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On ”Games and Dynamic Games” by A. Haurie, J.B. Krawczyk and G. Zaccour

100%

Więcek P.

Mathematica Applicanda

2013

tom 41

nr 1

Although the title, ''Games and Dynamic Games" of the new book by Alain Haurie, Jacek B. Krawczyk and Georges Zaccour suggests just another textbook on general game theory, with a little more stress made on dynamic aspect of games, the book appears to be something of novelty both in the scope and the treatment of the material. From the very first chapters of the book, it is clear that we will not be given another standard course of game theory, and that the dynamics of games will be the main issue. It starts with the simplest treatment of game dynamics, which is the extensive form of the game, and, having defined and explained the basic notions of static game theory, progresses to more complex dynamic models. Those start with deterministic repeated games, continue with variable state discrete time (or multistage) dynamic games and deterministic differential (that is, continuous time) games, and finish with stochastic models extending the ones presented before. Each part presents in a self-contained and comprehensive way, the main notions and solution concepts (such as open- and closed-loop strategies, Nash and correlated equilibria, subgame-perfectness, using threats to improve the performance of the equilibrium strategies) and theoretical results. It complements them with a number of economic applications, presented in detail. These applications relate to many fields of interest of economists, such as exploitation of renewable resources, environmental and energy issues, macroeconomics and finance. Each chapter ends with a set of excercices and a {\it game engineering} part, where a detailed presentation of a real-life application of some relevant game model is presented. Just this part is a novelty in the game-theoretic literature, as it shows to what extent the game-theoretic models can be applicable. Although the book presents the mathematical tools to model the situations of conflict in a rather rigorous way, it is clear that its target reader is someone with a background in economics or managemant science rather than mathematics. All the basic mathematical concepts used in the book are carefully explained, with those more complex ones (which appear in the latter chapters of the book) approached mainly through examples and intuitions. I believe this is a good decision of the authors, as in that way it makes the book really accessible to its potential readers. In fact, even though the tools (even those most advanced) of dynamic game theory have now been for many years used to model the economic interactions, surprisingly there has not been a single textbook addressed to more advanced students and researchers in economics and management science, presenting the state-of-art in the theory and applications of dynamic games in a comprehensive way. Those already existing required mathematical expertise well beyond what one could expect from typical economics/management students, and could not serve as textbooks for this type of readers. Moreover, those existing books concentrated without exception on some particular classes of dynamic games (such as differential games and their subclasses, stochastic games, repeated games). The book of Haurie, Krawczyk and Zaccour fills well the existing gap.Another important advantage of the book is its stress made not only on theoretical concepts and their adoption in real-life models, but on computation issues. The authors present a host of techniques to compute the solutions to the considered games either symbolically or numerically. They also provide information about software that can be used to solve similar games. Also the examples presented in game engineering sections are not abstract. They are based on real data, and issues of the credibility of the solutions obtained is discussed.The clarity of the exposition of often complicated topics is in general a big advantage of the book, although in latter chapters this clarity is sometimes unfortunatelly lost. The best example is the part about games over event trees with a confusing introduction concerning game with finite action sets and then the whole chapter concerning games with continuous ones. Also the last chapter, about stochastic-diffusion games, seems not very well conceived, as it jumps forth and back between an example and some more general considerations, leaving the reader confused about the generality of what is presented. These are certainly minor problems, but unfortunatelly there are other, more upseting issues. In general, it seems that the manuscript was not proofread carefully enough. The number of typos is enormous (the errata added to the book enumerates just a small fraction). Also some other faults, such as references to the colors on one of the figures, which is in black and white (p. 187), self-contradicting sentences like ''We use $\beta$ for a discount factor because we use $\beta$ to denote a strategy'' (p. 233), faults in references between chapters of the book (p. 372, footnote on p. 419) could have been easily eliminated if the book was properly proofread. What is worse though, is that there are also some more serious mistakes: one of the assumptions of the Kakutani fixed point theorem is missing, $\epsilon$-Nash equilibrium in Lemma 10.1 should be $A\epsilon$-equilibrium. And finally---it seems that most of the section 11.6.1 is missing from the manuscript. This significantly reduces the value of the book. Nevertheless, although I believe that all these errors should make the reader more cautious when reading the book, they do not unmake its main quality, which is acquainting all those interested in describing some real-life multi-agent economic situations in a rigorous way with the theory which provides the tools for doing it.

Simple equilibria in finite games with convexity properties

63%

Radzik T., Więcek P.

Applicationes Mathematicae

2015

tom 42

nr 1

83-109

This review paper gives a characterization of non-coalitional zero-sum and non-zero-sum games with finite strategy spaces and payoff functions having some concavity or convexity properties. The characterization is given in terms of the existence of two-point Nash equilibria, that is, equilibria consisting of mixed strategies with spectra consisting of at most two pure strategies. The structure of such simple equilibria is discussed in various cases. In particular, many of the results discussed can be seen as discrete counterparts of classical theorems about the existence of pure (or "almost pure") Nash equilibria in continuous concave (convex) games with compact convex spaces of pure strategies. The paper provides many examples illustrating the results presented and ends with four related open problems.

Ograniczanie wyników

1 Applicationes Mathematicae

1 Mathematica Applicanda

2 Więcek P.

1 Radzik T.

1 2015

1 2013

Wyniki wyszukiwania

On ”Games and Dynamic Games” by A. Haurie, J.B. Krawczyk and G. Zaccour

Simple equilibria in finite games with convexity properties