Originally Posted by Giggleswick

Trenitalia offers restricted Mini fares on many trains, with discounts of 10 to 70% on the full fare. Availability is limited, they're not available on Regionale trains (the old-fashioned, nonreserved trains), the amount of the discount is smaller the closer you get to the date of travel, and they're never available less than 2 days in advance. The main restriction--other than availability--is that Mini tickets are not changeable or refundable on or after the day of travel, though they can be can be changed once or refunded before that, with a penalty (50% for cancellation, but less for changing).

The site doesn't show you Mini fares on the first screen you get after searching for trains, which is probably why you're not seeing them. In my experience, you have to select a train and then click on Continue to see what, if any, Mini fare is available on that train. You can then click on Information to see the official word on restrictions.

I hope this helps (and I hope I've got it all correct!).