The problem every admits - people search on one device and may very well likely end up converting on a different device. Be it they are searching for a new pair of shoes while on the bus ride home from work and then end up purchasing it when they get home from their desktop.
This is fairly common, in fact, Google shared metrics on how common it is by vertical and what the estimated conversions are by vertical on the AdWords blog. Google said, "A September 2013 study of multi-device consumers found that over 90% move sequentially between several screens for everyday activities like booking a hotel or shopping for electronics."
Google launched, estimated cross-device conversions to help online marketers track and predict conversions.
Here is a graphic:
For more on how this works, see this page.
Forum discussion at Google+.