This question is inspired by one of the classic ways of breaking ties: the "I'm thinking of a number" game. In this game, one person thinks of a number in some range, say from $0$ to $100$ (inclusive), while the others go around guessing numbers in the range (one number per person). Whoever is closest to the chosen number wins.
If there are $p$ total people, including you, and you have to guess first what is the optimal strategy? You can assume that everyone else will first and foremost be trying to maximize their odds of winning, but if two numbers give equal chances, they will play whatever is worst for you.
For $p = 1$, the strategy doesn't matter: every number will guarantee you a win.
For $p = 2$, it's best to guess $50$. That way, in the worst case scenario, you still have a $\frac{50}{101}$ chance of winning.
However, for $p = 3$, the game becomes quite a bit more difficult to analyze. For instance, choosing $50$ (which initially seems like the best bet) will leave you with only a $\frac{1}{101}$ chance of success. The next obvious choice is $49$ or $51$. Since they're equivalent, let's say you choose $49$, in which case the second player will choose $52$, leaving $48$ for player $3$. Congrats -- you've upped your odds to $\frac{2}{101}$! However, you can do better; my conjecture is that $32$ is optimal, but I have been unable to verify it. A calc-bash strategy may be the best way of solving this, but I would certainly hope for something more elegant.
Once we move on to $p \ge 4$, though, things become very chaotic indeed. I suspect there might be something that can be done with induction, but I haven't made any progress.
Note: It's not actually that hard to combo-bash some of these, so consider the interval from $0$ to $n$ for arbitrary $n$. It might be even more interesting (and general) to explore choosing real numbers, rather than integers, in $[0,1]$, but that will inevitably require some notion of smallest positive number, which the reals don't have.