Imagine a game where each player generates $r$ random integers in a fixed range $[1, m]$, and their score $s$ is the smallest number generated. Players are sorted by $s$, and the player with the smallest number wins.
Each player can have a different $r$ value (ie. how many "rolls of the dice" they get). Obviously, the higher your $r$ value, the higher your chances of winning.
Is it possible to achieve the same result (same winning probabilities) by only generating a single random number per player and applying a scaling factor (based on $r$)?
The motivation here is to be able to apply this formula to non-integer $r$ values.
Updated to clarify: the desired outcome is a sorted list of all players scores, not just a single winner.
Assuming ties are rerolled among the tying dice: since each die is equally likely to be the smallest, this is equivalent to a lottery where each die is a ticket. Thus you can simply select the winning ticket by generating a single random number for the entire contest. This also makes the generalization from integers to reals trivial: each player's chance of winning is still their number of tickets divided by the total number of tickets in the game.
If you really do want to assign a single score to each player: The Elo rating system with a logistic curve also effectively models contests as lotteries, though usually in a 1v1 context. Here each player's ticket count is exponential in their Elo rating; equivalently, their Elo rating is logarithmic in their ticket count. Specifically, using the most common Elo scaling, a player's ticket count is
$$ r = 10^{\text{Elo rating} / 400} \approx e^{\text{Elo rating} / 173.7} $$
In this case you can have each player roll on a Gumbel distribution with the same scaling and add it to their Elo rating; highest score wins. This is known as the "Gumbel-max trick".