Cuban Playing Cards

There are two decks of cards, one with $n$ playing cards and the other with $m$ playing cards. Players $A$ and $B$ play alternately, performing one of the following operations in each turn:

(i) Take out one card from one deck.
(ii) Take out one card from each deck.
(iii) Move one card from one deck to the other.

Player $A$ begins the game, and the player that takes the last card wins.

Is there a winning strategy as a function of $m$ and $n$ such that one of the players following these rules can always win?