New Characterizations for Largeness of the Core

In this paper, we provide three new characterizations of largeness of the core. The first characterization is based on minimal covers of the grand coalition and associated inequalities. The second characterization shows the relation between the bases that provide core elements of the game and the bases that provide core elements of the games that are obtained from the original one by increasing the value of the grand coalition. The third characterization is based on the idea that if a base of the grand coalition does not provide a core element of the game, it should not provide a core element of a game which differs from the original one only by an increase of the value of the grand coalition. Based on these new characterizations, we show the equivalence between largeness of the core and stability of the core for games with at most 5 players. © 2012 Elsevier Inc.

Characterizing core stability with fuzzy games

This paper investigates core stability of cooperative, TU games via a fuzzy extension of the totally balanced cover of a TU game. The stability of the core of the fuzzy extension of a game, the concave extension, is shown to reflect the core stability of the original game and vice versa. Stability of the core is then shown to be equivalent to the existence of an equilibrium of a certain correspondence.cooperative game, core, stable set, fuzzy coalition, fuzzy game, core stability

Love, Will, and the Intellectual Ascents

Augustine’s accounts of his so-called mystical experiences in conf. 7.10.16, 17.23, and 9.10.24 are puzzling. The primary problem is that, although in all three accounts he claims to have seen “that which is,” we have no satisfactory account of what “that which is” is supposed to be. I shall be arguing that, contrary to a common interpretation, Augustine’s intellectual “seeing” of “being” in Books 7 and 9 was not a vision of the Christian God as a whole, nor of one of the divine persons, each of whom is equally God, according to Augustine. This becomes clear when we attend to the fact that Augustine is appropriating a specific meaning of “that which is” or “being” used by Plotinus in his account of the lover of Beauty. This resolution, however, leads to a second question. Is there anything distinctively Christian about any, or all, of Augustine’s ascents? On the one hand, it would be odd if there were not, given that the Confessions are addressed to the Christian God. On the other hand, upon close inspection we find that the allegedly specific “Christian” characteristics that modern commentators have identified in the ascents of conf. 7 and 9 also occur in the Neoplatonists. I will argue that there is in fact one important difference between Augustine and the Neoplatonists here that has not been pointed out in these prior interpretations

Lexicographic allocations and extreme core payoffs: the case of assignment games

We consider various lexicographic allocation procedures for coalitional games with transferable utility where the payoffs are computed in an externally given order of the players. The common feature of the methods is that if the allocation is in the core, it is an extreme point of the core. We first investigate the general relationship between these allocations and obtain two hierarchies on the class of balanced games. Secondly, we focus on assignment games and sharpen some of these general relationship. Our main result is the coincidence of the sets of lemarals (vectors of lexicographic maxima over the set of dual coalitionally rational payoff vectors), lemacols (vectors of lexicographic maxima over the core) and extreme core points. As byproducts, we show that, similarly to the core and the coalitionally rational payoff set, also the dual coalitionally rational payoff set of an assignment game is determined by the individual and mixed-pair coalitions, and present an efficient and elementary way to compute these basic dual coalitional values. This provides a way to compute the Alexia value (the average of all lemacols) with no need to obtain the whole coalitional function of the dual assignment game