The coordination game is very simple. You see it in any interaction where being on the same side or page as someone IS the intended goal.
When you don't know what to do in the situation, you ask. You ask your players what they think about an idea and if value their input and attempt to match them, that is the Coordination Game.
What happens when both needs are different but equal strong. Then you have "the battle of the sexes". In this situation, both are truly equal that you cannot reach a resolution. Any action you take in this situation: Changes the game to a coordination game.
When you "bribe" your players to go along with something. The bribe re-arrange the pay-offs to be more like the coordination game. If your players, make it clear that they won't enjoy your intended game: they make you see options where getting along is the higher pay-off.
We encounter these most basic and elementary of games in almost all human interactions. In fact the MOST basic act of manipulation is one party trying to achieve a Coordination Game without the awareness of the other party.
Now how does this relate to your game? These two basic games teach the fundamentals of interactive strategy (in my opinion). When I learned that information and my actions have the power to change the game, I have a greater power in influencing outcomes.
It may appear simple as common sense, but if you've read up on cognitive bias you'll learn that there is no such thing as common sense. No one has ever has a "perfect" understanding of logic and is able to divorce their biases to their own judgments without employing critical thinking techniques. Game theory and its little games are one of those powerful techniques we can employ to systemically improve our decision making and, consequently, our strategic thinking.