Higher-Order Decision Theory

  title={Higher-Order Decision Theory},
  author={Jules Hedges and Paulo Oliva and Evguenia Shprits and Viktor Winschel and Philipp Zahn},
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… 
Higher-Order Game Theory
This paper introduces a representation of games where players' goals are modeled based on so-called higher-order functions, and shows that equilibrium conditions based on Nash can be easily adapted to this framework.
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It is shown, surprisingly, no non-deterministic selection function exists which computes the set of all subgame perfect Nash equilibrium plays, but it is shown that there are selection functions corresponding to sequential versions of the iterated removal of strictly dominated strategies.
Compositional Game Theory
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Bar recursion is not computable via iteration
  • J. Longley
  • Mathematics, Computer Science
  • 2019
We show that the bar recursion operators of Spector and Kohlenbach, considered as third-order functionals acting on total arguments, are not computable in Goedel's System T plus minimization, which


Selection Equilibria of Higher-Order Games
It is shown that for a special class of games these two notions coincide, but that in general, the notion of selection equilibrium seems to be the right notion to consider, as illustrated through variants of coordination games where agents are modelled via fixed-point operators.
Expected utility theory without the completeness axiom
Higher-Order Game Theory
This paper introduces a representation of games where players' goals are modeled based on so-called higher-order functions, and shows that equilibrium conditions based on Nash can be easily adapted to this framework.
A generalization of Nash's theorem with higher-order functionals
  • Julian Hedges
  • Economics, Mathematics
    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2013
The Nash existence theorem for mixed-strategy equilibria of finite games is generalized to games defined by selection functions, and a normal form construction is given, which generalizes the game-theoretic normal form, and its soundness is proved.
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Selection functions, bar recursion and backward induction
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