Subgame Perfection for Equilibria in Quantitative Reachability Games

  • Thomas Brihaye
  • Véronique Bruyère
  • Julie De Pril
  • Hugo Gimbert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7213)


We study turn-based quantitative multiplayer non zero-sum games played on finite graphs with reachability objectives. In such games, each player aims at reaching his own goal set of states as soon as possible. A previous work on this model showed that Nash equilibria (resp. secure equilibria) are guaranteed to exist in the multiplayer (resp. two-player) case. The existence of secure equilibria in the multiplayer case remained, and is still an open problem. In this paper, we focus our study on the concept of subgame perfect equilibrium, a refinement of Nash equilibrium well-suited in the framework of games played on graphs. We also introduce the new concept of subgame perfect secure equilibrium. We prove the existence of subgame perfect equilibria (resp. subgame perfect secure equilibria) in multiplayer (resp. two-player) quantitative reachability games. Moreover, we provide an algorithm deciding the existence of secure equilibria in the multiplayer case.


Nash Equilibrium Subgame Perfect Equilibrium Initial Vertex Subgame Perfection Quantitative Objective 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Thomas Brihaye
    • 1
  • Véronique Bruyère
    • 1
  • Julie De Pril
    • 1
  • Hugo Gimbert
    • 2
  1. 1.University of Mons - UMONSMonsBelgium
  2. 2.LaBRI & CNRSBordeauxFrance

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