Pareto-Nash-Stackelberg game and control theory. Intelligent paradigms and applications (Q2315451)

Game springs from human interactives, and game theory springs from conflicts and cooperations. Game theory can be characterized by its dominant features as a theory of mathematical models of decision making during processes of conflicts and cooperations among rational players. There are at least eleven game theorists, laureates of Nobel Memorial Prize in Economic Sciences. Other than economics and micros economics, game theory has been applied into social sciences, computer sciences, public policy, manufacturing logistics, linguistics, biology, climate change, etc. Emile Borel coined the term ``game theory'' in 1921, \textit{J. von Neumann} proved the minimax theorem for the two-matrix zero-sum games in 1926, and he and \textit{O. Morgenstern} created the foundation and published seminar monograph [Theory of games and economic behavior. Princeton, New Jersey: Princeton University Press (1944; Zbl 0063.05930)]. It is inevitable for game theorists to consider hybrid/integrating models with noncooperative and cooperative features. The monograph under review mainly focus on noncooperative games with integrative features of Nash and Stackelberg games in terms of multi-objective optimization problems. Nash game is referred to play a simultaneous game, and Stackelberg game is to play a sequential game, and Pareto dominated is to each player optimizing simultaneously more than one criterion. Therefore, in a Pareto-Nash-Stackelberg game, each player is involved simultaneously both in Nash and Stackelberg game from the perspective of multi-objective optimization problems. The game types of even noncooperative game model have static game, simultaneous game, Nash game, dynamic game, Markovian game, Stackelberg game, etc with Pareto optimality named as Nash equilibrium, correlated equilibrium, Bayesian Nash equilibrium, Evolutionary stable strategy, Stackelberg equilibrium, Pareto equilibrium, etc. Pareto-Nash-Stackelberg game and control theory may be viewed as an extension for both initial domains in which dynamical system led by a multi-agent game and control is studied. The essential methods of the optimal control theory are the Hamilton-Jacobi-Bellman equation as a sufficient condition and Pontryagin maximum principle as a necessary condition. The monograph introduces basic notions and multi-agent decision problems to outline the framework of game theory and control theory, and explains Prisoner's dilemma game to show the Pareto dominated Nash equilibrium. Coordinate game sets the conflict situation with more than one Nash equilibria. Chapter 1 also explains strategic form games, simultaneous/Nash game, sequential/Stackelberg game with four basic axioms, the optimal control theory with Euler-Lagrange equation and transversality with different variables, optimality with respect to control process, stationarity, complementary slackness and dual feasibility. Part I of the monograph from Chapter 2 to Chapter 8 devotes to noncooperative games, their solution principle and methods of finding the full set of solutions. A Nash equilibrium can be a fixed point of a best response correspondence, a fixed point of a function (functional), a solution of a nonlinear complementary problem, a solution of a stationary point problem, a minimum/maximum of a function on a polytope, an element of a semi-algebraic set, the intersection of the graphs of Pareto-optimal response multivalued mappings, etc. The Karush-Kuhn-Tucker type theorems and their extensions on multi-criteria strategic form games for multi-criteria optimization problems are presented in Chapter 2 with details. The set of Nash equilibria in a polymatrix mixed-strategy game is described with upper bound components of specified types, importantly, an exponential algorithm for Nash equilibrium set is given with examples to illustrate how to solve systems of multi-linear equations and inequalities, and represents a serious obstacle to compute efficiently Nash equilibrium sets in Chapter 3. Algorithm with a particular \(2 \times 3\) games on a triangular prism is simplified from Chapter 3 to characterize the explicit Nash equilibrium set and the difference between different dominated players in Chapter 4, in the Wolfram language. Chapter 5 repeats the previous Chapter to work on three-matrix \(2\times 2 \times 2\) mixed strategy games with not only an algorithm but also a set-valued Nash equilibrium set function (Nash function) NES \((A, B)\) again with codes in the Wolfram language, and Chapter 6 on Dyadic mixed-strategy games proves main theoretic results with Wolfram language and the set of Nash equilibrium may be formed by a point, two points, three points, a segment, two connected segments, three connected segments, union of non-connected one point and one segment, the unit square. Stackelberg games in polymatrix mixed-strategy are dealt with in Chapter 7, to determine a Stackelberg equilibrium set by reducing the graph of best response mapping of the last player to the Stackelberg equilibrium set via a series of optimization problem solving. For every mixed-strategy game, the set of unsafe (safe) Stackelberg equilibria is non-empty and the set of unsafe (safe) Stackelberg equilibria is compact for bimatrix mixed-strategy Stackelberg games. For a single Stackelberg equilibrium, a simplified polynomial algorithm is given in Section 7.3 where the computational complexity of such algorithms is not exponential but polynomial. The strategic form games on digraphs are solved in Chapter 8 to explore the maximin solutions with necessary and sufficient conditions, and solvable matrix games on digraph are listed. The maximin (minimax) cost flow problem is an NP-hard problem. Part II of the monograph is dedicated to Pareto-Nash-Stackelberg games. The unsafe Stackelberg equilibrium notion and the Nash equilibrium notion are not equivalent, and the safe Stackelberg equilibria set is non empty provided in Chapter 9. Chapter 10 starts to compute Pareto-Nash equilibrium sets in finite multi-objective mixed-strategy games with Pareto optimality efficient, and shows that the problem of a Pareto-Nash equilibrium set computing is computationally very hard unless \(P=NP\). With the scalarization technique, one can reduce the Pareto-Nash equilibrium on a finite multi-criteria strategy game to a Nash equilibrium on a single objective game, thus earlier algorithm can be applied. Therefore, the set of Pareto-Nash equilibria in polymatrix mixed-strategy games is partitioned into a finite number of polytopes with upper bound components of specified types. Chapter 11 works on Pareto-Nash equilibrium in Dyadic two criterion mixed-strategy games as a continuation of the precedent chapter to explicitly find the set of Pareto-Nash equilibria as an intersection of players' graphs. Normal form game and axioms of rationality, knowledge, simultaneity and payoff are discussed in Chapter 11, the maximin-Nash (Stackelberg) taxon and the maximin-Nash (Stackelberg) solution principle are given for bimatrix games. Part III is to understand the Pareto-Nash-Stackelberg control of the systems governed by linear discrete-time laws, and the Nash-Stackelberg equilibrium control is equivalent to the optimal in linear programming problem in Chapter 13, the Pareto-Nash-Stackelberg equilibrium control is equivalent to the efficient solution of the multi-criteria linear programming problem in linear discrete time processes. Similarly, it works for Pareto-Nash-Stackelberg set-valued control problem of linear discrete time system in Chapter 14. The monograph adapts optimal control theory into game theory, and mainly finds the equilibrium of games for various simple and special cases. Finding explicit solutions and providing algorithms are the highlight of the monograph to distinguish other textbooks and monographs in game theory. The code in Wolfram language is given through the monograph. It will be certainly much better to see those algorithms in python or R.