Computational Economics

, Volume 37, Issue 4, pp 375–410 | Cite as

A Numerical Toolbox to Solve N-Player Affine LQ Open-Loop Differential Games

  • Tomasz Michalak
  • Jacob Engwerda
  • Joseph Plasmans
Open Access


We present an algorithm and a corresponding MATLAB numerical toolbox to solve any form of infinite-planning horizon affine linear quadratic open-loop differential games. By rewriting a specific application into the standard framework one can use the toolbox to calculate and verify the existence of both the open-loop non-cooperative Nash equilibrium (equilibria) and cooperative Pareto equilibrium (equilibria). In case there is more than one equilibrium for the non-cooperative case, the toolbox determines all solutions that can be implemented as a feedback strategy. Alternatively, the toolbox can apply a number of choice methods in order to discriminate between multiple equilibria. The user can predefine a set of coalition structures for which they would like to calculate the non-cooperative Nash solution(s). It is also possible to specify the relative importance of each player in any coalition structure. Furthermore, the toolbox offers plotting facilities as well as other options to analyse the outcome of the game. For instance, it is possible to disaggregate each player’s total loss into its contributing elements. The toolbox is available as a freeware from the authors of this paper.


Linear quadratic differential games Riccati equations Cooperative games Noncooperative games Coalition structures Numerical computation 



Tomasz Michalak acknowledges support from (a) the EPSRC under the project ALADDIN (Autonomous Learning Agents for Decentralised Data and Information Systems) project and is jointly funded by a BAE Systems and EPSRC strategic partnership; and (b) the FWO (Fonds voor Wetenschappelijk Onderzoek Vlaanderen, Belgium).

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2011

Authors and Affiliations

  • Tomasz Michalak
    • 1
  • Jacob Engwerda
    • 2
  • Joseph Plasmans
    • 2
    • 3
  1. 1.Institute of InformaticsUniversity of WarsawWarszawaPoland
  2. 2.CentERUniversity of TilburgTilburgThe Netherlands
  3. 3.University of AntwerpAntwerpenBelgium

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