Type space on a purely measurable parameter space

Several game theoretical topics require the analysis of hierarchical beliefs, particularly in incomplete information situations. For the problem of incomplete information, Hars´anyi suggested the concept of the type space. Later Mertens & Zamir gave a construction of such a type space under topological assumptions imposed on the parameter space. The topological assumptions were weakened by Heifetz, and by Brandenburger & Dekel. In this paper we show that at very natural assumptions upon the structure of the beliefs, the universal type space does exist. We construct a universal type space, which employs purely a measurable parameter space structure.

The non-existence of a universal topological type space

The concept of types was introduced by Harsányi[8]. In the literature there are two approaches for formalizing types, type spaces: the purely measurable and the topological models. In the former framework Heifetz and Samet [11] showed that the universal type space exists and later Meier[13] proved that it is complete. In this paper we examine the topological approach and conclude that there is no universal topological type space in the category of topological type spaces.

On a Class of Fractional Type Integral Equations in Variable Exponent Spaces

2000 Mathematics Subject Classification: 45A05, 45B05, 45E05,45P05, 46E30

Density of Polynomials in the L^2 Space on the Real and the Imaginary Axes and in a Sobolev Space

2000 Mathematics Subject Classification: 41A10, 30E10, 41A65.

Common priors for generalized type spaces

The notion of common prior is well-understood and widely-used in the incomplete information games literature. For ordinary type spaces the common prior is de�fined. Pint�er and Udvari (2011) introduce the notion of generalized type space. Generalized type spaces are models for various bonded rationality issues, for �nite belief hierarchies, unawareness among others. In this paper we de�ne the notion of common prior for generalized types spaces. Our results are as follows: the generalization (1) suggests a new form of common prior for ordinary type spaces, (2) shows some quantum game theoretic results (Brandenburger and La Mura, 2011) in new light.

Every hierarchy of beliefs is a type

When modeling game situations of incomplete information one usually considers the players’ hierarchies of beliefs, a source of all sorts of complications. Harsányi (1967-68)’s idea henceforth referred to as the ”Harsányi program” is that hierarchies of beliefs can be replaced by ”types”. The types constitute the ”type space”. In the purely measurable framework Heifetz and Samet (1998) formalize the concept of type spaces and prove the existence and the uniqueness of a universal type space. Meier (2001) shows that the purely measurable universal type space is complete, i.e., it is a consistent object. With the aim of adding the finishing touch to these results, we will prove in this paper that in the purely measurable framework every hierarchy of beliefs can be represented by a unique element of the complete universal type space.

Invariance under type morphisms: the bayesian Nash equilibrium

Ely and Peski (2006) and Friedenberg and Meier (2010) provide examples when changing the type space behind a game, taking a "bigger" type space, induces changes of Bayesian Nash Equilibria, in other words, the Bayesian Nash Equilibrium is not invariant under type morphisms. In this paper we introduce the notion of strong type morphism. Strong type morphisms are stronger than ordinary and conditional type morphisms (Ely and Peski, 2006), and we show that Bayesian Nash Equilibria are not invariant under strong type morphisms either. We present our results in a very simple, finite setting, and conclude that there is no chance to get reasonable assumptions for Bayesian Nash Equilibria to be invariant under any kind of reasonable type morphisms.

Generalized type spaces

Ordinary type spaces (Heifetz and Samet, 1998) are essential ingredients of incomplete information games. With ordinary type spaces one can grab the notions of beliefs, belief hierarchies and common prior etc. However, ordinary type spaces cannot handle the notions of finite belief hierarchy and unawareness among others. In this paper we consider a generalization of ordinary type spaces, and introduce the so called generalized type spaces which can grab all notions ordinary type spaces can and more, finite belief hierarchies and unawareness among others. We also demonstrate that the universal generalized type space exists.

Common priors for generalized type spaces

The notion of common prior is well-understood and widely-used in the incomplete information games literature. For ordinary type spaces the common prior is defined. Pinter and Udvari (2011) introduce the notion of generalized type space. Generalized type spaces are models for various bonded rationality issues, for nite belief hierarchies, unawareness among others. In this paper we dene the notion of common prior for generalized types spaces. Our results are as follows: the generalization (1) suggests a new form of common prior for ordinary type spaces, (2) shows some quantum game theoretic results (Brandenburger and La Mura, 2011) in new light.

Bulk structures of PtO and PtO(2) from density functional calculations

Platinum plays an important role in catalysis and electrochemistry, and it is known that the direct interaction of oxygen with Pt surfaces can lead to the formation of platinum oxides (PtO(x)), which can affect the reactivity. To contribute to the atomistic understanding of the atomic structure of PtO(x), we report a density functional theory study of the atomic structure of bulk PtO(x) (1 <= x <= 2). From our calculations, we identified a lowest-energy structure (GeS type, space group Pnma) for PtO, which is 0.181 eV lower in energy than the structure suggested by W. J. Moore and L. Pauling [J. Am. Chem. Soc. 63, 1392 (1941)] (PtS type). Furthermore, two atomic structures were identified for PtO(2), which are almost degenerate in energy with the lowest-energy structure reported so far for PtO(2) (CaCl(2) type). Based on our results and analysis, we suggest that Pt and O atoms tend to form octahedron motifs in PtO(x) even at lower O composition by the formation of Pt-Pt bonds.

Quotient spaces of boundedly rational types

By identifying types whose low-order beliefs up to level li about the state of nature coincide, weobtain quotient type spaces that are typically smaller than the original ones, preserve basic topologicalproperties, and allow standard equilibrium analysis even under bounded reasoning. Our Bayesian Nash(li; l-i)-equilibria capture players inability to distinguish types belonging to the same equivalence class.The case with uncertainty about the vector of levels (li; l-i) is also analyzed. Two examples illustratethe constructions.

Synthesis, characterisation and thermoelectric properties of the oxytelluride Bi2O2Te

Bi2O2Te was synthesised from a stoichiometric mixture of Bi, Bi2O3 and Te by a solid state reaction. Analysis of powder X-ray diffraction data indicates that this material crystallises in the anti-ThCr2Si2 structure type (space group I4/mmm), with lattice parameters a = 3.98025(4) and c = 12.70391(16) Å. The electrical and thermal transport properties of Bi2O2Te were investigated as a function of temperature over the temperature range 300 ≤ T/K ≤ 665. These measurements indicate that Bi2O2Te is an n-type semiconductor, with a band gap of 0.23 eV. The thermal conductivity of Bi2O2Te is remarkably low for a crystalline material, with a value of only 0.91 W m-1 K-1 at room temperature.

Boron site preference in ternary Ta and Nb boron suicides

X-ray single crystal (XSC) and neutron powder diffraction data (NPD) were used to elucidate boron site preference for five ternary phases. Ta3Si1-xBx (x=0.112(4)) crystallizes with the Ti3P-type (space group P4(2)/n) with B-atoms sharing the 8g site with Si atoms. Ta5Si3-x (x=0.03(1); Cr5B3- type) crystallizes with space group 14/mcm, exhibiting a small amount of vacancies on the 4 alpha site. Both, Ta-5(Si1-xBx)(3), X=0.568(3), and Nb-5(Si1-xBx)(3), x=0.59(2), are part of solid solutions of M5Si3 with Cr5B3-type into the ternary M-Si-B systems (M=Nb or Ta) with B replacing Si on the 8h site. The D8(8)-phase in the Nb-Si-B system crystallizes with the Ti5Ga4-type revealing the formula Nb5Si3B1-x (x=0.292(3)) with B partially filling the voids in the 2b site of the Mn5Si3 parent type. (C) 2012 Elsevier Inc. All rights reserved.

Generalized Strichartz Inequalities for the Wave Equation on the Laguerre Hypergroup

Mathematics Subject Classification: 42B35, 35L35, 35K35

Polyharmonic Hardy Spaces on the Klein-Dirac Quadric with Application to Polyharmonic Interpolation and Cubature Formulas

2010 Mathematics Subject Classification: Primary 65D30, 32A35, Secondary 41A55.