  We study multi-structural games, played on two sets $\mathcal{A}$ and
$\mathcal{B}$ of structures. These games generalize Ehrenfeucht-Fra\"{i}ss\'{e}
games. Whereas Ehrenfeucht-Fra\"{i}ss\'{e} games capture the quantifier rank of
a first-order sentence, multi-structural games capture the number of
quantifiers, in the sense that Spoiler wins the $r$-round game if and only if
there is a first-order sentence $\phi$ with at most $r$ quantifiers, where
every structure in $\mathcal{A}$ satisfies $\phi$ and no structure in
$\mathcal{B}$ satisfies $\phi$. We use these games to give a complete
characterization of the number of quantifiers required to distinguish linear
orders of different sizes, and develop machinery for analyzing structures
beyond linear orders.
