I recently watched a TV show where 5 contestants had to play a game. The strategies they took didn't look optimal to me. The game is as follows:
There are 100 identical looking buttons that are shared in front of the group of contestants. That is one set of 100 buttons in total. We are told five of the buttons are "bad" and if a contestant presses one of those they are out. At each turn a contestant must press between 1 and 10 buttons that have not been pressed before. They choose which buttons to press and how many. At the end of the game when all the contestants are out, the winner is the contestant who pressed the largest number of buttons.
What is the optimal strategy?
I can see that once you are the last contestant you just press buttons continuously until you are out. At all stages it doesn't matter which buttons you press as you never press a button that has been pressed before by anyone. The only question is how many you should press at each turn.
Since everything is unknown we can have four of the contestants always choose the first available button from the left while our fifth contestant will always choose from the right. Obviously nothing any of the four will decide to do is going to affect how many buttons the fifth contestant will manage to press.
There is no "optimal" strategy.