I do play games, but not because of your reasons. Because those calls are usually boring as hell and so I am able to pay more attention to them if I'm also doing something else less soul-crushing while listening to it. It has to be some kind of game that you can play on automatic mode, no thinking required. Any game that makes you think with more than 10% of your brain will detract you from the meeting.
Things that work for me:
- Knetwalk
- Mine Sweeper
- Teg (Risk) (sudo apt-get install teg)
- Pindleskin/Meph/Kurast farming in Diablo 2
- Completing easy mindless daily quests in mobile games
- Solitarie
- Trimming my fingernails
- Passing Clorox all over my home office surfaces