Consider a game of two players: Player A and Player B. Each of them is assigned with the same number of soldiers. There is a battlefield (like a board game) with 7 tiles numbered 1 through 7 (a single line in an arithmetic order), with Player A's home-base at 'tile 1' and player B's home-base at 'tile 7'. At the start of the game, both's armies are on the middle tile 'tile 4'. A player wins the whole game if and only if her/his army reaches the other player's home-base. On any given turn, both simultaneously decide what fraction of one's total soldiers to be deployed. Whoever deploys more soldiers wins the battle and the winner's army advances by 1 tile (while the loser's army retreats by 1 tile). However, the winning army loses all soldiers but the losing army loses no soldiers. In addition, if both deploy the same number of soldiers, Player A loses that battle. What fraction of soldiers should Player A deploy at the 1st battle? (assuming fractional soldiers is possible)
My thought: A strategy that guarantees Player A to win cannot exist, because otherwise, such strategy would also exist for Player B, then contradiction. So my best guess is there exists a strategy for Player A such that she/he is guaranteed not to lose the game.