The star avoidance game

Let $n, k$ be positive integers. The $(k+1)$-star avoidance game on $K_n$ is played as follows. Two players take it in turn to claim a (previously unclaimed) edge of the complete graph on $n$ vertices. The first player to claim all edges of a subgraph isomorphic to a $(k+1)$-star loses. Equivalently, each player must keep all degrees in the subgraph formed by his edges at most $k$. If all edges have been chosen and neither player has lost, the game is declared a draw. We prove that, for each fixed $k$, the game is a win for the second player for all $n$ sufficiently large.