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

Abstract

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.

Keywords:

• consistency
• minimal consistent extension
• maximal consistent subsolution

AMS Subject Classifications:

• 91A

Acknowledgments

The authors are grateful to the associate editor and the anonymous referees for very helpful suggestions and comments.

Notes

1 A multichoice TU game, introduced by Hsiao and Raghavan [21], is a generalization of a standard TU game.

2 The axiom was originally introduced by Harsanyi [22] under the name of bilateral equilibrium. For discussion of this axiom, please see Thomson [15].

3 On standard games, several solutions have been characterized by means of converse consistency properties, such as Peleg [5, 6], Tadenuma [12], Serrano and Volij [14], and so on.

4 This axiom is a weakening of CDMCON, since it requires that be individually rational as well.

5 In order to show the logical independence of the used axioms is needed.

6 Since the axiom MLIR is different from the axiom DMLIR, for some .

7 Let be a function on which assigns to each an elementSuch a function is called a single-valued solution. Here is the power index or the value of the player when he takes action to play game .