Suppose a game where you can bet an amount $a$ on a "signum" vector $v$ of length $n$ where the elements $v_i$ of the vector $v$ are drawn from $\{-1,1\}$.
On each round you try to predict the next vector.
You can bet an amount $a_i$(which can also be negative) on each element $v_i$; for example, if the next vector $v$ is $$ \begin{matrix} -1 \\ 1 \\ 1 \\ \end{matrix} $$ and you have distributed $$\sum_{i=0}^n |a_i|=a$$ like $$ \begin{matrix} -0.3 \\ 0.5 \\ -0.2 \\ \end{matrix} $$ then your reward $r$ would be $$\sum_{i=0}^n v_i\times a_i = -1\times(-0.3)+1\times 0.5+1\times(-0.2)=0.3+0.5-0.2=0.6$$
Suppose this game is not pure random, which means that some vectors will never occur in $V^n$.
Let denote this set of vectors which will not occur with $N$ and the vectors which do occur as $Y$.
Suppose further that all vectors of $Y$ occur with same probability.
Is there an optimal betting strategy (distribution of $a$ on $v$ at each round) for this game where the expected reward $r$ is $>0$ in the long run?