Minimax Theorems
Minimax Theorems

Author: Michel Willem

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 168

ISBN-13: 1461241464

DOWNLOAD EBOOK

Many boundary value problems are equivalent to Au=O (1) where A : X --+ Y is a mapping between two Banach spaces. When the problem is variational, there exists a differentiable functional rand inf.

Minimax and Applications
Minimax and Applications

Author: Ding-Zhu Du

Publisher: Springer Science & Business Media

Published: 2013-12-01

Total Pages: 300

ISBN-13: 1461335574

DOWNLOAD EBOOK

Techniques and principles of minimax theory play a key role in many areas of research, including game theory, optimization, and computational complexity. In general, a minimax problem can be formulated as min max f(x, y) (1) ",EX !lEY where f(x, y) is a function defined on the product of X and Y spaces. There are two basic issues regarding minimax problems: The first issue concerns the establishment of sufficient and necessary conditions for equality minmaxf(x,y) = maxminf(x,y). (2) "'EX !lEY !lEY "'EX The classical minimax theorem of von Neumann is a result of this type. Duality theory in linear and convex quadratic programming interprets minimax theory in a different way. The second issue concerns the establishment of sufficient and necessary conditions for values of the variables x and y that achieve the global minimax function value f(x*, y*) = minmaxf(x, y). (3) "'EX !lEY There are two developments in minimax theory that we would like to mention.

Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities
Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities

Author: Dumitru Motreanu

Publisher: Springer Science & Business Media

Published: 2013-12-01

Total Pages: 320

ISBN-13: 146154064X

DOWNLOAD EBOOK

Boundary value problems which have variational expressions in form of inequal ities can be divided into two main classes. The class of boundary value prob lems (BVPs) leading to variational inequalities and the class of BVPs leading to hemivariational inequalities. The first class is related to convex energy functions and has being studied over the last forty years and the second class is related to nonconvex energy functions and has a shorter research "life" beginning with the works of the second author of the present book in the year 1981. Nevertheless a variety of important results have been produced within the framework of the theory of hemivariational inequalities and their numerical treatment, both in Mathematics and in Applied Sciences, especially in Engineering. It is worth noting that inequality problems, i. e. BVPs leading to variational or to hemivariational inequalities, have within a very short time had a remarkable and precipitate development in both Pure and Applied Mathematics, as well as in Mechanics and the Engineering Sciences, largely because of the possibility of applying and further developing new and efficient mathematical methods in this field, taken generally from convex and/or nonconvex Nonsmooth Analy sis. The evolution of these areas of Mathematics has facilitated the solution of many open questions in Applied Sciences generally, and also allowed the formu lation and the definitive mathematical and numerical study of new classes of interesting problems.

The Minimax Theorem for U.S.C. (Uppersemicontinuous) - L.S.C. (Lowersemicontinuous) Payoff Functions
The Minimax Theorem for U.S.C. (Uppersemicontinuous) - L.S.C. (Lowersemicontinuous) Payoff Functions

Author: J. F. Mertens

Publisher:

Published: 1983

Total Pages: 34

ISBN-13:

DOWNLOAD EBOOK

Our aim is to get a general minimax theorem whose assumptions and conclusions are phrased only in terms of the data of the problem, i.e. the pair of pure strategy sets S and T and the payoff function on S x T. For the assumptions, this means that we want to avoid any assumption of the type there exists a topology (or a measurable structure) on S and (or) T such that ... For the conclusions, we are led to require that players have epsilon-optimal strategies with finite support, both because those are the easiest to describe in intrinsic terms, and because in any game where the value would not exist in strategies with finite support, all known general minimax theorems implicitly select as value either the sup inf or the inf sup by in effect restricting either player I or player II arbitrarily to strategies with finite support - so that the resulting value is completely arbitrary and misleading.

Minimax Theory and Applications
Minimax Theory and Applications

Author: Biagio Ricceri

Publisher: Springer Science & Business Media

Published: 2013-06-29

Total Pages: 278

ISBN-13: 940159113X

DOWNLOAD EBOOK

The present volume contains the proceedings of the workshop on "Minimax Theory and Applications" that was held during the week 30 September - 6 October 1996 at the "G. Stampacchia" International School of Mathematics of the "E. Majorana" Centre for Scientific Cul ture in Erice (Italy) . The main theme of the workshop was minimax theory in its most classical meaning. That is to say, given a real-valued function f on a product space X x Y , one tries to find conditions that ensure the validity of the equality sup inf f(x,y) = inf sup f(x, y). yEY xEX xEX yEY This is not an appropriate place to enter into the technical details of the proofs of minimax theorems, or into the history of the contribu tions to the solution of this basic problem in the last 7 decades. But we do want to stress its intrinsic interest and point out that, in spite of its extremely simple formulation, it conceals a great wealth of ideas. This is clearly shown by the large variety of methods and tools that have been used to study it. The applications of minimax theory are also extremely interesting. In fact, the need for the ability to "switch quantifiers" arises in a seemingly boundless range of different situations. So, the good quality of a minimax theorem can also be judged by its applicability. We hope that this volume will offer a rather complete account of the state of the art of the subject.

Game Theory, Alive
Game Theory, Alive

Author: Anna R. Karlin

Publisher: American Mathematical Soc.

Published: 2017-04-27

Total Pages: 400

ISBN-13: 1470419823

DOWNLOAD EBOOK

We live in a highly connected world with multiple self-interested agents interacting and myriad opportunities for conflict and cooperation. The goal of game theory is to understand these opportunities. This book presents a rigorous introduction to the mathematics of game theory without losing sight of the joy of the subject. This is done by focusing on theoretical highlights (e.g., at least six Nobel Prize winning results are developed from scratch) and by presenting exciting connections of game theory to other fields such as computer science (algorithmic game theory), economics (auctions and matching markets), social choice (voting theory), biology (signaling and evolutionary stability), and learning theory. Both classical topics, such as zero-sum games, and modern topics, such as sponsored search auctions, are covered. Along the way, beautiful mathematical tools used in game theory are introduced, including convexity, fixed-point theorems, and probabilistic arguments. The book is appropriate for a first course in game theory at either the undergraduate or graduate level, whether in mathematics, economics, computer science, or statistics. The importance of game-theoretic thinking transcends the academic setting—for every action we take, we must consider not only its direct effects, but also how it influences the incentives of others.