The problems of contextual equivalence and approximation are studied for the third-order fragment of Idealized Algol with iteration (IA ^*₃^*_{3}). They are approached via a combination of game semantics and language theory. It is shown that for each (IA ^*₃^{*}_{3})-term one can construct a pushdown automaton recognizing a representation of the strategy induced by the term. The automata have some additional properties ensuring that the associated equivalence and inclusion problems are solvable in Ptime. This gives an Exptime decision procedure for contextual equivalence and approximation for β-normal terms. Exptime-hardness is also shown in this case, even in the absence of iteration.