# gale

Let $\nu$ be a probability measure on Cantor space $\mathbf{C}$, and let $s\in[0,\infty)$.

1. 1.

A $\nu$-$s$-supergale is a function $d:\{0,1\}^{*}\rightarrow[0,\infty)$ that satisfies the condition

 $d(w)\nu(w)^{s}\geq d(w0)\nu(w0)^{s}+d(w1)\nu(w1)^{s}$ (1)

for all $w\in\{0,1\}^{*}$, the set of all finite strings of $0$’s and $1$’s (including $e$, the empty string).

2. 2.

A $\nu$-$s$-gale is a $\nu$-$s$-supergale that satisfies the condition with equality for all $w\in\{0,1\}^{*}$.

3. 3.

A $\nu$- is a $\nu$-1-supergale.

4. 4.

A $\nu$-martingale is a $\nu$-1-gale.

5. 5.

An $s$-supergale is a $\mu$-$s$-supergale, where $\mu$ is the uniform probability measure.

6. 6.

An $s$-gale is a $\mu$-$s$-gale.

7. 7.

A supermartingale is a 1-supergale.

8. 8.

A martingale is a 1-gale.

Put in another way, a martingale is a function $d:\{0,1\}^{*}\rightarrow[0,\infty)$ such that, for all $w\in\{0,1\}^{*}$, $d(w)=(d(w0)+d(w1))/2$.

Let $d$ be a $\nu$-$s$-supergale, where $\nu$ is a probability measure on $\mathbf{C}$ and $s\in[0,\infty)$. We say that $d$ succeeds on a sequence $S\in\mathbf{C}$ if

 $\limsup_{n\rightarrow\infty}d(S[0..n-1])=\infty.$

The success set of $d$ is $S^{\infty}[d]=\{S\in\mathbf{C}\bigl{|}d\text{ succeeds on }S\}$. $d$ succeeds on a language $A\subseteq\{0,1\}^{*}$ if $d$ succeeds on the characteristic sequence $\chi_{A}$ of $A$. We say that $d$ succeeds strongly on a sequence $S\in\mathbf{C}$ if

 $\liminf_{n\rightarrow\infty}d(S[0..n-1])=\infty.$

The strong success set of $d$ is $S^{\infty}_{\text{str}}[d]=\{S\in\mathbf{C}\bigl{|}d\text{ succeeds strongly % on }S\}$.

Intuitively, a supergale $d$ is a betting strategy that bets on the next bit of a sequence when the previous bits are known. $s$ is the parameter that tunes the fairness of the betting. The smaller $s$ is, the less fair the betting is. If $d$ succeeds on a sequence, then the bonus we can get from applying $d$ as the betting strategy on the sequence is unbounded. If $d$ succeeds strongly on a sequence, then the bonus goes to infinity.

 Title gale Canonical name Gale Date of creation 2013-03-22 16:43:37 Last modified on 2013-03-22 16:43:37 Owner skubeedooo (5401) Last modified by skubeedooo (5401) Numerical id 5 Author skubeedooo (5401) Entry type Definition Classification msc 60G46 Classification msc 60G44 Classification msc 60G42 Defines supergale Defines gale Defines supermartingale Defines succeed Defines succeed strongly Defines success set Defines strong success set