Solving the „Tracking Game“

Guest article from Doris A. Behrens


Games taking place in a shared environment are characterized by the fact that the effectiveness of individual decisions heavily depends on the decisions of other players. Our algorithm OPTGAME 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. We call this a „tracking game“, since is an extension of the linear regulator problem (also known as „tracking problem“) that is well known from LQ optimal control theory.
OPTGAME is a tool that 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 such a 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.

European Monetary Union

For instance, if a single country aims at economic growth, one option could be to increase the demand for goods and services from the public sector. This increases production in response to demand, which in turn increases incomes. However, within an open economy the future success strongly depends on the situation and behavior of the other member countries. In order to find a solution for this problem it is necessary to estimate the countries’ individual parameters as well as the degree of economic interdependencies (like trade) between countries.
Such models, in order to be accurate are inherently nonlinear, which cannot be solved analytically by a linear model such as the LQ game. In our work we apply OPTGAME to a monetary union macroeconomic model based on the nonlinear MUMOD1 model. In this model, there are basically two groups of countries, one economically stronger than the other, all experiencing a brief period of recession.

Doris A. Behrens is a senior re-
searcher working on optimization
in techno-socio-economic systems
at the Alpen-Adria-Universität
Klagenfurt

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).

Literature:

Doris A. Behrens, Reinhard Neck, Approximating Solutions for Nonlinear Dynamic Tracking Games, Computational Economics, Springer, February 2014. DOI: 10.1007/s10614-014-9420-4

Reinhard Neck, Doris A. Behrens, A macroeconomic policy game for a monetary union with adaptive expectations. Atlantic Economic Journal, 37(4), 335–349, 2009. DOI: 10.1007/s11293-009-9186-6f