I don't know if you can plan in advance or not. From my experience I never find a significant difference in prices between the alliances. That is if you're flexible on the routing. Eg. Direct flights with OW can be more expensive than indirect flights on other alliances.

I usually start planning 4 to 6 months out (I know this is not possible for business travel but as your question is regarding pricing I think it's regarding leisure trips). I compare rates to find a reasonable price and then I start monitoring OW rates to reach this price level.
Over the last years I never had to pay more for a OW flight.

I'm based in EU and my experience is mainly with AA, BA, AB and AY.