Can Multi-Armed Bandit pick the winning push by purchases instead of opens?

Yes. In the Multi-Armed Bandit journey element, the Reward section sets what counts as a win for a variant:

Push opened: a variant wins a send when the user opens that push.
Event happened: a variant wins a send when the user triggers an event after receiving that push, for example a purchase. Select the event in Reward event.

Set Reward window (hours), how long Pushwoosh waits after a send for the open or the event:

Reward	Default window	Allowed window
Push opened	2 hours	1–72 hours
Event happened	24 hours	1–168 hours (7 days)

Things to know:

With Event happened, the event is credited to the last push this element sent to the user, if it happens within that push's window. Several events within the window count as one conversion.
Opens or events after the window still appear in regular statistics, but don't count toward optimization.
Pushwoosh compares variants only after the window has passed since the first sends, so a longer window means slower learning.
In a saved element, changing the reward keeps the current window value, so check that it suits the new reward.

Multi-Armed Bandit is experimental and supports only push presets as variants (at least two).