Axioms for Rational Reinforcement Learning

  • Peter Sunehag
  • Marcus Hutter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6925)


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].


Rationality Probability Utility Banach Space Linear Functional 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [All53]
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [Arr70]
    Arrow, K.: Essays in the Theory of Risk-Bearing. North-Holland, Amsterdam (1970)zbMATHGoogle Scholar
  3. [Cox46]
    Cox, R.T.: Probability, frequency and reasonable expectation. Am. Jour. Phys 14, 1–13 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [deF37]
    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)Google Scholar
  5. [Die84]
    Diestel, J.: Sequences and series in \(\text{Banach}\) spaces. Springer, Heidelberg (1984)CrossRefGoogle Scholar
  6. [Ell61]
    Ellsberg, D.: Risk, Ambiguity, and the Savage Axioms. The Quarterly Journal of Economics 75(4), 643–669 (1961)CrossRefzbMATHGoogle Scholar
  7. [Hal99]
    Halpern, J.Y.: A counterexample to theorems of Cox and Fine. Journal of AI research 10, 67–85 (1999)MathSciNetzbMATHGoogle Scholar
  8. [Hut05]
    Hutter, M.: Universal Artificial Intelligence: Sequential Decisions based on Algorithmic Probability. Springer, Berlin (2005)CrossRefzbMATHGoogle Scholar
  9. [Jay03]
    Jaynes, E.T.: Probability theory: the logic of science. Cambridge University Press, Cambridge (2003)CrossRefzbMATHGoogle Scholar
  10. [Kre89]
    Kreyszig, E.: Introductory Functional Analysis With Applications. Wiley, Chichester (1989)zbMATHGoogle Scholar
  11. [NB97]
    Naricia, L., Beckenstein, E.: The Hahn-Banach theorem: the life and times. Topology and its Applications 77(2), 193–211 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [NM44]
    Neumann, J., Morgenstern, O.: Theory of Games and Economic Behavior. Princeton University Press, Princeton (1944)zbMATHGoogle Scholar
  13. [Par94]
    Paris, J.B.: The uncertain reasoner’s companion: a mathematical perspective. Cambridge University Press, New York (1994)zbMATHGoogle Scholar
  14. [Ram31]
    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)Google Scholar
  15. [Sav54]
    Savage, L.: The Foundations of Statistics. Wiley, New York (1954)zbMATHGoogle Scholar
  16. [Sug91]
    Sugden, R.: Rational choice: A survey of contributions from economics and philosophy. Economic Journal 101(407), 751–785 (1991)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Peter Sunehag
    • 1
  • Marcus Hutter
    • 1
  1. 1.Research School of Computer ScienceAustralian National UniversityCanberraAustralia

Personalised recommendations