The „tracking game“ is a nonlinear extension of the linear regulator problem (also known as „tracking problem“), which is well known from LQ optimal control theory. Such a game taking place in a shared environment is characterized by the fact that the effectiveness of individual decisions heavily depends on the decisions of other players.

Such models cannot be solved analytically by a linear model such as the LQ game. The OPTGAME algorithm is able to approximate the evolution of choices to be made if a number of independent decision makers seek to reach individually desirable states.

The evolution of states subject to control is described by a system of nonlinear difference equations. OPTGAME steers the control and state paths towards desired outcomes. It is novel in a way that it works for game theoretic systems with nonlinear constraints. It searches for equilibrium solutions by iteratively applying a sequence of local linearization and optimization over the entire planning horizon. The tool yields three types of non-cooperative equilibrium solutions (open-loop Nash equilibrium, feedback Nash equilibrium, feedback Stackelberg equilibrium) plus one cooperative solution (Pareto-optimal strategy).

An example for its application in a tracking game could be the decision-making within a monetary union such as the European Monetary Union (EMU). In this game all but one player represent countries with intentions for economic growth, employment and limited budget deficit and one player represents the European Central Bank, aiming solely at price stability. Besides trade-offs between state variables, for example the well-known trade-off between unemployment and price stability (see Phillips curve), there are strong economic interdependencies due to international trade. Without policy intervention all countries would experience a deep recession and an enormous increase in public debt. By applying OPTGAME for different solution concepts we learn that macroeconomic properties like public debt, economic growth, inflation, etc. can be significantly improved with system-aware control actions of players.

The OPTGAME tool is available as MATLAB implementation upon request (Contact Doris A. Behrens).

### Citations to this discussion:

- Doris A. Behrens, Solving the “Tracking Game”, Self-Organizing Networked Systems Blog, March 2014
- Reinhard Neck, Doris A. Behrens, A macroeconomic policy game for a monetary union with adaptive expectations, Atlantic Economic Journal, 37(4), 335–349. DOI: 10.1007/s11293-009-9186-6f, 2009
- Doris A. Behrens, Reinhard Neck, Approximating Solutions for Nonlinear Dynamic Tracking Games, Computational Economics, DOI: 10.1007/s10614-014-9420-4, February 2014