We estimate the maximal length of interactions between strategies in HO/N
game semantics, in the spirit of the work by Schwichtenberg and Beckmann for
the length of reduction in simply typed lambdacalculus. Because of the
operational content of game semantics, the bounds presented here also apply to
head linear reduction on lambda-terms and to the execution of programs by
abstract machines (PAM/KAM), including in presence of computational effects
such as non-determinism or ground type references. The proof proceeds by
extracting from the games model a combinatorial rewriting rule on trees of
natural numbers, which can then be analyzed independently of game semantics or
lambda-calculus.