A characterization of the multi-choice Shapley value with partially consistent property (Q971524)

In 1993 C. R. Hsiao and T. E. S. Raghavan extended the notion of the classical coopearative games (in characteristic function form) to multi-choice cooperative games, and they axiomatically derived the Shapley value for them (called by some authors later as the H\&R Shapley value). The authors of this paper study a slight modification of multi-choice cooperative games where each player can have a different number of choices, and they extend the H\&R Shapley value to this wider class \(G\) of games. In particular, the notion of \(w\)-potential function for multi-choice games in \(G\) and its properties are discussed. Next, reduced games for multi-choice games in \(G\) and two possible properties (consistency and partial consistency) of values on \(G\) are introduced. Using the \(w\)-potential functions, the authors show that the H\&R Shapley value is consistent. In their second main result they characterize the H\&R Shapley value with the help of three properties of a value on \(G\): Pareto optimality, \(w\)-proportionality for multi-choice two-person games, and partial consistency. This generalizes the known result of Hart and Mas-Collel for classical cooperative games. An interesting part of the paper is the appendix, where the authors give a wide discussion why they study multi-choice games.