 If \(A\) and \(B\) are back-and-forth equivalent, then certainly they are elementarily equivalent; but in fact back-and-forth equivalence turns out to be the same as elementary equivalence in an  infinitary logic  which is much more expressive than first-order logic. There are many adjustments of the game that give other kinds of equivalence. For example Barwise, Immerman and Bruno Poizat independently described a game in which the two players have exactly \(p\) numbered pebbles each; each player has to label his or her choices with a pebble, and the two choices in the same step must be labelled with pebbles carrying the same number. As the game proceeds, the players will run out of pebbles and so they will have to re-use pebbles that were already used. The condition for Spoiler to win at a position (and all subsequent positions) is the same as before, except that only the elements carrying labels at that position count. The existence of a winning strategy for Duplicator in this game means that the two structures agree for sentences which use at most \(p\) variables (allowing these variables to occur any number of times).