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Convergence method, properties and computational complexity for Lyapunov games

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Clempner J., Poznyak A.
International Journal of Applied Mathematics and Computer Science
|
2011
|
tom 21
|
nr 2
349-361
EN
We introduce the concept of a Lyapunov game as a subclass of strictly dominated games and potential games. The advantage of this approach is that every ergodic system (repeated game) can be represented by a Lyapunov-like function. A direct acyclic graph is associated with a game. The graph structure represents the dependencies existing between the strategy profiles. By definition, a Lyapunov-like function monotonically decreases and converges to a single Lyapunov equilibrium point identified by the sink of the game graph. It is important to note that in previous works this convergence has not been guaranteed even if the Nash equilibrium point exists. The best reply dynamics result in a natural implementation of the behavior of a Lyapunov-like function. Therefore, a Lyapunov game has also the benefit that it is common knowledge of the players that only best replies are chosen. By the natural evolution of a Lyapunov-like function, no matter what, a strategy played once is not played again. As a construction example, we show that, for repeated games with bounded nonnegative cost functions within the class of differentiable vector functions whose derivatives satisfy the Lipschitz condition, a complex vector-function can be built, where each component is a function of the corresponding cost value and satisfies the condition of the Lyapunov-like function. The resulting vector Lyapunov-like function is a monotonic function which can only decrease over time. Then, a repeated game can be represented by a one-shot game. The functionality of the suggested method is successfully demonstrated by a simulated experiment.
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Modeling shortest path games with Petri nets: a Lyapunov based theory

86%
Clempner J.
International Journal of Applied Mathematics and Computer Science
|
2006
|
tom 16
|
nr 3
387-397
EN
In this paper we introduce a new modeling paradigm for shortest path games representation with Petri nets. Whereas previous works have restricted attention to tracking the net using Bellman's equation as a utility function, this work uses a Lyapunov-like function. In this sense, we change the traditional cost function by a trajectory-tracking function which is also an optimal cost-to-target function. This makes a significant difference in the conceptualization of the problem domain, allowing the replacement of the Nash equilibrium point by the Lyapunov equilibrium point in game theory. We show that the Lyapunov equilibrium point coincides with the Nash equilibrium point. As a consequence, all properties of equilibrium and stability are preserved in game theory. This is the most important contribution of this work. The potential of this approach remains in its formal proof simplicity for the existence of an equilibrium point.
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