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? (Since Ramsey type questions tend to be very hard, even non trivial heuristics are appreciated!)

Additional infrmation: For $k=3, n=6$ (it is different from the game called Sim!) we played it for a few hours, and found that the second player wins a lot more than the first. After that we found out that the very similar game Sim is solved by computer and the second player has a winning strategy (not presented in the paper): http://arxiv.org/pdf/cs/9911004v1.pdf
When playing with $k=3, n=5$ i think that there is no winning strategy for anyone, but i can't prove it yet. I think that this must be present somewhere in the literature. Interesting observations are: The maximal red subgraphs without a triangle are: a star, a cycle of length four plus a vertex adjacent to two nonadjacent vertices in the 4-cycle, and a cycle of length $5$. (Their maximal degrees are 4,3,2 respectively) I think that given enough time we could prove that the first player who makes a vertex with $3$ monochromatic edges loses. (Maybe with maintaining that the red and blue subgraphs are isomorphic.)