Capacities and Games on Lattices: A Survey of Results

摘要

Capacities, introduced by Choquet, or fuzzy measures, introduced by Sugeno , are set functions vanishing on the empty set and being monotone w.r.t. set inclusion. They have been widely used in decision making and related areas, e.g., pattern recognition (see a compilation of applications in the monograph ). Cooperative games in characteristic form are set functions vanishing on the empty set, but not requiring monotonicity; they are sometimes called non monotonic fuzzy measures. They model the power or worth of coalitions of players. Recently, many variations or generalizations have been proposed around the above definition of a game, and to a less extent for capacities also. The primary aim of such generalizations is to model reality in a more accurate way. A first example in game theory is the case of ternary voting games. Compared to classical voting games, a ternary voting game allows for abstention. Similarly, bi-cooperative games proposed by Bilbao model a situation where a coalition of players is playing against another coalition, the remaining players not participating to the game. Multichoice games allow different levels of participation for each player, these levels being totally ordered. A rather different way has been explored by Gilboa and Lehrer with global games, where instead of coalitions, partitions of players are considered. Lastly we may cite Faigle and Kern, who introduced games with precedence constraints, which means that a coalition of players is valid only if it is consistent with some order relation among the players.