An example of an entirely cyclic game (from the Transitive cyclic decomposition paper) is the disk game.

Let W={x∈R2:∥x∥22≤k}W = \{x \in \mathbb{R}^2 : \|x\|_ 2^2 \le k \}

\phi(v,w) = v^T \cdot \left( \begin{smallmatrix} 0&-1\\ 1&0 \end{smallmatrix} \right0)\cdot w

Note that Rock Paper Scissors embeds in the disk game:

disk_game

Here is another time Deepmind shows off this game in their blog.

blog_disk_game