I claim that losing positions are the the tuples $(a,b,c,d)$ in which the minimum element appears three or four times, and the winning positions are those in which the minimum appears once or twice. If we have $(a,a,a,c)$ with $c>a$, by making a move we have to change one of the $a$ heaps, and thus change either to $(b,d,a,c)$, $(b,b,a,c)$, $(a,b,a,b)$ or $(a,b,a,d)$ with $b$ minimum. After any of these moves, we can get $(b,b,b,c)$, $(b,b,b,b)$, $(b,b,b,b)$ or $(b,b,b,d)$ respectively. If we have $(a,a,a,a)$, then we can change it to $(b,c,a,a)$ or $(b,b,a,a)$. After that we can go to $(b,b,b,a)$ or $(b,b,b,b)$ respectively. This proves the claim. Therefore, the first player has a winning strategy.