Structural holes in social networks

We consider a setting where every pair of players that undertake a transaction (e.g. exchange goods or information) creates a unit surplus. A transaction can take place only if the players involved have a connection. If the connection is direct the two players split the surplus equally, while if it is indirect then intermediate players also get an equal share of the surplus. Thus, individuals form links with others to create surplus, to gain intermediation rents, and to circumvent others who are trying to become intermediary.Our analysis clarifies the interplay between these forces in the process of strategic network formation. First, we show that, in the absence of capacity constraints on links, it leads to the emergence of a star network where a single agent acts as an intermediary for all transactions and enjoys significantly higher payoffs. Second, we study the implications of capacity constraints in the ability of agents to form links. In this case, distances between players must be long, which induces players who are “far off” to connect in order to avoid paying large intermediation rents. A cycle network then emerges, payoffs being equal across all players.     

Problems of fair division and the Egalitarian solution

The problem of fair division is considered in situations where the number of agents among whom the division is to take place may vary. The approach is axiomatic. Along with four familiar axioms, Weak Pareto-optimality, Symmetry, Independence of Irrelevant Alternatives, and Continuity, a new axiom, named Monotonicity with Respect to Changes in the Number of Agents, is imposed, expressing a certain form of solidarity among the agents as their number varies but the resources at their disposal remain fixed. The only solution to satisfy these axioms is the Egalitarian solution, which selects the only feasible alternative that yields equal utilities to all agents and is undominated by any other feasible alternative.     

A game-theoretic model of coalition formation among primates

This paper models the coalition formation process among primates as a sequential game. The population consists of individuals having distinct social ranks which is determined by the individual’s resource holding potential. Each member of the population is interested in gaining access to a food resource, either individually or via a coalition. At any given stage of the game, a player can either propose a specific coalition or he can be proposed to in order to join one. Hence, the strategy of a player consists of a sequence of decisions regarding who to propose to for the formation of a coalition and which proposals to accept or reject. We derive the preferences of the players over the various coalition structures under the assumption that the probability of a coalition to obtain the resource is given by a logistic distribution as a function of relative strengths of the players. We show that, given the primates’ strategic behavior, a variety of different coalition structures can emerge in equilibrium.      

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.     

Forming and Dissolving Partnerships in Cooperative Game Situations

A group of players in a cooperative game are partners (e.g., as in the form of a union or a joint ownership) if the prospects for cooperation are restricted such that cooperation with players outside the partnership requires the accept of all the partners. The formation of such partnerships through binding agreements may change the game implying that players could have incentives to manipulate a game by forming or dissolving partnerships. The present paper seeks to explore the existence of allocation rules that are immune to this type of manipulation. An allocation rule that distributes the worth of the grand coalition among players is called partnership formation‐proof if it ensures that it is never jointly profitable for any group of players to form a partnership and partnership dissolution‐proof if no group can ever profit from dissolving a partnership. The paper provides results on the existence of such allocation rules for general classes of games as well as more specific results concerning well‐known allocation rules.     

An experimental analysis of ultimatum bargaining

There are many experimental studies of bargaining behavior, but suprisingly enough nearly no attempt has been made to investigate the so-called ultimatum bargaining behavior experimentally. The special property of ultimatum bargaining games is that on every stage of the bargaining process only one player has to decide and that before the last stage the set of outcomes is already restricted to only two results. To make the ultimatum aspect obvious we concentrated on situations with two players and two stages. In the ‘easy games’ a given amount c has to be distributed among the two players, whereas in the ‘complicated games’ the players have to allocate a bundle of black and white chips with different values for both players. We performed two main experiments for easy games as well as for complicated games. By a special experiment it was investigated how the demands of subjects as player 1 are related to their acceptance decisions as player 2.     

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.     

Valuations and dynamics of negotiations

This paper analyzes three‐party negotiations in the presence of externalities. We obtain a closed‐form solution for the Markov perfect equilibrium of a multilateral non‐cooperative bargaining model, yielding an equilibrium value and dynamics of negotiations that are supported by experimental studies. Players’ values are monotonically increasing (or decreasing) in the amount of negative (or positive) externalities that they impose on others. Moreover, players’ values are continuous and piecewise linear on the worth of bilateral coalitions, and are inextricably related to their negotiation strategies: the equilibrium value is the Nash bargaining solution when no bilateral coalitions form; the Shapley value when all bilateral coalitions form; or the nucleolus, when either one bilateral coalition among “natural partners” or two bilateral coalitions including a “pivotal player” form.     

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.     

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.     

Majority Rule in a Stochastic Model of Bargaining

In this paper we consider multilateral stochastic bargaining models with general agreement rules. For n-player games where in each period a player is randomly selected to allocate a stochastic level of surplus and q?n players have to agree on a proposal to induce its acceptance, we characterize the set of stationary subgame perfect equilibrium payoffs and establish their existence. We show that for agreement rules other than the unanimity rule, the equilibrium payoffs need not be unique. Furthermore, even when the equilibrium is unique, it need not be efficient. Journal of Economic Literature Classification Numbers: C73, C78, D70.     

Conservative belief and rationality

Playersʼ beliefs may be incompatible, in the sense that player i can assign probability 1 to an event E to which player j assigns probability 0. One way to block incompatibility is to assume a common prior. We consider here a different approach: we require playersʼ beliefs to be conservative, in the sense that all players must ascribe the actual world positive probability. We show that common conservative belief of rationality (CCBR) characterizes strategies in the support of a subjective correlated equilibrium where all playersʼ beliefs have common support. We also define a notion of strong rationalizability, and show that it is characterized by CCBR.     

Representing equilibrium aggregates in aggregate games with applications to common agency

An aggregate game is a normal-form game with the property that each playerʼs payoff is a function of only his own strategy and an aggregate of the strategy profile of all players. Such games possess properties that can often yield simple characterizations of equilibrium aggregates without requiring that one solves for the equilibrium strategy profile. When payoffs have a quasi-linear structure and a degree of symmetry, we construct a self-generating maximization program over the space of aggregates with the property that the solution set corresponds to the set of equilibrium aggregates of the original n-player game. We illustrate the value of this approach in common-agency games where the playersʼ strategy space is an infinite-dimensional space of nonlinear contracts. We derive equilibrium existence and characterization theorems for both the adverse selection and moral hazard versions of these games.     

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.     

Gambling in contests

This paper presents a strategic model of risk-taking behavior in contests. Formally, we analyze an n-player winner-take-all contest in which each player decides when to stop a privately observed Brownian motion with drift. A player whose process reaches zero has to stop. The player with the highest stopping point wins. Unlike the explicit cost for a higher stopping time in a war of attrition, here, higher stopping times are riskier, because players can go bankrupt. We derive a closed-form solution of a Nash equilibrium outcome. In equilibrium, highest expected losses occur at an intermediate negative value of the drift.     

Measuring Inequalities without Linearity in Envy: Choquet Integrals for Symmetric Capacities

The (generalized) Gini indices rely on the social welfare function of a decision maker who behaves in accordance with Yaari's model, with a function f that transforms frequencies. This SWF can also be represented as the weighted sum of the welfare of all the possible coalitions in the society, where the welfare of a coalition is defined as the income of the worst-off member of that coalition. We provide a set of axioms (Ak) and prove that the three following statements are equivalent: (i) the decision maker respects (Ak), (ii) f is a polynomial of degree k, (iii) the weight of all coalitions with more than k members is equal to zero. Journal of Economic Literature Classification Number: D63.     

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.     

On choosing which game to play when ignorant of the rules

This paper suggests a theory of choice among strategic situations when the rules of play are not properly specified. We take the view that a “strategic situation” is adequately described by a TU game since it specifies what is feasible for each coalition but is silent on the procedures that are used to allocate the surplus. We model the choice problem facing a decision maker (DM) as having to choose from finitely many “actions”. The known “consequence” of the ith action is a coalition from game f _i over a fixed set of players \(N_i\cup\{d\}\) (where d stands for the DM). Axioms are imposed on her choice as the list of consequences (f ₁,..., f _m ) from the m actions varies. We characterize choice rules that are based on marginal contributions of the DM in general and on the Shapley Value in particular.     

Multilateral Bargaining Problems

In many situations in economics and political science there are gains from forming coalitions but conflict over which coalition to form and how to distribute the gains. This paper presents an approach to suchmultilateral bargaining problems. Asolutionto a multilateral bargaining problem specifies an agreement for each coalition that is consistent with the bargaining process in every coalition. We establish the existence of such solutions, show that they are determined by reservation prices, and characterize these reservation prices as the payoffs ofsubgame perfect equilibrium outcomesof a non-cooperative bargaining model.Journal of Economic LiteratureClassification Numbers: C71, C72, C78.     

Collusive dominant-strategy truthfulness

We show that collusion and wrong beliefs may cause a dramatic efficiency loss in the Vickrey mechanism for auctioning a single good in limited supply. We thus put forward a new mechanism guaranteeing efficiency in a very adversarial collusion model, where the players can partition themselves into arbitrarily many coalitions, exchange money with each other, and perfectly coordinate their actions. Our mechanism bypasses classic impossibility results (such as those of Green and Laffont, and of Schummer) by providing the players with a richer set of strategies, making it dominant for every coalition C to instruct each of its members to report truthfully not only his own valuation, but also his belonging to C. Our mechanism is coalitionally rational, which implies being individually rational for independent players.