Biprobabilistic values for bicooperative games

Abstract

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.

The selectope for bicooperative games

“…This result can also be proved with an alternative proof which is closely related to the proof of the inclusion of the core in the Weber set for cooperative games given by Derks [9] and it is similar to the proof of the analogous result in [4].…”

“…There are some special collections of games in BG The relevance of these collections of games is made clear in the following result (see [3,4]). …”

“…A sharing system is a collection (1) and (2) the summation is taken only over consistent selectors, then the corresponding sharing value is the compatible-order value [4]. These values fill up the Weber set.…”

A Survey of Bicooperative Games

“…It is easy to prove (see [2]) that all the above collections are bases of BG N . A value on BG N is a function Φ : BG N → R n , which associates to each bicooperative game b a vector (Φ 1 (b) , .…”

Anonymous yes–no voting with abstention and multiple levels of approval

“…See, for example Bolger (1986), Fishburn (1973), Rubenstein (1980). More recently, there has been considerable attention (in, for example, Amer et al, 1998;Bilbao, 2000, Bilbao et al, 2008a, 2008bBolger, 1993Bolger, , 2000Machover, 1997, 1998;Freixas, 2005aFreixas, , 2005bLindner, 2005;Tchantcho et al, 2008) paid to the question of how to modify various power indices so as to take account of abstention. In Freixas and Zwicker (2003), we introduce a class of structures -the ( j,k) simple games, or ( j,k) games, for shortthat generalize simple games by allowing j ordered levels of approval in the input and k ordered levels in the output.…”