Multi-choice Games Research Articles

Consistent extensions and subsolutions of the core for the multichoice transferable-utility games

We extend the reduced games introduced by Davis and Maschler (Naval Res. Log. Q. 12:223–259, 1965) and Moulin (J. Econ. Theory. 36:120–148, 1985) to multichoice transferable-utility games. First, we provide an example to illustrate that the core proposed by van den Nouweland et al. (Math Methods Oper. Res. 41:289–311, 1995) violates related consistency properties. Further, we propose the minimal consistent extensions of the core and the maximal consistent subsolutions of the core. We also provide an axiomatization based on related consistency properties and its converse.

Reduction and dynamic approach for the multi-choice Shapley value

In the framework of multi-choice games, we propose a specific reduction to construct a dynamic process for the multi-choice Shapley value introduced by Nouweland et al. [8].

Two-step coalition values for multichoice games

We introduce and compare several coalition values for multichoice games. Albizuri defined coalition structures and an extension of the Owen coalition value for multichoice games using the average marginal contribution of a player over a set of orderings of the player’s representatives. Following an approach used for cooperative games, we introduce a set of nested or two-step coalition values on multichoice games which measure the value of each coalition and then divide this among the players in the coalition using either a Shapley or Banzhaf value at each step. We show that when a Shapley value is used in both steps, the resulting coalition value coincides with that of Albizuri. We axiomatize the three new coalition values and show that each set of axioms, including that of Albizuri, is independent. Further we show how the multilinear extension can be used to compute the coalition values. We conclude with a brief discussion about the applicability of the different values.

The duplicate extension for the equal allocation of nonseparable costs

Different from the multi-choice extension of the equal allocation of nonseparable costs (EANSC) due to Liao (Econ Bull 3:1–8, 2008), this paper proposes an alternative extension of the EANSC on multi-choice transferable-utility (TU) games. It turns out that the EANSC of a multi-choice TU game coincides with the EANSC of a corresponding “replicated” TU game. Further, we introduce an extended reduced game to characterize this extended EANSC by means of related consistency. Based on the reduced games, a dynamic process is also proposed.

The average tree solution for multi-choice forest games

In this article we study cooperative multi-choice games with limited cooperation possibilities, represented by an undirected forest on the player set. Players in the game can cooperate if they are connected in the forest. We introduce a new (single-valued) solution concept which is a generalization of the average tree solution defined and characterized by Herings et al. (Games Econ. Behav. 62:77-92, 2008) for TU-games played on a forest. Our solution is characterized by component efficiency, component fairness and independence on the greatest activity level. It belongs to the precore of a restricted multi-choice game whenever the underlying multi-choice game is superadditive and isotone. We also link our solution with the hierarchical outcomes (Demange in J. Polit. Econ. 112:754-778, 2004) of some particular TU-games played on trees. Finally, we propose two possible economic applications of our average tree solution.

The construction and characterization of egalitarian solutions for multi-choice NTU games

In this article we construct a procedure to define the egalitarian solutions in the context of multi-choice non-transferable utility (NTU) games. Also, we show that in the presence of other weak axioms the egalitarian solutions are the only monotonic ones.

The multi-core, balancedness and axiomatizations for multi-choice games

This note extends the solution concept of the core for cooperative games to multi-choice games. We propose an extension of the theorem of Bondareva (Problemy Kybernetiki 10:119–139, 1963) and Shapley (Nav Res Logist Q 14:453–460, 1967) to multi-choice games. Also, we introduce a notion of reduced games for multi-choice games and provide an axiomatization of the core on multi-choice games by means of corresponding notion of consistency and its converse.

Converse consistent enlargements of the unit-level-core of the multi-choice games

We extend the reduced games introduced by Moulin (J Econ Theory 36:120–148, 1985) to multi-choice games, and define related properties of consistency and its converse. Since the unit-level-core proposed by Hwang and Liao (J Glob Optim 2009) violates (weak) converse consistency, we propose to minimally enlarge the unit-level-core so as to recover (weak) converse consistency.

Dynamic Approaches for Multichoice Solutions

Based on alternative reduced games, several dynamic approaches are proposed to show how the three extended Shapley values can be reached dynamically from arbitrary efficient payoff vectors on multichoice games.

Multilinear extensions and values for multichoice games

We define multilinear extensions for multichoice games and relate them to probabilistic values and semivalues. We apply multilinear extensions to show that the Banzhaf value for a compound multichoice game is not the product of the Banzhaf values of the component games, in contrast to the behavior in simple games. Following Owen (Manag Sci 18:64–79, 1972), we integrate the multilinear extension over a simplex to construct a version of the Shapley value for multichoice games. We compare this new Shapley value to other extensions of the Shapley value to multichoice games. We also show how the probabilistic value (resp. semivalue, Banzhaf value, Shapley value) of a multichoice game is equal to the probabilistic value (resp. semivalue, Banzhaf value, Shapley value) of an appropriately defined TU decomposition game. Finally, we explain how semivalues, probabilistic values, the Banzhaf value, and this Shapley value may be viewed as the probability that a player makes a difference to the outcome of a simple multichoice game.

The multichoice coalition value

In this paper we define a solution for multichoice games which is a generalization of the Owen coalition value (Lecture Notes in Economics and Mathematical Systems: Essays in Honor of Oskar Morgenstern, Springer, New York, pp. 76–88, 1977) for transferable utility cooperative games and the Egalitarian solution (Peters and Zanks, Ann. Oper. Res. 137, 399–409, 2005) for multichoice games. We also prove that this solution can be seen as a generalization of the configuration value and the dual configuration value (Albizuri et al., Games Econ. Behav. 57, 1–17, 2006) for transferable utility cooperative games.

Convex multi-choice games: Characterizations and monotonic allocation schemes

This paper focuses on new characterizations of convex multi-choice games using the notions of exactness and superadditivity. Furthermore, level-increase monotonic allocation schemes (limas) on the class of convex multi-choice games are introduced and studied. It turns out that each element of the Weber set of such a game is extendable to a limas, and the (total) Shapley value for multi-choice games generates a limas for each convex multi-choice game.

MOULIN REDUCTION, V-N REDUCTION AND AXIOMATIZATIONS OF THE DUPLICATE CORE

In this paper, we extend to the multi-choice games cases the reduced games introduced by Moulin (1985) and Voorneveld and van den Nouweland (1998), respectively. Also, we provide several axiomatizations of the duplicate core on multi-choice games by means of the properties of consistency and its converse.

The unit-level-core for multi-choice games: the replicated core for TU games

This note extends the solution concept of the core for traditional transferable-utility (TU) games to multi-choice TU games, which we name the unit-level-core. It turns out that the unit-level-core of a multi-choice TU game is a "replicated subset" of the core of a corresponding "replicated" TU game. We propose an extension of the theorem of Bondareva (Probl Kybern 10:119---139, 1963) and Shapley (Nav Res Logist Q 14:453---460, 1967) to multi-choice games. Also, we introduce the reduced games for multi-choice TU games and provide an axiomatization of the unit-level-core on multi-choice TU games by means of consistency and its converse.

Consistent extensions and subsolutions of the core of multi-choice NTU games

We extend the reduced games introduced by Davis and Maschler (Naval Res Log Q 12:223–259, 1965) and Moulin (J Econ Theory 36:120–148, 1985) to multi-choice non-transferable utility games and define two related properties of consistency. We also show that the core proposed by Hwang and Li (Math Methods Oper Res 61:33–40, 2005) violates these two consistency properties. In order to investigate how seriously it violates these two consistency properties, we provide consistent extensions and consistent subsolutions of the core.

Equivalence theorem, consistency and axiomatizations of a multi-choice value

This paper is devoted to the study of solutions for multi-choice games which admit a potential, such as the potential associated with the extended Shapley value proposed by Hsiao and Raghavan (Int J Game Theory 21:301---302, 1992; Games Econ Behav 5:240---256, 1993). Several axiomatizations of the family of all solutions that admit a potential are offered and, as a main result, it is shown that each of these solutions can be obtained by applying the extended Shapley value to an appropriately modified game. In the framework of multi-choice games, we also provide an extension of the reduced game introduced by Hart and Mas-Colell (Econometrica 57:589---614, 1989). Different from the works of Hsiao and Raghavan (1992, 1993), we provide two types of axiomatizations, one is the analogue of Myerson's (Int J Game Theory 9:169---182, 1980) axiomatization of the Shapley value based on the property of balanced contributions. The other axiomatization is obtained by means of the property of consistency.

POTENTIAL IN MULTI-CHOICE COOPERATIVE TU GAMES

This paper is devoted to the study of solutions for multi-choice transferable-utility (TU) games which admit a potential, such as the potential associated with a solution in the context of multi-choice TU games. Several axiomatizations of the family of all solutions that admit a potential are offered and, as a main result, it is shown that each of these solutions can be obtained by applying the weighted associated consistent value proposed in this paper to an appropriately modified game. We also characterize the weighted associated consistent value by means of the weighted balanced contributions and the associated consistency.

Potential approach and characterizations of a Shapley value in multi-choice games

The main focus of this paper is on the restricted Shapley value for multi-choice games introduced by Derks and Peters [Derks, J., Peters, H., 1993. A Shapley value for games with restricted coalitions. International Journal of Game Theory 21, 351–360] and studied by Klijn et al. [Klijn, F., Slikker, M., Zazuelo, J., 1999. Characterizations of a multi-choice value. International Journal of Game Theory 28, 521–532]. We adopt several characterizations from TU game theory and reinterpret them in the framework of multi-choice games. We generalize the potential approach and show that this solution can be formulated as the vector of marginal contributions of a potential function. Also, we characterize the family of all solutions for multi-choice games that admit a potential. Further, a consistency result is reported.

Potentializability and consistency for multi-choice solutions

The paper introduces and discusses several characterizations of the multi-choice Shapley value (introduced by Nouweland et al. in ZOR-Math Methods Oper Res 41:289–311, 1995), on the domain of multi-choice games. These characterizations generalize the potential approach and results of Calvo and Santos (Math Soc Sci 34:175–190, 1997) into this domain, and the relations are treated with the axiomatic approach in Calvo and Santos (Math Soc Sci 40:341–354, 2000). Also, based on the axioms of two-person standardness and consistency, we offer an axiomatization of the multi-choice Shapley value.

A new approach to the core and Weber set of multichoice games

Multichoice games have been introduced by Hsiao and Raghavan as a generalization of classical cooperative games. An important notion in cooperative game theory is the core of the game, as it contains the rational imputations for players. We propose two definitions for the core of a multichoice game, the first one is called the precore and is a direct generalization of the classical definition. We show that the precore coincides with the definition proposed by Faigle, and that the set of imputations may be unbounded, which makes its application questionable. A second definition is proposed, imposing normalization at each level, causing the core to be a convex compact set. We study its properties, introducing balancedness and marginal worth vectors, and defining the Weber set and the pre-Weber set. We show that the classical properties of inclusion of the (pre)core into the (pre)-Weber set as well as their coincidence in the convex case remain valid. A last section makes a comparison with the core defined by van den Nouweland et al.