Suppose we have a game with the following: the player loses $\$1$ with probability $\alpha = 0.52$,wins $\$1$ with probability $\beta = 0.45$ and $\$2$ with probability $\gamma= 0.03$.
At each round, the player bets one dollar and that $\{X_k\}$ is the amount won/lost on the $k$-th round. I am trying to find $x \in \mathbb{R}$ such that $M_n = x^{S_n}$ is a martingale.
My strategy is the following: My definition of martingales, we have that:
$$ E[x^{X_1+X_2}|x^{X_1}] = x^{X_1} \implies E[x^{X_1}\cdot x^{X_2}|x^{X_1}] = x^{X_1} \implies x^{X_1}E[x^{X_2}|x^{X_1}] = x^{X_1} \implies E[x^{X_2}|x^{X_1}] = 1. $$
From this, is it fair to say that by law of iterated expectations that:
$E[x^{X_2}]=E[E[x^{X_2}|x^{X_1}]] = E[1] = 1 \implies E[x^{X_2}]= 1$
Hence, because each round has the same distribution, we have that $E[x^{X_2}]= x^{-1}\alpha + x^{1}\beta + x^{2}\gamma = 1$? Also, is there an implicit iid assumption here as well? thanks.