Let $(\Omega, \mathcal F, P)$ be a probability space, and $\{\mathcal F_n\}_{n=1}^\infty$ be a filtration of $\mathcal F$, that is, each $\mathcal F_i$ is a sub-$\sigma$-algebra of $\mathcal F$ with $\mathcal F_1\subseteq \mathcal F_2\subseteq \mathcal F_3\cdots$.
Given the above data, a martingale is a sequence $(X_n)_{n=1}^\infty$ of random variables on $\Omega$ such that
1) $E[|X_i|]<\infty$ for each $i$.
2) $X_n$ is $\mathcal F_n$-measurable for each $n$.
3) $E[X_n|\mathcal F_{n-1}]=X_{n-1}$ for all $n>1$.
I want to understand what is the analogy of this with gambling. Here is what I think.
The probability space $(\Omega, \mathcal, P)$ is the game that the gambler plays. For concreteness, it could be the roll of a die. The gambler chooses a number from the six faces of the die, and he wins if his chosen number appears.
The random variables $X_i$'s is the amount the gambler wins based on some strategy he has in mind. That is, the gambler has decided that in the first game he will pick $i_1$, in the second game he will pick $i_2$, and so on. This determines the $X_i$'s.
But I am not able to see the role of the filtration in the definition, and the role of condition $(3)$. Maybe my analogy is wrong, and maybe what I have considered so far is too crude to capture the glory of a martingale.
Can somebody throw some light on the formal definition of a martingale via a rich gambling analogy, and also, if possible, discuss more standard concepts like previsibility.