Suppose $X_n$ is the fortune of a gambler after $n$ th game. Then the game is called fair (Breiman 1968) if
$$E[X_{n+1} \mid X_1, \dots, X_n] = X_n \forall n$$
My question is why a fair game is not defined as the following
$$E[X_{n+1}] = E[X_n] \forall n$$ i.e. $$E[X_{n+1}- X_n]=0$$. This should be the proper definition as a fair game is where avg. gain is zero. Nothing conditioning should be there.
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A definition of a fair game is one where the expectation value is zero, so people who like risk would always play it.
Playing heads or tails with a fair coin with a friend under the rules that the cost for the game is 1 dollar and the winner gets $2$ dollars. Lottery, for example isn't a fair game, since the probability you win multiplied by the prize is smaller than the cost of the ticket.
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I guess the reason lies how to compare two dependent (in random sense) entities. Suppose today's fortune is $Y$ and tomorrow's is $X$. Then a fair game should be $E[X|Y ] = Y$ a.e. i.e. we fix a $Y=y$ and then calculate the expectation of $X$. So, we remove the randomness of $Y$ in this way. In some sense, it is a way to make both the fortunes independent.
If we take the unconditional average then tomorrow's expected fortune will contain the randomness of today's. In some sense, this is not a fair comparison.
This argument is not much technical. Please comment so that I can evaluate my understanding.
$$\operatorname{E}[X_{n+1}\mid X_1,X_2,\ldots, X_n] = X_n\;, \forall n$$
That's an ... interesting definition. It says the game's odds can be adjusted given knowledge of the gambler's past fortune, but will be considered fair if that adjustment always means that the gambler's fortune can be expected not to change after each game (whatever it may be at the time).
Of course, since $\operatorname{E}[Y] = \operatorname{E}\!\left[\operatorname{E}[Y\mid X]\right]$ then the familiar definition follows:
$$\begin{align}\operatorname{E}[X_{n+1}] & = \operatorname{E}\left[\operatorname{E}[X_{n+1}\mid X_1,X_2,\ldots, X_n]\right] \\ & = \operatorname{E}[X_{n}]\;, \forall n \end{align}$$
Which is an equivalent, but somewhat weaker, statement that a game is fair if the gambler's expected fortune does not depend on how many games are played.