Abstract
We provide a formal, simple and intuitive theory of rational decision making including sequential decisions that affect the environment. The theory has a geometric flavor, which makes the arguments easy to visualize and understand. Our theory is for complete decision makers, which means that they have a complete set of preferences. Our main result shows that a complete rational decision maker implicitly has a probabilistic model of the environment. We have a countable version of this result that brings light on the issue of countable vs finite additivity by showing how it depends on the geometry of the space which we have preferences over. This is achieved through fruitfully connecting rationality with the Hahn-Banach Theorem. The theory presented here can be viewed as a formalization and extension of the betting odds approach to probability of Ramsey and De Finetti [Ram31, deF37].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Allais, M.: Le comportement de l’homme rationnel devant le risque: Critique des postulats et axiomes de l’ecole americaine. Econometrica 21(4), 503–546 (1953)
Arrow, K.: Essays in the Theory of Risk-Bearing. North-Holland, Amsterdam (1970)
Cox, R.T.: Probability, frequency and reasonable expectation. Am. Jour. Phys 14, 1–13 (1946)
de Finetti, B.: La prévision: Ses lois logiques, ses sources subjectives. In: Annales de l’Institut Henri Poincaré, Paris, vol. 7, pp. 1–68 (1937)
Diestel, J.: Sequences and series in \(\text{Banach}\) spaces. Springer, Heidelberg (1984)
Ellsberg, D.: Risk, Ambiguity, and the Savage Axioms. The Quarterly Journal of Economics 75(4), 643–669 (1961)
Halpern, J.Y.: A counterexample to theorems of Cox and Fine. Journal of AI research 10, 67–85 (1999)
Hutter, M.: Universal Artificial Intelligence: Sequential Decisions based on Algorithmic Probability. Springer, Berlin (2005)
Jaynes, E.T.: Probability theory: the logic of science. Cambridge University Press, Cambridge (2003)
Kreyszig, E.: Introductory Functional Analysis With Applications. Wiley, Chichester (1989)
Naricia, L., Beckenstein, E.: The Hahn-Banach theorem: the life and times. Topology and its Applications 77(2), 193–211 (1997)
Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)
Paris, J.B.: The uncertain reasoner’s companion: a mathematical perspective. Cambridge University Press, New York (1994)
Ramsey, F.P.: Truth and probability. In: Braithwaite, R.B. (ed.) The Foundations of Mathematics and other Logical Essays, ch.7, pp. 156–198. Brace & Co. (1931)
Savage, L.: The Foundations of Statistics. Wiley, New York (1954)
Sugden, R.: Rational choice: A survey of contributions from economics and philosophy. Economic Journal 101(407), 751–785 (1991)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sunehag, P., Hutter, M. (2011). Axioms for Rational Reinforcement Learning. In: Kivinen, J., Szepesvári, C., Ukkonen, E., Zeugmann, T. (eds) Algorithmic Learning Theory. ALT 2011. Lecture Notes in Computer Science(), vol 6925. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24412-4_27
Download citation
DOI: https://doi.org/10.1007/978-3-642-24412-4_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24411-7
Online ISBN: 978-3-642-24412-4
eBook Packages: Computer ScienceComputer Science (R0)