The aim of the present paper is to study a one-point solution concept for bicooperative games. For these games introduced by Bilbao (Cooperative Games on Combinatorial Structures, 2000), we define a one-point solution called the Shapley value, since this value can be interpreted in a similar way to the classical Shapley value for cooperative games. The main result of the paper is an axiomatic characterization of this value.
The Shapley-Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. The Banzhaf power index depends on the number of ways in which each voter can effect a swing. If there are n players in a voting situation, then the function which measures the worst case running time for computing these indices is in Oðn2 n Þ. We present a combinatorial method based in generating functions to compute these power indices efficiently in weighted double or triple majority games and we study the time complexity of the algorithms. Moreover, we calculate these power indices for the countries in the Council of Ministers of the European Union under the new decision rules prescribed by the Treaty of Nice.
The present paper introduces bicooperative games and develops some general values on the vector space of these games. First, we define biprobabilistic values for bicooperative games and observe in detail the axioms that characterize such values. Following the work of Weber [R.J. Weber, Probabilistic values for games, in: A.E. Roth (Ed.), The Shapley Value: Essays in Honor of Lloyd S. Shapley Cambridge University Press, Cambridge, 1988, pp. 101-119], these axioms are sequentially introduced observing the repercussions they have on the value expression. Moreover, compatible-order values are introduced and there is shown the relationship between these values and efficient values such that their components are biprobabilistic values.
a b s t r a c tIn this paper we analyze ternary bicooperative games, which are a refinement of the concept of a ternary voting game introduced by Felsenthal and Machover. Furthermore, majority voting rules based on the difference of votes are simple bicooperative games. First, we define the concepts of the defender and detractor swings for a player. Next, we introduce the Banzhaf power index and the normalized Banzhaf power index. The main result of the paper is an axiomatization of the Banzhaf power index for the class of ternary bicooperative games. Moreover, we study ternary bicooperative games with two lists of weights and compute the Banzhaf power index using generating functions.
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