20 research outputs found

    Cooperative Games on Antimatroids

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    AMS classification: 90D12;game theory;cooperative games;antimatroids

    Decomposition of the space of TU-games, Strong Transfer Invariance and the Banzhaf value

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    We provide a new and concise characterization of the Banzhaf value on the (linear) space of all TU-games on a fixed player set by means of two transparent axioms. The first one is the well-known Dummy player axiom. The second axiom, called Strong transfer invariance, indicates that a player's payoff is invariant to a transfer of worth between two coalitions he or she belongs to. To prove this result we derive direct-sum decompositions of the space of all TU-games. We show that, for each player, the space of all TU-games is the direct sum of the subspace of TU-games where this player is dummy and the subspace spanned by the TU-games used to construct the transfers of worth. This decomposition method has several advantages listed as concluding remarks

    Axiomatizations of two types of Shapley values for games on union closed systems

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    A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utility, or simply a TU-game. A (single-valued) solution for TU-games assigns a payoff distribution to every TU-game. A well-known solution is the Shapley value. In the literature various models of games with restricted cooperation can be found. So, instead of allowing all subsets of the player set N to form, it is assumed that the set of feasible coalitions is a subset of the power set of N. In this paper, we consider such sets of feasible coalitions that are closed under union, i.e. for any two feasible coalitions also their union is feasible. We consider and axiomatize two solutions or rules for these games that generalize the Shapley value: one is obtained as the conjunctive permission value using a corresponding superior graph, the other is defined as the Shapley value of a modified game similar as the Myerson value for games with limited communication. © 2010 The Author(s)

    Indices of criticality in simple games

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    3sìThe correct reference for the aricle is Vol. 21, No. 1 (2019) 1940003partially_openWe generalize the notion of power index for simple games to different orders of criticality, where the order of criticality represents the possibility for players to gain more power over the members of a coalition thanks to the collusion with other players. We study the behaviour of these criticality indices to compare the power of dierent players within a single voting situation, and that of the same player with varying weight across different voting situations. In both cases we establish monotonicity results in line with those of Turnovec (1998). Finally, we examine which properties characterizing the indices of Shapley-Shubik and Banzhaf are shared by these new indices.embargoed_20200515Marco Dall'Aglio, Vito Fragnelli, Stefano MorettiDall'Aglio, Marco; Vito, Fragnelli; Stefano, Morett

    The core of games on ordered structures and graphs

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    In cooperative games, the core is the most popular solution concept, and its properties are well known. In the classical setting of cooperative games, it is generally assumed that all coalitions can form, i.e., they are all feasible. In many situations, this assumption is too strong and one has to deal with some unfeasible coalitions. Defining a game on a subcollection of the power set of the set of players has many implications on the mathematical structure of the core, depending on the precise structure of the subcollection of feasible coalitions. Many authors have contributed to this topic, and we give a unified view of these different results
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