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We study multi-structural games, played on two sets A and B of structures. These games generalize Ehrenfeucht-Fra\iss\'e games. Whereas Ehrenfeucht-Fra\iss\'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 with at most r quantifiers, where every structure in A satisfies and no structure in B satisfies. 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.
Fagin et al. (Wed,) studied this question.