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.     

Scoring rules: A cooperative game-theoretic approach

In this work we define the game of the alternatives for each preference profile, and establish relations between scoring rules and cooperative solution concepts for that game, such as the family of semivalues and the family of least square values. Received: 17 March 1997/Accepted: 12 May 1998     

Minimal winning coalitions in weighted-majority voting games

Riker's size principle for n-person zero-sum games predicts that winning coalitions that form will be minimal in that any player's defection will negate the coalition's winning status. Brams and Fishburn (1995) applied Riker's principle to weighted-majority voting games in which players have voting weights w ₁≥w ₂≥...≥w _n,and a coalition is winning if its members' weights sum to more than half the total weight. We showed that players' bargaining power tends to decrease as their weights decrease when minimal winning coalitions obtain, but that the opposite trend occurs when the minimal winning coalitions that form are “weight-minimal”, referred to as least winning coalitions. In such coalitions, large size may be more harmful than helpful. The present paper extends and refines our earlier analysis by providing mathematical foundations for minimal and least winning coalitions, developing new data to examine relationships between voting weight and voting power, and applying more sophisticated measures of power to these data. We identify all sets of minimal and least winning coalitions that arise from different voting weights for n≤6 and characterize all coalitions that are minimal winning and least winning for every n. While our new analysis supports our earlier findings, it also indicates there to be less negative correlation between voting weight and voting power when least winning coalitions form. In this context, players' powers are fairly insensitive to their voting weights, so being large or small is not particularly important for inclusion in a least winning coalition.     

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.     

Criticality in games with multiple levels of approval

In this article, criticality within a voting game is rigorously defined and examined. Criticality forms the basis of the traditional voting power measures frequently employed to analyse voting games; therefore understanding criticality is a pre-requisite to understanding any such analysis. The concept of criticality is extended to encompass games in which players are allowed to express multiple levels of approval. This seemingly innocuous extension raises some important questions, forcing us to re-evaluate exactly what it means to be critical. These issues have been largely side-stepped by the main body of research as they focus almost exclusively on ‘yes/no’ voting games, the so called single level approval voting games. The generalisation to multilevel approval voting games is much more than just a theoretical extension, as any single level approval game in which a player can abstain is in effect a multilevel approval voting game.     

Relationship admitting families of candidates

A central theme in social choice is to determine when must there be a relationship among a group's sincere election rankings of several different subsets of candidates. This issue is completely resolved here for positional voting methods. Namely, necessary and sufficient conditions are derived for a family of subsets of candidates to determine when there is a choice of positional voting methods so that there are relationships among the election rankings. The same issue is resolved for a related family of social choice mappings. Then, in part, these necessary and sufficient conditions are used i) to analyze sequential voting procedures, ii) to show how to create new classes of axiomatic representations for social choice mappings that uniquely characterize the Borda Count, and iii) to determine the limits of indeterminacy for positional voting election outcomes.This research was supported in part by NSF grant IRI-8803505 and a Fellowship from the Guggenheim Memorial Foundation.     

Equilibrium in abstract political economies: with an application to a public good economy with voting

Consider an abstract political economy which has a collective choice rule together with strategic interactions among players. We prove that there exists an equilibrium in such an economy by synthesizing an equilibrium existence theorem in generalized games by Shafer and Sonnenschein (1975) and a voting core existence theorem in simple games by Schofield (1984, 1989). The theorem can be applied to a public good economy where public good provisions are determined by a class of voting rule.Thanks are due to David Austen-Smith, Jeffrey Banks, Marcus Berliant, Steve Ching, Ryo-Ichi Nagahisa, Mary Beth Savio, Norman Schofield, and Tomoichi Shinotsuka. Detailed comments from an anonymous referee of the journal are gratefully acknowledged. Errors are, of course, my own.     

Condorcet efficiencies under the maximal culture condition

The Condorcet winner in an election is a candidate that could defeat each other candidate in a series of pairwise majority rule elections. The Condorcet efficiency of a voting rule is the conditional probability that the voting rule will elect the Condorcet winner, given that such a winner exists. The study considers the Condorcet efficiency of basic voting rules under various assumptions about how voter preference rankings are obtained. Particular attention is given to situations in which the maximal culture condition is used as a basis for obtaining voter preferences. Received: 4 February 1998/Accepted: 13 April 1998     

Sophisticated voting and equilibrium refinements under plurality rule

In this paper we show in the context of voting games with plurality rule that the “perfect” equilibrium concept does not appear restrictive enough, since, independently of preferences, it can exclude at most the election of only one candidate. Furthermore, some examples show that there are “perfect” equilibria that are not “proper”. However, also some “proper” outcome is eliminated by sophisticated voting, while Mertens' stable set fully satisfies such criterium, for generic plurality games. Moreover, we highlight a weakness of the simple sophisticated voting principle. Finally, we find that, for some games, sophisticated voting (and strategic stability) does not elect the Condorcet winner, neither it respects Duverger's law, even with a large number of voters. Received: 16 March 1999/Accepted: 25 September 1999     

Scoring rule voting games and dominance solvability

This article studies the dominance solvability (by iterated deletion of weakly dominated strategies) of general scoring rule voting games when there are three alternatives. The scoring rules we study include Plurality rule, Approval voting, Borda rule, and Relative Utilitarianism. We provide sufficient conditions for dominance solvability of general scoring rule voting games. The sufficient conditions that we provide for dominance solvability are in terms of one statistic of the game: sufficient agreement on the best alternative or on the worst alternative. We also show that the solutions coincide with the set of Condorcet Winners whenever the sufficient conditions for dominance solvability are satisfied. Approval Voting performs the best in terms of our criteria.     

The Shapley-Shubik index,the donation paradox and ternary games

In this paper, we show that although the Shapley-Shubik index is immune to the donation paradox in weighted binary games, extension of the index to ternary games along the direction suggested in Felsenthal and Machover (1996, 1997) will cause it to be vulnerable to the paradox and this is the case as long as the number of players in the game exceeds three. This undermines the attractiveness of the Shapley-Shubik index as a measure of a priori voting power. Received: 23 November 2001/Accepted: 3 April 2002     

Millions of election outcomes from a single profile

By changing the choice of a positional voting method, different election rankings can result from a fixed profile. A geometric theory is developed to explain why this occurs, to completely characterize all possible sets of rankings that can arise in this manner, to determine the number of rankings and other properties of these sets of rankings, to design profiles that cause the different conclusions, to develop elementary tools to analyze actual data, and to compare new types of social choice solutions that are based on the set of rankings admitted by a profile. A secondary theme is to indicate how results for voting theory can be obtained with (relative) ease when they are analyzed with a geometric approach.This research was supported by NSF Grant IRI-9103180.     

Endogenous agenda formation in three-person committees

This paper analyzes a 3-person voting game in which two or three players have the ability to choose alternatives to be considered. Once the set of possible alternatives and the structure of the voting procedure are known, the players can solve for the outcome. Thus, the actual choice over outcomes takes place in the choice of alternatives to be voted on, i.e., the agenda. An equilibrium to this agenda-formation game in shown to exist under different assumptions about the information relative to the order of the players in the voting game. Further, this equilibrium is computed and found to possess certain features which are attractive from a normative point of view.Prepared for delivery at the 1985 Annual Meeting of the American Political Science Association, The New Orleans Hilton, August 29–September 1, 1985. We would like to thank Richard McKelvey, Norman Schofield and two anonymous referees for valuable comments and suggestions. We retain responsibility for remaining errors.     

Irrational thinking among slot machine players

According to the cognitive perspective on gambling, regular gamblers persist in trying to win money at gambling because they hold a set of false beliefs about the nature of gambling, the likelihood of winning, and their own expertise. In order to investigate this claim, twenty seven university students were recruited who played one of three types of games at least twice a week: slot machines, video draw poker, and video amusement games. Subjects played their preferred machines on site (clubs, hotels and amusement arcades) first for at least thirty minutes and then the other two games for a minimum of twenty minutes each. During play, each subject spoke aloud into a microphone describing what he or she was doing or thinking about in the game. It was hypothesised that slot machine players would verbalise more irrational thinking than video poker or video amusement players and that slot machines would elicit more irrational thinking than video poker or video amusement machines. Most importantly, it was hypothesised that slot machine players would exhibit relatively greater amounts of irrational thinking when playing their preferred game. The data supported all three hypotheses. Out of all of the statements made by slot machine players when playing slot machines, 38% were categorised irrational. Furthermore, 80% of the strategic statements made by slot machine players while playing slot machines were categorised as irrational. These results are consistent with earlier work which showed high levels of irrational thinking in artificial gambling games. Together, the results provide support for a cognitive view of the origins of gambling problems.     

BECAUSE IT'S FREE POKER! A QUALITATIVE ANALYSIS OF FREE TEXAS HOLD'EM POKER TOURNAMENTS

The game of poker has dramatically increased in popularity in America over the past decade, and includes a new trend in poker in which tournaments are played with no monetary wagering. These “free” poker tournaments are the primary focus of this project. Our interest in these free poker tournaments was twofold: first, we wished to identify what would motivate a person to play poker when there is no profit motive; second, we wanted to learn how the social dynamics of the game itself may or may not change when monetary risk is not present. Through covert participant observation of free poker tournaments at four different locations we were able to discover three main motivations among the players: sharpening skills for later cash games, sociability, and gaining the status of a “regular player.” We also discovered that the tenor and tempo of the game tended to vary not as a function of the location, but rather because of the social class of the players. We offer several hypotheses to guide future research endeavors in this area.     

The partnered core of a game with side payments

An outcome of a game is partnered if there are no asymmetric dependencies between any two players. For a cooperative game, a payoff is in the partnered core of the game if it is partnered, feasible and cannot be improved upon by any coalition of players. We show that the relative interior of the core of a game with side payments is contained in the partnered core. For quasi-strictly convex games the partnered core coincides with the relative interior of the core. When there are no more than three partnerships, the sums of the payoffs to partnerships are constant across all core payoffs. When there are no more than three players, the partnered core satisfies additional properties. We also illustrate our main result for coalition structure games.     

Problem gambling in poker: money,rationality and control in a skill-based social game

The article explores problem gambling in poker. The distinctions between chance and skill and between bank games and social games are applied to demonstrate how poker is structurally different from most other gambling games. Bank games are organised around a central actor such as the house, the casino or the bookmaker. In social games, players compete against each other on equal statistical footing. Poker is a skill-based social game where players with superior skills may be expected to win even in the long run. Fourteen poker playing problem gamblers were recruited through a treatment program and 15 professional and recreational poker players were recruited through snowball sampling (N = 29). Through qualitative interviews, the paper shows that the particular structural composition of poker has implications for the ways problem gambling arises in poker players. It is argued that poker challenges existing theoretical conceptions about problem gambling relating to money, rationality and control.     

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.     

100‐Percenting It: Videogame Play Through the Eyes of Devoted Gamers1

We describe salient aspects of the idioculture of videogame play, based on interviews conducted with 20 devoted videogame players and several hours of observation of small‐group game play. We explore the meaningfulness of videogame play for participants, specifically through an examination of the social quality of play, interaction during play, the enactment of status differences in and through play, and players’ desire to play games perfectly and/or completely. We elicit players’ comments about acceptable and unacceptable forms of cheating, and we explore their management of the stigma attached to playing violent games and playing excessively. We conclude by highlighting game players’ penchant for finding moral content in their favorite games and game characters.     

Losses Disguised as Wins Affect Game Selection on Multiline Slots

Multiline slots are exciting games that contain features which make them alluring. One such feature is a loss disguised as a win (LDW); wherein, players win less than they wager (e.g., bet 2 dollars, win back 50 cents), but this net loss is disguised by flashing graphics and winning sounds. Research to date concludes that LDWs are both rewarding and reinforcing. Here, we investigated whether LDWs affect players’ game selection. Thirty-two undergraduate students with experience playing slot machines played 100 spins on four games—two had positive payback percentages (115%) and two had negative payback percentages (85%) after 100 spins. For each payback percentage condition, there was a game with no LDWs and a game with a moderate number of LDWs. For the 100 spins, players could choose to play whichever game they wished. They then rated their preference for each game following the 100-spins and chose a game to continue playing. The majority of players preferred playing the positive payback percentage game with LDWs and chose to continue playing this game over the three other games. We conclude that in addition to LDWs being reinforcing and rewarding, LDWs do in fact influence game selection. We conclude that responsible gambling initiatives should educate players about LDWs.