Bi-cooperative Games Research Articles

The Shapley value for bicooperative games based on the semi-tensor product

In cooperative games, players can choose whether or not to participate in the coalition based on their own strategies. Bicooperative games are the generalization of cooperative games. In bicooperative games, players have the option of abstention in addition to “yes” and “no”. Therefore, in cooperative games, distributing the total profit obtained by the participant alliance is one of the most important issues in bicooperative game theory. In this paper, the calculation of the Shapley value for bicooperative games is discussed by using the semi-tensor product of matrices. Firstly, the Shapley matrix of bicooperative games is constructed. Secondly, the Shapley value formula for a bicooperative game is transformed into the product of the characteristic function matrix and the Shapley matrix. Finally, an example is given to demonstrate the main results. The matrix form of Shapley value obtained in this paper simplifies the calculation and provides a new tool for researching bicooperative games.

The Unique Characterization of the Shapley Value for Bi-cooperative Games

The aim of the present paper is to study the unique characterization for bicooperative games. Shapley value is the expected marginal contribution of the alliance. We will introduce some properties for bicooperative games. Our first characterization is based on the classical axioms determining the Shapley value with the symmetry axiom replaced excluded null axiom. In our second axiomatization we use structural axiom and a zero excluded axiom instead of effective axiom in classical cooperative games. Finally, We provide here linearity, anonymity, dummy and efficiency and structural axioms to study a one-point solution concept for Bi-cooperative games.

Weak null, necessary defender and necessary detractor players: characterizations of the Banzhaf and the Shapley bisemivalues

We focus on bicooperative games, a variation of the classic cooperative games and, in particular, on the Banzhaf and the Shapley bisemivalues defined on these games. We consider three special classes of players: weak null, necessary defender and necessary detractor players. By introducing new properties related to this kind of players, we provide new axiomatic characterizations of the Banzhaf and the Shapley bisemivalues giving, in both cases, a set of independent properties that univocally determine them. We also provide a computational procedure to calculate the allocations given by the Shapley bisemivalue in terms of the generalized multilinear extension of the game.

The Banzhaf interaction index for bi-cooperative games

The bipolar Möbius transform is an extension of dividends of cooperative games to that of bi-cooperative games. In this paper, we introduce the bipolar co-Möbius transform as an extension of co-Möbius transform of the cooperative game to that of bi-cooperative game. Then, we propose the Banzhaf interaction index among players in bi-cooperative games (or among criteria in bi-capacities ). Lastly, we give the expression of the Banzhaf interaction index in terms of each of the bipolar Möbius transform and bipolar co-Möbius transform as it was done for a cooperative game.

Some Properties for Bisemivalues on Bicooperative Games

In this work, we focus on bicooperative games, a variation of the classic cooperative games, and investigate the conditions for the coefficients of the bisemivalues—a generalization of semivalues for cooperative games—necessary and / or sufficient in order to satisfy some properties, including among others, desirability relation, balanced contributions, null player exclusion property and block property. Moreover, a computational procedure to calculate bisemivalues in terms of the multilinear extension of the game is given.

Bisemivalues for bicooperative games

We introduce bisemivalues for bicooperative games and we also provide an interesting characterization of this kind of values by means of weighting coefficients in a similar way as it was given for semivalues in the context of cooperative games. Moreover, the notion of induced bisemivalues on lower cardinalities also makes sense and an adaptation of Dragan’s recurrence formula is obtained. For the particular case of (p, q)-bisemivalues, a computational procedure in terms of the multilinear extension of the game is given.

Bi-cooperative games in bipolar fuzzy settings

In this paper, we introduce the notion of a bi-cooperative game with Bipolar Fuzzy Bi-coalitions and discuss the related properties. In many decision-making situations, players show bipolar motives while cooperating among themselves. This is modelled in both crisp and fuzzy environments. Bi-cooperative games with fuzzy bi-coalitions have already been proposed under the product order of bi-coalitions where one had memberships in [0, 1]. In the present paper, we adopt the alternative ordering: ordering by monotonicity and account for players’ memberships in , a break from the previous formulation. This simplifies the model to a great extent. The corresponding Shapley axioms are proposed. An explicit form of the Shapley value to a particular class of such games is also obtained. Our study is supplemented with an illustrative example.

A multilinear extension of a class of fuzzy bi-cooperative games

In this paper, a new class of bi-cooperative games with fuzzy bi-coalitions is proposed in multilinear extension form. The extension is shown to be unique. The solution concept discussed in [3] is investigated and characterized for this class of games.

Fuzzy Bi-cooperative games in multilinear extension form

In this paper, we introduce the notion of a fuzzy Bi-cooperative game in multilinear extension form. An LG value as a possible solution concept is obtained using standard fuzzy game theoretic axioms.

Equal Sharing Solutions for Bicooperative Games

Equal Sharing Solutions for Bicooperative Games

Bi-cooperative games with fuzzy bi-coalitions

In this paper, we introduce the notion of a bi-cooperative game with fuzzy bi-coalitions and discuss the related properties. In real game theoretic decision making problems, many criteria concerning the formation of coalitions have bipolar motives. Our model tries to explore such bipolarity in fuzzy environment. The corresponding Shapley axioms are proposed. An explicit form of the Shapley value as a possible solution concept to a particular class of such games is also obtained. Our study is supplemented with an illustrative example.

The selectope for bicooperative games

A bicooperative game is defined by a worth function on the set of ordered pairs of disjoint coalitions of players. The aim of this paper is to analyze the selectope for bicooperative games. This solution concept was introduced by Hammer et al. (1977) [20] and studied by Derks et al. (2000) [10] for cooperative games. We show the relations between the selectope, the core and the Weber set and obtain a characterization of almost positive bicooperative games as bicooperative games such that the core, the Weber set and the selectope coincide. Moreover, an axiomatic characterization of the elements of the selectope is obtained.

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 value for bi-cooperative games

Bi-cooperative games were introduced by Bilbao et al. as a generalization of TU cooperative games, in which each player can participate positively, negatively, or not at all. In this paper, we propose a definition of a share of the worth obtained by some players after they decided on their participation in the game. It turns out that the cost allocation rule does not look for a given player to her contribution at the opposite participation option to the one she chooses. The relevance of the value is discussed on several examples.

Biprobabilistic values for bicooperative games

The present paper introduces bicooperative games and develops some general values on the vector space of these games. First, we define biprobabilistic values for bicooperative games and observe in detail the axioms that characterize such values. Following the work of Weber [R.J. Weber, Probabilistic values for games, in: A.E. Roth (Ed.), The Shapley Value: Essays in Honor of Lloyd S. Shapley Cambridge University Press, Cambridge, 1988, pp. 101–119], these axioms are sequentially introduced observing the repercussions they have on the value expression. Moreover, compatible-order values are introduced and there is shown the relationship between these values and efficient values such that their components are biprobabilistic values.

Bipolarization of posets and natural interpolation

The Choquet integral w.r.t. a capacity can be seen in the finite case as a parsimonious linear interpolator between vertices of [ 0 , 1 ] n . We take this basic fact as a starting point to define the Choquet integral in a very general way, using the geometric realization of lattices and their natural triangulation, as in the work of Koshevoy. A second aim of the paper is to define a general mechanism for the bipolarization of ordered structures. Bisets (or signed sets), as well as bisubmodular functions, bicapacities, bicooperative games, as well as the Choquet integral defined for them can be seen as particular instances of this scheme. Lastly, an application to multicriteria aggregation with multiple reference levels illustrates all the results presented in the paper.

The Shapley value for bicooperative games

The aim of the present paper is to study a one-point solution concept for bicooperative games. For these games introduced by Bilbao (Cooperative Games on Combinatorial Structures, 2000) , we define a one-point solution called the Shapley value, since this value can be interpreted in a similar way to the classical Shapley value for cooperative games. The main result of the paper is an axiomatic characterization of this value.

The core and the Weber set for bicooperative games

This paper studies two classical solution concepts for the structure of bicooperative games. First, we define the core and the Weber set of a bicooperative game and prove that the core is always contained in the Weber set. Next, we introduce a special class of bicooperative games, the so-called bisupermodular games, and show that these games are the only ones in which the core and the Weber set coincide.

The core of bicapacities and bipolar games

Bicooperative games generalize classical cooperative games in the sense that a player is allowed to play in favor or against some aim, besides nonparticipation. Bicapacities are monotonic bicooperative games, they are useful in decision making where underlying scales are of bipolar nature, i.e., they distinguish between good/satisfactory values and bad/unsatisfactory ones. We propose here a more general framework to represent such situations, called bipolar game. We study the problem of finding the core of such games, i.e., the set of additive dominating games.

Interaction transform for bi-set functions over a finite set

Set functions appear as a useful tool in many areas of decision making and operations research, and several linear invertible transformations have been introduced for set functions, such as the Möbius transform and the interaction transform. The present paper establish similar transforms and their relationships for bi-set functions, i.e. functions of two disjoint subsets. Bi-set functions have been recently introduced in decision making (bi-capacities) and game theory (bi-cooperative games), and appear to open new areas in these fields.