Information patterns and Nash equilibria in extensive games — II

In Part I of this paper we introduced extensive games with a non-atomic continuum of players. It was shown that the Nash plays (outcomes) are invariant of the information patterns on the game, provided that no player's unilateral change in moves can be observed by others. This led to an enormous reduction in the Nash plays of the these games, as exemplified in the anti-folk theorem. Our concern in this sequel is to develop a finite version of these results.     

Finite-size scaling in a microcanonical ensemble

The finite-size scaling technique is extended to a microcanonical ensemble. As an application, equilibrium magnetic properties of anL×L square lattice Ising model are computed using the microcanonical ensemble simulation technique of Creutz, and the results are analyzed using the microcanonical ensemble finite-size scaling. The computations were done on the multitransputer system of the Condensed Matter Theory Group at the University of Mainz.     

Peripheral phonons and phonon conductivity of the doped semiconductor

The percentage contribution of the peripheral phonons towards the total phonon conductivity of P-doped Ge has been studied with the help of the Ziman expression of the electron-phonon scattering relaxation rate and the Callaway integral of the lattice thermal conductivity. It is found that the percentage contribution of the peripheral phonons increases with the increase of temperature. The entire study is limited to the temperature range 1–5 K.     

On the uniqueness of the Shapley value

L.S. Shapley [1953] showed that there is a unique value defined on the classD of all superadditive cooperative games in characteristic function form (over a finite player setN) which satisfies certain intuitively plausible axioms. Moreover, he raised the question whether an axiomatic foundation could be obtained for a value (not necessarily theShapley value) in the context of the subclassC (respectivelyC′, C″) of simple (respectively simple monotonic, simple superadditive) gamesalone. This paper shows that it is possible to do this. Theorem I gives a new simple proof ofShapley's theorem for the classG ofall games (not necessarily superadditive) overN. The proof contains a procedure for showing that the axioms also uniquely specify theShapley value when they are restricted to certain subclasses ofG, e.g.,C. In addition it provides insight intoShapley's theorem forD itself. Restricted toC′ orC″, Shapley's axioms donot specify a unique value. However it is shown in theorem II that, with a reasonable variant of one of his axioms, a unique value is obtained and, fortunately, it is just theShapley value again.     

Ising thermodynamics on the Sierpinski gasket

We report a calculation of the thermodynamic properties of an Ising system on a fractal lattice, the Sierpinski gasket (SG). The scale-invariant geometry of SG leads to a wider critical region compared to that in translationally invariant systems. We calculate exactly the near-neighbor correlation function and specific heat and discuss their critical behaviour.     

Diffusion-induced modulational instability in magnetised semiconductor plasmas: effect of carrier heating

The present paper is aimed to the exploration of modulational amplification in transversely magnetised diffusive semiconductors. By considering that the origin of modulational interaction lies in the third-order optical susceptibility c⁽³⁾\chi^{(3)} arising from the nonlinear diffusion current density and using coupled mode theory, an analytical investigation of frequency modulational interaction between co-propagating laser beams and internally generated acoustic mode is presented. We have studied the steady state and transient amplification characteristics of modulated waves arising in the transversely magnetised semiconductor plasmas. The effect of carrier heating adds new dimensions to the present study. The heating effect reduces the required threshold amplitude of wave and enhances steady-state as well as transient gain of the generated acoustic mode.     

Ligand-induced distortion of a tetranuclear manganese butterfly complex

The reaction of the pentadentate Schiff-base ligand 1,3-bis(salicylideneamino)-2-propanol (salproH3) with [Mn3O(O2CR)6(py)3] (R = Me, Et, But) gives the corresponding tetranuclear manganese product [Mn4O2O2CR)5(salpro)] (4Mn(III)). The syntheses, structure and magnetochemical characterization of these complexes are reported. The structure of the [Mn4(mu3-O)2]8+ is butterfly-like much more closed than in previous complexes with this core as a result of the alkoxide oxygen of the salpro ligand bridging the two wingtip Mn atoms. Variable-temperature, solid-state magnetic susceptibility studies reveal that these complexes possess S = 0 ground state spins. Fitting of the magnetic susceptibility data to the theoretical chiMT vs. T expression derived for a C2v symmetry complex, assuming an isotropic Heisenberg spin-Hamiltonian and using the Van Vleck equation, revealed that the various exchange parameters are all antiferromagnetic, and the core thus experiences spin frustration effects.