Battle is particle, so it will, without a doubt, handle heavy oil conditions, and give you more control on more difficult tournament patterns.
Odyssey is solid reactive. BTM gave it a monstrous 17.5 backend rating, which is tied for the highest ever. Should be more of a medium-heavy (closer to heavy) oil ball, weaker than the Battle in carrydown and in oil, but MUCH stronger off the dry. Recovery should be fantastic...but of course, nobody has thrown one yet.
Both have the same core. All you have to do is pick a cover. Personally, I would go with the Battle, since it is a proven winner on oily lanes. The Odyssey might struggle on a pattern where you need the ball to roll pretty early.
Hope this helps.
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I am the University of Virginia's Bowling Team. After all, one All-American is all you need.