I'll quickly explain the rules of a game i'm interested in.
- The server determines a decimal number with 1.00 as lower bound and no upper bound. This number is unknown to the player.
- The player bets 1 money before beginning.
- The player sees a number on board, which starts from 1.00 and increments until it reaches the number determined by the server (could be instantly).
- The player can chose to stop and take whatever number was written on the board at the time as a winning.
- If the player doesn't stop until the counter reaches it's maximum, he loses the bet.
What algorithm could the server use to determine the number, so that the distribution is fair? As in, no matter what number the player chooses to bet all the time on, he won't win too much?
Sorry if this is a common game, I couldn't describe it well enough to find anything using search engines.
Presumably the player pays the $1$ win or lose, and if betting at point $x$ would win that amount for a net gain of $x-1$ if the player wins, so sensibly $x \ge 1$.
If the Server then turns this into a fair game for all such $x$, then it will have a random variable $S$ such that for $x \ge 1$: $$x \, \mathbb P (S \gt x) = 1$$ i.e. a cumulative distribution function $$ \mathbb P (S \le x) = 1 -\frac{1}{x}$$ and a density function $$ f_S(x)= \frac{1}{x^2}$$
This is a Pareto distribution with $\alpha=1$ and $x_{\min}=1$. Curiously it has $\mathbb E[S]=+\infty$.
The server could choose a real number $U$ uniformly on $[0,1)$ and then let $S=\dfrac{1}{1-U}$