We study models of the untyped lambda calculus in the setting of game
semantics. In particular, we show that, in the category of games
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. These are the theory H*, the theory which identifies two terms
if and only if they have the same Boehm tree and the theory which
identifies all the terms which have the same Levy-Longo tree.
