What do you think will sell more lifetime?
Playstation Portable has currently sold ≈81M Global and won't sell anymore.
Xbox One has currently sold ≈11.5M Global and has still at least 5 years to sell. 

My guess:
I think it will be a very hard fight where Playstation Portable will win.
I can see Xbox One doing:
40M in North America (X360 did ≈48M)
25M in Europe (X360 did ≈26M)
10M in Rest of the World (X360 did 9M)
0.3M in Japan (X360 did 1.7M)
This means the Xbox One will do ≈75.3M Global and Playstation Portable will be bigger!

PSP is my guess.

XB1 has a good amount of time to continue to right the ship and get a solid userbase. Microsoft has also shown they are willing to do whatever it takes to get consoles into the hands of gamers. All that said though, breaking the 80 mil plateau could be difficult for them. I will say they edge out the PSP though.

My money is on the PSP. That machine managed to compete with the best selling handheld in the world. I don't think the XBone can put up that kind of fight, specially if most games are going to be multiplat and looking better on the PS4.

Also, the PSP might just have the most well balanced console in terms of sales: It sold around 20 million in every region, while the XBone is only going to shine in North America.

At this rate the X1 can catch the 360. However they need to announce some big exclusives to gain and keep momentum. The Order scores open a chance for the X1 to steal some thunder, but that performance gap will be it's biggest enemy.

Gonna go with X1 cause I feel MS will keep pushing even if it loses money for brand recognition and set up for next gen.

The 360 barely outsold the PSP. I don't think Xbone stands a chance.

Why would you think that X1 will sell as much in Europe and ROW as 360 when these territories are even more dominated by Playstation now?

What? I am saying that Xbox One will do worse than X360 in Europe.
In Rest of the World Xbox One will win thanks China (Xbox One can do there 1-2M imo).

How is the X1 tracking in EU and RotW compared to 360 over same time frame?

I think Xbox one can do better number than X360, The consoles is a success and selling well, they also don't have to worry much about competition from Nintendo, no lost sales to the Wii anymore. Xbox One is on pace to outsell there 2014 numbers and at this rate Xbox One will surpass the Xbox originial and have over 25M+ sales by the end of the year. Halo 5 will see to it.

http://www.vgchartz.com/article/253161/xbox-one-vs-xbox-360vgchartz-gap-chartsjanuary-2015-update/