I purchased a PSP in 2014 and still own it till date. I have played a ton of games on it, and now, finally, I conclude that you should not buy a PSP in 2018( or late 2017) .
1) The PSP is the oldest handheld console. The strongest part of a gaming console is the variety of games it offers. This is an area where the PSP lags. The last game to be made for PSP was in 2012. No new games have been released since then. Thus, the number of games you can play is very less. Only around 300 titles have been released.
2) This makes me come to my second point. Because this an old device, the games released are also old. Thus, the graphics are bad compared to this era. The PSP costs around Rupees 10000 and a similar priced phone like the Moto G5 will give you much higher graphics. Also, the number of games you can play on a phone from PlayStore is massive. Moreover, many of the good titles are absolutely free and those which do cost money, are comparitively very cheap compared to PSP games.
3) The PSP screen is only 640x480. This is a very outdated resolution and trust me, you are actually able to see individual pixels on the screen.
4) When you buy a 10000 rupee PSP, you buy an outdated machine, which has a bad display, very slow processor, and cannot do much apart from gaming, and that too, the games are expensive and less in number. A similar priced smartphone will not only provide you with much better features, but will also give you a large variety of games.
Some other drawbacks I would state are :
1) PSP is only suited for AAA games like God of War or Call of Duty. Playing racing games with its controls is actually frustrating. Playing racing games on a phone is much better. Plus there are some really good racing titles like No Limits, Real Racing and Asphalt series on android.
2) The PSP has no internal storage. If you want some, you have to buy a PSP exclusive card which costs a lot of money. A smartphone comes with 16-32gb storage at starting price.
My solution and best alternative to a PSP is an android tablet. The Mi Pad is a good one. Some from Lenovo are great too. You can also go for a second hand iPad. Not only will you be able to play a wide variety of high graphic games but also do other stuff such as listen to music, watch movies and browse the web.