2014 | 24 | 1 | 71-96
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Probabilities on streams and reflexive games

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Probability measures on streams (e.g. on hypernumbers and p-adic numbers) have been defined. It was shown that these probabilities can be used for simulations of reflexive games. In particular, it can be proved that Aumann’s agreement theorem does not hold for these probabilities. Instead of this theorem, there is a statement that is called the reflexion disagreement theorem. Based on this theorem, probabilistic and knowledge conditions can be defined for reflexive games at various reflexion levels up to the infinite level.
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  • University of Information Technology and Management in Rzeszow, ul. Sucharskiego 2, 35-225 Rzeszów, Poland,
  • ABREU D., PEARCE D., STACCHETTI E., Toward a theory of discounted repeated games with imperfect monitoring, Econometrica, 1990, 58 (5), 1041–1063.
  • ACZEL P., Non-Well-Founded Sets, Stanford University, Stanford 1988.
  • AUMANN R.J., Agreeing to Disagree, The Annals of Statistics, 1976, 4 (6), 1236–1239.
  • AUMANN R.J., Notes on Interactive Epistemology, Mimeo, Hebrew University of Jerusalem, Jerusalem 1989
  • BERGER S., The Foundations of Non-Equilibrium Economics: The Principle of Circular Cumulative Causation, Routledge Chapman & Hall, 2009.
  • BRANDENBURGER A., Epistemic game theory: an overview, [in:] The New Palgrave Dictionary of Economics, 2nd Ed., S. Darlauf, L. Blume (Eds.), Macmillan 2008.
  • BRANDENBURGER A., FRIEDENBERG A., KEISLER H.J., Fixed points in epistemic game theory, [in:] Mathematical Foundations of Information Flow, S. Abramsky, M. Mislove (Eds.), American Mathematical Society, Tulane University, New Orleans, Louisiana, 2012.
  • BRANDENBURGER A., KEISLER H.J., An impossibility theorem on beliefs in games, Studia Logica, 2006, 84 (2), 211–240.
  • CAPRETTA V., Common Knowledge as a Coinductive Modality, [in:] Reflections on Type Theory, Lambda Calculus, and the Mind. Essays Dedicated to Henk Barendregt on the Occasion of His 60th Birthday, E. Barendsen, H. Geuvers, V. Capretta, M. Niqui (Eds.), Radboud University, Nijmegen 2007.
  • CHEVALLEY C., BELZUNG C., Emotional behaviour as the result of stochastic interactions: a process crucial for cognition, Behavioral Processes, 2002, 60 (2), 115–132.
  • CHKHARTISHVILI A.G., Bayes–Nash equilibrium: Infinite-depth point information structures, Automation and Remote Control, 2003, 64 (12), 1922–1927.
  • CHKHARTISHVILI A.G., Concordant informational control, Automation and Remote Control, 2012, 73 (8), 1401–1409.
  • CHKHARTISHVILI A.G., NOVIKOV D.A., Models of reflexive decision-making, Systems Science, 2004, 30 (2), 45–59.
  • CHKHARTISHVILI A.G., Reflexive games: Transformation of awareness structure, Automation and Remote Control, 2010, 71 (6), 1208–1216.
  • FIORE M.P., A coinduction principle for recursive data types based on bisimulation, [in:] Proc. 8th Conf. Logic in Computer Science (LICS), IEEE XPlore, 1993, 110–119.
  • HEIFETZ A., Non-well-founded-type spaces, Games and Economic Behavior, 1996, 16 (2), 202–217.
  • HOWARD N., General metagames: an extension of the metagame concept. Game Theory as a Theory of Conflict Resolution, Reidel, Dordrecht 1974.
  • JACOBS B., RUTTEN J., A tutorial on (co)algebras and (co)induction, EATCS Bulletin, 1997, 62, 222–259.
  • KHRENNIKOV A.YU., Interpretations of probability, VSP Int. Sc. Publishers, Utrecht 1999.
  • KHRENNIKOV A.YU., Modeling of processes of thinking in p-adic coordinates, Nauka, Fizmatlit, Moscow, 2004 (in Russian).
  • KHRENNIKOV A.YU., p-Adic quantum mechanics with p-adic valued functions, Journal of Mathematical Physics, 1991, 32 (4), 932–937
  • KHRENNIKOV A.YU., p-Adic valued distributions in mathematical physics, Kluwer Academic Publishers, Dordrecht 1994.
  • KHRENNIKOV A.YU., SCHUMANN A., Logical approach to p-adic probabilities, Bulletin of the Section of Logic, 2006, 35 (1), 49–57.
  • KNUDSEN CH., Equilibrium, perfect rationality and the problem of self-reference in economics, [in:] Rationality, Institutions and Economic Methodology, Uskali Maki (Ed.), Routledge Chapman & Hall, 1993.
  • LEFEBVRE V.A., The structure of awareness: Toward a symbolic language of human reflexion, Sage Publications, New York 1977.
  • LEFEBVRE V.A., Algebra of conscience, D. Reidel, Dordrecht 1982.
  • LEFEBVRE V.A., Lectures on reflexive game theory, Leaf & Oaks, Los Angeles 2010.
  • LEFEBVRE V.A., The basic ideas of reflexive game’s logic, [in:] Problems of systems and structures research, AS USSR Publishers, Moscow 1965, 73–79 (in Russian).
  • LESCANNE P., PERRINEL M., Backward coinduction, Nash equilibrium and the rationality of escalation, Acta Informatica, 2012, 49 (3), 117–137.
  • MAILATH G.J., SAMUELSON L., Repeated games and reputations. Long-run relationships, Oxford University Press, Oxford 2006.
  • MILNER R., Communication and Concurrency, Prentice-Hall, Upper Saddle River, New York 1989.
  • MOSS L.S., Coalgebraic logic, Ann. Pure & Appl. Logic, 1999, 96 (1–3), 77–317. Erratum in Ann. Pure & Appl. Logic, 1999, 99 (1–3), 241–259.
  • NOVIKOV D.A., CHKHARTISHVILI A.G., Stability of information equilibrium in reflexive games, Automation and Remote Control, 2005, 66 (3), 441–448.
  • NOVIKOV D.A., CKHARTISHVILI A.G., Reflexive games, Sinteg, Moscow 2003 (in Russian).
  • PAVLOVIC D., ESCARDÓ M.H., Calculus in coinductive form, [in:] Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science, IEEE XPlore, 1998, 408–417.
  • Coalgebraic methods in computer science, H. Reichel, J. Rutten (Eds.), Elect. Notes in TCS 11, Elsevier, 1998.
  • ROBINSON A., Non-standard analysis. Studies in logic and the foundations of mathematics, North-Holland, Amsterdam 1966.
  • RUTTEN J.J.M.M., A coinductive calculus of streams, Math. Struct. in Comp. Sci., 2005, 15 (1), 93–147.
  • RUTTEN J.J.M.M., Universal coalgebra: a theory of systems, Theor. Comput. Sci., 2000, 249 (1), 3–80.
  • RUTTEN J., Behavioral differential equations: a coinductive calculus of streams, automata, and power series, Theor. Comput. Sci., 2003, 308, 1–53.
  • SCHUMANN A., Non-Archimedean fuzzy and probability logic, Journal of Applied Non-Classical Logics, 2008, 18 (1), 29–48.
  • SCHUMANN A., Non-Archimedean valued and p-adic valued fuzzy cellular automata, Journal of Cellular Automata, 2008, 3 (4), 337–354.
  • SCHUMANN A., Non-Archimedean valued extension of logic LП and p-adic valued extension of logic BL, Journal of Uncertain Systems, 2010, 4 (2), 99–115.
  • SCHUMANN A., Non-Archimedean valued predicate logic, Bulletin of the Section of Logic, 2007, 36 (1–2), 67–78.
  • SCHUMANN A., Non-Archimedean valued sequent logic, Eighth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2006), IEEE Press, 2006, 89–92.
  • SCHUMANN A., Non-well-founded probabilities and coinductive probability logic, Eighth International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC 2008), IEEE Computer Society Press, 2008, 54–57.
  • SCHUMANN A., Non-well-founded probabilities on streams, [in:] Soft Methods for Handling Variability and Imprecision, D. Dubois, M.A. Lubiano, H. Prade (Eds.), Var. and Impresion, ASC 48, Springer-Verlag, 2008, 59–65.
  • SCHUMANN A., p-Adic multiple-validity and p-adic valued logical calculi, Journal of Multiple- Valued Logic and Soft Computing, 2007, 13 (1–2), 29–60.
  • SCHUMANN A., Prooftheoretic cellular automata as logic of unconventional computing, Journal of Unconventional Computing, 2012, 8 (3), 263–280.
  • SEARLE J.R., Expression and meaning: studies in the theory of speech acts, Cambridge University Press, Cambridge 1979.
  • SEARLE J.R., Speech acts, an essay in the philosophy of language, Cambridge University Press, Cambridge 1969.
  • SEARLE J.R., VANDERVEKEN D., Foundations of illocutionary logic, Cambridge University Press, Cambridge 1984.
  • [53] WINRICH J.S., Self-reference and the incomplete structure of neoclassical economics, Journal of Economic Issues, 1984, 18 (4), 987–1005.
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