 For simplicity we disallow function and constant symbols as well as function variables in this section. Suppose \(\ma\) and \(\mb\) are two models of the same finite relational vocabulary. In the game which we denote by \(G^2_n(\ma,\mb)\) two players I and II pick one at a time subsets (or elements) of A or subsets (or elements) of B. During round i of the game player I can pick a relation \(A_i\) on A (or an element \(a_i\) of A) and then player II has to pick a relation \(B_i\) on B of the same arity as \(A_i\) (or an element \(b_i\) of on B) and vice versa: Player I can instead pick a relation \(B_i\) on B (or an element \(b_i\) of B) and then II picks a relation \(A_i\) on A of the same arity as \(B_i\) (or an element \(a_i\)) of A. After n rounds the pairs of played elements \((a_i,b_i)\) form a binary relation R on \(A\times B\). If this relation is a partial isomorphism of the structures \(\ma\) and \(\mb\) expanded by the played relations \(A_i\) and \(B_i\), i.e., it preserves atomic formulas and their negations, we say that II has won. The models \(\ma\) and \(\mb\) satisfy the same second-order sentences if and only if for each \(n\in \oN\) Player II has a winning strategy in \(G^2_n(\ma,\mb)\). A model class (i.e. a class of models of a fixed vocabulary, closed under isomorphisms) K is definable in second-order logic if and only if there is an n such that if \(\ma\in K\) and Player II has a winning strategy in the \(G^2_n(\ma,\mb)\), then \(\mb\in K\).