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The authors referred 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, they focus their study on the concept of sub-game perfect equilibrium, a refinement of Nash equilibrium well-suited in the framework of games played on graphs. They also introduce the new concept of sub-game perfect secure equilibrium.
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