The theme is also important in the opening sequence; not only does it establish the tone, but also the entire history and idea of repetitive histories in the series.
Let me explain:
The first thing I notice in the title sequence is the eternal image of the sun, the idea that no matter what happens on this earth and in this world, that the sun will keep turning. It represents the idea of the gods, that they are eternal, it will exist even after the events of this epic. It could bring together the whole theme of futility and tragedy to the 'players' of this game; that, in the end, it will all be for nought. The first image we can see is that the sun is comprised of rings: Rings of time, or, wheels of time.
There we glance at several other images, Aegon's Conquering of Westeros from the First Men, and then further on to Robert's rebellion against that conquest. We see that through all the castles, lands and war, that time is repetitive and has always been about one thing: Power. So the questions we then ask ourselves is, what will the next ring of the Sun's circle be?
I may also note:
Games are reference quite a bit both in the shows and the book series, and it's here we can conclude that most rulers see this bloody and inhumane struggle for power as a game. A board game, in fact:
- Robb and Stannis are constantly shown with a large board, displaying the whole of armies (as pieces) across vast and tragically complex lands. It both desensitizes them and their situation within the realities of war and life.
- In ADWD and perhaps in ASOIAF, the game cyvasse is played particularly by kings (or would be kings) and in that they learn the lessons of strategy.
There are most likely innumerable examples and connections, but these are simply a few reasons why the show-runners chose this as a display of "The Game of Thrones".