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Modeling shortest path games with Petri nets: A Lyapunov based theory

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Clempner J.

International Journal of Applied Mathematics and Computer Science

2006

tom Vol. 16, no 3

387-397

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.

The Bruss-Robertson Inequality: Elaborations, Extensions, and Applications

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Steele J. M.

2016

tom Vol. 44, No. 1

3--16

The Bruss-Robertson inequality gives a bound on the maximal number of elements of a random sample whose sum is less than a specified value. The extension of that inequality which is given here neither requires the independence of the summands nor requires the equality of their marginal distributions. A review is also given of the applications of the Bruss-Robertson inequality, especially the applications to problems of combinatorial optimization such as the sequential knapsack problem and the sequential monotone subsequence selection problem.

Nierówność Bruss-Robertson szacuje maksymalną liczbę elementów w próbie, której suma jest ograniczona przez zadaną liczbę. Uogólnienia tej nierówności podane w tej pracy nie wymagają założenia niezależności składników sumy ani tego, by były o tym samym rozkładzie. Podano także przegląd zastosowań nierówności Brussa-Robertsona, a zwłaszcza zastosowania do problemów kombinatorycznych, takich jak sekwencyjny problem upakowania i wybór monotonicznego podciągu.

The machine learning approach: analysis of experimental results

100%

Poliscuk J. E.

tom Vol. 11, nr 1

61-76

The article analyses a reinforcement learning method in which the subject of learning is defined. The essence of this method is the selection of activities by a try and fail process and awarding deferred rewards. Theoretical analyses were supplemented by the practical studies, with reference to implementation of the Sarsa( Lambda) algorithm, with replacing eligibility traces and the Epsilon greedy policy.