  We propose a new version of formula size game for modal logic. The game
characterizes the equivalence of pointed Kripke-models up to formulas of given
numbers of modal operators and binary connectives. Our game is similar to the
well-known Adler-Immerman game. However, due to a crucial difference in the
definition of positions of the game, its winning condition is simpler, and the
second player does not have a trivial optimal strategy. Thus, unlike the
Adler-Immerman game, our game is a genuine two-person game. We illustrate the
use of the game by proving a non-elementary succinctness gap between
bisimulation invariant first-order logic $\mathrm{FO}$ and (basic) modal logic
$\mathrm{ML}$. We also present a version of the game for the modal
$\mu$-calculus $\mathrm{L}_\mu$ and show that $\mathrm{FO}$ is also
non-elementarily more succinct than $\mathrm{L}_\mu$.
