scholarly journals Selection functions, bar recursion and backward induction

2010 ◽  
Vol 20 (2) ◽  
pp. 127-168 ◽  
Author(s):  
MARTÍN ESCARDÓ ◽  
PAULO OLIVA
Keyword(s):  
Proof Theory ◽  
Optimal Strategies ◽  
Backward Induction ◽  
Constructive Mathematics ◽  
Infinite Games ◽  
Sequential Games ◽  
Finite Games ◽  
Conceptual Explanation ◽  
Cartesian Closed ◽  
Countably Infinite

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.

scholarly journals Generalized Backward Induction: Justification for a Folk Algorithm

Games ◽  
2019 ◽  
Vol 10 (3) ◽  
pp. 34
Author(s):  
Marek Mikolaj Kaminski
Keyword(s):  
Imperfect Information ◽  
Infinite Horizon ◽  
Backward Induction ◽  
Behavioral Strategies ◽  
Infinite Games ◽  
Sequential Games ◽  
Finite Games ◽  
Pure Strategies ◽  
Subgame Perfect Equilibria ◽  
Perfect Equilibria

I introduce axiomatically infinite sequential games that extend Kuhn’s classical framework. Infinite games allow for (a) imperfect information, (b) an infinite horizon, and (c) infinite action sets. A generalized backward induction (GBI) procedure is defined for all such games over the roots of subgames. A strategy profile that survives backward pruning is called a backward induction solution (BIS). The main result of this paper finds that, similar to finite games of perfect information, the sets of BIS and subgame perfect equilibria (SPE) coincide for both pure strategies and for behavioral strategies that satisfy the conditions of finite support and finite crossing. Additionally, I discuss five examples of well-known games and political economy models that can be solved with GBI but not classic backward induction (BI). The contributions of this paper include (a) the axiomatization of a class of infinite games, (b) the extension of backward induction to infinite games, and (c) the proof that BIS and SPEs are identical for infinite games.

scholarly journals A generalization of Nash's theorem with higher-order functionals

2013 ◽  
Vol 469 (2154) ◽  
pp. 20130041 ◽  
Author(s):  
Julian Hedges
Keyword(s):  
Normal Form ◽  
Proof Theory ◽  
Mixed Strategy ◽  
Recent Theory ◽  
Sequential Games ◽  
New Class ◽  
Finite Games ◽  
Minimax Strategies ◽  
Game Theoretic ◽  
Axiom Of Countable Choice

The recent theory of sequential games and selection functions by Escardó & Oliva is extended to games in which players move simultaneously. The Nash existence theorem for mixed-strategy equilibria of finite games is generalized to games defined by selection functions. A normal form construction is given, which generalizes the game-theoretic normal form, and its soundness is proved. Minimax strategies also generalize to the new class of games, and are computed by the Berardi–Bezem–Coquand functional, studied in proof theory as an interpretation of the axiom of countable choice.

scholarly journals Computing Nash Equilibria of Unbounded Games

10.29007/1wpl ◽  
2018 ◽  
Author(s):  
Martin Escardo ◽  
Paulo Oliva
Keyword(s):  
Nash Equilibria ◽  
Proof Theory ◽  
A Priori ◽  
Subgame Perfect Equilibrium ◽  
Backward Induction ◽  
Closed Formula ◽  
Perfect Equilibrium ◽  
Finite Games ◽  
Game Theoretic ◽  
Theoretic Technique

Using techniques from higher-type computability theory and proof theory we extend the well-known game-theoretic technique of backward induction to finite games of unbounded length. The main application is a closed formula for calculating strategy profiles in Nash equilibrium and subgame perfect equilibrium even in the case of games where the length of play is not a-priori fixed.

scholarly journals Sequential games and optimal strategies

2010 ◽  
Vol 467 (2130) ◽  
pp. 1519-1545 ◽  
Author(s):  
Martín Escardó ◽  
Paulo Oliva
Keyword(s):  
Game Theory ◽  
Fixed Point ◽  
Recent Work ◽  
Proof Theory ◽  
Fixed Point Theory ◽  
Optimal Strategies ◽  
Point Theory ◽  
General Notion ◽  
Sequential Games ◽  
Research Areas

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.

Interpreting Knowledge in the Backward Induction Problem

Episteme ◽  
2011 ◽  
Vol 8 (3) ◽  
pp. 248-261 ◽  
Author(s):  
Ken Binmore
Keyword(s):  
Common Knowledge ◽  
Perfect Information ◽  
Backward Induction ◽  
Finite Games ◽  
Common Knowledge Of Rationality

AbstractRobert Aumann argues that common knowledge of rationality implies backward induction in finite games of perfect information. I have argued that it does not. A literature now exists in which various formal arguments are offered in support of both positions. This paper argues that Aumann's claim can be justified if knowledge is suitably reinterpreted.

Mathematical Logic: Proof Theory, Constructive Mathematics

2008 ◽  
pp. 907-952
Author(s):  
Samuel Buss ◽  
Helmut Schwichtenberg ◽  
Ulrich Kohlenbach
Keyword(s):  
Mathematical Logic ◽  
Proof Theory ◽  
Constructive Mathematics

Mathematical Logic: Proof Theory, Constructive Mathematics

2011 ◽  
Vol 8 (4) ◽  
pp. 2963-3002 ◽  
Author(s):  
Samuel Buss ◽  
Ulrich Kohlenbach ◽  
Michael Rathjen
Keyword(s):  
Mathematical Logic ◽  
Proof Theory ◽  
Constructive Mathematics

Mathematical Logic: Proof Theory, Type Theory and Constructive Mathematics

2005 ◽  
pp. 779-813
Author(s):  
Helmut Schwichtenberg ◽  
Vladimir Keilis-Borok ◽  
Samuel Buss
Keyword(s):  
Mathematical Logic ◽  
Type Theory ◽  
Proof Theory ◽  
Constructive Mathematics

Parallel strategies

2003 ◽  
Vol 68 (4) ◽  
pp. 1242-1250
Author(s):  
Pavel Pudlák
Keyword(s):  
Bounded Arithmetic ◽  
Infinite Games ◽  
Open Problems ◽  
Finite Games ◽  
Combinatorial Principles ◽  
Axiomatic Systems

AbstractWe consider combinatorial principles based on playing several two person games simultaneously. We call strategies for playing two or more games simultaneously parallel. The principles are easy consequences of the determinacy of games, in particular they are true for all finite games. We shall show that the principles fail for infinite games. The statements of these principles are of lower logical complexity than the sentence expressing the determinacy of games, therefore, they can be studied in weak axiomatic systems for arithmetic (Bounded Arithmetic). We pose several open problems about the provability of these statements in Bounded Arithmetic and related computational problems.

Mathematical Logic: Proof Theory, Constructive Mathematics

2021 ◽  
Vol 17 (4) ◽  
pp. 1693-1757
Author(s):  
Samuel Buss ◽  
Rosalie Iemhoff ◽  
Ulrich Kohlenbach ◽  
Michael Rathjen
Keyword(s):  
Mathematical Logic ◽  
Proof Theory ◽  
Constructive Mathematics
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