Higher-Order Decision Theory

@inproceedings{Hedges2017HigherOrderDT,
  title={Higher-Order Decision Theory},
  author={Jules Hedges and Paulo Oliva and Evguenia Shprits and Viktor Winschel and Philipp Zahn},
  booktitle={ADT},
  year={2017}
}
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… 
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References

SHOWING 1-10 OF 31 REFERENCES
Selection Equilibria of Higher-Order Games
TLDR
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
TLDR
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
TLDR
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.
Regret Theory: An alternative theory of rational choice under uncertainty Review of Economic Studies
The main body of current economic analysis of choice under uncertainty is built upon a small number of basic axioms, formulated in slightly different ways by von Neumann and Morgenstern (I 947),
Rationalising Choice with Multi�?Self Models
This paper studies a class of multi-self decision-making models proposed in economics, psychology, and marketing. In this class, choices arise from the set-dependent aggregation of a collection of
String diagrams for game theory
TLDR
A monoidal category whose morphisms are games (in the sense of game theory, not game semantics) and an associated diagrammatic language is presented and a definition of Nash equilibrium is given which is recursive on the causal structure of the game.
A compositional approach to economic game theory
TLDR
An new and original model of economic games based upon the computer science idea of compositionality is built, and the usual lexicon of games is augmented with a new concept of coutility, which guarantees the model is mathematically rigourous.
Utility Representation of an Incomplete Preference Relation
TLDR
These results generalize some of the classical utility representation theorems of the theory of individual choice and paves the way towards developing a consumer theory that realistically allows individuals to exhibit some “indecisiveness” on occasion.
Selection functions, bar recursion and backward induction
TLDR
This work explains bar recursion in terms of sequential games, and shows how it can be naturally understood as a generalisation of the principle of backward induction that arises in game theory.
...
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