A Sensitivity Based Approach to Determining Non-Cooperative Solutions in Nash and Stackelberg Games

2012 ◽  
Author(s):  
Ehsan Ghotbi ◽  
Wilkistar A. Otieno ◽  
Anoop K. Dhingra
Keyword(s):  
Response Surface Methodology ◽  
Computational Effort ◽  
Stackelberg Games ◽  
Nash Solution ◽  
Numerical Errors ◽  
Multiobjective Problems ◽  
Cooperative Solutions ◽  
The One ◽  
Rational Reaction Set ◽  
The Nash Solution

A sensitivity based approach is presented to determine Nash solution(s) in multiobjective problems modeled as a non-cooperative game. The proposed approach provides an approximation to the rational reaction set (RRS) for each player. An intersection of these sets yields the Nash solution for the game. An alternate approach for generating the RRS based on design of experiments (DOE) combined with response surface methodology (RSM) is also explored. The two approaches for generating RRS are compared on three example problems to find Nash and Stackelberg solutions. It is seen that the proposed sensitivity based approach (i) requires less computational effort than a RSM-DOE approach, (ii) is less prone to numerical errors than the RSM-DOE approach, (iii) is able to find all Nash solutions when the Nash solution is not a singleton, (iv) is able to approximate non linear RRS, and (v) is able to find better a Nash solution on an example problem than the one reported in the literature.

Structure Information From Electron Diffraction Intensities

1990 ◽  
Vol 48 (2) ◽  
pp. 516-517
Author(s):  
J. Gjønnes ◽  
N. Bøe ◽  
K. Gjønnes
Keyword(s):  
Computational Effort ◽  
Unit Cell ◽  
Small Unit ◽  
Structure Information ◽  
Dispersion Surface ◽  
Integrated Intensity ◽  
The One ◽  
Image Position ◽  
Convergent Beam ◽  
Beam Patterns

Structure information of high precision can be extracted from intentsity details in convergent beam patterns like the one reproduced in Fig 1. From low order reflections for small unit cell crystals,bonding charges, ionicities and atomic parameters can be derived, (Zuo, Spence and O’Keefe, 1988; Zuo, Spence and Høier 1989; Gjønnes, Matsuhata and Taftø, 1989) , but extension to larger unit cell ma seem difficult. The disks must then be reduced in order to avoid overlap calculations will become more complex and intensity features often less distinct Several avenues may be then explored: increased computational effort in order to handle the necessary many-parameter dynamical calculations; use of zone axis intensities at symmetry positions within the CBED disks, as in Figure 2 measurement of integrated intensity across K-line segments. In the last case measurable quantities which are well defined also from a theoretical viewpoint can be related to a two-beam like expression for the intensity profile:With as an effective Fourier potential equated to a gap at the dispersion surface, this intensity can be integrated across the line, with kinematical and dynamical limits proportional to and at low and high thickness respctively (Blackman, 1939).

Bargaining and Justice

1985 ◽  
Vol 2 (2) ◽  
pp. 29-47 ◽  
Author(s):  
David Gauthier
Keyword(s):  
Bargaining Problem ◽  
Nash Solution ◽  
Bargaining Process ◽  
Independence Of Irrelevant Alternatives ◽  
Principles Of Justice ◽  
The Nash Solution

My concern in this paper is with the illumination that the theory of rational bargaining sheds on the formulation of principles of justice. I shall first set out the bargaining problem, as treated in the theory of games, and the Nash solution, or solution F. I shall then argue against the axiom, labeled “independence of irrelevant alternatives,” which distinguished solution F, and also against the Zeuthen model of the bargaining process which F formalizes.

scholarly journals The Nash Solution as a von Neumann–Morgenstern Utility Function on Bargaining Games

2020 ◽  
Vol 37 (1-2) ◽  
pp. 87-104
Author(s):  
Anke Gerber
Keyword(s):  
Utility Function ◽  
Status Quo ◽  
Nash Solution ◽  
Bargaining Games ◽  
Von Neumann ◽  
Risk Neutral ◽  
The Nash Solution

AbstractIn this paper we prove that the symmetric Nash solution is a risk neutral von Neumann–Morgenstern utility function on the class of pure bargaining games. Our result corrects an error in Roth (Econometrica 46:587–594, 983, 1978) and generalizes Roth’s result to bargaining games with arbitrary status quo.

Hydrogen in Electrodynamics. VII The Pauli Principle

1991 ◽  
Vol 46 (10) ◽  
pp. 869-872 ◽  
Author(s):  
Hans Sallhofer
Keyword(s):  
Pauli Principle ◽  
Computational Effort ◽  
The One

AbstractAfter a discussion of the one-component Schrödinger (1926) and the four-component Dirac (1928) representation of hydrogen it is shown that the six-component electrodynamic picture turns out to be considerably simpler and clearer. The computational effort is reduced to a fraction.

scholarly journals XII. The kinetic theory of a gas constituted of spherically symmetrical molecules

1912 ◽  
Vol 211 (471-483) ◽  
pp. 433-483 ◽  
Keyword(s):  
Kinetic Theory ◽  
The Other ◽  
Mathematical Form ◽  
The Real ◽  
Numerical Errors ◽  
Diffusion Of Knowledge ◽  
Descriptive Theory ◽  
The One ◽  
Physical Reasons ◽  
High Degree

The principal kinetic theories of a gas proceed either on the hypothesis that the molecules are rigid elastic spheres, or that they are point centres of forces which vary inversely as the fifth power of the distance. Maxwell has worked out the consequences of the letter hypothesis in his well-known theory, which is unrivalled in its high degree of accuracy and (after some improvements by Boltzmann) in its perfection of mathematical form. All the quantities not taken account of in the theory (such as the time occupied by molecular encounters, and the effect of collisions in which more than two molecules take part) are properly negligible under ordinary conditions. The theory has the disadvantage, however, that the underlying hypothesis is highly artificial (being chosen chiefly on account of mathematical simplifications connected with it, rather than from any physical reasons), and does not represent the real facts at all adequately. The other hypothesis referred to seems to be much more in agreement with fact, but its consequences have been worked out less accurately. The method which has almost always been used is the one originally devised by Clausius and Maxwell; Maxwell abandoned it later, however, as it had “led him at times into grave error.” In spite of its apparent simplicity, numerical errors of large amount may undoubtedly creep in in a very subtle way. Hence the theory of a gas whose molecules are elastic spheres remains in a rather unsatisfactory state. As a “descriptive” theory (to use Meyer’s apt term) it has, however, served a useful purpose; the general laws of gaseous phenomena have been developed by its aid in an elementary way, which has conduced to a wider diffusion of knowledge of the kinetic theory than would have been possible if the sole line of development had been by the more mathematical and accurate methods used by Maxwell and Boltzmann.

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