Optgame 2.0: An Algorithm For Equilibrium Solutions Of N-Person Discrete-Time (Non-)Linear Dynamic Games

We present the algorithm OPTGAME 2.0 to solve N-person discrete-time LQ games exactly, and discrete-time non-linear quadratic games approximately by means of an appropriate linearization procedure, where N>2. I.e., the objective function is assumed to be quadratic in the deviations of states and control variables from their respective desired target-values, and will be optimized for a pre-specified period of time subject to a nonlinear autonomous system. OPTGAME 2.0 allows the calculation of the Nash and Stackelberg equilibrium solutions, and the cooperative Pareto-optimal solutions for any number of players.Emanating from the informational basis of each player's decision, we distinguish between open-loop information patterns, where the player's strategies depend only on the initial state of the dynamic system, and feedback information patterns, where the strategies depend on the current state of the system (but not on the initial conditions). Deterministic dynamic game models can be solved by using essentially the same techniques as for solving deterministic optimal control models, but the choice of the solution technique determines the information pattern. I.e., the application of the minimum principle generates the open-loop solutions, while the application of the dynamic programming technique determines the feedback solutions.The algorithm OPTGAME 2.0 starts from computing a tentative path of the state vector from the nonlinear system equations - using the Gauss-Seidl algorithm - with a given tentative path for the control variables. Then the algorithm linearizes the system equations at the reference values obtained before, replacing the nonlinear autonomous system by a linear non-autonomous one. Then, the algorithm calculates numerically the Nash, Stackelberg, and Pareto solutions of non-linear, quadratic deterministic games (with a finite planing horizon) under open-loop and feedback information structure. This is done by use of the appropriate optimization technique corresponding to the desired information structure of the game, and yields so-called Riccati equations which can be solved by backward integration. Then, we derive by forward iteration the so-called feedback matrices, where further substitution of these matrices into linear relations in the preceding state variable delivers the values of the optimal control variables (expressed in feedback form) - as well as the optimal state values.The term äOPTGAME" denotes both, the computer algorithm and its implementation, where the implementation part consists of a set of procedures which are implemented in the programming language GAUSS. GAUSS is a high level matrix programming language specializing in commands, functions, and procedures for data analysis and statistical applications. This interplay is of special interest for the application of OPTGAME 2.0 in the field of optimal short-run and long-run fiscal policies towards the EMU. Furthermore, GAUSS includes a variety of routines which perform standard matrix operations, e.g. routines to calculate determinants, matrix inverses, decompositions, eigenvalues and eigenvectors, and condition numbers.