Higher-Order Decision Theory

  • Jules Hedges
  • Paulo Oliva
  • Evguenia Shprits
  • Viktor Winschel
  • Philipp ZahnEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10576)


This paper investigates a surprising relationship between decision theory and proof theory. Using constructions originating in proof theory based on higher-order functions, so called quantifiers and selection functions, we show that these functionals model choice behavior of individual agents. Our framework is expressive, it captures classical theories such as utility functions and preference relations but it can also be used to faithfully model abstract goals such as coordination. It is directly implementable in functional programming languages. Lastly, modeling an agent with selection functions and quantifiers is modular and thereby allows to seamlessly combine agents bridging decision theory and game theory.


Decision theory Utility functions Preferences Higher-order functions Quantifiers 


  1. 1.
    Ambrus, A., Rozen, K.: Rationalising choice with multi-self models. Econ. J. 125(585), 1136–1156 (2015)CrossRefGoogle Scholar
  2. 2.
    Camerer, C.F.: Behavioral Game Theory: Experiments in Strategic Interaction. The Roundtable Series in Behavioral Economics. Princeton University Press, Princeton (2011)zbMATHGoogle Scholar
  3. 3.
    Escardó, M., Oliva, P.: Selection functions, bar recursion and backward induction. Math. Struct. Comput. Sci. 20(2), 127–168 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Escardo, M., Oliva, P.: Sequential games and optimal strategies. Proc. R. Soc. A Math. Phys. Eng. Sci. 467(2130), 1519–1545 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ghani, N., Hedges, J.: A compositional approach to economic game theory. CoRR, abs/1603.04641 (2016)Google Scholar
  6. 6.
    Hedges, J.: A generalisation of Nash’s theorem with higher-order functionals. Proc. R. Soc. A 469(2154), 1–18 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hedges, J.: String diagrams for game theory. CoRR, abs/1503.06072 (2015)Google Scholar
  8. 8.
    Hedges, J., Oliva, P., Shprits, E., Winschel, V., Zahn, P.: Selection equilibria of higher-order games. In: Lierler, Y., Taha, W. (eds.) PADL 2017. LNCS, vol. 10137, pp. 136–151. Springer, Cham (2017). doi: 10.1007/978-3-319-51676-9_9 CrossRefGoogle Scholar
  9. 9.
    Hedges, J., Oliva, P., Sprits, E., Winschel, V., Zahn, P.: Higher-order game theory. CoRR, abs/1506.01002 (2015)Google Scholar
  10. 10.
    Hedges, J., Shprits, E., Winschel, V., Zahn, P.: Compositionality and string diagrams for game theory. CoRR, abs/1604.06061 (2016)Google Scholar
  11. 11.
    Kahneman, D.: Thinking, Fast and Slow. Farrar, Straus and Giroux, New York (2011)Google Scholar
  12. 12.
    Kalai, G., Rubinstein, A., Spiegler, R.: Rationalizing choice functions by multiple rationales. Econometrica 70(6), 2481–2488 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Loomes, G., Sugden, R.: Regret theory: An alternative theory of rational choice under uncertainty. Econ. J. 92(368), 805–824 (1982)CrossRefGoogle Scholar
  14. 14.
    Mostowski, A.: On a generalization of quantifiers. Fund. Math. 44(1), 12–36 (1957)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Savage, L.J.: The theory of statistical decision. J. Am. Stat. Assoc. 46(253), 55–67 (1951)CrossRefzbMATHGoogle Scholar
  16. 16.
    Wald, A.: Statistical Decision Functions. Wiley, Oxford (1950)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Jules Hedges
    • 1
  • Paulo Oliva
    • 2
  • Evguenia Shprits
    • 3
  • Viktor Winschel
    • 4
  • Philipp Zahn
    • 5
    Email author
  1. 1.Department of Computer ScienceUniversity of OxfordOxfordUK
  2. 2.Department of Electronic Engineering and Computer ScienceQueen Mary University of LondonLondonUK
  3. 3.Department of EconomicsUniversity of MannheimMannheimGermany
  4. 4.OICOS GmbHMannheimGermany
  5. 5.Department of EconomicsUniversity of St. GallenSt. GallenSwitzerland

Personalised recommendations