Advertisement

Robustness of Fictitious Play in a Resource Allocation Game

  • Michalis SmyrnakisEmail author
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 91)

Abstract

Nowadays it is well known that decentralised optimisation tasks can be represented as so-called “potential games”. An example of a resource allocation problem that can be cast as a game is the “vehicle-target assignment problem” originally proposed by Marden et al.

In this article we use fictitious play as “negotiation” mechanism between the agents, and we examine its robustness in the case where a fraction of non-cooperative players, s, choose a random action. This addresses situations in which there is, e.g., a malfunction of some units. In our simulations we consider cases where the non-cooperative agents communicate their proposed action to the other agents and cases in which they do not announce their actions (e.g. in the case of a breakdown of communication). We observe that the performance of fictitious play is the same as if all players were able to fully coordinate, when the fraction of the non-coordinating agents, s, is smaller than a critical value \(\tilde{s}\). Moreover in both cases, where non-cooperative agents shared and did not share their action with others, the critical value was the same. Above this critical value the performance of fictious play is always affected. Also even in the case where only the 40% of the agents manage to cooperate and share their information the final reward is the 85% of the reference case’s reward, where every agent cooperates.

Keywords

Nash Equilibrium Mixed Strategy Resource Allocation Problem Final Reward Potential Game 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This work is supported by The Engineering and Physical Sciences Research Council EPSRC (grant number EP/I005765/1).

References

  1. 1.
    Kho, J., Rogers, A. and Jennings, N. R. (2009) Decentralise control of adaptive sampling in wireless sensor networks. ACM Transactions on Sensor Networks, 5 (3).Google Scholar
  2. 2.
    van Leeuwen, P. (2002) Scheduling aircraft using constraint satisfaction. In Electronic Notes in Theoretical Computer Science. Elsevier, 252–268.Google Scholar
  3. 3.
    Zhang, Y. and Meng, Y. (2010) A decentralized multi-robot system for intruder detection in security defense. In Intelligent Robots and Systems (IROS).Google Scholar
  4. 4.
    Kallitsis, M.G., Michailidis, G. and Devetsikiotis, M. (2011) A decentralized algorithm for optimal resource allocation in smartgrids with communication network externalities. Smart Grid Communications (SmartGridComm), 2011.Google Scholar
  5. 5.
    Rogers, A., Ramchurn, S. and Jennings, N.R. (2012) Delivering the smart grid: Challenges for autonomous agents and multi-agent systems research. Twenty-Sixth AAAI Conference on Artificial Intelligence (AAAI-12).Google Scholar
  6. 6.
    Kitano, H., Todokoro, S., Noda, I., Matsubara, H., Takahashi, T., Shinjou, A. and Shimada, S. (1999) Robocup rescue: Search and rescue in large-scale disaster as a domain for autonomous agents research. In IEEE International Conference on Systems, Man, and Cybernetics (SMC ’99), 6, 739–743.Google Scholar
  7. 7.
    Stranjak, A., Dutta, P.S., Rogers, A. and Vytelingum, P.V. (2008) A multi-agent simulation system for prediction and sceduling of aero engine overhaul. In Proceedings of the 7th International Conference on Autonomous Agents and Multiagent Systems (AAMAS ’08) Google Scholar
  8. 8.
    Bowling, M., anf Veloso, M. (2002). Multiagent learning using a variable learning rate. Artificial Intelligence, 136 (2), 215–250.Google Scholar
  9. 9.
    Herrmann, J. W. (1999). A genetic algorithm for minimax optimization problems. In Evolutionary Computation on Proceedings of the 1999 Congress.Google Scholar
  10. 10.
    Dorigo, M., Birattari, M., and Stutzle, T. (2006) Ant colony optimization. Computational Intelligence Magazine, IEEE, 1(4), 28–39.CrossRefGoogle Scholar
  11. 11.
    Nash, J. (1950) Equilibrium Points in n-Person Games Proceedings of the National Academy of Science, USA, 36, 48–49.Google Scholar
  12. 12.
    Bertsekas, D. (1982) Distributed dynamic programming. IEEE Transactions on Automatic Control, 27(3), 610–616.CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Silva, C.A., Sousa, J.M.C., Runkler, T.A. and Sa da Costa, J.M.G. (2009) Distributed supply chain management using ant colony optimization. European Journal of Operational Research, 199(2), 349–358.CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Monderer, D. and Shapley, L. (1996) Potential Games. Games and Economic Behavior, 14, 124–143.CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Wolpert, D. and Tumer, K. (1999) An overview of collective intelligence. In J. M. Bradshaw, editor, Handbook of Agent Technology. AAAI Press/MIT Press.Google Scholar
  16. 16.
    Arslan, G., Marden, J. and Shamma, J. (2007) Autonomous Vehicle-Target Assignment: A Game Theoretical Formulation. Journal of Dynamic Systems, Measurement, and Control, 129, 584–596.CrossRefGoogle Scholar
  17. 17.
    Brown, G.W. (1951) Iterative Solutions of Games by Fictitious Play. In Activity Analysis of Production and Allocation, T.C. Koopmans (Ed.). New York: Wiley.Google Scholar
  18. 18.
    Fudenberg, D. and Levine, D. (1998) The theory of Learning in Games. The MIT PressGoogle Scholar
  19. 19.
    Rezek, I., Leslie, D. S., Reece, S. and Roberts, S. J., Rogers, A., Dash, R. K. and Jennings, N.R. (2008) On Similarities between Inference in Game Theory and Machine Learning. Journal of Artificial Intelligence Research, 33, 259–283.zbMATHMathSciNetGoogle Scholar
  20. 20.
    MacKay, D. J. (1992) The evidence framework applied to classification networks. Neural computation, 4(5), 720–736.CrossRefGoogle Scholar
  21. 21.
    Berger U. (2005) Fictitious play in 2xn games. Journal of Economic Theory, 120 (2), 139–154.CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Miyasawa, K. (1961) On the convergence of learning process in a 2x2 non-zero-sum two person game. Research Memorandum No 33, Princeton Unniversity.Google Scholar
  23. 23.
    Robinson, J. (1951) An iterative Method of solving a game. Annals of Mathematics, 54, 269–301.CrossRefGoogle Scholar
  24. 24.
    Nachbar, J. (1990) “Evolutionary” selection dynamics in games: Convergence and limit properties. International Journal of Game Theory, 19, 58–89.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Automatic Control and Systems EngineeringSheffieldUK

Personalised recommendations