Games and dynamic games (Q2921625)

This book addresses a comprehensive study of the theory of static and dynamic games. It contains the most studied models appearing in the literature and at the same time provides interesting applications. The book is well written and is mainly addressed to graduate students of engineering, management science and operations research.NEWLINENEWLINEAfter an introductory chapter, the contents of the book can be divided into three parts; namely, static games, deterministic dynamic games and stochastic dynamic games.NEWLINENEWLINEThe first part regards to the classical theory of games in the static case. This part is divided into three chapters. NEWLINENEWLINENEWLINEChapter 2 (``Description of the game'') provides a complete description of the elements of a game, such as the concepts of players, actions, strategies, information structure, payoffs, among others. Furthermore, the two different representations of a game are introduced; namely, the extensive form (represented by a graph) and the normal form (posed in terms of the payoff and the strategies of the players). Some empirical methods to deduce ``optimality'' for the players in a game are discussed as well.NEWLINENEWLINE In Chapter 3 (``Equilibrium solutions for noncooperative games''), the aim of the authors is to obtain noncooperative equilibria in different types of games. To this end, the concept of Nash equilibrium is introduced first, and then two different types of games are introduced: the so-called matrix and bimatrix games. The analysis of these games, does not only focus on the existence of saddle points (for matrix games) or Nash equilibria (for bimatrix games), but also useful algorithms are deeply analyzed to obtain these classes of equilibria. Some shortcomings of the Nash equilibrium concept are also discussed; in particular, its inefficiency as reflected in the prisoner's dilemma. Finally, the last part of this chapter studies the concept of the Stackelberg equilibrium, showing the existence of this type of equilibrium by means of an illustrative example. The question when it is better to use this kind of equilibria rather than Nash equilibria is argued.NEWLINENEWLINEIn Chapter 4 (``Extensions and refinements of the equilibrium concepts''), the authors study some refinements of the concept of the Nash equilibrium in order to overcome the drawbacks and shortcomings that this latter concept may cause. Such refinements studied in this section are the concepts of correlated equilibria, Bayesian equilibra with incomplete information, subgame perfectness, quantal-response equilibria, and the so-called supermodular games. Each of this refinements is clearly analyzed, most of them in detail.NEWLINENEWLINEThe second part of the book is about deterministic dynamic games. This part consists of three chapters. Chapter 5 (``Repeated games and memory strategies'') is devoted to the study of repeated games, which are games the nature of which changes over time through the actions of the players (by means of the accumulative information they acquire) and there is no influence on the periodic form of the game to be played in the future. Along this chapter, the concepts of repeated matrix games and repeated concave games are introduced. Two important types of strategies arise in this class of games: open-loop and closed loop (and, as a particular case, feedback) strategies. The concepts of cumulative payoff criteria, more specifically, the finite horizon criterion and the discounted and average infinite horizon criteria are also introduced, and the definition of Nash equilibrium based on these type of criteria is established. Thus, based on the use of all these ingredients, the main contribution of the chapter is Folk's theorem which ensures the existence of Nash equilibria for the class of repeated matrix games.NEWLINENEWLINEIn Chapter 6 (``Multistage games''), the authors analyze optimality results on this type of games, which are similar to repeated games, but now there is a (discrete-time) dynamical system that is influenced by the actions of the players and vice versa. This section can be essentially divided into six parts. The first part regards an overview of the optimal control theory for discrete-time deterministic systems. Parts two and three are devoted to the statement of the multistage game as well as to the information structure allowed for this class of games; namely, open-loop, closed loop and feedback strategies. Part four, which is the barycenter of the chapter, is devoted to show the existence of both open-loop and feedback Nash equilibria. Within this analysis, the authors introduce two important methods that provide necessary and/or sufficient conditions for the existence of Nash equilibria: the so-called maximum principle and dynamic programming. These two methods are widely discussed. The rest of the chapter is devoted to the study of particular cases of games, where the existence of Nash equilibra is guaranteed; although in the last part of this chapter, Nash equilibra are studied for a large class of strategies, the so-called memory strategies, in which players are allowed to do threats and alliances in order to get better outcomes.NEWLINENEWLINEChapter 7 (``Differential games'') can be seen as the continuous-time counterpart of multistage games. Indeed, in the present case, the associated dynamic system evolves as a deterministic ordinary differential equation. The main results in this chapter are very similar to those given Chapter 6, although the mathematical methods are considerably different. Specifically, this chapter mainly addresses to the existence of open-loop and feedback Nash equilibria for either the finite horizon case and the infinite horizon discounted payoffs. The tools under consideration are either the maximum principle and the dynamic programming techniques. Several special cases and examples are studied. Besides to the analysis of Nash equilibrium, the authors also study the Stackelberg equilibrium for this class of games, providing sufficient conditions for the existence of such equilibria. At the end of the chapter, the authors define the continuous-time version of the aforementioned memory strategies, providing existence in some special cases.NEWLINENEWLINEThe third part of the book regards stochastic games, in which there is a stochastic device affecting the evolution of the game. This last part consists of four chapters; that is, Chapters 8 to 11.NEWLINENEWLINEWe start this last part with Chapter 8 (``Equilibria in games played over event trees''), which is devoted to games represented as an uncontrolled event tree. The characteristic of these games is the actions of the players do not affect the randomness of the game and it plays the role only in each player's information. Hence, the transition from one state (node) to another is caused only by the Nature. The concept of \(S\)-adapted Nash equilibrium is introduced (\(S\) being the sample random realizations of the random process), and its existence is guaranteed under concavity and finiteness assumptions (for a special case) and through the use of the maximum principle (for the general case).NEWLINENEWLINEIn Chapter 9 (``Markov games''), the authors analyze the case when the (discrete-time) dynamical system evolves as a stochastic process but now its evolution depends on the strategies employed by players and vice versa. This chapter can be divided into three different sections; namely, Shapley games (or zero-sum games with finite state and action spaces), nonzero-sum games (with finite state and action spaces) and nonzero-sum games with continuous state and action spaces. For the case of games with finite state and action spaces, the existence of feedback Nash equilibria is possible from the use of dynamic programming arguments. Such existence is based on the theory of contraction operators as well as the well-known fixed point Banach theorems. Some useful methods to compute Nash equilibra in the case of Shapley games (policy and value iteration methods) are discussed, and several references to complement the details are also provided. As for the third part of this chapter, the existence of \(\varepsilon\)- and correlated equilibria is mentioned, providing only references for further details. Finally, the case of memory strategies is analyzed by means of an interesting example.NEWLINENEWLINEChapter 10 (``Piecewise deterministic differential games''). This class of games mixes both a continuous and a discrete process in the dynamical system. Along the chapter, the authors define the model and the concept of open-loop Nash equilibrium associated to this type of games. The existence of a \(\varepsilon\)-open-loop Nash equilibrium is provided by means of an example regarding a duopoly model. A numerical example illustrates the results.NEWLINENEWLINEFinally, Chapter 11 (``Stochastic diffusion games''), addresses games whose dynamical system is driven by means of a (controlled) stochastic differential equation. The analysis begins with a brief review of Ito's calculus. Then, the concept of the feedback Nash equilibrium is introduced for this class of games and the existence of this equilibrium is provided thanks to the use of a dynamic programming method. A great analysis on memory strategies is also discussed.NEWLINENEWLINEIt is worth to mention that every chapter of this book contains a summary, explaining the most important characteristics of the chapter, as well as a list of exercises with different levels of difficulty. Furthermore, there is an interesting real-life application at the end of each chapter, so-called \(\mathbb{G}\mathbb{E}\) illustration, so that the reader can apply all the theory proposed in the chapter under study to very interesting real-life problems such as those involving commodity markets, competition between firms or countries, fishery-management, climate change, among others.NEWLINENEWLINEThe reader must be warned on some typos the book contains along its pages.