[UprightCreature]

… or at least I find these things fun.

Imagine a heads up coin flipping tournament where each player starts with T100. There is some number of pre-determined rounds, in this case say 2. The players sit at the same table, but wager against the house not each other. At the beginning of each round each player secretly writes down an amount of their remaining stack to wager. This amount may be any real number >= 0 and <= the players T chips. Once all wagers are recorded they are revealed and each player flips a fair coin. If the player’s coin is tails they lose their wager. If the coin is heads they win 4 to 5 on their money. At the end of all rounds the person with the most T chips wins the tournament.

You’ll note that every wager a player makes is –EV, so to maximize ones T chip EV each round would be to always wager 0. This strategy however would result in losing the tournament 75% of the time against a halfway intelligent opponent that was aware of your strategy, and is clearly not the optimal strategy.

I think the problem is more interesting with more opponents and more rounds, but we can start with 2 people and 2 rounds.

What is the optimal strategy for this tournament?