Got an interesting interview/game question today:
You have the opportunity to participate in an auction for a treasure chest. The seller anonymously flips 200 coins, and for each head he adds \$1 to the treasure chest, and nobody else knows how much is in this chest. Everyone bids a price simultaneously, and the person with the highest bid wins. (If more than one person chooses the same highest number, the winner is chosen randomly.) What should your optimal strategy be?
Now, I’ve seen a few auction questions on this site, but they still boggle me as to how an optimal strategy would work. I only know that:
- The expected value of the chest is \$100.
- This auction is symmetric, and everyone should have the same strategy for Nash equilibrium.
- The probability of having the winning bid is $\frac1{n}$ where $n$ is the number of people. (I imagine it as arranging $n$ numbers on a line and the probability of being assigned the first one.)
How should one piece these together? Should I be aiming to maximise profit or minimise losses? Would bidding \$50 be better than \$100, and should I even bid at all? Edit: the comments say bidding \$99 is the best bid, why is this so? Is this the case for any number of people?
Mathematically speaking is it correct to say would if I bid \$$b$, my expected value is $(100-b)/n$? Still it feels odd that the strategy is just bidding \$100 — not sure if this is too risky...
All help is greatly appreciated!