  Multi-structural (MS) games are combinatorial games that capture the number
of quantifiers of first-order sentences. On the face of their definition, MS
games differ from Ehrenfeucht-Fraisse (EF) games in two ways: first, MS games
are played on two sets of structures, while EF games are played on a pair of
structures; second, in MS games, Duplicator can make any number of copies of
structures. In the first part of this paper, we perform a finer analysis of MS
games and develop a closer comparison of MS games with EF games. In particular,
we point out that the use of sets of structures is of the essence and that when
MS games are played on pairs of structures, they capture Boolean combinations
of first-order sentences with a fixed number of quantifiers. After this, we
focus on another important difference between MS games and EF games, namely,
the necessity for Spoiler to play on top of a previous move in order to win
some MS games. Via an analysis of the types realized during MS games, we
delineate the expressive power of the variant of MS games in which Spoiler
never plays on top of a previous move. In the second part we focus on
simultaneously capturing number of quantifiers and number of variables in
first-order logic. We show that natural variants of the MS game do *not*
achieve this. We then introduce a new game, the quantifier-variable tree game,
and show that it simultaneously captures the number of quantifiers and number
of variables. We conclude by generalizing this game to a family of games, the
*syntactic games*, that simultaneously capture reasonable syntactic measures
and the number of variables.
