Imagine $n > 100$ participants in a (toy) casino. In one round each of the $n$ participants simultaneously toss a different independent coin. If a participant gets a head he/she gets a $\$1$ profit and otherwise he/she suffers a $\$1$ loss in that round. The expected profit/loss if the coins are all unbiased is $\$0$.
Now imagine that $1 \leq m \leq n/2$ players are cheats and working together. They each have independent coins with probability $1/2+\epsilon$ of getting a head. This subgroup of players has an expected positive profit from this game. The casino would like to detect this and expel the cheats.
Assuming all the players play in every round, after $r$ rounds how confident can the casino be that:
- That some people are cheating?
- They can identify which people are cheating?
Note that $\epsilon$ is the same for all the cheaters.
Half-baked thoughts The two simplest strategies for the casino are: a) The casino could just count the number of heads in all the rounds so far and see it is far $rn/2$ b) The casino could look at each individual person and count how many heads they have so far and see if it is far $r/2$.