Power indices and minimal winning coalitions

The Penrose–Banzhaf index and the Shapley–Shubik index are the best-known and the most used tools to measure political power of voters in simple voting games. Most methods to calculate these power indices are based on counting winning coalitions, in particular those coalitions a voter is decisive for. We present a new combinatorial formula how to calculate both indices solely using the set of minimal winning coalitions.     

On the structure of minimal winning coalitions in simple voting games

According to Coleman’s index of collective power, a decision rule that generates a larger number of winning coalitions imparts the collectivity a higher a priori power to act. By the virtue of the monotonicity conditions, a decision rule is totally characterized by the set of minimal winning coalitions. In this paper, we investigate the structure of the families of minimal winning coalitions corresponding to maximal and proper simple voting games (SVG). We show that if the proper and maximal SVG is swap robust and all the minimal winning coalitions are of the same size, then the SVG is a specific (up to an isomorphism) system. We also provide examples of proper SVGs to show that the number of winning coalitions is not monotone with respect to the intuitively appealing system parameters like the number of blockers, number of non-dummies or the size of the minimal blocking set.     

Agenda control in coalition formation

Theoretical models of government formation in political science usually assume that the head of state is non-strategic. In this paper, we analyze the power of an agenda setter who chooses the order in which players are recognized to form coalitions in simple games. We characterize those sets of players which can be imposed in the equilibrium coalition and show that the only decisive structures where the agenda setter can impose the presence of any minimal winning coalition are apex games, where a large player forms a winning coalition with any of the small players. Keywords: Government Formation, Agenda Control, Coalitional Bargaining, Finite Bargaining Rules. Received: 26 January 2001/Accepted: 31 July 2001     

A note on the paradox of smaller coalitions

We consider hedonic coalition formation games that are induced by a simple TU-game and a cooperative solution. For such models, Shenoy’s (Int J Game Theory 8:133–164, 1979) absence of the paradox of smaller coalitions provides a sufficient condition for core existence. We present three different versions of his condition in order to compare it to the top coalition property of Banerjee et al. (Social Choice Welfare 18:135–153, 2001) that guarantees nonemptiness of the core in more general models. As it turns out, the top coalition property implies a condition in which Shenoy’s paradox is not present for at least one minimal winning coalition. Conversely, if for each non-null player Shenoy’s paradox is not present for at least one minimal winning coalition containing that player, then the induced hedonic game satisfies the top coalition property.     

Power indices expressed in terms of minimal winning coalitions

A voting situation is given by a set of voters and the rules of legislation that determine minimal requirements for a group of voters to pass a motion. A priori measures of voting power, such as the Shapley–Shubik index and the Banzhaf value, show the influence of the individual players in a voting situation and are calculated by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the legislative rules. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions and derive explicit formulae for the Shapley–Shubik and Banzhaf values. This new approach logically appealing as it writes measures as functions of the rules of the legislation. For certain classes of games that arise naturally in applications the logical shortcut drastically simplifies the numerical calculations to obtain the indices. The technique generalises directly to all semivalues.     

Hierarchy of players in swap robust voting games

Ordinarily, the process of decision making by a committee through voting is modeled by a monotonic game the range of whose characteristic function is restricted to {0, 1}. The decision rule that governs the collective action of a voting body induces a hierarchy in the set of players in terms of the a-priori influence that the players have over the decision making process. In order to determine this hierarchy in a swap robust game, one has to either evaluate a power index (e.g., the Shapley–Shubik index, the Banzhaf–Coleman index) for each player or conduct a pairwise comparison between players, whereby a player i is ranked higher than another player j if there exists a coalition in which i is more desirable as a coalition partner than j. In this paper, we outline an alternative mechanism to determine the ranking of players in terms of their a-priori power. This simple and elegant method uses only minimal winning coalitions, rather than the entire set of winning coalitions.     

Stable sets and cores for multi-criteria simple games and for their extensions

In this article we deal with multi-criteria simple games which constitute an extension of the basic framework of voting systems and related social-choice situations. We generalize two closely related solution concepts for these games, the stable sets and the core. A relationship between stable sets and minimal winning coalitions is established. Two generalizations of the core for multi-criteria simple games are characterized in terms of veto players. In addition, extended multi-criteria simple games obtained from different aggregation operations, such as union, intersection, marginalization, and composition, are introduced. It is shown that a voting system can be established when these operations are applied to complex voting systems.     

A value for games with coalition structures

This paper presents an axiomatization of a value for games with coalition structures which is an alternative to the Owen Value. The motor of this new axiomatization is a consistency axiom based on an associated game, which is not a reduced game. The new value of an n-player unanimity game is the compound average of the new values of all the (n-1)-player unanimity games. The new value of a unanimity game allocates to bigger coalitions a larger share of the total wealth. Note that the Owen value allocates to all the coalitions the same share independently of their size.     

How the size of a coalition affects its chances to influence an election

Since voting rules are prototypes for many aggregation procedures, they also illuminate problems faced by economics and decision sciences. In this paper we are trying to answer the question: How large should a coalition be to have a chance to influence an election? We answer this question for all scoring rules and multistage elimination rules, under the Impartial Anonymous Culture assumption. We show that, when the number of participating agents n tends to infinity, the ratio of voting situations that can be influenced by a coalition of k voters to all voting situations is no greater than $D_{m} \frac{k}{n}Since voting rules are prototypes for many aggregation procedures, they also illuminate problems faced by economics and decision sciences. In this paper we are trying to answer the question: How large should a coalition be to have a chance to influence an election? We answer this question for all scoring rules and multistage elimination rules, under the Impartial Anonymous Culture assumption. We show that, when the number of participating agents n tends to infinity, the ratio of voting situations that can be influenced by a coalition of k voters to all voting situations is no greater than , where D _m is a constant which depends only on the number m of alternatives but not on k and n. Recent results on individual manipulability in three alternative elections show that this estimate is exact for k=1 and m=3.

Arkadii SlinkoEmail:

     

Kingmakers and leaders in coalition formation

Assume that players strictly rank each other as coalition partners. We propose a procedure whereby they “fall back” on their preferences, yielding internally compatible, or coherent, majority coalition(s), which we call fallback coalitions. If there is more than one fallback coalition, the players common to them, or kingmakers, determine which fallback coalition will form. The first player(s) acceptable to all other members of a fallback coalition are the leader(s) of that coalition. The effects of different preference assumptions—particularly, different kinds of single-peakedness—on the number of coherent coalitions, their connectedness, and which players become kingmakers and leaders are investigated. The fallback procedure may be used (i) empirically to identify kingmakers and leaders or (ii) normatively to select them. We illustrate the model using data from the U.S. Supreme Court, 2005–2009.     

Ordinal Bayesian incentive compatible representations of committees

We consider the problem of strategic manipulation for decision schemes that provide an adequate representation (in some sense) of the distribution of power within a committee. Strategy-proof representation is very restrictive: it implies that the committee contains exactly one minimal winning coalition. So we introduce the weaker concept of Ordinally Bayesian Incentive Compatible representation and prove the existence of such representations for weak games under some conditions. Finally, constructing examples, we show first how necessary these conditions are—including the use of chance in the voting procedure — and second that we cannot avoid Condorcet's paradox.     

In quest of the banks set in spatial voting games

The Banks set (1(4):295–306, 1985) is one of the more important concepts in voting theory since it tells us about the sophisticated outcomes of standard amendment voting procedures commonly in use throughout the English speaking world (and elsewhere as well). While the properties of the Banks set for finite voting games have been extensively studied, little is known about how to find members of this set for majority rule spatial voting games involving possibly infinite agendas. We look at this question for two-dimensional games where voters have Euclidean preferences, and offer a variety of new results that delimit areas of the space that can be shown to lie within the Banks set, such as the Schattschneider set, the tri-median set, and the Banks line set—geometric constructs which we show to be nested within one another.     

Classification theorem for smooth social choice on a manifold

A classification theorem for voting rules on a smooth choice space W of dimension w is presented. It is shown that, for any non-collegial voting rule, σ, there exist integers v ^*(σ), w ^*(σ) (with v ^*(σ)<w ^*(σ)) such that

1. structurally stable σ-voting cycles may always be constructed when w ? v ^*(σ) + 1
2. a structurally stable σ-core (or voting equilibrium) may be constructed when w ? v ^*(σ) ? 1

Finally, it is shown that for an anonymous q-rule, a structurally stable core exists in dimension \(\frac{{n - 2}}{{n - q}}\) , where n is the cardinality of the society.     

The Effects of Different Group Sex Compositions on Support Rates and Coalition Formation

Une réplique des expériences de T. M. Mills sur des coalitions dans les triades, en variant la composition par sexe du groupe, met en lumière le fait que les femmes apportent un meilleur appui que les hommes, qu'elles établissent un plus grand nombre de coalitions et répartissent leur appui de f aeon plus équitable parmi les autres membres lorsqu'elles entrent en interaction avec des personnes de meme sexe. Tandis que le membre le plus inactif de la triade masculine devient un isole, le moins actif de la triade féminine regoit le meilleur appui émotif. Le modele de coalition typique dans la triade masculine est une coalition de deux membres et un isolé: dans le groupe féminin les trois membres du groupe etablissent des coalitions. Dans les triades mixtes, l'impact de l'interaction des membres d'un sexe avec ceux du sexe opposé est de faconner leurs comportements à l'image de ceux des autres. On suggère que le processus qui influence la constitution de coalitions dans les triades mixtes est reliéà la concurrence que se livrent les deux membres du meme sexe pour obtenir l'appui du membre du sexe opposé. Dans une telle situation il n'est pas rare de constater que le membre du sexe minoritaire devient un mediateur entre les deux membres majoritaires du sexe opposé. II s'ensuit une formation de deux coalitions composée du membre minoritaire et d'un membre majoritaire et parfois, des coalitions entre les trois membres du groupe. A replication of T. M. Mills' experiments dealing with coalition-formation in triads, varying the sex composition of the group, reveals that females are more supportive, form more coalitions and distribute their support more equally among others when interacting with members of the same sex, than males do. While the least active member of the triad becomes an isolate in the 3-male groups, the least active member in 3-female groups receives more support than any other group member. The typical coalition pattern in the 3-male groups is a coalition between two, with the third being an isolate: in the 3-female groups, coalitions develop between all three members of the group. In the mixed-sex groups, the effect of interacting with members of the opposite sex on both men and women is to make their behaviour more like the other's. It is suggested that the process that governs the formation of coalitions in mixed-sex triads is that of competition between the two majority-sex members for the support of the one member of the other sex, which often leads to the minority-sex member mediating between the competing pair of the opposite sex, and this results in the formation of two coalitions between the minority-sex member and each of the majority-sex members in the group, and sometimes even in the eventual establishment of coalitions between all three members of the group.     

Yes-no voting

Yes-No (Y-N) voting is a voting method for choosing a governing coalition in a parliament after the seating of its members. Each member can designate a party to be Y (it must be included in the governing coalition), N (it must be excluded from the governing coalition), or neither (it may be either in or out of the governing coalition). The majority coalition acceptable to the most voters, because it includes all parties that these voters designate Y and no parties that they designate N, is given the first opportunity to form a government. Possible combinations of majority coalitions that a member might vote for are derived, including ones based on consistent and interval voting strategies. Examples illustrate a number of different phenomena, such as when a rational voter might not be loyal to his or her party by designating it Y.     

On power distribution in weighted voting

We study here the strategic possibilities in a voting system where, in weight terms, there is a single main agent and the others are equivalent. As an alternative to the coalition formation, we suggest for the minor agents a more discreet, anonymous and solidary behavior based on commitments referred to the size of the coalitions that they would agree to form. In order to analyze, in a coherent way, the effects of such commitments on the a priori power distribution, we use as measures of power the Shapley value and a certain class of semivalues closely related to it.Research partially supported by Grant BFM 2003–01314 of the Science and Technology Spanish Ministry. The authors wish to thank Professor Maurice Salles and two anonymous referees for their stimulating comments.     

Coordinated Action in the Massively Multiplayer Online Game World of Warcraft

In tandem with the technological advancement of immersive virtual environments, digital games have evolved into complex social worlds where people play collaboratively to achieve individual‐ and group‐oriented goals. The massively multiplayer online games genre has received international attention for the large numbers of users that commit a significant amount of time and effort to online play. Play styles within such game worlds are diverse, but “raiding” is generally considered among gamers and scholars alike to be the most challenging form of collaborative play. Like other forms of social activity, raiding is predicated on players' abilities to successfully coordinate individual lines of action. Yet the vast majority of raiders are not physically copresent during their collaborative efforts. Using Couch's theory of coordinated action as an interpretive frame, we explore the computer‐mediated process of raiding in the massively multiplayer online game World of Warcraft by analyzing ethnographic data and audio/visual recordings of gameplay. Our study expands Couch's theory of coordinated action in two ways. First, we demonstrate how individuals and groups simultaneously engage in multiple forms of coordinated activity (e.g., conflict and cooperation). Second, we reveal two semiotic layers of coordinated action—the “subject‐subject” (i.e., intersubjective) layer that is facilitated through the game's user interface, and the “subject‐object” layer that comprises players' interactions with the user interface itself. Our study thus considers the potential of Couch's theory for the study of computer‐mediated communication and nonhumans in interaction.     

Membership rules and stability of coalition structures in positive externality games

This paper compares the stability of coalition structures under six institutional settings of treaty formation in positive externality games. We argue that many economic problems belong to the class of positive externality games (i.e. outsiders benefit from the formation of coalitions) and hence our results can be applied to many situations. It is shown that it is easier to sustain agreements under exclusive than under open membership and the higher the “degree of consensus” necessary to form a coalition.     

A General Equilibrium Model of Multi-Party Competition

The paper introduces a generalized spatial model that is motivated by the frequent changes in party identity and electoral laws that characterize transitional party systems. In this model, parties may (1) change their platforms, (2) their identities through coalitions and splits and (3) if they form a winning coalition, the electoral law. The equilibrium is defined as a state such that no party or coalition can strictly benefit from changing the electoral law, its platform, or from splitting or coalescing. The results show that while there are games with no institutional or coalitional-split equilibria, such equilibria do exist under relatively undemanding conditions. The main finding is that once an institutional and identity equilibrium is achieved, it is generically robust against small trembles in party platforms or voter preferences. This robustness facilitates greater stability in terms of institutions and party identities in mature party systems where such trembles are smaller than in transitional systems.     

Measuring influence in command games

In the paper, we study a relation between command games proposed by Hu and Shapley and an influence model. We show that our framework of influence is more general than the framework of the command games. We define several influence functions which capture the command structure. These functions are compatible with the command games, in the sense that each commandable player for a coalition in the command game is a follower of the coalition under the command influence function. Some of the presented influence functions are equivalent to the command games, that is, they are compatible with the command games, and additionally each follower of a coalition under the command influence function is also a commandable player for that coalition in the command games. For some influence functions we define the equivalent command games. We show that not for all influence functions the compatible command games exist. Moreover, we propose a more general definition of the influence index and show that under some assumptions, some power indices, which can be used in the command games, coincide with some expressions of the weighted influence indices. Both the Shapley–Shubik index and the Banzhaf index are equal to a difference between the weighted influence indices under some influence functions, and the only difference between these two power indices lies in the weights for the influence indices. An example of the Confucian model of society is broadly examined. The authors wish to gratefully thank two anonymous referees for useful suggestions concerning this paper.