I think the PS3 has its own internet or something.  When I search for a device, the PS3 always shows up as an option.  I own a PSP and a Vita and no matter how I connect to the PS3, whether internet or directly to the console, there's some noticeable lag.  Maybe the PS4 is different.  One thing you'll agree with me on is despite the short range, the Wii U is virtually lag free.  There's videos of people actually using the Wii U to test the lag on their HDTV's because the Wii U controller is so fast.

I'll get a PS4 soon.  Maybe tomorrow if I'm not too sleepy and they're actually in stock.  Then, I can find out for myself.

Just thought I might make a recommendation in case you have never had the Wii before.
http://www.gamestop.com/wii/games/xenoblade-chronicles-gamestop-exclusive/98535
Get this game. $60.00 for this game is a deal you cannot pass up. This game may not have the best graphics, but holy crap...I cannot get enough of this game. I'm constantly craving the game even after I beat it...

Also, it is the best RPG of last generation. Period.

Also, I can recommend the Pro Controller to you. It will probably match someone at an FPS game, and the battery life is 80 hours.

I never understood the whole inferiority deal people have with the GamePad, as it's basically a regular dual analog controller plus a touchscreen in the middle. It's not like it's the Wiimote or anything, which did lack a 2nd stick plus a lot of buttons.

I love my WiiU and PS4, one complement the other very well both are excellent systems.

Pick up a pro controller, that thing lasts 80 hours, it's insane.

The signal is sent via a wireless router in PS4/Vita remote play. It can have a very wide range (dependend on the router), but it is never lag free.

The Wii U sends the signal directly to the GamePad and it has almost no lag (less than usual televisions), but the range is lower. It isn't designed to be used it in the entire house. It works with a very high bandwidth, the video stream that is sent to the GamePad has up to 40 Mbps and the audio is uncompressed. I guess that virtually lag free transmission over a wide range wasn't possible with affordable tech of 2012.

The chip that handles the communication with the GamePad is upgradeable, therefore we might see some small improvements in the future. I wouldn't expect much.

He is.

But to be fair to the GamePad it receives signals that are up to 40 MB per second, and uncompressed sound. So I think it's Battery Life is fair, plus the GaneOad controller can get its own Firmware Updates for Battery Life increase. Expect maybe no more then an additional 1 to 2 hours after 3 years on the market, but like the 3DS it will go a long way for such a small increase in Battery Life.