Let $P_1(k)$, $k\in \mathbb{Z}$ be a probability distribution representing the probability of a gambler winning (if $k\geq 0$) or losing (if $k<0$) $k$ dollars in a game. The gambler starts with a fortune of $M$ and will continue playing until they either go broke (fortune reaches 0) or reach their target of $T$ dollars. $P_1(k)$ does not change, i.e. the gambler does not adapt their strategy. Denote this game by $G_1$.
Let there be a second game with same starting fortune and same target, but with a different distribution over winnings. Specifically let $k_0\in\mathbb{N}$ and let $P_2(k)$ be given by
$P_2(0)=\sum_{k=-k_0+1}^{k_0-1}P_1(k)$
$P_2(k_0)=\sum_{k\geq k_0}P_1(k)$
$P_2(-k_0)=\sum_{k\leq -k_0}P_1(k)$
$P_2(k)=0$ otherwise
Denote this game by $G_2$.
I am not interested in probabilities of ruin or reaching the target, just in the duration.
Claims:
- The duration of $G_1$ is shorter than $G_2$.
- The probability that the game is still going after $N$ steps is always larger for $G_2$ than for $G_1$.
The second claim would imply the first. I am happy to assume $T=2M$ if this makes the problem easier.
Intuitively I believe this to be true: I am basically shortening the stepsizes so I'd expect the game to last longer. However I do not know how to prove this since the solutions for generalized Gambler's ruin games (i.e. with different possible winnings) seem to be quite complicated and only numerically computable.