The Banzhaf power index for political games

The Banzhaf index of a voting game is a measure of a priori power of the voters. The model on which the index is based treats the voters symmetrically, i.e. the ideology, outlook, etc., of the voters influencing their voting behavior is ignored. Here we present a nonsymmetric generalization of the Banzhaf index in which the ideology of the voters affecting their voting behavior is taken into account. A model of ideologies and issues is presented. The conditions under which our model gives the Shapley-Shubik index (another index of a priori power of the voters) are given. Finally several examples are presented and some qualitative results are given for straight majority and pure bargaining games.     

The Banzhaf power index for ternary bicooperative games

In this paper we analyze ternary bicooperative games, which are a refinement of the concept of a ternary voting game introduced by Felsenthal and Machover. Furthermore, majority voting rules based on the difference of votes are simple bicooperative games. First, we define the concepts of the defender and detractor swings for a player. Next, we introduce the Banzhaf power index and the normalized Banzhaf power index. The main result of the paper is an axiomatization of the Banzhaf power index for the class of ternary bicooperative games. Moreover, we study ternary bicooperative games with two lists of weights and compute the Banzhaf power index using generating functions.     

A decisiveness index for simple games

The decisiveness index introduced in this paper is designed to provide a normalized measure of the agility of all simple games, primarily viewed as collective decision-making mechanisms. We study the mathematical properties of the index and derive different axiomatic characterizations for it. Moreover, a close relationship is shown to the Banzhaf index of power––for which twice the decisiveness index plays the role of potential function––that gives rise to an effective computational procedure. Some real-world examples illustrate the usefulness of the decisiveness index, together with the Banzhaf power index, in applications to political science.     

A class of power indices for voting games

A class of power indices is discussed which includes not only the usual power indices but also the class of semivalues and the indices introduced byDeegan/Packel [1978]. Sufficient conditions for members of this class to be symmetric, support-independent, and self-dual are given. We then discuss various conditions under which members of this class coincide with the ShapleyShubik and Banzhaf indices.     

A class of non-normalized power indices for simple games

Weights are attached to the coalitions of a finite set of players in order to reflect the fact that some coalitions are more likely to be formed than others. A new power index for simple games which depends only on these weights and the minimal winning coalitions of a game is introduced. Two axiomatic characterizations of this index are given. Symmetry, dummy player and quarrel properties of the index are treated. Two special forms of the index arising from certain assumptions about the weights are described.     

Harsanyi power solutions for graph-restricted games

We consider cooperative transferable utility games, or simply TU-games, with limited communication structure in which players can cooperate if and only if they are connected in the communication graph. Solutions for such graph games can be obtained by applying standard solutions to a modified or restricted game that takes account of the cooperation restrictions. We discuss Harsanyi solutions which distribute dividends such that the dividend shares of players in a coalition are based on power measures for nodes in corresponding communication graphs. We provide axiomatic characterizations of the Harsanyi power solutions on the class of cycle-free graph games and on the class of all graph games. Special attention is given to the Harsanyi degree solution which equals the Shapley value on the class of complete graph games and equals the position value on the class of cycle-free graph games. The Myerson value is the Harsanyi power solution that is based on the equal power measure. Finally, various applications are discussed.     

A new value for games without transferable utilities

We define axiomatically a unique concept of value for games without transferable utilities, which is a generalization ofNash's bargaining model. Unlike other concepts, it does not coincide with theShapley value in the case of transferable utilities.     

A new index for polytopes

A new index for convex polytopes is introduced. It is a vector whose length is the dimension of the linear span of the flag vectors of polytopes. The existence of this index is equivalent to the generalized Dehn-Sommerville equations. It can be computed via a shelling of the polytope. The ranks of the middle perversity intersection homology of the associated toric variety are computed from the index. This gives a proof of a result of Kalai on the relationship between the Betti numbers of a polytope and those of its dual. Margaret M. Bayer was supported in part by a National Science Foundation grant, by a Northeastern University Junior Research Fellowship, and by the Institute for Mathematics and Its Applications.     

The exact lower bound for the Coleman index of the power of a collectivity for a special class of simple majority games

Coleman [Coleman, J.S., 1971. Control of collectivities and the power of a collectivity to act. In: Leberman, B. (Ed.), Social Choice. Gordon and Breach, New York] proposed a measure of the power that a decision-making body has to pass any bill that comes before it. He termed it “the power of a collectivity to act”. In this paper we provide some numerical bounds on the values that this measure can take when the collectivity takes decisions under simple majority voting rule.     

Semisimple games,enactment power and generalized power indices

By allowing for individual abstention, the classC of simplen-person games is extended to the class ofsemisimple gamesS. Using this extension, any given index of individual power onC gives rise to a measure of individual power onS in the form of a vector function with 2 ⁿ ?1 components. After developing an axiomatic characterization of Coleman's notion of collective power, thisenactment power is combined with any index of individual power to provide a general nonnormalized meausre of individual power. Using these results, enactment and individual power in different games can be meaningfully compared. In the presence of abstention, various “paradoxes” associated with power indices lose some of the impact.     

A new index of poverty

This paper introduces a new index of poverty. The index satisfies all the axioms for ‘a good index of poverty’.     

The class of microarray games and the relevance index for genes

Nowadays, microarray technology is available to generate a huge amount of information on gene expression. This information must be statistically processed and analyzed, in particular, to identify those genes which are useful for the diagnosis and prognosis of specific diseases. We discuss the possibility of applying game-theoretical tools, like the Shapley value, to the analysis of gene expression data. Via a “truncation” technique, we build a coalitional game whose aim is to stress the relevance (“sufficiency”) of groups of genes for the specific disease we are interested in. The Shapley value of this game is used to select those genes which deserve further investigation. To justify the use of the Shapley value in this context, we axiomatically characterize it using properties with a genetic interpretation. The authors are grateful to two anonymous referees for their extremely helpful comments. An earlier version of this paper was presented at the VI Spanish Meeting on Game Theory and Practice, July 12–14, 2004, Elche, Spain. S. Moretti gratefully acknowledges the financial support of the EU project NewGeneris, European Union 6th FP (FOOD-CT-2005-016320).     

A new index of creditworthiness for retail credit products

This paper introduces a novel family of indexes to describe borrowers’ creditworthiness in retail credit products, both for fixed term loans and for open-ended products such as credit cards. Each index is the ratio at a given time of the net present value of actually received cashflows to the contractual ones. Some interpretations of the indexes are given and it is also described how to link them to the profitability of the credit financial operation. For open-ended products, a competing risks survival analysis methodology is proposed to estimate the cashflow returns and illustrated with a simulation.     

Hedonic coalition formation games: A new stability notion

This paper studies hedonic coalition formation games where each player’s preferences rely only upon the members of her coalition. A new stability notion under free exit-free entry membership rights, referred to as strong Nash stability, is introduced which is stronger than both core and Nash stabilities studied earlier in the literature. Strong Nash stability has an analogue in non-cooperative games and it is the strongest stability notion appropriate to the context of hedonic coalition formation games. The weak top-choice property is introduced and shown to be sufficient for the existence of a strongly Nash stable partition. It is also shown that descending separable preferences guarantee the existence of a strongly Nash stable partition. Strong Nash stability under different membership rights is also studied.     

The Shapley–Shubik index for simple games with multiple alternatives

When analyzing mathematically decision mechanisms ruled by voting it is sometimes convenient to include abstention as a possible alternative for the voters. In classical simple games, abstention, if considered, is formally equivalent to voting against the proposal. Simple games with alternatives are useful to study voting systems where abstention does not favour any of the options. In this work, we axiomatically characterize the Shapley–Shubik index for simple games with alternatives and apply it to an example taken from real life. This work has been partially supported by Grant MTM 2006–06064 of the Education and Science Spanish Ministry and the European Regional Development Fund, and Grant SGR 2005–00651 of the Catalonia Government.     

Communication games with partially soft power constraints

We consider a class of communication games which involves the transmission of a Gaussian random variable through a conditionally Gaussian memoryless channel in the presence of an intelligent jammer. The jammer is allowed to tap the channel and feed a correlated signal back into it. The transmitter-receiver pair is assumed to cooperate in minimizing some quadratic fidelity criterion while the jammer maximizes this same criterion. Security strategies which protect against irrational jammer behavior and which yield an upper bound on the cost are shown to exist for the transmitter-receiver pair over a class of fidelity criteria. Closed-form expressions for these strategies are provided in the paper, which are, in all cases but one, linear in the available information.This work was supported in part by the US Air Force under Grant No. AFOSR-84-0056 and in part by the Joint Services Electronics Program under Contract No. N00014-84-C-0149. An earlier version of this paper was presented at the 1986 IEEE Symposium on Information Theory, Ann Arbor, Michigan, 1986.     

A new characteristic index of chaos

In this paper, k step chaometry (k SCM) is defined based on monopolized sphere and instantaneous chaometry (ICM), and the convergent theorem of asymptotical periodic orbit is also proved. The 400 SCM of the discrete model of Lorenz system is calculated and results disclose that 400 SCM can clearly identify the parameters of chaotic dynamic system. The EEG instrument is applied to measure time series of EEG, and it is observed that the instantaneous chaometry of the EEG and the data generated from Lorenz attractor produce similar results.     

A value for multichoice games

A multichoice game is a generalization of a cooperative TU game in which each player has several activity levels. We study the solution for these games proposed by Van Den Nouweland et al. (1995) [Van Den Nouweland, A., Potters, J., Tijs, S., Zarzuelo, J.M., 1995. Cores and related solution concepts for multi-choice games. ZOR-Mathematical Methods of Operations Research 41, 289–311]. We show that this solution applied to the discrete cost sharing model coincides with the Aumann-Shapley method proposed by Moulin (1995) [Moulin, H., 1995. On additive methods to share joint costs. The Japanese Economic Review 46, 303–332]. Also, we show that the Aumann-Shapley value for continuum games can be obtained as the limit of multichoice values for admissible convergence sequences of multichoice games. Finally, we characterize this solution by using the axioms of balanced contributions and efficiency.     

A solution for noncooperative games

In this paper, we study solutions of strict noncooperative games that are played just once. The players are not allowed to communicate with each other. The main ingredient of our theory is the concept of rationalizing a set of strategies for each player of a game. We state an axiom based on this concept that every solution of a noncooperative game is required to satisfy. Strong Nash solvability is shown to be a sufficient condition for the rationalizing set to exist, but it is not necessary. Also, Nash solvability is neither necessary nor sufficient for the existence of the rationalizing set of a game. For a game with no solution (in our sense), a player is assumed to recourse to a standard of behavior. Some standards of behavior are examined and discussed.This work was sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and by the National Science Foundation under Grant No. MCS-75-17385-A01. The author is grateful to J. C. Harsanyi for his comments and to S. M. Robinson for suggesting the problem.