Games are a big part of our lifestyle and permeate our home. My husband works in the game industry, which took a big hit this last year, so he is currently unemployed but games are how we make a living and our household motto is: "The family that plays together, stays together."
We play age appropriate games with our four year old daughter. Her favorite game right now is Lego Star Wars. It can be played by two people who work as a team to defeat obstacles. We'll play for forty minutes to an hour and then stop at a save point to continue another day.
She also does not like to stop when it's time and will unfurl her lower lip in a bid to get more time. Sometimes we give it and other times not. Being completely ridged about it isn't any fun. If she pitches a fit, which has only happened a few times. She is marched straight up to bed for a nap. When she throws fits like that, it's completely obvious she's over tired and she usually passes out in her bed shortly after her head hits the pillow.
It's good to have passion for something, even if it's the dreaded "video games", but make sure to teach him to temper his passion and divert him to other things that are also enjoyable. Whip out some play doh and make creatures to hop around and collect hidden marbles you hide under magazines on the coffee table. Find out what game is his favorite and print out some free coloring pages that are found all over the internet of his favorite game characters then stick them up on the fridge.
Pull out a board game and show him that not all games are on the TV and he can have fun with you. I know my kid will dump her controller faster than anything if I pull out chutes and ladders to play with her. She's still the reigning champ in the house as the spinner seems to favor her for every game. I almost won once and she looked like she was prepared for devastation but uh oh... mommy hit a big ladder for pulling a cats tail and down down down she went. Lucky little stinker...
Now my only problem is trying to teach her to be a good sport when she loses. We play so many team games that we are always the victors! And when we play board games for her age, she's just stinking lucky and always wins...
Maybe I should take her to Vegas and rub the top of her head for good luck...
Nah.
Anyhow, as for arguing with your three year old... don't do it. You are the law in your home and if he argues with you just tell him that this is the way it has to be. Turn off the game, take the controller and if he won't be reasonable. Stick him in his bed for a nap letting him know that you understand he's angry but playing video-games for more than x minutes or hours (whatever you decide), isn't good and he needs to find something else to do now.
Do not negotiate with three year old holy terrors. I give my daughter three options if she starts giving me grief. I tell her, "You can color a printout, you can play with (whatever) or you can take a nap. You pick, if you do not pick you will be taking a nap."
If he can't be reasonable about it at all and his bad behavior continues despite knowing he's going to end up taking a lot of naps, put the video games up until he's four. We didn't really start letting our daughter play on the actual game systems until she was a bit over four years. At three the closest she got was her leapfrog system or helping daddy push the a button to jump when he was playing something. She had a lot less control over herself at three.