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.     

ON THE NEUTRALITY OF COALITION FORMATION IN A PURE BARGAINING PROBLEM*

We consider the sequential bargaining game à la Stahl–Binmore–Rubinstein with random proposers, juxtaposing an ex ante coalition formation stage to their bargaining game. On the basis of the expected outcomes in the negotiation over how to split a dollar, players can form coalitions in a sequential manner, within each of which they can redistribute their payoffs. It turns out that the grand coalition does form, and that each player receives his discounted expected payoff, which is obtained by playing as a single player in the negotiation, although there could be many equilibria in the bargaining stage.     

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.     

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.     

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.     

Mixed-strategy equilibria in the Nash Demand Game

In the Nash Demand Game, each of the two players announces the share he demands of an amount of money that may be split between them. If the demands can be satisfied, they are; otherwise, neither player receives any money. This game has many pure-strategy equilibria. This paper characterizes mixed-strategy equilibria. The condition critical for an equilibrium is that players’ sets of possible demands be balanced. Two sets of demands are balanced if each demand in one set can be matched with a demand in the other set such that they sum to one. For Nash’s original game, a complete characterization is given of the equilibria in which both players’ expected payoffs are strictly positive. The findings are applied to the private provision of a discrete public good.     

A decomposition algorithm for N-player games

An N-player game can be decomposed by adding a coordinator who interacts bilaterally with each player. The coordinator proposes profiles of strategies to the players, and his payoff is maximized when players’ optimal replies agree with his proposal. When the feasible set of proposals is finite, a solution of an associated linear complementarity problem yields an equilibrium of the approximate game and thus an approximate equilibrium of the original game. Computational efficiency is improved by using vertices of a triangulation of the players’ strategy space for the coordinator’s pure strategies. Computational experience is reported.     

‘Give me a chance!’ An experiment in social decision under risk

This paper reports the results of a ‘probabilistic dictator game’ experiment in which subjects were given an option to share chances to win a prize with a dummy player. Using a within-subject design we manipulated two aspects of the decision, the relative cost of sharing and the nature of the lottery: the draws were either independent for the two players (‘noncompetitive’ condition) or one’s success meant other’s failure (‘competitive’ condition). We also asked for decisions in a standard, non-probabilistic, setting. The main results can be summarized as follows: first, a substantial fraction of subjects do share chances to win, also in the competitive treatments, thus showing concern for the other player that cannot be explained by outcome-based models. Second, subjects share less in the competitive treatment than in other treatments, indicating that procedural fairness alone cannot explain the data. Overall, these results suggest that models aiming at generalizing social concerns to risky environments will have to rely on a mix of distributive and procedural fairness.     

Coordination of strategic responses to security threats: Laboratory evidence

Laboratory experiments are used to evaluate the extent to which players in games can coordinate investments that diminish the probability of losses due to security breaches or terrorist attacks. In this environment, economically sensible investments may be foregone if their potential benefits are negated by failures to invest in security at other sites. The result is a coordination game with a desirable high-payoff, high-security equilibrium and an undesirable low-security equilibrium that may result if players do not expect others to invest in security. One unique feature of this coordination situation is that investment in security by one player generates a positive externality such that all other players’ expected payoffs are increased, regardless of those other players’ investment decisions. Coordination failures are pervasive in a baseline experiment with simultaneous decisions, but coordination is improved if players are allowed to move in an endogenously determined sequence. In addition, coordinated security investments are observed more often when the largest single security threat to individuals is preventable by their own decisions to invest in security. The security coordination game is a “potential game,” and the success of coordination on the more secure equilibrium is related to the notion of potential function maximization and basin of attraction.      

Veto power in committees: an experimental study

Veto power consists of the right of one or more players to unilaterally block decisions but without the ability to unilaterally secure their preferred outcome. Our experiment shows that (i) committees with a veto player take longer to reach decisions (are less efficient) and generate less consensus than without a veto player, (ii) veto power substantially enhances proposer’s power, and (iii) non-veto players are substantially more willing to compromise than veto players. We relate our results to the theoretical literature on the impact of veto power as well as to concerns about the impact of veto power in real-life committees.     

The monoclus of a coalitional game

The analysis of single-valued solution concepts, providing payoffs to players for the grand coalition only, has a long tradition. Opposed to most of this literature we analyze allocation scheme rules, which assign payoffs to all players in all coalitions. We introduce several closely related allocation scheme rules, each resulting in a population monotonic allocation scheme (PMAS) whenever the underlying coalitional game with transferable utilities has a PMAS. Monotonicities, which measure the payoff difference for a player between two nested coalitions, are the driving force. These monotonicities can best be compared with the excesses in the definition of the (pre-)nucleolus. Variants are obtained by considering different domains and/or different collections of monotonicities. We deal with nonemptiness, uniqueness, and continuity, followed by an analysis of conditions for (some of) the rules to coincide. We then focus on characterizing the rules in terms of subbalanced weights. Finally, we deal with computational issues.     

Values for Markovian coalition processes

Time series of coalitions (so-called scenarios) are studied that describe processes of coalition formation where several players may enter or leave the current coalition at any point in (discrete) time and convergence to the grand coalition is not necessarily prescribed. Transitions from one coalition to the next are assumed to be random and to yield a Markov chain. Three examples of such processes (the Shapley-Weber process, the Metropolis process, and an example of a voting situation) and their properties are presented. A main contribution includes notions of value for such series, i.e., schemes for the evaluation of the expected contribution of a player to the coalition process relative to a given cooperative game. Particular processes permit to recover the classical Shapley value. This methodology??s power is illustrated with well-known examples from exchange economies due to Shafer (Econometrica 48:467?C476, 1980) and Scafuri and Yannelis (Econometrica 52:1365?C1368, 1984), where the classical Shapley value leads to counterintuitive allocations. The Markovian process value avoids these drawbacks and provides plausible results.     

On the core of economies with multilateral environmental externalities

We revisit the cooperative model of coalition formation in economies with environmental externalities. Motivated by recent concerns over the true behavior and incentives of key players in international negotiations over the climate and the environment, we construct a cooperative game where the members of each coalition have uncertainty over the behavior of the nonmembers, and in particular they face uncertainty over their coalition structure. As a result, a coalition assigns various probability distributions over the set of partitions the outsiders can form. We compute the payoff of each coalition under this assumption and we derive conditions under which the core of the induced cooperative game is nonempty.     

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.     

Strongly stable networks

We analyze the formation of networks among individuals. In particular, we examine the existence of networks that are stable against changes in links by any coalition of individuals. We show that to investigate the existence of such strongly stable networks one can restrict focus on a component-wise egalitarian allocation of value. We show that when such strongly stable networks exist they coincide with the set of efficient networks (those maximizing the total productive value). We show that the existence of strongly stable networks is equivalent to core existence in a derived cooperative game and use that result to characterize the class of value functions for which there exist strongly stable networks via a “top convexity” condition on the value function on networks. We also consider a variation on strong stability where players can make side payments, and examine situations where value functions may be non-anonymous—depending on player labels.     

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     

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.     

Equilibrium play in matches: Binary Markov games

We study two-person extensive form games, or “matches,” in which the only possible outcomes (if the game terminates) are that one player or the other is declared the winner. The winner of the match is determined by the winning of points, in “point games.” We call these matches binary Markov games. We show that if a simple monotonicity condition is satisfied, then (a) it is a Nash equilibrium of the match for the players, at each point, to play a Nash equilibrium of the point game; (b) it is a minimax behavior strategy in the match for a player to play minimax in each point game; and (c) when the point games all have unique Nash equilibria, the only Nash equilibrium of the binary Markov game consists of minimax play at each point. An application to tennis is provided.     

Core-stable rings in auctions with independent private values

We propose a semi-cooperative game theoretic approach to check whether a given coalition is stable in a Bayesian game with independent private values. The ex ante expected utilities of coalitions, at an incentive compatible (noncooperative) coalitional equilibrium, describe a (cooperative) partition form game. A coalition is core-stable if the core of a suitable characteristic function, derived from the partition form game, is not empty. As an application, we study collusion in auctions in which the bidders? final utility possibly depends on the winner?s identity. We show that such direct externalities offer a possible explanation for cartels? structures (not) observed in practice.     

Null or nullifying players: The difference between the Shapley value and equal division solutions

A famous solution for cooperative transferable utility games is the Shapley value. Most axiomatic characterizations of this value use some axiom related to null players, i.e. players who contribute zero to any coalition. We show that replacing null players with nullifying players characterizes the equal division solution distributing the worth of the ‘grand coalition’ equally among all players. A player is nullifying if every coalition containing this player earns zero worth. Using invariance we provide similar characterizations of the equal surplus division solution assigning to every player its own worth, and distributing the remaining surplus equally among all players.