How do game and program developers solve the pinyin input problem in their programs
This question does not belong in Chinese language section, nevertheless.

What you're talking are Chinese language input tools. There is actually nothing to solve here for game developers. Any game user can use Google pinyin, Sogou, Baidu input tools in a full screen application for Chinese input. A user types Chinese words in pinyin and gets a list of corresponding words. Some games may independently implement different language input switches via os IME API:



On Linux machine a user may use scim, fcitx or ibus input tools which work in the similar manner.