The real purpose behind enclosing a speaker is so that you have the proper air volume on the backside of the cone to achieve the optimal response from the speaker based on it's Vas, Fs, and Qts specs.
By enclosing a speaker, what you are really doing is increasing it's Q value. So the smaller the enclosure, the higher the Q. If the enclosure is too small (and the Q too high), you lose considerable low-bass response and get a boomy mid-bass. As you increase the volume of the enclosure, you extend the low bass, but at a lower SPL compared to the midrange. Optimally, you shoot for a Q of around .7 for optimum bass response. But unless you know the Qts of your speaker and build the correct size enclosure to achieve a Q of .7, you are just shooting in the dark.
For most arcade applications a free-air woofer will work just fine. But if you are an audiophile geek and have the specs and proper charts that dictate the volume of the enclosure for your speaker, then you can achieve much better response. Even then, you are still dealing mostly with beeps, voices, explosions, etc on the video games, and the river can't rise above it's source...