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.

翻译：多结构（MS）博弈是捕捉一阶语句量词数量的组合博弈。从其定义表面来看，MS博弈与Ehrenfeucht-Fraisse（EF）博弈在两方面存在差异：首先，MS博弈在两个结构集合上进行，而EF博弈在成对结构上进行；其次，在MS博弈中，复制者可以制作任意数量的结构副本。本文第一部分对MS博弈进行更精细的分析，并与EF博弈展开更深入的比较。特别指出，使用结构集合具有本质意义，当MS博弈在成对结构上进行时，它们捕捉的是具有固定量词数量的一阶语句的布尔组合。此后，我们聚焦于MS博弈与EF博弈的另一个重要差异，即欺骗者为赢得某些MS博弈必须在先前移动基础上进行操作的必然性。通过对MS博弈过程中实现类型的分析，我们界定了欺骗者永不基于先前移动的MS博弈变体的表达能力。第二部分重点研究同时捕捉一阶逻辑中量词数量和变量数量的问题。我们证明MS博弈的自然变体*无法*实现这一目标。随后引入一种新博弈——量词-变量树博弈，并证明其能同时捕捉量词数量和变量数量。最后通过将该博弈推广至*语法博弈*家族进行总结，该家族博弈能同时捕捉合理的语法度量与变量数量。