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Simple games. Desirability relations, trading, pseudoweightings. (English) Zbl 0943.91005

Princeton, NJ: Princeton University Press. 246 p. (1999).
Let \(P\) be a finite set and \(W\) a family of subsets of \(P\). \((P,W)\) is a hypergraph. If, furthermore, \(W\) satisfies a monotonicity property on set inclusion (if \(A\in W\) and \(A\subset B\), then \(B\in W\)), the authors call this structure \(a\) simple game. Though this is a common assumption, some authors do not distinguish simple games from hypergraphs, i.e., do not impose this monotonicity property to define a simple game. This structure is known, of course, in game theory, but also, under various names, in threshold logic and operations research (under the name of ‘coherent structure’). The authors have obviously chosen to use a political science vocabulary: winning coalitions (the elements of \(W\)), losing, blocking coalitions, weighted games, voting system etc.
An important development of simple games (also called voting games) in political science and economic theory concerns the association to the simple game structure of a voting or social choice rule. Given a set of options, a domination or social preference relation is defined by the existence of a winning coalition in which all the players have the same preference over two alternatives (alternative \(a\) dominates alternative \(b\) if there is a winning coalition in which everyone prefers \(a\) to \(b\)). Various solution concepts based on this domination relation or its refinements have been introduced and extensively studied according to different mathematical structures over the set of alternatives and/or players’ preferences. Simple games, in this case, play a rôle similar to voting rules such as majority rule or weighted majority rules. This book does not deal with this kind of topic. It has a very algebraic or order-theoretic flavor and as such includes many insights rarely seen in other books, not only because they are entirely new, but also because they were not in the mainstream of the current research trends. For this reason and also because the results will reveal to be important for the future applications to political science, this book is major contribution to our knowledge of this apparently simple structure.
Let me give some examples of the book main results. Rudin-Keisler ordering taken from ultrafilter theory is adapted to the simple game structure. A notion of trading in weighted games is introduced (this trading occurs in the form of players’ exchanges between winning coalitions). Weighted games are characterized by using this notion of trading. A desirability relation is defined to take account of individual or coalitional power. The individual desirability refers to the intuitive idea that for a coalition player 1 is more desirable than player 2 if player 1’s addition to the coalition would make it winning, but player 2’s would not. The desirability relation is shown to be connected with what the authors call pseudoweights, via an acyclicity property. This acyclicity is further studied for coalitional desirability relations. In particular, weighted games are characterized by the acyclicity of certain desirability relations on coalitions.
To conclude, this book is a major addition to the literature on cooperative game theory, to be read by game theorists, social choice theorists and mathematically-inclined political scientists. I recommend it also to theoretical economists who seem, sometimes, to have forgotten the very existence of cooperative game theory.
Reviewer: M.Salles (Caen)

MSC:

91A12 Cooperative games
91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance
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