e., they respond only to their own previous experience and ignore others' payoff information) and behavior is not sensitive to the way in which players are matched. Empirical evidence suggests otherwise. In this paper, we extend our adaptive experience-weighted attraction (EWA) learning model to capture sophisticated learning and strategic teaching in repeated games. The generalized model assumes there is a mixture of adaptive learners and sophisticated players. An adaptive learner adjusts his behavior the EWA way. A sophisticated player rationally best-responds to her forecasts of all other behaviors. A sophisticated player can be either myopic or farsighted. A farsighted player develops multiple-period rather than single-period forecasts of others' behaviors and chooses to “teach” the other players by choosing a strategy scenario that gives her the highest discounted net present value. We estimate the model using data from p-beauty contests and repeated trust games with incomplete information. The generalized model is better than the adaptive EWA model in describing and predicting behavior. Including teaching also allows an empirical learning-based approach to reputation formation which predicts better than a quantal-response extension of the standard type-based approach. Journal of Economic Literature Classification Numbers: C72, C91.     

Pillage and property

This paper introduces a class of coalitional games, called pillage games, as a model of Hobbesian anarchy. Any coalition can pillage, costlessly and with certainty, any less powerful coalition. Power is endogenous, so a pillage game does not have a characteristic function, but pillage provides a domination concept that defines a stable set, which represents an endogenous balance of power. Every stable set contains only finitely many allocations, and can be represented as a farsighted core. Additional results are obtained for particular games, including the game in which the power of each coalition is determined by its total wealth.     

RISKY VERSUS RISKLESS BARGAINING PROCEDURES: THE AUMANN‐ROTH CONTROVERSY REVISITED*

In a series of papers, Aumann and Roth discussed a game in which players can cooperate in pairs and two of them prefer to form a coalition with each other. Roth argued that the only rational outcome is that the players who prefer each other form a coalition; Aumann argued that all three coalitions are possible because the players have a problem of expectation coordination. A non‐cooperative analysis provides additional support for Aumann's arguments and shows that the difference between Aumann's and Roth's views can be traced back to a difference (risky versus/riskless) in the bargaining procedure.