## Sequential Games and Optimal Strategies

Citations: | 3 - 3 self |

### BibTeX

@MISC{Escardó_sequentialgames,

author = {Martín Escardó and Paulo Oliva},

title = {Sequential Games and Optimal Strategies},

year = {}

}

### OpenURL

### Abstract

This article gives an overview of recent work on the theory of selection functions. We explain the intuition behind these higher-type objects, and define a general notion of sequential game whose optimal strategies can be computed via a certain product of selection functions. Several instances of this game are considered in a variety of areas such as fixed point theory, topology, game theory, higher-type computability, and proof theory. These examples are intended to illustrate how the fundamental construction of optimal strategies based on products of selection functions permeates several research areas.

### Citations

731 | Notions of computation and monads
- Moggi
- 1991
(Show Context)
Citation Context ...tion functions turns out to be simply the canonical map that makes any strong monad into a monoidal monad. Monads are widely used in programming language semantics as a way to interpret side-effects (=-=Moggi 1991-=-) and in functional programming, particularly in the language Haskell, more generally as a way of structuring programs. As it turns out – see (Escardó & Oliva 2010d) – the ability to iterate this bina... |

235 | A formulae-as-types notion of control
- Griffin
- 1990
(Show Context)
Citation Context ...wn in (Escardó & Oliva 2010c) that the construction JR over any cartesian closed category gives rise to a strong monad, with the monad morphism (·): JR → KR into the well-known continuation monad KR (=-=Griffin 1990-=-). This monad morphism assigns the quantifier ε ∈ KRA defined by Equation (5.1) to a given selection function ε ∈ JRA. Moreover, the case n = 2 of the product of selection functions turns out to be si... |

142 |
Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica 12
- Gödel
- 1958
(Show Context)
Citation Context ...f ) Proof theory – Analysis The product of selection functions also appears in proof theory, in the form of bar recursion (Spector 1962). More precisely, in order to extend Gödel’s consistency proof (=-=Gödel 1958-=-) from arithmetic to analysis, Spector arrived at the following Article submitted to Royal Society24 Martín Escardó and Paulo Oliva system of equations: Given a family of (selection) functions εi : J... |

71 |
On a generalization of quantifiers
- Mostowski
(Show Context)
Citation Context ...s that Article submitted to Royal SocietySequential Games and Optimal Strategies 3 outcome. Note that our concept of “quantifier” is more general than Mostowski’s notion of a generalised quantifier (=-=Mostowski 1957-=-), which is the case when R is the set of truth-values or booleans. For the rest of this review we shall explore the connection between quantifiers and selection functions, and define algebraic operat... |

71 |
Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics
- Spector
- 1962
(Show Context)
Citation Context ...le principle, but here it is given an intuitive game-theoretic explanation. (f ) Proof theory – Analysis The product of selection functions also appears in proof theory, in the form of bar recursion (=-=Spector 1962-=-). More precisely, in order to extend Gödel’s consistency proof (Gödel 1958) from arithmetic to analysis, Spector arrived at the following Article submitted to Royal Society24 Martín Escardó and Paul... |

68 | Gödel’s functional (‘Dialectica’) interpretation
- Avigad, Feferman
- 1998
(Show Context)
Citation Context ... ≤ t∀m∃kA0(n, m, k). The negative translation (assuming Markov principle) of this principle is equivalent to ∀b∃n ≤ t∀m ≤ b∃kA0(n, m, k) → ¬¬∃n ≤ t∀m∃kA0(n, m, k). Its dialectica interpretation, see (=-=Avigad & Feferman 1998-=-), is ∃f, g∀b(fb ≤ t ∧ ∀m ≤ bA0(fb, m, gbm)) → ∀ε∃n ≤ t∃pA0(n, εnp, p(εnp)). In other words, given f, g and ε we must produce n, b and p such that ∀b(fb ≤ t ∧ ∀m ≤ bA0(fb, m, gbm)) → (n ≤ t ∧ A0(n, εn... |

43 |
A semantics of evidence for classical arithmetic
- Coquand
- 1995
(Show Context)
Citation Context ...a 2006) is primitive recursive equivalent to the product of selection functions from Definition 15. The second form of bar recursion arose from a game computational interpretation of classical logic (=-=Coquand 1995-=-), and it would be interesting to investigate in further details how our notion of games corresponds to the one used to interpret classical logic. See also (Aczel 2001) for more connections between st... |

40 |
On a number-theoretic choice schema and its relation to induction, in: Intuitionism and Proof Theory, edited by
- Parsons
- 1970
(Show Context)
Citation Context ...or existential formulas, i.e. ∀b∃n ≤ t∀m ≤ bA(n, m) → ∃n ≤ t∀mA(n, m). where A(n, m) is a Σ 0 1-formula. It is well-known that such principle lies in between Σ 0 2-induction and Σ 0 1-induction, see (=-=Parsons 1970-=-, Kohlenbach 2008). For simplicity, let us consider the case when A(n, m) = ∃kA0(n, m, k) where A0(n, m, k) is quantifier-free, i.e. ∀b∃n ≤ t∀m ≤ b∃kA0(n, m, k) → ∃n ≤ t∀m∃kA0(n, m, k). The negative t... |

36 |
Strongly majorizable functionals of finite type: a model of bar recursion containing discontinuous functionals
- Bezem
- 1985
(Show Context)
Citation Context ...tely iterated product is well-defined. The model we should bear in mind in this case is that of the continuous functionals (Normann 1999), and in some cases also the model of majorisable functionals (=-=Bezem 1985-=-). 2. The Product of Quantifiers Continuing from the examples given in the introduction, suppose now that we could only choose which petrol station to stop at, and the owner of the petrol stations the... |

34 | On the computational content of the axiom of choice - Berardi, Bezem, et al. - 1998 |

25 | Modified Bar Recursion and Classical Dependent Choice
- Berger, Oliva
- 2005
(Show Context)
Citation Context ... in fact Spector’s bar recursion is primitive recursive equivalent to the product of selection functions from Definition 18, and that a different form of bar recursion (Berardi, Bezem & Coquand 1998, =-=Berger & Oliva 2005-=-, Berger & Oliva 2006) is primitive recursive equivalent to the product of selection functions from Definition 15. The second form of bar recursion arose from a game computational interpretation of cl... |

13 | Exhaustible sets in higher-type computation
- Escardó
- 2008
(Show Context)
Citation Context ...functions εi : JRXi define their unbounded product by simply iterating the binary product as ∞⊗ i=k εi JRΠ ∞ i=kXi ( ∞⊗ = εk ⊗ i=k+1 † E.g. the natural numbers or more generally the types defined in (=-=Escardó 2008-=-, Definition 4.12). εi ) Article submitted to Royal Society14 Martín Escardó and Paulo Oliva where, for clarity, the type of the final product is shown above the equality sign. It is perhaps surprisi... |

12 | The Russell-Prawitz modality
- Aczel
(Show Context)
Citation Context ...rpretation of classical logic (Coquand 1995), and it would be interesting to investigate in further details how our notion of games corresponds to the one used to interpret classical logic. See also (=-=Aczel 2001-=-) for more connections between strong monads and classical logic. Article submitted to Royal SocietySequential Games and Optimal Strategies 25 (g) Functional programming We have also shown in (Escard... |

10 |
The continuous functionals
- Normann
- 1999
(Show Context)
Citation Context ...k with (non-standard) models of functionals, so as to ensure that the infinitely iterated product is well-defined. The model we should bear in mind in this case is that of the continuous functionals (=-=Normann 1999-=-), and in some cases also the model of majorisable functionals (Bezem 1985). 2. The Product of Quantifiers Continuing from the examples given in the introduction, suppose now that we could only choose... |

9 | Computational interpretations of analysis via products of selection functions - Escardó, Oliva - 2010 |

9 |
Understanding and using Spector’s bar recursive interpretation of classical analysis
- Oliva
- 2006
(Show Context)
Citation Context ...ns (7.1). By definition n = fb and m = an = εnpn = εnp. By the definition of q we also have pn(εn(pn)) = q(⃗a) = g(max⃗a)(a f(max ⃗a)) = gbafb = gban = gbm. This is essentially the solution given in (=-=Oliva 2006-=-) for the dialectica interpretation of the infinite pigeon-hole principle, but here it is given an intuitive game-theoretic explanation. (f ) Proof theory – Analysis The product of selection functions... |

8 | Selection functions, bar recursion, and backward induction - Escardó, Oliva |

8 |
A model of bar recursion of higher types
- Scarpellini
- 1971
(Show Context)
Citation Context ... clarity, the type of the final product is shown above the equality sign. It is perhaps surprising that such product functional is in fact not only welldefined in the model of continuous functionals (=-=Scarpellini 1971-=-) but also computable and part of the standard installation of Haskell (see Section 7(g) for more details). This is in stark contrast with the iterated product of quantifiers which is not well-defined... |

7 | Modified bar recursion
- Berger, Oliva
- 2006
(Show Context)
Citation Context ...r recursion is primitive recursive equivalent to the product of selection functions from Definition 18, and that a different form of bar recursion (Berardi, Bezem & Coquand 1998, Berger & Oliva 2005, =-=Berger & Oliva 2006-=-) is primitive recursive equivalent to the product of selection functions from Definition 15. The second form of bar recursion arose from a game computational interpretation of classical logic (Coquan... |

5 | The Peirce translation and the double negation shift - Escardó, Oliva - 2010 |

4 |
Programming languages and their definition
- Bekič
- 1984
(Show Context)
Citation Context ...e whole construction shows that if each space Xi has a fixed point operator, then the product space Π n−1 i=0 Xi must also have a fixed point operator. That is the essential content of Bekič’s lemma (=-=Bekič 1984-=-), a celebrated result in fixed point theory, and frequently used in domain theory. ) . Article submitted to Royal SocietySequential Games and Optimal Strategies 21 (c) Game theory – Nash equilibrium... |

2 | What sequential games, the Tychnoff theorem and the double-negation shift have - Escardó, Oliva - 2010 |

1 |
Applied Proof Theory
- Kohlenbach
- 2008
(Show Context)
Citation Context ... formulas, i.e. ∀b∃n ≤ t∀m ≤ bA(n, m) → ∃n ≤ t∀mA(n, m). where A(n, m) is a Σ 0 1-formula. It is well-known that such principle lies in between Σ 0 2-induction and Σ 0 1-induction, see (Parsons 1970, =-=Kohlenbach 2008-=-). For simplicity, let us consider the case when A(n, m) = ∃kA0(n, m, k) where A0(n, m, k) is quantifier-free, i.e. ∀b∃n ≤ t∀m ≤ b∃kA0(n, m, k) → ∃n ≤ t∀m∃kA0(n, m, k). The negative translation (assum... |

1 | Article submitted to Royal Society Games and Optimal Strategies 27 - Nisan - 2007 |