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

1. A $\nu$ -$s$ -supergale is a function $d:\lbrace 0,1\rbrace ^{*}\rightarrow [0,\infty)$ that satisfies the condition \begin{equation} d(w)\nu(w)^{s}\geq d(w0)\nu(w0)^{s}+d(w1)\nu(w1)^{s} \end{equation}for all $w\in\lbrace 0,1\rbrace^{*}$ , the set of all finite strings of $0$ 's and $1$ 's (including $e$ , the empty string).
2. A $\nu$ -$s$ -gale is a $\nu$ -$s$ -supergale that satisfies the condition with equality for all $w\in\lbrace 0,1\rbrace^{*}$ .
3. A $\nu$ -supermartingale is a $\nu$ -1-supergale.
4. A $\nu$ -martingale is a $\nu$ -1-gale.
5. An $s$ -supergale is a $\mu$ -$s$ -supergale, where $\mu$ is the uniform probability measure.
6. An $s$ -gale is a $\mu$ -$s$ -gale.
7. A supermartingale is a 1-supergale.
8. A martingale is a 1-gale.

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

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 $S\in\mathbf{C}$ i $$ \limsup_{n\rightarrow \infty} d(S[0..n-1])=\infty $$ The success set of $d$ is $S^{\infty}[d]=\lbrace S\in\mathbf{C}\bigl | d{ succeeds on }S\rbrace$ . $d$ succeeds on a language $A\subseteq \lbrace 0,1 \rbrace^{*}$ if $d$ succeeds on the characteristic sequence $\chi_A$ of $A$ . We say that $d$ succeeds strongly on a sequence $S\in\mathbf{C}$ i $$ \liminf_{n\rightarrow\infty}d(S[0..n-1])=\infty $$ The strong success set of $d$ is $S^{\infty}_{{str}}[d]=\lbrace S\in\mathbf{C}\bigl | d{ succeeds strongly on }S\rbrace$ .

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.