The following is an extension of 100 Dwarves and a tiny room. The text of this puzzles spoils that one in a minor way, so go read that excellent puzzle first!
An evil elf puts $100$ dwarves in a room and gives each a shirt with a number from $0$ to $99$ (repetitions allowed). The dwarves can see everyone's number except their own, but otherwise can't communicate in any way. Their goal is to all simultaneously shout out the same number, and for this shouted number to appear on the shirt of at least one dwarf. Thus, they lose if more than one number is shouted at that instant, or if the single number shouted is not on any dwarf's shirt.
It is up to you to decide their strategy: in particular, you will give instructions to each dwarf, and the dwarf will follow your instructions exactly. However, the evil elf will also read these instructions, and assign shirts to minimize the probability of the dwarves' success.
What strategy would you give them, and what is the probability they win?
A few notes:
In the room there is a large screen, which all the dwarves can see, that can display whatever random number(s) your instructions require. These random number(s) are chosen after the shirts are distributed. This is to help the dwarves implement some sort of strategy involving randomness.
The only major difference between this puzzle and the original is that I'm asking not about whether a 100% win rate is possible, but how high you can make the win percentage.
For our purposes, if a dwarf has a shirt with a 29 on it, this doesn't match a shouted answer of "9" or "2".
I don't know the answer; I'll accept the answer with highest probability after a reasonable amount of time. I do know it is possible to do better than 1/100.