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Selection functions, bar recursion and backward induction

Published online by Cambridge University Press:  25 March 2010

MARTÍN ESCARDÓ  and
PAULO OLIVA
MARTÍN ESCARDÓ
Affiliation:
University of Birmingham, Birmingham B15 2TT, U.K. Email: m.escardo@cs.bham.ac.uk
PAULO OLIVA
Affiliation:
Queen Mary University of London, London E1 4NS, U.K. Email: paulo.oliva@eecs.qmul.ac.uk
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Abstract

Bar recursion arises in constructive mathematics, logic, proof theory and higher-type computability theory. We explain bar recursion in terms of sequential games, and show how it can be naturally understood as a generalisation of the principle of backward induction that arises in game theory. In summary, bar recursion calculates optimal plays and optimal strategies, which, for particular games of interest, amount to equilibria. We consider finite games and continuous countably infinite games, and relate the two. The above development is followed by a conceptual explanation of how the finite version of the main form of bar recursion considered here arises from a strong monad of selections functions that can be defined in any cartesian closed category. Finite bar recursion turns out to be a well-known morphism available in any strong monad, specialised to the selection monad.

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Paper
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Mathematical Structures in Computer Science , Volume 20 , Special Issue 2: Domains , April 2010 , pp. 127 - 168
Copyright
Copyright © Cambridge University Press 2010

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