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Licensed Unlicensed Requires Authentication Published by De Gruyter Oldenbourg March 27, 2018

Slow and Steady Wins the Race: Approximating Nash Equilibria in Nonlinear Quadratic Tracking Games Steter Tropfen höhlt den Stein: Approximation von Nash Gleichgewichten in Nicht-linearen Dynamischen Spielen

  • Ivan Savin EMAIL logo , Dmitri Blueschke and Viktoria Blueschke-Nikolaeva


We propose a new method for solving nonlinear dynamic tracking games using a meta-heuristic approach. In contrast to ‘traditional’ methods based on linear-quadratic (LQ) techniques, this derivative-free method is very flexible with regard to the objective function specification. The proposed method is applied to a three-player dynamic game and tested versus a derivative-dependent method in approximating solutions of different game specifications. In particular, we consider a dynamic game between fiscal (played by national governments) and monetary policy (played by a central bank) in a monetary union. Apart from replicating results of the LQ-based techniques in a standard setting, we solve two ‘non-standard’ extensions of this game (dealing with an inequality constraint in a control variable and introducing asymmetry in penalties of the objective function), identifying both a cooperative Pareto and a non-cooperative open-loop Nash equilibria, where the traditional methods are not applicable. We, thus, demonstrate that the proposed method allows one to study more realistic problems and gain better insights for economic policy.

JEL Classification: C61; C63; C72; C73; E61


This work has benefited from a presentation at the WEHIA workshop in Nice. IS acknowledges support from the German Science Foundation (DFG RTG 1411), the Helmholtz Association (HIRG-0069), Projex CSES, Initiative d’Excellence, Université de Strasbourg and RFBR grant Nr. 18-010-01190.


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As a stochastic optimization algorithm DE does not claim to identify “the global optimum” but just a good approximation of it. This is because it only asymptotically converges towards the global optimum using the allocated number of restarts and iterations. We report our results based on ten restarts (Table 6) and demonstrate very small differences between them. However, similar values of the aggregated objective value do not necessarily result from the same equilibrium (i.e. similar approximations of it). Figure 6 provides graphical illustration of the optimal paths from different restarts for the control variables. We concentrate on the control variables only as those define the outcome of the dynamic system. Our results unequivocally indicate that we get similar approximations of the same equilibrium. In addition, in Table 8 the Euclidian distance for the normalized (subtracted mean value and divided by standard deviation) control variable paths is reported for each restart from the “best” restart (the row of the best restart is then 0).

Table 8

Euclidian distances between the normalized values of control variables resulting from 10 restarts and the best solution.

Figure 6 Optimal paths from 10 restarts for the control variables RE$RE$ and gi$g_i$. The left panel reports results for the Pareto equilibrium and the right panel for the Nash equlibrium. A scenario with asymmetry and inequality constraints combined is considered.
Figure 6

Optimal paths from 10 restarts for the control variables RE and gi. The left panel reports results for the Pareto equilibrium and the right panel for the Nash equlibrium. A scenario with asymmetry and inequality constraints combined is considered.

Code and Datasets

The author(s) published code and data associated with this article in the ZBW Journal Data Archive, a storage platform for datasets. See:

Received: 2017-08-08
Revised: 2017-12-18
Accepted: 2018-03-04
Published Online: 2018-03-27
Published in Print: 2018-10-25

© 2018 Oldenbourg Wissenschaftsverlag GmbH, Published by De Gruyter Oldenbourg, Berlin/Boston

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