Consider the following game: Given $K_n$ the complete graph on $n$ vertices, two players take turns coloring its edges. Initially no edges are colored. At his turn a player can color a prevoiusly not colored edge blue or red. The goal is to avoid monochromatic $K_k$. The first player who completes a monochromatic $K_k$ loses. If $n$ is large enough the Ramsey theorem ensures that there can be no draw. Therefore someone has a winning strategy. My question is that:

Is there any pair $(n,k)$ such that $n<R(k,k)$ but in the corresponding game one of the players has a winning strategy?