There are $25$ coins with values $1,2,\dots,25$. Two persons, $A$ and $B$, play the following game with these coins. Person $A$ chooses a coin and person $B$ decides whether to keep the coin for himself or to hand over the coin to him. Each player with the most coins must choose the next coin and the other player must decide whether to keep the coin or give it to another player. If the number of coins is equal, the previous state is repeated. The game continues in the same way until all the coins are selected. At the end, the player with the most valuable coins wins the game. Which player has a winning strategy?
I think $B$ has the winning strategy, but I can't prove it. I will be grateful if someone helps me to solve this problem.