I am trying to prove a result using forward induction and I get the following expressions for each round:
- first: $a$
- second: $a+b(1-a)$
- third: $a^2(1-b) +b^2(1-a) + ab$
- fourth: (a bit too cumbersome and I stopped to figure if I already had something concrete)
Alternatively, we can get the following (equivalent) expressions for the respective rounds:
- first: $a$
- second: $a^2 + (1-a)(b+a)$
- third: $a^2 + (1-a)(b+a)b$
These theoretical results are in line with simulations. Does it follow some traditional math form/expansion? (something like a binomial-like expansion or so...) Not a mathematician here...
Further info (if needed):
- $\frac{1}{2} \leq a \leq 1$,
- $0 \leq b \leq \frac{1}{2}$,
- $a = (1-\mu)\theta + \mu\gamma$,
- $b=(1-\mu)\theta + \mu(1-\gamma)$.
- $0 \leq \mu \leq1$, $0 \leq \theta \leq 1$ and $\frac{1}{2} < \gamma \leq 1$.