Game of Thrones

Which position in Westeros would I want to hold?

Prove every bracket of 2^n entries is good, where n is a positive integer, while not every bracket outside 2^n entries is good.

Suppose n is a positive integer and a bracket has 2^n entries. Since (2^n)/2 = 2^(n-1), which is still an integer (I don't want to prove it), 2^n is even and every game in the first round has two participants i.e. is a good game. If we enforce one winner per game (coin flip tiebreakers), the next round contains (2^n)/2 = 2^(n-1) entries. 2^(n-1) will be even by the same proof as 2^n (I don't want to show it), so the next round is good as well. This will progress as n steps down to 1, the lowest positive integer, in which case 2^n = 2, which I already said was a good game, and one with one winner, who wins the bracket cleanly.

Now suppose a bracket has 3 entries. 3/2 is not an integer, so it can't be in 2^n (I don't want to prove it). By axiom, a game can only involve 2 entries, so there's a remainder. The same can be said of brackets of any odd positive integer (I don't want to prove it). If there's a remainder, and a game can't be formulated, the bracket is bad. So not every bracket outside 2^n entries is good. [square]

Seeds:

1. King of Westeros
2. Hand of Robert
3. Hand of Dany
4. Hand of Bran
5. King in the North
6. Lord of some house+castle
7. Lord Commander of the Nights Watch
8. Younger brother of some ruler
9. Knight
10. Maester
11. Master of Coin
12. Lead Ranger
13. Septon
14. Brother of the Brotherhood
15. Common bannerman
16. Small folk

1v16: Small folk. The King is dethroned instantly. King of Westeros involves so much pressure, so many enemies, control of so many problematic and inter-disputing houses. The pleasure in the power can never offset the pressure in the power for someone like me, nor can I sleep easy thinking I'm saving Westeros, as it'd be hard for me to be a great king, without applying so much pressure to myself to offset said pleasure. Maybe I'd be a good king without much pressure on myself, but it'd be a hurtle in itself to learn to accept I'm a good king and I don't need to be great and I don't need to stress too much.

2v15: Hand of Robert. I just have to go with my gut here. A Hand represents so much of what I covet: competence, intimate respect from the people who matter, and an interesting, substantial occupation. My occupation right now isn't very substantial. The outcomes aren't tangible, the passion is artificial.

3v14: Brotherhood. If you take Dany of Season 6-7 (the real Dany?) I may change my answer. Aren't the Brotherhood always on the lam? I've never been interested in a wanted life, though I enjoy the badassery of ex-knights who look like peasants. I already stated my case for Hand, but Dany in the end is intolerable, and I never liked her too much beforehand anyway. She has good moments, where I'd be proud to serve her, and I could be her close friend, but her proud moments are never pretty.

4v13: Hand of Bran. I already stated my case for Hand, and Bran is a sound king. Not the best king to be Hand for (it must be weird advising a god) but the septons such in GoT. Actually monastic life is not furthest from me, but the septons get a bad rap, and religion in GoT is usually cringe. I doubt the Seven would be my choice.

5v12: King in the North. As in the NCAA, 5v12 is fierce. I love the image of the Lead Ranger, be it Benjen, Jon, or otherwise. Ranger is the coolest job in the Watch, and Lead Ranger is the best of the cool. At the same time it's a miserable life, in the cold, with no women (okay even if I'm sworn to celibacy it's nice to have some variety around), in rundown castles, with little respect from the rest of the realm, and just waiting for horror to arrive. Is King in the North any better? Didn't I already denounce kingship? I did at King's Landing, but Winterfell is different. Northfolk are different. They're more loyal because they can feel winter coming at all times. They're like Wisconsinites. I'm probably biased because I loved the Starks and didn't love many important southerners. But I'm going to ride the Winterfell/Wisconsin image.

6v11: Lord of some house+castle. So many good options, whereas Master of Coin sounds so unheroic.

7v10: Lord Commander of the Nights Watch. Look, some people might call me a nerd. That's not what I want to be. I want to be a light leader, not King of Westeros, but a light leader, charismatic, versatile not just in brains. The maesters of GoT don't give maesterhood a great name, even Samwell, who is such a nerd. I could be more well-rounded. But I'd still be treated as one of the nerds. As I mentioned, Nights Watch is a miserable life, and this was a tough decision. I just went with my gut, because I like the Old Bear and I like Jon, and I respect the position. Maester is a better life, but Lord Commander is more respectable... I probably should have chosen Maester, but oh well, neither of these will win the big dance.

8v9: Younger brother of some ruler. Knights are so stale. All they do is stand around, feigning purity, until it's time to get sliced open. If you're a great fighter, and the crown has many enemies, you could be doing more fighting than standing around, but that's no consolation, however valiant. Fighting and waiting silently to fight is not a great life. If my brother ruled, I could be a trusted advisor and trusted commander without all the weight on my shoulders. And I could live, when I'm not fighting.

New seeds:

1. Small folk
2. Hand of Robert
3. Brotherhood
4. Hand of Bran
5. King in the North
6. Lord of some house+castle
7. Lord Commander of the Nights Watch
8. Younger brother of some ruler

1v8: Younger brother of some ruler. I want to be consequential.

2v7: Hand of Robert. However I admire the Watch, symbolized in Jon and Benjen, it's an unhappy life. However volatile Robert, he's a good man with the right council.

3v6: Lord of some house+castle. So many good options, while the Brotherhood is always running.

4v5: King in the North. Bran doesn't need a Hand other than to make decisions he doesn't have time to make. I'd rather be an important complement to the king, not just a worse second king for when the king doesn't have time. I'd rather be king of a good land than bitch second king of a bad land.

New seeds:

1. Younger brother of some ruler
2. Hand of Robert
3. Lord of some house+castle
4. King in the North

Now we're talkin. These are all options I would have considered if you asked me out of the blue which position I'd like to hold in Westeros. Is Robert the ideal monarch to be Hand for? No, but he may be the best in GoT.

1v4: 

2v3: