How can I determine whether two decks are equally strong?
Playtesting is by far the best option (and how all major (and minor) trading card games) have been designed. Do not only pitch the two decks against each other, but also against other decks.

Other than that, if the rules are simple enough, you can do some mathematical analysis. E.g. for Magic: The Gathering, compare the mana curves of the decks, and if both are Aggro decks, you can compare the average Power/Toughness of the creatures. But given Magic's rich ruleset, that is bound to fail. Still, it might work for simpler examples, such as when comparing two non-standard chess armies by relative piece value.