 The Ehrenfeucht-Fraïssé game is a game-theoretic tool for investigating to what extent two models are similar (see entry on  logic and games  for a general introduction to Ehrenfeucht-Fraïssé games). Two isomorphic models would be very similar but normally we are interested in similarity of models that are not isomorphic. The Ehrenfeucht-Fraïssé game of second-order logic characterizes \(\ma\equiv_{L^2}\mn\), i.e. the proposition that exactly the same second-order sentences are true in \(\ma\) and \(\mb\), just as the Ehrenfeucht-Fraïssé game of first order logic characterizes the first order elementary equivalence \(\ma\equiv\mn\). See the entry on  first-order model theory  for more on elementary equivalence.