Axiomatizations of the normalized Banzhaf value and the Shapley value

A cooperative game with transferable utilities– or simply a TU-game – describes a situation in which players can obtain certain payoffs by cooperation. A solution concept for these games is a function which assigns to every such a game a distribution of payoffs over the players in the game. Famous solution concepts for TU-games are the Shapley value and the Banzhaf value. Both solution concepts have been axiomatized in various ways. An important difference between these two solution concepts is the fact that the Shapley value always distributes the payoff that can be obtained by the `grand coalition' consisting of all players cooperating together while the Banzhaf value does not satisfy this property, i.e., the Banzhaf value is not efficient. In this paper we consider the normalized Banzhaf value which distributes the payoff that can be obtained by the `grand coalition' proportional to the Banzhaf values of the players. This value does not satisfy certain axioms underlying the Banzhaf value. In this paper we introduce some new axioms that characterize the normalized Banzhaf value. We also provide an axiomatization of the Shapley value using similar axioms. Received: 10 April 1996 / Accepted: 2 June 1997     

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.     

Government Sanctioned “Tight” and “Loose” Slot Machines: How Having Multiple Versions of the Same Slot Machine Game May Impact Problem Gambling

In Ontario, Canada, the regulator approves identical looking slot machine games with different payback percentages. We gained access to the design documents (called PAR Sheets) used to program these different versions of the same slots game and ran Gambler’s Ruin simulations of 2,000 first-time players who each arrived with a $100 bankroll and played either the 85 or 98% version of the same game until broke. Simulations revealed that the typical (median) player’s experience did not differ significantly between versions. However the payback percentage affected the experience of players in the upper tails of the distributions with those in the 98% version having dramatically more total spins, winning spins, entries into the “bonus mode”, and “hand pays” (a win of $100 bankroll and played either the 85 or 98% version of the same game until broke. Simulations revealed that the typical (median) player’s experience did not differ significantly between versions. However the payback percentage affected the experience of players in the upper tails of the distributions with those in the 98% version having dramatically more total spins, winning spins, entries into the “bonus mode”, and “hand pays” (a win of 125 or more on a given spin). Most importantly, the number of simulated players who had a maximum peak balance in excess of $1,000 rose tenfold—from 5 in the 85% version to 54 in the 98% version. The results are discussed in terms of the Pathways Model of Problem and Pathological Gambling especially in terms of behavioural conditioning, cognitive beliefs, and early big wins. It may well be that those machines that are on the surface the “fairest” to the gambler, actually pose the most risk for ensuing gambling problems.     

Core concepts for share vectors

A value mapping for cooperative games with transferable utilities is a mapping that assigns to every game a set of vectors each representing a distribution of the payoffs. A value mapping is efficient if to every game it assigns a set of vectors which components all sum up to the worth that can be obtained by all players cooperating together.? An approach to efficiently allocate the worth of the ‘grand coalition’ is using share mappings which assign to every game a set of share vectors being vectors which components sum up to one. Every component of a share vector is the corresponding players' share in the total payoff that is to be distributed among the players. In this paper we discuss a class of share mappings containing the (Shapley) share-core, the Banzhaf share-core and the Large Banzhaf share-core, and provide characterizations of this class of share mappings. Received: 9 August 1999/Accepted: 25 April 2000     

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.     

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.     

Gamson’s law and hedonic games

This note adds one celebrated coalition formation game due to Gamson (Am Soc Rev 26:373–382 1961a, Am Soc Rev 26:565–573, 1961b) in the list of applications of the theory of hedonic games explored by Banerjee et al. (Soc Choice Welf 18:135–153, 2001) and Bogomolnaia and Jackson (Games Econ Behav 38: 204–230, 2002). We apply their results to study the original Gamson game and offer extensions both to a multi-dimensional characteristics space and to an infinite number of players.     

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.     

Losses disguised as wins in multiline slots: using an educational animation to reduce erroneous win overestimates

Slot machines are available in several countries, with multiline games growing in popularity. Interestingly, many audiovisually reinforced small ‘wins’ in multiline games are in fact monetary losses – outcomes referred to as losses disguised as wins (LDWs). Research suggests that LDWs cause players to overestimate how many times they remember actually winning during a playing session. The study sought to replicate this finding and see if a short educational animation about LDWs could significantly reduce this LDW-triggered win overestimation effect. It employed a mixed design, with animation viewed (LDW, control) as the between-subjects factor, and game played (200 spins on a few LDW or many LDW game; game order counterbalanced) as the within-subjects factor. Fifty-four novice participants estimated how many times they won more than they wagered in each game. In the control animation group, the study replicated the LDW-triggered win overestimation effect for participants playing the many LDW game. Crucially, win overestimates were significantly reduced in this many LDW game for players exposed to the LDW animation. The study concludes that LDWs can lead novice gamblers to remember winning more often than they actually do during a playing session, but educating participants about LDWs can reduce these erroneous win overestimates.     

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.     

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     

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.     

Axiomatizations of public good indices with a priori unions

We provide axiomatizations for six variants of the Public Good Index (PGI) for games with a priori unions. Two such coalitional PGIs have been introduced and alternatively axiomatized in Alonso-Meijide et al. (Working paper 18, 2008b). They assign power in two steps. In the first step, power is distributed between unions according to the PGI of the quotient game. In a second step, the Solidarity PGI splits power equally among union members while the Owen Extended PGI takes into account so-called essential parts. The other four coalitional PGIs have been introduced in Holler and Nohn (Homo Oeconomicus 26, 2009). The first variant elaborates the original idea of Holler (Political Studies 30:262–271, 1982) that the coalitional value is a public good and only minimal winning coalitions of the quotient game are relevant. The remaining three variants also use the two-step distribution where, however, on the member stage they take into account the possibilities of players to threaten their partners through leaving their union.     

Subject-specific performance information can worsen the tragedy of the commons: Experimental evidence

The main aim of this article is to investigate the behavioral consequences of the provision of subject-specific information in the group effort levels chosen by players in an experimental CPR game. We examine two basic treatments, one with incomplete information and the other with complete information. In the former, subjects are informed only about their own individual payoffs and the aggregate extraction effort level of the group, and in the latter they are also informed about the individual effort levels and payoffs of each subject. Given this setting, the basic question we attempt to answer is: Will the provision of subject-specific performance information (i.e. individual’s effort levels and payoffs) improve or worsen the tragedy of the commons (i.e. an exploitation effort level greater than the socially optimum level)? In order to motivate our hypotheses and explain our experimental results at the individual level, we make use of the theory of learning in games, which goes beyond standard non-cooperative game theory, allowing us to explore the three basic benchmarks in the commons context: Nash equilibrium, Pareto efficient, and open access outcomes. We use several learning and imitation theoretical models that are based on contrasting assumptions about the level of rationality and the information available to subjects, namely: best response, imitate the average, mix of best response and imitate the average, imitate the best and follow the exemplary learning rules. Finally, in order to econometrically test the hypotheses formulated from the theoretical predictions we use a random-effects model to assess the explanatory power of the different selected behavioral learning and imitation rules.     

A Quality-of-Play Model of a Professional Sports League

Assuming that consumers value both the absolute and relative quality of play, I compare the choice of ticket prices, team qualities, and number of games played in a noncooperative outcome versus that chosen by a social planner. I find that the nature of consumer preferences regarding the quality of play determines whether the demand for talented players are strategic complements or substitutes. A strong preference by fans for a superior team makes players strategic substitutes, and a concern for a high quality of play and competitive balance make players strategic complements. Moreover, when fans only value the relative quality of play, there is an overemployment of talented players in the noncooperative outcome versus the socially optimal outcome; when they only value the absolute quality of play then there is an underemployment of talented players in the noncooperative outcome.     

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.     

Perceived harmony,similarity and cooperation in 2 × 2 games: An experimental study

Game harmony is a generic game property describing how conflictual or non-conflictual the interests of players are. Simple and general game harmony measures can predict mean cooperation in 2 × 2 games such as the Prisoner’s Dilemma, the Chicken and trust games. Two measures can be simply computed from monetary payoffs; another, the similarity index, can also be justified by theories of similarity-based reasoning. When data from Oxford and Frankfurt–Oder are disaggregated across experiments, countries and learning history, and when the similarity index is a valid measure, parsimonious regressions can explain around half of the variance in mean cooperation rates.     

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.     

STRUCTURAL ESTIMATION OF SEQUENTIAL GAMES OF COMPLETE INFORMATION

In models of strategic interaction, there may be important order of entry effects if one player can credibly commit to an action (e.g., entry) before other players. If one estimates a simultaneous‐move model, then the move‐order effects will be confounded with the payoffs. This article considers nonparametric identification and simulation‐based estimation of sequential games of complete information. Relative to simultaneous‐move games, these models avoid the problem of multiple equilibria and require fewer payoff normalizations. We apply the estimator in several Monte Carlo experiments and to study entry‐order effects using data from the airline industry. (JEL C57, C15, L93)     

An axiomatization of the core for finite and continuum games

We provide a new axiomatization of the core of games in characteristic form. The games may have either finite sets of players or continuum sets of players and finite coalitions. Our research is based on Peleg's axiomatization for finite games and on the notions of measurement-consistent partitions and the f-core introduced by Kaneko and Wooders. Since coalitions are finite in both finite games and in continuum games, we can use the reduced game property and the converse reduced game property for our axiomatization. Both properties are particularly appealing in large economies.This paper is a revision of University of Bonn Sonderforschungsbereich 303 Discussion Paper No. B-149, with the same title.