Integer Valued Betting strategies and Turing Degrees

TL;DR

This paper explores integer-valued betting strategies modeled as martingales, examining their role in defining a form of algorithmic randomness and analyzing their computational properties within computability theory.

Contribution

It introduces and studies the concept of integer-valued randomness, connecting it with genericity and Turing degrees, and investigates its computational strength.

Findings

Integer-valued randomness interacts with genericity and computably enumerable degrees.

Effective integer-valued martingales define a new form of algorithmic randomness.

The computational power of integer-valued random reals is characterized within classical computability theory.

Abstract

Betting strategies are often expressed formally as martingales. A martingale is called integer-valued if each bet must be an integer value. Integer-valued strategies correspond to the fact that in most betting situations, there is a minimum amount that a player can bet. According to a well known paradigm, algorithmic randomness can be founded on the notion of betting strategies. A real X is called integer-valued random if no effective integer-valued martingale succeeds on X. It turns out that this notion of randomness has interesting interactions with genericity and the computably enumerable degrees. We investigate the computational power of the integer-valued random reals in terms of standard notions from computability theory.