Online Ramsey numbers: Long versus short cycles

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.