When you're working on a problem having a short break and thinking about something unrelated allows you to return to the problem with a fresh mind. People often report this kind of short break is a powerful tool in problem solving. This short break could easily be a game of something.

When you're working in a company sometimes people are in different silos. Opportunities to discuss things across the company are lacking. But these discussions are often useful; a problem in one department might be a simple fix from another department. People hate meetings. People like games. A games afternoon once a fortnight would help people discuss this stuff.

You could turn outages into a gane. Have someone who knows the systems design the root cause scenario and pick responses to your actions.