Under Arriva Trains Wales, it used to be that Chester ticket machines offered the Saveway and there was a notice stuck on the machines saying 'travelling off peak, buy an all day rover, just search for NorthWirralLines' or something like that and it would print and while it didn't say Saveaway, it was recognised generally by bus drivers that 'northwirrallines' meant saveaway.
Sadly I couldn't find it on the new machines last time I checked, and they also don't sell the Merseyrail only day ticket, so you have to end up getting the more expensive tickets for no good reason, or use the ticket office.
Merseyrail and TFW working together to shaft passengers and stop them accessing the best value fares.