Abstract
We introduce a very complex game based on an approximate solution of a NP-hard problem, so that the probability of victory grows monotonically, but of an unknown amount, with the resources each player employs. We formulate this model in the computational learning framework and focus on the problem of computing a confidence interval for the losing probability. We deal with the problem of reducing the width of this interval under a given threshold in both batch and on-line modality. While the former leads to a feasible polynomial complexity, the on-line learning strategy may get stuck in an indeterminacy trap: the more we play the game the broader becomes the confidence interval. In order to avoid this indeterminacy we organise in a better way the knowledge, introducing the notion of virtual game to achieve the goal efficiently. Then we extend the one-player to a team mode game. Namely, we improve the success of a team by redistributing the resources among the players and exploiting their mutual cooperation to treat the indeterminacy phenomenon suitably.
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References
Nash, J.: Non-cooperative games. Annals of Mathematics 54, 286–295 (1951)
Angluin, D.: Queries and concept learning. Machine Learning, 319–342 (1988)
Ben-David, S., Kushilevitz, E., Mansour, Y.: Online learning versus offline learning. Machine Learning 29, 45–63 (1997)
Blackwell, D., Girshick, M.A.: Theory of Games and Statistical Decisions. Dover Publications, Inc., New York (1979)
Sahni, S.: Some related problems from network flows, game theory, and integer programming. In: Proceedings of the 13th Annual IEEE Symposium of Switching and Automata Theory, pp. 130–138 (1972)
Martello, S., Toth, P.: The 0–1 knapsack problem. In: Combinatorial Optimization, pp. 237–279. Wiley, Chichester (1979)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1994)
Sahni, S.: Approximate algorithms for the 0/1 knapsack problem. Journal of the Association of Computing Machinery 22, 115–124 (1975)
Apolloni, B., Bassis, S., Malchiodi, D., Gaito, S.: The statistical bases of learning. In: From Synapses to Rules: Disovering Symbolic Rules from Neural Processed Data, International School on Neural Nets “E.R. Caianiello”, 5th course (2002)
Apolloni, B., Ferretti, C., Mauri, G.: Approximation of optimization problems and learnability. In: Di Pace, L. (ed.) Atti del Terzo Workshop del Gruppo AI*IA di Interesse Speciale su Apprendimento Automatico (1992)
Apolloni, B., Chiaravalli, S.: Pac learning of concept classes through the boundaries of their items. Theoretical Computer Science 172, 91–120 (1997)
Tukey, J.: Nonparametric estimation II. Statistically equivalent blocks and multivariate tolerance regions. the continuous case. Annals of Mathematical Statistics 18, 529–539 (1947)
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© 2003 Springer-Verlag Berlin Heidelberg
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Apolloni, B., Bassis, S., Gaito, S., Malchiodi, D. (2003). Cooperative Games in a Stochastic Environment. In: Apolloni, B., Marinaro, M., Tagliaferri, R. (eds) Neural Nets. WIRN 2003. Lecture Notes in Computer Science, vol 2859. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45216-4_2
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DOI: https://doi.org/10.1007/978-3-540-45216-4_2
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