Playing games can discharge the battery very quickly.
The Kindle works on the approach that it only has to do something every time you press a button, and when you are reading that's a page change button only every 30 seconds or minute or so. The rest of the time it's almost asleep.
This is helped of course by the e-ink screen that doesn't use power except when changing.
If you play a game you are typically pressing buttons much more often, and the screen is having to refresh much more often. This eats up power.
If the game is web based (as some are), the wireless is using up power as well.
I'd suggest you try using it for reading only for a few days with the wireless off, and see how it gets on. If it still needs charging every other day it could well be faulty - if the battery life improves then that's the explanation, and you'll either have to play less games or live with charging it regularly.