A note on the multiple partners assignment game

In the assignment game of Shapley and Shubik [Shapley, L.S., Shubik, M., 1972. The assignment game. I. The core, International journal of Game Theory 1, 11-130] agents are allowed to form one partnership at most. That paper proves that, in the context of firms and workers, given two stable payoffs for the firms there is a stable payoff which gives each firm the larger of the two amounts and also one which gives each of them the smaller amount. Analogous result applies to the workers. Sotomayor [Sotomayor, M., 1992. The multiple partners game. In: Majumdar, M. (Ed.), Dynamics and Equilibrium: Essays in Honor to D. Gale. Mcmillian, pp. 322-336] extends this analysis to the case where both types of agents may form more than one partnership and an agent`s payoff is multi-dimensional. Instead, this note concentrates in the total payoff of the agents. It is then proved the rather unexpected result that again the maximum of any pair of stable payoffs for the firms is stable but the minimum need not be, even if we restrict the multiplicity of partnerships to one of the sides. (C) 2009 Elsevier B.V. All rights reserved.

Stability property of matchings is a natural solution concept in coalitional market games

Stability of matchings was proved to be a new cooperative equilibrium concept in Sotomayor (Dynamics and equilibrium: essays in honor to D. Gale, 1992). That paper introduces the innovation of treating as multi-dimensional the payoff of a player with a quota greater than one. This is done for the many-to-many matching model with additively separable utilities, for which the stability concept is defined. It is then proved, via linear programming, that the set of stable outcomes is nonempty and it may be strictly bigger than the set of dual solutions and strictly smaller than the core. The present paper defines a general concept of stability and shows that this concept is a natural solution concept, stronger than the core concept, for a much more general coalitional game than a matching game. Instead of mutual agreements inside partnerships, the players are allowed to make collective agreements inside coalitions of any size and to distribute his labor among them. A collective agreement determines the level of labor at which the coalition operates and the division, among its members, of the income generated by the coalition. An allocation specifies a set of collective agreements for each player.

Menezesite, the first natural heteropolyniobate, from Cajati, Sao Paulo, Brazil: Description and crystal structure

Menezesite, ideally Ba2MgZr4(BaNb12O42)center dot 12H(2)O, occurs as a vug mineral in the contact zone between dolomite carbonatite and ""jacupirangite"" (=a pyroxenite) at the Jacupiranga mine, in Cajati county, Sao Paulo state, Brazil, associated with dolomite, calcite, magnetite, clinohumite, phlogopite, ancylite-(Ce), strontianite, pyrite, and tochilinite. This is also the type locality for quintinite-2H. The mineral forms rhombododecahedra up to I mm, isolated or in aggregates. Menezesite is transparent and displays a vitreous luster; it is reddish brown with a white streak. It is non-fluorescent. Mohs hardness is about 4. Calculated density derived from the empirical formula is 4.181 g/cm(3). It is isotropic, 1.93(1) (white light); n(calc) = 2.034. Menezesite exhibits weak anomalous birefringence. The empirical formula is (Ba1.47K0.53Ca0.3,Ce0.17Nd0.10Na0.06La0.02)(Sigma 2.66)(Mg0.94Mn0.23Fe0.23Al0.03)(Sigma 1.43)(Zr2.75Ti0.96Th0.29)(Sigma 4.00)[(Ba0.72Th0.26U0.02)(Sigma 1.00)(Nb9.23Ti2.29Ta0.36Si0.12)Sigma O-12.00(42)]center dot 12H(2)O. The mineral is cubic, space group 10 (204), a = 13.017(1) angstrom, V = 2206(1) angstrom(3), Z = 2. Menezesite is isostructural with the synthetic compound Mg-7[MgW12O42](OH)(4)center dot 8H(2)O. The mineral was named in honor of Luiz Alberto Dias Menezes Filho (born 1950), mining engineer, mineral collector and merchant. Both the description and the name were approved by the CNMMN-IMA (Nomenclature Proposal 2005-023). Menezesite is the first natural heteropolyniobate. Heteropolyanions have been employed in a range of applications that include virus-binding inorganic drugs (including the AIDs virus), homogeneous and heterogeneous catalysts, electro-optic and electrochromic materials, metal and protein binding, and as building blocks for nanostructuring of materials.