What is a fair game?

Solution 1:

$$\operatorname{E}[X_{n+1}\mid X_1,X_2,\ldots, X_n] = X_n\;, \forall n$$

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.

Solution 2:

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.