Characteristic Function Form Research Articles

On Milnor's classes “L” and “D”

A twenty-one player counterexample is presented which disposes of two questions raised by J. W. Milnor in 1952 concerning the existence of certain pre-solutions, based on plausible lower and upper bounds to what a coalition should expect to receive in a cooperative game in characteristic function form. In the counterexample, the lower-bound set, known as “L”, is empty, and the upper-bound set, known as “D”, contains no efficient outcomes.

On strong representations of games by social choice functions

Games in characteristic function form are used to model an allocation of decision power among individuals involved in a voting situation. The problem of strong representation is to find a strategically acceptable social choice function that entails the allocation of power prescribed by a given game. Within the class of non-weak characteristic function games, we fully characterize the games that admit a strong representation. We apply this result to Peleg's problem of strong representation of simple games. Our results indicate that a strong representation requires significantly more than has been recognized in the literature.

A solution concept in a cooperative n‐person game based on the complaint differences and its application

AbstractThis paper proposes a new solution concept, based on the minimization, following the lexicographical order, of the complaint differences among coalitions in a cooperative n‐person game in characteristic function form. Such a solution minimizing the maximum complaint difference can be obtained as the pay‐off vector as the inequality constraints sequentially. A discussion is made on the pay‐off vector, which minimizes in the lexicographical order the complaint differences, taking the characteristics of the coalitions into consideration. Then as a method which can solve the formulated linear programming problem with less computational burden, an algorithm involving the relaxation method is proposed. A computer program was constructed and is applied to the cooperative water resource development program.

Random walk over a hypersphere

In a recent paper the author had shown that a special case of S. M. Joshi transform (so named after the author′s reverent father) of distributions urn:x-wiley:01611712:media:ijmm357149:ijmm357149-math-0001 is a characteristic function of a spherical distribution. Using the methods developed in that paper; the problem of distribution of the distance CD, where C and D are points niformly distributed in a hypersphere, has been discussed in the present paper. The form of characteristic function has also been obtained by the method of projected distribution. A generalization of Hammersley′s result has also been developed. The main purpose of the paper is to show that although the use of characteristic functions, using the method of Bochner, is available in problems of random walk yet distributional S. M. Joshi transform can be used as a natural tool has been proved for the first time in the paper.

A Core Existence Theorem for Games Without Ordered Preferences

[Introduction] To a large extent the cooperative theory of games has an altogether different appearance from the noncooperative theory. The noncooperative theory generally deals with games in either extensive form or normal form, while the cooperative theory is usually described in characteristic function form. One of the central concepts in the cooperative theory is that of the core, which is the set of utility allocations which no coalition can improve upon. This notion of the core and of the characteristic function form of a game depends heavily on the existence of a utility representation for players' preferences. Recently Gale and Mas-Colell [3] and Shafer and Sonnenschein [6] have proven theorems on the existence of a Nash equilibrium for noncooperative games in normal form in which the players' preferences over strategy vectors are not necessarily complete or transitive and so may fail to have a utility representation. Thus it might appear that the noncooperative theory is applicable in environments where the cooperative theory is not. In order to formulate theorems in the cooperative theory of games which can be applied to environments in which players may have nonordered preferences, the characteristic function must be reformulated in terms of physical outcomes as opposed to utility outcomes. The players' preferences can then be expressed in terms of the physical outcomes without the use of a utility function.

Totally balanced games arising from controlled programming problems

A cooperative game in characteristic-function form is obtained by allowing a number of individuals to esercise partial control over the constraints of a (generally nonlinear) mathematical programming problem, either directly or through committee voting. Conditions are imposed on the functions defining the programming problem and the control system which suffice to make the game totally balanced. This assures a nonempty core and hence a stable allocation of the full value of the programming problem among the controlling palyers. In the linear case the core is closely related to the solutions of the dual problem. Applications are made to a variety of economic models, including the transferable utility trading economies of Shapley and Shubik and a multishipper one-commodity transshipment model with convex cost functions and concave revenue functions. Dropping the assumption of transferable utility leads to a class of controlled multiobjective or ‘Pareto programming’ problems, which again yield totally balanced games.

A Unique Bargaining Solution Based on Competitive Commitments

In this paper, a bargaining solution is defined based on the ability of bargainers to commit themselves to various minimum utility levels. The solution is descriptive rather than axiomatic. Bargainers must spend resources for commitments, and they argue about the respective merits of the various outcomes, making objections and counterobjections. The solution is based on Aumann and Maschler’s bargaining set for cooperative games in characteristic function form. Existence and uniqueness (in utilities) are shown using assumptions weak enough to be satisfied in most applications. The solution is illustrated by a simple example.

Array beam noise characterization

The formulation of the stochastic process for beam noise of a horizontal array operating at low frequency is analyzed based on the array's response to the distant ship-induced noise field. The general form of beam noise characteristic function obtained with nonhomogeneous compound Poisson process assumption for shipping distribution is related to three major operation environment parameters: the mean number of shipping density, the array aperture length and beam pattern, and random fluctuation and range dependence in the transmission loss function. This characteristic function can be used to derive the probability of observing low noise region in terms of frequency-azimuth-time representation of noise surface for some specific operation scenarios. Computer modeling for ship lane dominant noise field and uniform shipping distributed noise field are performed for investigating the sensitivity of the relevant parameters to the array's beam noise statistical property.

The occurrence of low beam noise level in ship induced noise field

The formulation of the stochastic process for beam noise of a horizontal array operating at low frequency is analyzed based on the array's response to the distant ship-induced noise field. The general form of beam noise characteristic function obtained with nonhomogeneous compound Poisson process assumption for shipping distribution is related to three major environmental parameters: the mean number of shipping density, the array aperture length and beam pattern, and random fluctuation and range dependence in the transmission loss function. This characteristic function can be used to derive the probability of observing low noise regions in terms of frequency-azimuth-time representation of noise surface for certain acoustic environments. Computer modeling for ship lane dominant noise field and uniform shipping distributed noise field were performed for investigating the relationship among the relevant parameters in the array's beamforming output. The results illustrate the sensitivity of beam noise statistical characteristics to the angular ship density.

Power, luck and the right index

We have pointed out the theoretical drawbacks of the traditional indices for measuring a priori voting power inasmuch as they are implied in considering the coalition value a private good. This criticism caused us to view the coalition outcome as a public good. From this aspect and additional considerations with respect to power, luck, and decisiveness, we obtained a “story” describing the characteristics of an adequate measure of a priori voting power. These characteristics were found to be fulfilled by an index presented by Holler (1978). Through the above analysis this index has received its theoretical justification. An independent view of this index was then provided by means of an axiomatic characterization. This characterization makes possible abstract comparison of the index with previously established “private good” indices. While we have restricted our attention to simple games, the index presented can be generalized to provide a “value” on games in characteristic function form. We leave this topic for future conideration.

Finite Solution Theory for Coalitional Games

In 1944 von Neumann and Morgenstern introduced a theory of solutions (stable sets) for multiperson cooperative games in characteristic function form. Some special classes of games are known to have solutions which are finite sets. These finite solutions give rise to interesting geometrical structures and basic combinatorial patterns. They have provided new insights into problems in the social sciences and they invite additional interpretations and uses in the mathematical and physical sciences. This paper provides an introduction and survey of finite solution theory.

Games with No Solutions and Empty Cores

Rather general existence theorems are known for two of the three classical models in game theory. These include the theorems by J. von Neumann and J. F. Nash on the existence of equilibria for 2-person, zero-sum and n-person, general-sum games in normal (strategic) form; as well as the theorems on information and the nature of optimal strategies by E. Zermelo, H. W. Kuhn, N. N. Vorob'ev and G. L. Thompson for games in extensive (tree) form. This paper discusses the lack of a general existence theorem for the von Neumann-Morgenstern theory of solutions (stable sets) for their multiperson cooperative games in characteristic function (coalitional) form. In particular, it demonstrates that there are games of 14 or more players for which no such solution exists, and for which the core (an alternate solution concept) is the empty set.

Mollifiers for games in normal form and the Harsanyi-Selten valuation function

The concepts of disruption and mollifiers ofCharnes/Rousseau/Seiford [1978] for games in characteristic function form are here extended to games in normal form.

Characterizing the Banzhaf and Shapley values assuming limited linearity

This paper presents characterizations of the Banzhaf-Coleman and Shapley-Shubik indices for monotonic simple games. The characterizations are obtained without explicitly requiring that the indices satisfy the linearity assumptionψ (v∧ w) +ψ (v ∨ w) =ψ (v) + ψ (w). The ideas developed are then used to obtain a characterization of the Banzhaf value for the class of alln-person games in characteristic function form.

Minimum cost spanning tree games

We consider the problem of cost allocation among users of a minimum cost spanning tree network. It is formulated as a cooperative game in characteristic function form, referred to as a minimum cost spanning tree (m.c.s.t.) game. We show that the core of a m.c.s.t. game is never empty. In fact, a point in the core can be read directly from any minimum cost spanning tree graph associated with the problem. For m.c.s.t. games with efficient coalition structures we define and construct m.c.s.t. games on the components of the structure. We show that the core and the nucleolus of the original game are the cartesian products of the cores and the nucleoli, respectively, of the induced games on the components of the efficient coalition structure.

A Stochastic Solution Concept for n-Person Games

Let X be a set of outcomes among which a set of N players, each having a preference relation on X, must choose. Let v: 2N\/Ø} → 2X be a game in generalized characteristic function form, where v(C) denotes the set of outcomes that a coalition C can guarantee its members regardless of actions by players outside of C. By letting c(x, y) denote the number of minimal coalitions via which y is directly accessible from x, a Markov chain model of outcome selection is developed. By establishing convergence results for both finite and spatial (X ⊆ Rm) outcome cases, a probability measure on X is obtained and referred to as the stochastic solution of the game. After presenting some initial results on the dependence of the stochastic solution on the initial outcome distribution, it is shown under reasonable assumptions that a (strong) core, if it exists, must occur with probability one. Generally, the results obtained have natural interpretations and proofs using the language and theory of Markov chains. Finally, some examples and previous experimental results are considered in terms of the model. Stochastic solution values obtained and their agreement with available experimental values appear to be very encouraging.

A Social Coalitional Equilibrium Existence Lemma

The purpose of this paper is to construct an abstract model of a society in which each member can cooperate with others by forming a coalition, but at the same time can be influenced by the members outside the coalition. A new concept of equilibrium, called here a social coalitional equilibrium, is proposed, and a sufficient condition for its existence provided. The social coalitional equilibrium may be considered a synthesis of the Nash equilibrium (a noncooperative solution concept) and the core (a cooperative solution concept). The paper provides a broadly applicable mathematical tool for proving existence of equilibrium for models of labor-managed market economies. THE PURPOSE OF THE PRESENT PAPER is to construct an abstract model of a society in which each member can cooperate with others by forming a coalition, but at the same time can be influenced by the members outside the coalition. A new concept of equilibrium, called here a social coalitional equilibrium, is proposed, and a sufficient condition for its existence provided. A coalition structure is endogenously determined in equilibrium. When the model is suitably specified, it describes noncooperative behavior of the members, and the social coalitional equilibrium is reduced to the Nash equilibrium. Given another extreme specification, the model is reduced to a cooperative game without sidepayments in characteristic function form, and the set of social coalitional equilibria corresponds precisely to the core. The need for study of the present model arose originally out of the author's recent investigation of labor-managed market economies, both the socialistic version and the capitalistic version, in [10, 11, 12]. Indeed, the model and the concept of social coalitional equilibrium were obtained by abstracting and generalizing the formal aspect of those studies. The socialistic labor-managed market economy has long been the object of study of various authors; see, e.g., Vanek [19]. The capitalistic labor-managed market economy seems to reflect the institutional aspect of the modern capitalistic society better than the economies formulated by other available general equilibrium models. It differs, for example, from the standard private ownership economy of Arrow and Debreu [1] (see also Debreu [6, Sec. 5.5]) in that (i) the ownership and control of a firm are separated; (ii) no particular firm is given a priori, but instead, the list of potential firms is given as a primitive datum; and (iii) the behavioral principle of the economic agents involves both noncooperative behavior and cooperative behavior. Because of the above feature (ii), any general equilibrium model of this economy has to formulate the economic mechanism that underlies the formation of firms in equilibrium. As

A dynamic solution concept for abstract games

Several solution concepts have been defined for abstract games. Some of these are the core, due to Gillies and Shapley, the Von Neumann-Morgenstern stable sets, and the subsolutions due to Roth. These solution concepts are rather static in nature. In this paper, we propose a new solution concept for abstract games, called the dynamic solution, that reflects the dynamic aspects of negotiation among the players. Some properties of the dynamic solution are studied. Also, the dynamic solution of abstract games arising fromn-person cooperative games in characteristic function form is investigated.

Logrolling and budget allocation games

In this paper we approach the concept of logrolling by examining a voting system where choices are made among sets of competing projects as a game in characteristic function form. We translate the question: “Will there be prices for votes on different projects which clear the market?” into a different, but equivalent question: “Is the formal game we have described amarket game?” We show that in general the answer is no, unless all voters have virtually the same preferences.

A new index of power for simplen-person games

Measures of (a priori) power play a useful role in assessing the character of interpersonal interaction found in collective decision making bodies. We propose and axiomatically characterize an alternative power index to the familiarShapley/Shubik andBanzhaf indices which can be used for such purposes. The index presented is shown to be unique for the class of simplen-person games. By subsequent generalization of the index and its axioms to the class ofn-person games in characteristic function form we obtain an analog to theShapley value.