After selling over 80 mill PSPs, Sony blew it with the Vita. Expensive memory cards, rear touch instead of L2 + R2, the name "Vita" instead of PSP 2, and last but not least, a weak game library. Both Sony's home and handheld consoles have always been sold primarily on the basis of 3rd party support (Gran Turismo is the exception, but ironically that series never released on Vita).
Sony also thought 3rd parties would (help) carry the Vita but 3rd parties were more interested in the mobile platform where they could earn at least as much money with much less effort, investment and risk. Even Nintendo, whose handheld consoles sell gangbusters, have learned that you cannot rely on 3rd parties to sell your console. At best, they'll arrive later when there's a guarantee for profits. Therefore, they made Switch to be a hybrid so they could combine the home and handheld libraries and support the Switch themselves.
With 3DS and Switch, Nintendo provides all of the best-selling games themselves with only Monster Hunter as the exception. Sony would also have to be able to provide the content themselves and the only way I see them capable of that is if they make a hybrid or a handheld that can run all PS4 games in downscaled form.
However, a "PSP 3" would still need an identity and exclusive games, and for this I think Sony would be wise to buy a smaller developer such as Sanzaru Games (Sly Cooper: Thieves in Time, Secret Agent Clank) or Level- 5 (Dark Cloud, Rogue Galaxy, Yo-Kai Watch, Ni no Kuni, Professor Layton, Inzuma Eleven). These games are not AAA games, but they fit the handheld format so well. It would also need an exclusive Gran Turismo, God of War and Daxter, and Sony funded third party games such as a smaller GTA, Fallout, Minecraft, Metal Gear Solid and Final Fantasy.
I've enjoyed my Vita a lot. Especially on long distance travels. Therefore, I might be interested in a "PSP 3" if it supported SD cards, had L2 + R2 and ran the entire PS4 library in downscaled form. Though I doubt they'll release a new handheld. I think they're more likely to introduce a new kind of controller for phones as an alternative to touch and tilt. Soon, phones are capable of running downscaled PS4 games and there's a lot of potential for Sony to grow in the mobile market and increase the number of subscribers of services such as PS Now and PS Vue.
