Suppose there are two betting games $G_1$ and $G_2$, the outcomes of which are entirely independent of player input.
The expected return for a single iteration of $G_1$ is $r_1$ and for $G_2$ is $r_2$, these returns being a percentage of your bet.
Design a game $G$ where you start by choosing either $G_1$ or $G_2$, and you have a chance of getting to play again, and you again choose between the two games.
Suppose if you choose $G_1$ the chance of playing again is $p_1$ and for $G_2$ is $p_2$.
If you take the strategy of always choosing $G_1$, then the overall expected return of $G$ is $$\frac{r_1}{1-p_1}.$$
If we assume that $$\frac{r_1}{1-p_1} = \frac{r_2}{1-p_2},$$
Is this enough to conclude that $G$ has no optimal strategy and that the overall return is the same regardless of your choice at each stage?
If not, what conditions can we place on $r_1,r_2,p_1,p_2$ to ensure that $G$ has no optimal strategy?