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.     

Baire generalized topological spaces, generalized metric spaces and infinite games

Different generalizations of topological Baire spaces to the case of generalized topological spaces are considered and the properties of such spaces are examined. In particular, these considerations are focused on the relationship between Baire generalized topological spaces and semi-continuous real functions and infinite games. The notion of generalized metric spaces corresponding to generalized topological spaces is introduced as an important tool in this discussion.     

Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games

In Pang and Fukushima (Comput Manage Sci 2:21–56, 2005), a sequential penalty approach was presented for a quasi-variational inequality (QVI) with particular application to the generalized Nash game. To test the computational performance of the penalty method, numerical results were reported with an example from a multi-leader-follower game in an electric power market. However, due to an inverted sign in the penalty term in the example and some missing terms in the derivatives of the firms’ Lagrangian functions, the reported numerical results in Pang and Fukushima (Comput Manage Sci 2:21–56, 2005) are incorrect. Since the numerical examples of this kind are scarce in the literature and this particular example may be useful in the future research, we report the corrected results. The online version of the original article can be found under doi:.     

Quasi-variational inequalities, generalized Nash equilibria, and multi-leader-follower games

The noncooperative multi-leader-follower game can be formulated as a generalized Nash equilibrium problem where each player solves a nonconvex mathematical program with equilibrium constraints. Two major deficiencies exist with such a formulation: One is that the resulting Nash equilibrium may not exist, due to the nonconvexity in each players problem; the other is that such a nonconvex Nash game is computationally intractable. In order to obtain a viable formulation that is amenable to practical solution, we introduce a class of remedial models for the multi-leader-follower game that can be formulated as generalized Nash games with convexified strategy sets. In turn, a game of the latter kind can be formulated as a quasi-variational inequality for whose solution we develop an iterative penalty method. We establish the convergence of the method, which involves solving a sequence of penalized variational inequalities, under a set of modest assumptions. We also discuss some oligopolistic competition models in electric power markets that lead to multi-leader-follower games.Jong-Shi Pang: The work of this authors research was partially supported by the National Science Foundation under grant CCR-0098013 and ECS-0080577 and by the Office of Naval Research under grant N00014-02-1-0286.Masao Fukushima: The work of this authors research was partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, Culture and Sports of Japan.     

Differential games,dynamic optimization,and generalized control theory

In this paper, we examine the problem of control in a generalized framework in which there are more than one criterion, more than one intelligent controller, each of which has access to different information. It is argued that optimal control, differential games, dynamic team theory, etc., can all be viewed as special cases of this generalized control theory. Useful concepts and potential difficulties are discussed.The work reported in this paper is a result of research support extended over a period of years by Joint Services Electronic Contract No. N00014-67-A-0298-0006, NASA Grant No. NGR-22-007-068, and AFOSR Grant No. 69-1768 administered through Harvard University. The author has benefited from discussions with numerous colleagues, in particular, K. C. Chu, A. Starr, W. Willman, H. Witsenhausen, and S. Mitter.     

Semisimple finite group algebra of a generalized strongly monomial group

The complete algebraic structure of semisimple finite group algebra of a generalized strongly monomial group is provided. This work extends the work of Broche and del Río on strongly monomial groups. The theory is complimented by an algorithm and is illustrated with an example.     

Power indices for multicandidate voting games

In Bolger (1983), a “Banzhaf-type” power index was used to measure the voting power of individual voters in an electoral college presidential election. It was pointed out that this power index could be affected by the addition of a “dummy” player. In this paper we introduce a modification which is dummy-independent. We also derive an efficient, linear index which has different properties than the index introduced by Myerson (1977).     

On generalized Nash games and variational inequalities

We show that for a large class of problems a generalized Nash equilibrium can be calculated by solving a variational inequality. We analyze what solutions are found by this reduction procedure and hint at possible applications.     

Constrained ratio games and generalized fractional programming

A certain constrained ratio game is shown to be equivalent to a pair of mutually dual generalized fractional programming problems. This also extends the concept of symmetric duality to min-max fractional programming.Research by this author was carried out while he was visiting I.I.T. Delhi, India, under an Australian Vice-Chancellors' Committee Visiting Fellowship.     

Decomposition algorithms for generalized potential games

We analyze some new decomposition schemes for the solution of generalized Nash equilibrium problems. We prove convergence for a particular class of generalized potential games that includes some interesting engineering problems. We show that some versions of our algorithms can deal also with problems lacking any convexity and consider separately the case of two players for which stronger results can be obtained.     

Collective fixed points, generalized games and systems of generalized quasi-variational inclusion problems in topological spaces

In this paper, by using a fixed point theorem for expansive set-valued mappings with noncompact and nonconvex domains and ranges in topological spaces due to the author, we first prove a collective fixed point theorem and an existence theorem of equilibrium points for a generalized game. As applications, some new existence theorems of solutions for systems of generalized quasi-variational inclusion problems are established in noncompact topological spaces. Our results are different from known results in the literature.     

Protectionism and blocking power indices

We study here the protectionist role of blocking coalitions in a voting game. More precisely, we first present necessary properties that a family of coalitions must satisfy in order to be the blocking family of some game and show that they are sufficient conditions too. Furthermore, a procedure to determine all games having a given blocking family is provided. With regard to uniqueness and multiplicity, (a) the blocking families that univocally determine the game are characterized by means of a separation condition, and (b) it is shown that in the nonseparating case at least three games share each nonempty blocking family, and an upper bound is given for the number of such games. Some numerical examples illustrate our results. Finally, power indices related to the blocking structure are discussed. Research partially supported by Grant SGR 2005–00651 of the Catalonia Government and Grant MTM 2006–06064 of the Education and Science Spanish Ministry and the European Regional Development Fund.