  An online Ramsey game is a game between Builder and Painter, alternating in
turns. They are given a graph $H$ and a graph $G$ of an infinite set of
independent vertices. In each round Builder draws an edge and Painter colors it
either red or blue. Builder wins if after some finite round there is a
monochromatic copy of the graph $H$, otherwise Painter wins. The online Ramsey
number $\widetilde{r}(H)$ is the minimum number of rounds such that Builder can
force a monochromatic copy of $H$ in $G$. This is an analogy to the size-Ramsey
number $\overline{r}(H)$ defined as the minimum number such that there exists
graph $G$ with $\overline{r}(H)$ edges where for any edge two-coloring $G$
contains a monochromatic copy of $H$.
  In this paper, we introduce the concept of induced online Ramsey numbers: the
induced online Ramsey number $\widetilde{r}_{ind}(H)$ is the minimum number of
rounds Builder can force an induced monochromatic copy of $H$ in $G$. We prove
asymptotically tight bounds on the induced online Ramsey numbers of paths,
cycles and two families of trees. Moreover, we provide a result analogous to
Conlon [On-line Ramsey Numbers, SIAM J. Discr. Math. 2009], showing that there
is an infinite family of trees $T_1,T_2,\dots$, $|T_i|<|T_{i+1}|$ for $i\ge1$,
such that \[
  \lim_{i\to\infty} \frac{\widetilde{r}(T_i)}{\overline{r}(T_i)} = 0. \]
