This is an interview question:
There are 200 1 dollar coins, each with an equal probability of going into a pot and not. You bid for that pot (get the money in the pot, but you don't know how many coins are in it). The person who offers the highest bid wins the auction. What would you bid (with 1 competitor, 10 competitors)? Now if we just two bid and both players bid optimally, but you have the advantage of knowing how many of the first 10 coins go into the pot, what will be the strategies? How much will you bid and what is your expected payoff?
I feel like in the first case it's 100 regardless of how many competitors there are based on the EV, but I also feel like I need to take into account the standard deviation of $\sqrt{50}$, since the scenario follows a Binomial distribution. I'm not sure how to approach the second part besides calculating the EV as well.
At first the number of competitors isn't matter, since you compete only the $Max$ one!
If it settle down like an Auction, so you must only bid one higher than current $Max$, not $100$ at first shot.
If you look for absolutely positive $EV$, you must bid up to $99$ and no more.
When you know $n$ of first $10$ went into the pot, you must bid up to $n+95$ or $n+94$ instead of $100$ or $99$ respectively.
Some more interested question:
How you bid if you have only one blind bid and no bided money return?
How you bid if you know $10$ first shot, and no knowledge about fairness of coin?