  Online Ramsey game is played between Builder and Painter on an infinite board
$K_{\mathbb N}$. In every round Builder selects an edge, then Painter colors it
red or blue. Both know target graphs $H_1$ and $H_2$. Builder aims to create
either a red copy of $H_1$ or a blue copy of $H_2$ in $K_{\mathbb N}$ as soon
as possible, and Painter tries to prevent it. The online Ramsey number
$\tilde{r}(H_1,H_2)$ is the minimum number of rounds such that the Builder
wins. We study $\tilde{r}(C_k,C_n)$ where $k$ is fixed and $n$ is large. We
show that $\tilde{r}(C_k,C_n)=2n+\mathcal O(k)$ for an absolute constant $c$ if
$k$ is even, while $\tilde{r}(C_k,C_n)\le 3n+o(n)$ if $k$ is odd.
