Unique stability in simple coalition formation games

We investigate the uniqueness of stable coalition structures in a simple coalition formation model, for which specific coalition formation games, such as the marriage and roommate models, are special cases that are obtained by restricting the coalitions that may form. The main result is a characterization of collections of permissible coalitions which ensure that there is a unique stable coalition structure in the corresponding coalition formation model. In particular, we show that only single-lapping coalition formation models have a unique stable coalition structure for each preference profile, where single-lapping means that two coalitions cannot have more than one member in common, and coalitions do not form cycles. We also give another characterization using a graph representation, explore the implications of our results for matching models, and examine the existence of strategyproof coalition formation rules.     

Uniqueness of stationary equilibrium payoffs in coalitional bargaining

We study a model of sequential bargaining in which, in each period before an agreement is reached, the proposer?s identity is randomly determined, the proposer suggests a division of a pie of size one, each other agent either approves or rejects the proposal, and the proposal is implemented if the set of approving agents is a winning coalition for the proposer. The theory of the fixed point index is used to show that stationary equilibrium expected payoffs of this coalitional bargaining game are unique. This generalizes Eraslan [34] insofar as: (a) there are no restrictions on the structure of sets of winning coalitions; (b) different proposers may have different sets of winning coalitions; (c) there may be a positive probability that no proposer is selected.     

Indicative bidding: An experimental analysis

Indicative bidding is a practice commonly used in sales of complex and very expensive assets. Theoretical analysis shows that efficient entry is not guaranteed under indicative bidding, since there is no equilibrium in which more qualified bidders are more likely to be selected for the final sale. Furthermore, there exist alternative bid procedures that, in theory at least, guarantee 100% efficiency and higher revenue for the seller. We employ experiments to compare actual performance between indicative bidding and one of these alternative procedures. The data shows that indicative bidding performs as well as the alternative procedure in terms of entry efficiency, while having other characteristics that favor it over the alternative procedure. Our results provide an explanation for the widespread use of indicative bidding despite the potential problem identified in the equilibrium analysis.     

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.     

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.     

Value Cores for Finite Agents*

The standard definition of the competitive equilibrium, at given prices, presents the difficulty that finite agents (i.e. agents with finite budget and production sets), and a fortiori coalitions of such agents, need not be price takers and will upset the equilibrium. Since 1963 an extensive literature has sprung up, which has dealt with the problem, essentially by considering infinite numbers of infinitesimal agents. The present article outlines two alternative approaches to deal with this problem, both of which, realistically, admit (a finite number of) finite agents and redefine the core in terms of the value structure of the economy, viz. prices in the presence of controls, and cost and revenue functions in their absence. It is shown that the notion of value core, here introduced, coincides with that of competitive equilibrium, which is formulated as a characteristic-function game.     

Coalition formation as a dynamic process

We study coalition formation as an ongoing, dynamic process, with payoffs generated as coalitions form, disintegrate, or regroup. A process of coalition formation (PCF) is an equilibrium if a coalitional move to some other state can be “justified” by the expectation of higher future value, compared to inaction. This future value, in turn, is endogenous: it depends on coalitional movements at each node. We study existence of equilibrium PCFs. We connect deterministic equilibrium PCFs with unique absorbing state to the core, and equilibrium PCFs with multiple absorbing states to the largest consistent set. In addition, we study cyclical as well as stochastic equilibrium PCFs.     

Retailers and consumers in sequential auctions of collectibles

Abstract.  We analyse an independent private-value model, where heterogeneous bidders compete for objects sold in sequential second-price auctions. In this heterogeneous game, bidders may have differently distributed valuations, and some have multi-unit demand with decreasing marginal values (retailers); others have a specific single-unit demand (consumers). By examining equilibrium bidding strategies and price sequences, we show that the presence of consumers leads to more aggressive bidding from the retailers on average and heterogeneous bidders is a plausible explanation of the price decline effect. The study of the expected revenue of the seller confirms the interest of auctioneers in inviting different types of bidders.     

REDISTRIBUTIVE POLITICS AND GOVERNMENT DEBT IN A BORROWING‐CONSTRAINED ECONOMY

We develop a two‐period, three‐class of income model where low‐income agents are borrowing constrained because of capital market imperfections, and where redistributive expenditure is financed by tax and government debt. When the degree of capital market imperfection is high, there is an ends‐against‐the‐middle equilibrium where the constrained low‐income and the unconstrained high‐income agents favour low levels of government debt and redistributive expenditure; these agents form a coalition against the middle. In this equilibrium, the levels of government debt and expenditure might be below the efficient levels, and the spread of income distribution results in a lower debt‐to‐GDP ratio.     

Coalition governments in a model of parliamentary democracy

We analyse a model of coalition government in a parliamentary democracy where parties care both for ideology and perks from office, and examine how the magnitude of this tradeoff affects the nature of coalitions that form. It is shown that equilibrium coalitions can be minimal winning, minority or surplus and they may be ideologically disconnected. The types of coalitions that emerge depend upon the relative importance of rents from office and the distribution of party ideologies. Further, there is a non-monotonic relationship between ideological connectedness of coalitions and rents from office.     

Contracting with externalities and outside options

This paper proposes a model of multilateral contracting where players are engaged in two parallel interactions: they dynamically form coalitions and play a repeated normal form game with temporary and permanent decisions. We show that when outside options are independent of the actions of other players all Markov perfect equilibrium without coordination failures are efficient, regardless of externalities created by interim actions. Otherwise, in the presence of externalities on outside options, all Markov perfect equilibrium may be inefficient. This formulation encompasses many economic models, and we analyze the distribution of coalitional gains and the dynamics of coalition formation in four illustrative applications.     

Credit contracting and bidding under wealth constraints

Summary. We model credit contracting and bidding in a first-price sealed-bid auction when bidder valuation and wealth are private information. An efficient separating equilibrium exists only if the wealth levels of both bidder types are sufficiently different. If not, high-valuation bidders signal by borrowing more and using less of their wealth – this is inefficient as wealth is a cheaper source of funds. An increase in the amount of borrowing required to signal does not necessarily decrease seller expected revenue. In contrast to separating equilibria, high-valuation bidders adopt pure strategy bids in pooling equilibria. Conditions are identified under which the lower bound on winning bids is higher in pooling than separating equilibria. Received: January 22, 2001; revised version: August 28, 2001     

Coalitions, agreements and efficiency

If agents negotiate openly and form coalitions, can they reach efficient agreements? We address this issue within a class of coalition formation games with externalities where agents’ preferences depend solely on the coalition structure they are associated with. We derive Ray and Vohra's [Equilibrium binding agreements, J. Econ. Theory 73 (1997) 30-78] notion of equilibrium binding agreements using von Neumann and Morgenstern [Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944] abstract stable set and then extend it to allow for arbitrary coalitional deviations (as opposed to nested deviations assumed originally). We show that, while the extended notion facilitates the attainment of efficient agreements, inefficient agreements can nevertheless arise, even if utility transfers are possible.     

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.     

Equilibrium binding agreements: a comment

To construct their Equilibrium Binding Agreements, Ray and Vohra (J. Econ. Theory, 73 (1997) 30-78) define a concept of an equilibrium between coalitions and prove its existence for any coalition structure. We show that this result crucially depends on the quasi-concavity of the utility functions, which in turn depends on the type of mixed strategies used by the coalitions. When coalitions use uncorrelated mixed strategies utility functions may not be quasi-concave and an equilibrium may not exist. However, if coalitions use correlated strategies, an equilibrium always exist.     

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.     

Partition function bargaining with public demands

In the partition function bargaining problem the value of a coalition depends on the coalition structure in which it is embedded. This paper applies the demand-making bargaining game of coalition formation to the three-player partition function bargaining problem. The values of some embedded coalitions appear to be strategically irrelevant. The strategically relevant values constitute a coalition function bargaining problem. A classification in terms of the associated coalition function bargaining problem is provided. For one of the three classes that are distinguished the set of equilibrium outcomes closely relates to the core of the associated coalitional problem. For another class of problems the equilibrium outcome corresponds to the Von Neumann-Morgenstern outcome. Received: February 14, 2000; revised version: December 13, 2002 RID="*" ID="*" I would like to thank Harold Houba and Gerard van der Laan for valuable discussions, and an anonymous referee for helpful comments. This research was conducted while I was at the Vrije Universiteit Amsterdam.     

A measure of aggregate power in organizations

We present axiomatically an index of aggregate power in organizations, by measuring the overall ability of the agents to induce a change through coalition formation. The axioms are closely related to those used for the Shapley Value. The resulting formula appears as the sum, taken over all possible coalitions, of the products of two probabilities for each coalition: its probability to form and its probability to become winning.     

Auctions with cross-shareholdings

Summary. We study the effect of cross-shareholding among two competing firms on their bidding behavior and the expected sales revenue for the seller in an auction environment. The bidders private signals are independent, and the model encompasses the private values model and a particular common value model as special cases. When cross-shareholding is symmetric, the bids decrease towards the collusive level as the degree of cross-shareholding increases. The Revenue Equivalence result no longer holds: the first-price auction generates higher expected revenue for the seller than the second-price auction.With asymmetric cross-shareholding, revenue comparisons are only possible in the common value setting. Expected revenue for the seller is again higher in the first-price than in the second price auction. Bidding behavior in the second-price auction is more sensitive to changes in cross-shareholding and the value environment than in the first-price auction.Received: 18 September 2000, Revised: 27 May 2003, JEL Classification Numbers: C72, D44.Correspondence to: Sudipto DasguptaWe thank Sugato Bhattacharyya, Paul Klemperer, Kunal Sengupta and Guofu Tan for helpful discussions, and an anonymous referee for suggestions that improved the paper. The usual disclaimer, of course, applies.     

Premium auctions and risk preferences: An experimental study

In premium auctions, the highest losing bidder receives a reward from the seller. This paper studies the private value English premium auction (EPA) for different risk attitudes of bidders. We explicitly derive the symmetric equilibrium for bidders with CARA utilities and conduct an experimental study to test the theoretical predictions. In our experiment, subjects are sorted into risk-averse and risk-loving groups. We find that revenues in the EPA are significantly higher when bidders are risk loving rather than risk averse. These results are partly consistent with theory and confirm the general view that bidders' risk preferences constitute an important factor that affects bidding behavior and consequently also the seller's expected revenue. However, individual subjects rarely follow the equilibrium strategy and revenue in our experiment is lower than in the symmetric equilibrium.