2 of 4 -3 bytes and output specification

Python 3, 135 bytes

def g(s,l):n=len(s);u=[i for i in l if s not in{*l}-{i}and i[:n]==s];return len(u)<2and~(len(u[0])-n)%2or 1-min(g(i[:n+1],l)for i in u)

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A 1 indicates the current player will win, whereas a 0 indicates they will lose.