Generating Functions for Coalitional Power Indices: An Application to the IMF

This paper provides “ready-to-apply” procedures, based on generating functions, which allow to compute power indices in weighted majority games restricted by an a priori system of unions. We illustrate these methods by an application to the International Monetary Fund. We compare the empirical properties of the coalitional and traditional power indices keeping the game fixed or allowing for variations in its set of parameters.     

Allocation rules for coalitional network games

Coalitional network games are real-valued functions defined on a set of players organized into a network and a coalition structure. We adopt a flexible approach assuming that players organize themselves the best way possible by forming the efficient coalitional network structure. We propose two allocation rules that distribute the value of the efficient coalitional network structure: the atom-based flexible coalitional network allocation rule and the player-based flexible coalitional network allocation rule.     

A Coalitional Power Value for Set Games

We propose the concept of a coalitional power value for set games, and present its axiomatic characterization of global efficiency, equal treatment property and coalitional power monotonicity. The coalitional power value is a generalization of the marginalistic value introduced by Aarts et al.     

A coalitional procedure leading to a family of bankruptcy rules

We provide a general coalitional procedure that characterizes a family of rules for bankruptcy problems inspired by the Talmud.     

The distribution of power in the European Constitution

The aim of this paper is to analyze the distribution of voting power in the Constitution for the enlarged European Union. By using generating functions, we calculate the Banzhaf power indices for the European countries in the Council of Ministers under the decision rules prescribed by the Treaty of Nice and the new rules proposed by the European Constitution Treaty. Moreover, we analyze the power of the European citizens under the egalitarian model proposed by Felsenthal and Machover [D.S. Felsenthal, M. Machover, The measurement of voting power: Theory and practice, problems and paradoxes, Edward Elgar, Cheltenham, 1998].     

Weakly unimodal domains,anti-exchange properties,and coalitional strategy-proofness of aggregation rules

It is shown that simple and coalitional strategy-proofness of an aggregation rule on any rich weakly unimodal domain of an idempotent interval space are equivalent properties if that space satisfies interval anti-exchange, a basic property also shared by a large class of convex geometries including–but not reducing to–trees and Euclidean convex spaces. Therefore, strategy-proof location problems in a vast class of networks fall under the scope of that proposition.It is also established that a much weaker minimalanti-exchangeproperty is necessary to ensure equivalence of simple and coalitional strategy-proofness in that setting. An immediate corollary to that result is that such equivalence fails to hold both in certain median interval spaces including those induced by bounded distributive lattices that are not chains, and in certain non-median interval spaces including those induced by partial cubes that are not trees.Thus, it turns out that anti-exchange properties of the relevant interval space provide a powerful general common principle that explains the varying relationship between simple and coalitional strategy-proofness of aggregation rules for rich weakly unimodal domains across different interval spaces, both median and non-median.     

Asymptotic vulnerability of positional voting rules to coalitional manipulation

Voting rule performances are sometimes evaluated according to their respective resistances to allow profitable misrepresentation of individual preferences. This seems to be a hard task when scoring systems with possibly non integer weights are involved. In this paper, it is shown how one can still obtain asymptotic results in these settings. Our analysis for three-candidate elections provides a characterization of unstable voting situations at which a positional voting rule is manipulable by some coalition not larger than an arbitrary proportion of the electorate. This allows us to address a conjecture by Pritchard and Wilson (2007). That is, under the Impartial Anonymous Culture (IAC), the plurality rule asymptotically minimizes the vulnerability to coalitional manipulation when the size of the manipulating coalition is unrestricted. This later result is no longer valid when only manipulation by small coalitions is considered: now, the Borda rule tends to outperform other rules. Furthermore, the vulnerability of a positional voting rule to coalitional manipulation is not affected by increasing the size of the manipulating coalition from 0.5 to 1.     

Robust dynamic cooperative games

Classical cooperative game theory is no longer a suitable tool for those situations where the values of coalitions are not known with certainty. We consider a dynamic context where at each point in time the coalitional values are unknown but bounded by a polyhedron. However, the average value of each coalition in the long run is known with certainty. We design “robust” allocation rules for this context, which are allocation rules that keep the coalition excess bounded while guaranteeing each player a certain average allocation (over time). We also present a joint replenishment application to motivate our model. We thank two anonymous referees for their valuable comments.     

Generalized coalitional semivalues

In this paper we change some axioms in the axiom system which defines coalitional semivalues [Albizuri, M.J., Zarzuelo J.M., 2004. On coalitional semivalues. Games and Economic Behavior 49, 221–243] and we define generalized coalitional semivalues. Generalized coalitional semivalues, like coalitional semivalues, are “compositions” of semivalues, but they form a broader set of “compositions”. Like coalitional semivalues, generalized coalitional semivalues are extensions to the coalitional context of semivalues for transferable utility games [Dubey, P., Neyman, A., Weber, R.J., 1981. Value theory without efficiency. Mathematics of Operations Research 6, 122–128].     

Voting power in the European Union enlargement

The Shapley–Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. The Banzhaf power index depends on the number of ways in which each voter can effect a swing. If there are n players in a voting situation, then the function which measures the worst case running time for computing these indices is in O(n2ⁿ). We present a combinatorial method based in generating functions to compute these power indices efficiently in weighted double or triple majority games and we study the time complexity of the algorithms. Moreover, we calculate these power indices for the countries in the Council of Ministers of the European Union under the new decision rules prescribed by the Treaty of Nice.     

Allocation rules for hypergraph communication situations

This paper considers hypergraph communication situations, where for a group of agents the economic possibilities are described by a coalitional game and the communication possibilities are described by a hypergraph in which the nodes are the agents and the edges are the subgroups of agents who can effect communication. Axiomatic characterizations are provided for two allocation rules, the Myerson value and the position value.     

On the core,the Weber set and convexity in games with a priori unions

This paper deals with the concepts of core and Weber set with a priori unions à la Owen. As far as we know, the Owen approach to games with a priori unions has never been studied from the coalitional stability point of view. Thus we introduce the coalitional core and coalitional Weber set and characterize the class of convex games with a priori unions by means of the relationships between both solution concepts.     

Robust design of rule-based fuzzy controllers

This paper deals with the design of stable and robust rule-based fuzzy control systems. New expressions to compute indices which provide a measure of the stability and robustness of fuzzy control systems are presented. The relation between the modification of the rules and the stability is studied through the so-called sensitivity indices. The paper presents procedures that make use of these indices to improve the design of fuzzy control systems, including the modification of the rules to obtain the global stability of an unstable system with multiple attractors, and to improve the dynamic behavior or the robustness of a non-linear plant. An example with a fuzzy controller for a system with non-linear damping and saturation in the actuation is presented to illustrate the design procedure.     

A fundamental study for partially defined cooperative games

The payoff of each coalition has been assumed to be known precisely in the conventional cooperative games. However, we may come across situations where some coalitional values remain unknown. This paper treats cooperative games whose coalitional values are not known completely. In the cooperative games it is assumed that some of coalitional values are known precisely but others remain unknown. Some complete games associated with such incomplete games are proposed. Solution concepts are studied in a special case where only values of the grand coalition and singleton coalitions are known. Through the investigations of solutions of complete games associated with the given incomplete game, we show a focal point solution suggested commonly from different viewpoints.     

A Tractable and Expressive Class of Marginal Contribution Nets and Its Applications

Coalitional games raise a number of important questions from the point of view of computer science, key among them being how to represent such games compactly, and how to efficiently compute solution concepts assuming such representations. Marginal contribution nets (MC‐nets), introduced by Ieong and Shoham, are one of the simplest and most influential representation schemes for coalitional games. MC‐nets are a rulebased formalism, in which rules take the form pattern → value, where “pattern ” is a Boolean condition over agents, and “value ” is a numeric value. Ieong and Shoham showed that, for a class of what we will call “basic” MC‐nets, where patterns are constrained to be a conjunction of literals, marginal contribution nets permit the easy computation of solution concepts such as the Shapley value. However, there are very natural classes of coalitional games that require an exponential number of such basic MC‐net rules. We present read‐once MC‐nets, a new class of MC‐nets that is provably more compact than basic MC‐nets, while retaining the attractive computational properties of basic MC‐nets. We show how the techniques we develop for read‐once MC‐nets can be applied to other domains, in particular, computing solution concepts in network flow games on series‐parallel networks (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modification of the Banzhaf Value for Games with a Coalition Structure

In this paper we introduce a new coalitional value in the context of TU games with an a priori system of unions, which it is called the symmetric coalitional Banzhaf value. This value satisfies the property of symmetry in the quotient game, the quotient game property, and it is a coalitional value of Banzhaf. Several characterizations are provided and two political examples illustrate the differences with respect to the Owen value and the Banzhaf–Owen value.     

Filtrations,values and voting discipline

Replication and condensation of games are shown to be partially reciprocal procedures, in which partnerships play an important role. Two extensions are studied: filtrations, that form partial replications by introducing any given set of partnerships while the quotient game is kept invariant, and partial condensations; they inherit full compatibility properties. The weighted Shapley value and the coalitional value, respectively associated with these procedures, are shown to exhibit some kind of parallelism, and their behavior under both extensions is also studied. When applied to simple games these values give a measure of the effect of voting discipline within parliamentary bodies. Some final examples, including two Spanish regional parliaments, illustrate both stable and unstable situations.     

Aggregate Wind Power Production via Coalitional Games and Optimal Control

In this paper, benefits from aggregating independent wind power producers are analyzed in a scenario, in which the producers willingly form coalitions to increase their expected profits. For every deviation from the declared contract, the coalition is penalized and a cost is paid, if the producers want to update their contract. The underlying idea is that coalitions reduce the risk of being penalized. The main contribution of this paper is a new market model and an allocation mechanism based on optimal control and coalitional games with transferable utilities. Optimal control is used to obtain the optimal contract size, while coalitional games provide an insight on stable revenue allocations, namely allocations, that make the grand coalition preferable to all producers.     

On the symmetric and weighted shapley values

We present new axiomatic characterizations of the symmetric Shapley value and of weighted Shapley values for transferable utility coalitional form games without imposing the axiom ofadditivity (Shapley [1953a,b]). Our main condition iscoalitional strategic equivalence, introduced by Chun [1989]. We show thatcoalitional strategic equivalence, together withefficiency, andsymmetry, characterizes the symmetric Shapley value, and this axiom, together withefficiency, positivity, homogeneity, andpartnership, characterizes weighted Shapley values.     

Hedonic coalition formation games with variable populations: core characterizations and (im)possibilities

We consider hedonic coalition formation games with variable sets of agents and extend the properties competition sensitivity and resource sensitivity (introduced by Klaus, Games Econ Behav 72:172–186, 2011, for roommate markets) to hedonic coalition formation games. Then, we show that on the domain of solvable hedonic coalition formation games, the Core is characterized by coalitional unanimity and Maskin monotonicity (see also Takamiya, Maskin monotonic coalition formation rules respecting group rights. Niigata University, Mimeo, 2010, Theorem 1). Next, we characterize the Core for solvable hedonic coalition formation games by unanimity, Maskin monotonicity, and either competition sensitivity or resource sensitivity (Corollary 2). Finally, and in contrast to roommate markets, we show that on the domain of solvable hedonic coalition formation games, there exists a solution not equal to the Core that satisfies coalitional unanimity, consistency, competition sensitivity, and resource sensitivity (Example 2).