 The first order version \(G^1_n(\ma,\mb)\), where players play only individual elements, i.e., no relations are played, is very useful in working with first order logic. Unfortunately the game \(G^2_n(\ma,\mb)\) is much more complex. It is very difficult to invent winning strategies for Player II except in the trivial case when \(\ma\cong\mb\). If Player I plays a binary relation R on A, Player II should be able to play a binary relation \(R'\) on B such that whatever challenges Player I makes in the rest of the game, even involving new binary relations, the relations R and \(R'\) look similar. If \(V=L\), then countable second-order equivalent models (with a finite vocabulary) are actually isomorphic (Ajtai 1979). For details, see  §10.  Thus consistently in countable models there is no other winning strategy for Player II than an isomorphism. Perhaps this explains why the game is not as useful in second-order logic as it is in first order logic. However, if we restrict to monadic second-order logic, which in terms of the Ehrenfeucht-Fraïssé game means restricting the game to unary predicates, the situation changes. A unary predicate just divides the model into two parts. If Player I divides A into two parts, Player II should find a similar division in B. This is more reasonable and there actually are useful strategies for Player II. For examples in the finite context see  §16.