Skip to main content
SpringerLink
Log in

A Simple P-Matrix Linear Complementarity Problem for Discounted Games

  • Conference paper
Logic and Theory of Algorithms (CiE 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5028))

Included in the following conference series:

Abstract

The values of a two-player zero-sum binary discounted game are characterized by a P-matrix linear complementarity problem (LCP). Simple formulas are given to describe the data of the LCP in terms of the game graph, discount factor, and rewards. Hence it is shown that the unique sink orientation (USO) associated with this LCP coincides with the strategy valuation USO associated with the discounted game. As an application of this fact, it is shown that Murty’s least-index method for P-matrix LCPs corresponds to both known and new variants of strategy improvement algorithms for discounted games.

This research was supported in part by EPSRC projects EP/D067170/1, EP/E022030/1, and EP/D063191/1 (DIMAP).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Condon, A.: The complexity of stochastic games. Information and Computation 96, 203–224 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Condon, A.: On algorithms for simple stochastic games. In: Advances in Computational Complexity Theory, pp. 51–73. American Mathematical Society (1993)

    Google Scholar 

  3. Cottle, R.W., Dantzig, G.B.: Complementary pivot theory of mathematical programming. Linear Algebra and Its Applications 1, 103–125 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic Press (1992)

    Google Scholar 

  5. Derman, C.: Finite State Markov Decision Processes. Academic Press (1972)

    Google Scholar 

  6. Filar, J., Vrieze, K.: Competitive Markov Decision Processes. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  7. Gärtner, B., Morris, W.D., Rüst, L.: Unique sink orientations of grids. In: Jünger, M., Kaibel, V. (eds.) IPCO 2005. LNCS, vol. 3509, pp. 210–224. Springer, Heidelberg (2005)

    Google Scholar 

  8. Gärtner, B., Rüst, L.: Simple stochastic games and P-matrix generalized linear complementarity problems. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 209–220. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  9. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press (1985)

    Google Scholar 

  10. Johnson, C.R., Tsatsomeros, M.J.: Convex sets of nonsingular and P-matrices. Linear and Multilinear Algebra 38, 233–239 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kojima, M., Noma, T., Megiddo, N., Yoshise, A. (eds.): A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems. LNCS, vol. 538. Springer, Heidelberg (1991)

    Google Scholar 

  12. Lemke, C.E.: Bimatrix equilibrium points and mathematical programming. Management Science 11, 681–689 (1965)

    MathSciNet  Google Scholar 

  13. Mangasarian, O.L.: Linear complementarity problems solvable by a single linear program. Mathematical Programming 10, 263–270 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  14. Puri, A.: Theory of Hybrid Systems and Discrete Event Systems. PhD thesis, University of California, Berkeley (1995)

    Google Scholar 

  15. Shapley, L.S.: Stochastic games. Proc. Nat. Acad. Sci. U.S.A. 39, 1095–1100 (1953)

    Article  MATH  MathSciNet  Google Scholar 

  16. Stickney, A., Watson, L.: Digraph models of Bard-type algorithms for the linear complementarity problem. Mathematics of Operations Research 3, 322–333 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  17. Svensson, O., Vorobyov, S.: Linear complementarity and P-matrices for stochastic games. In: Virbitskaite, I., Voronkov, A. (eds.) PSI 2006. LNCS, vol. 4378, pp. 408–421. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  18. Szabó, T., Schurr, I.: Jumping doesn’t help in abstract cubes. In: Integer Programming and Combinatorial Optimization (IPCO). LNCS, vol. 11, pp. 225–235. Springer, Heidelberg (2005)

    Google Scholar 

  19. Szabó, T., Welzl, E.: Unique sink orientations of cubes. In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 547–555 (2001)

    Google Scholar 

  20. Vöge, J., Jurdziński, M.: A discrete strategy improvement algorithm for solving parity games (Extended abstract). In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 202–215. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  21. Zwick, U., Paterson, M.: The complexity of mean payoff games on graphs. Theoretical Computer Science 158, 343–359 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Warwick, UK

    Marcin Jurdziński & Rahul Savani

Authors
  1. Marcin Jurdziński
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rahul Savani
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Arnold Beckmann Costas Dimitracopoulos Benedikt Löwe

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Jurdziński, M., Savani, R. (2008). A Simple P-Matrix Linear Complementarity Problem for Discounted Games. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds) Logic and Theory of Algorithms. CiE 2008. Lecture Notes in Computer Science, vol 5028. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69407-6_32

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-69407-6_32

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-69405-2

  • Online ISBN: 978-3-540-69407-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation