### Abstract

We study models of the untyped lambda calculus in the setting of game
semantics. In particular, we show that, in the category of
games *G*, introduced by Abramsky, Jagadeesan and Malacaria, all
lambda-models can be partitioned in three disjoint classes, and
each
model in a class induces the same theory, *i.e.* the set of
equations between terms, that is: the theory H*, the theory which
identifies all the terms which have the same Boehm tree and the
theory
which identifies all the terms which have the same Levy-Longo tree.