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2 International Journal of Applied Mathematics and Computer Science

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Convergence method, properties and computational complexity for Lyapunov games

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Clempner J. B. , Poznyak A. S.

2011

tom Vol. 21, no 2

349-361

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.

Negotiating transfer pricing using the Nash bargaining solution

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Clempner J. B. , Poznyak A. S.

2017

tom Vol. 27, no. 4

853--864

This paper analyzes and proposes a solution to the transfer pricing problem from the point of view of the Nash bargaining game theory approach. We consider a firm consisting of several divisions with sequential transfers, in which central management provides a transfer price decision that enables maximization of operating profits. Price transferring between divisions is negotiable throughout the bargaining approach. Initially, we consider a disagreement point (status quo) between the divisions of the firm, which plays the role of a deterrent. We propose a framework and a method based on the Nash equilibrium approach for computing the disagreement point. Then, we introduce a bargaining solution, which is a single-valued function that selects an outcome from the feasible pay-offs for each bargaining problem that is a result of cooperation of the divisions of the firm involved in the transfer pricing problem. The agreement reached by the divisions in the game is the most preferred alternative within the set of feasible outcomes, which produces a profit-maximizing allocation of the transfer price between divisions. For computing the bargaining solution, we propose an optimization method. An example illustrating the usefulness of the method is presented.