As a followup. Also contain power from first principles.
I ask myself; what is the president of America doing that's useful? What could he be doing? I have one interesting answer, but setting that aside...
For the most part,* he's just a figurehead. I found a blog once which, among other useful things, discussed all the productive things G.W. Bush tried to do. They were all shot down, of course, president or no president. Most republican presidents are like this, while the democratic presidents know better than to try anything. (The dem faction has workarounds anyway.) So they're not actually doing anything.
*(Let's say 70-80%)
But, constructing an imaginary polity where da Prez actually had power, what could they be doing? It turns out this requires a rather large perturbation - it's not like the president can just draft a law and sign it in. But, even going through that, it looks like the best thing the president could do would be to leave well enough alone. The existing over-regulation has kind of taken on a life of its own and the economy has grown over it like a mold. Trying to fix it now is just likely to make a mess.
And, aside from the economy's inherent structural flaws, such as factional reserve, which can't be fixed by a president anyway, it works pretty well. It's not like it needs a whole lot of messing with.
So, answer: the president should basically be sitting on their thumbs. (Again, say 70-80%.)
That figurehead position is exactly what the president should be, and very likely where the head of any country with a functioning high-tech economy would be. Ultimately the function of a modern head of state is just a coordination point - because discipline beats numbers every time, to the point where it hardly matters what everyone is coordinating on.
Which makes the obsession with who gets to fill that spot ironic. King, priest, dictator, elected official, CEO...it doesn't matter, because your options are; sit on your thumbs, or make things worse. (Hello, North Korea!) And while military culling has been suppressed recently, every real war is a pissing contest - the one with the bigger economy wins. For instance, imagine South Korea's backers decided tomorrow to let South Korea mobilize and reunite the peninsula - what, exactly, is Kim Jong Il supposed to do about it? Throw pitchforks at their jets? Modulo some nukes, here, of course...but I suspect the necessary artful diplomacy would have already neutralized even those. And without his own backers, KJ would never have gotten nukes in the first place.
To try to check this answer, I'm going to look at power from first principles.
A person has power when they can influence the actions of another. Hierarchies start to form economically, with some kind of contract exchanging benefits for servitude of some scope.
Consider a stone age tribe, with some kind of primitive economy, such as barter or simply communism. Also assume that for some reason, there's an opening for tribal elder.
Someone who knows where the mammoths roam exchanges his knowledge for extra pointy things from the spear-knapper and more money from the cockleshell guy. They all go out and down a mammoth, and the three enjoy delicious mammoth steak; everyone benefits. Eventually, a couple more of the tribe crave meaty mammoth, and trade extra clothes and roots.
Eventually, though, the everyone knows where the mammoths roam, because they've been on so many hunts; mammoth-guy isn't really contributing anymore. So, let's say at the next generation, the son of spear-knapper realizes that he really doesn't need the son of mammoth-guy. He tries to get son of shell-guy to break with him so they can go mammoth hunting with the dead weight of mammoth-guy, and thus the labour can get a 'fairer' share of the fruits.
It doesn't work. Shell-guy is all like, "But if I go with you, I'll lose the extra clothes and roots." The same goes for the others; they would lose out on shells and clothes, and roots and shells. Individually, each has a positive motive to stay in the agreement, and mammoth-guy can use his existing agreement-wealth against any dissenter. In the end, spear-guy probably has to give up more spear points just to keep access to the in-group.
Yes, it does seem that if the children of everyone but mammoth-guy all stand up together and cast off mammoth-guy, the only real downside is that if mammoth-guy wasn't retarded, he will have stockpiled a bunch of points and shells, and so they will temporarily lose a store of capital. But even with four people, the necessary coordination is nontrivial, not to mention that overturning the existing agreement means that all have an opportunity and motive to squabble for advantage in the new agreement.
With thousands of people, it effectively becomes impossible. Historical revolutions originate from competing leaders, not anything resembling class consciousness.
So, power from first principles; it can form noncoercively, but once formed, its main advantages are those of a coordination point - it preempts conflict and imparts coordination - while breaking it reopens the conflict and requires even more coordination. It has both a positive and negative feedback promoting stability.
Moreover, a good leader can use their economic clout to channel their followers to more productive pursuits. If son of mammoth-guy is canny, he will specialize in being more of a merchant; while he still goes on hunts, his main pursuit is trading his stockpiled wealth for positive-sum labours by his followers. Given enough time, son of mammoth guy will be able to buy the whole tribe, and, most likely, even the tribe next door.
(This probably never happened - humans prefer to build up economies to make armies, and then simply conquer the next tribe over. All the kingdoms and nations we are familiar with were united by conquest; as a result, it's absurd to be against modern conquest.)
The president, no matter who they are, is pretty much in the same boat - they can't truly offer anything much of value as a person, and are instead interchangeable. However, the existence of -some- president is of great value. And while it is true that being president is pretty swank, providing incentive for people to take the job, in a modern economy there's lots of opportunity for nearly equally swank jobs elsewhere.
And what about my point that, even more so than the tribal merchant-prince-elder, they should be sitting on their thumbs?
Beyond some critical point of wealth, mammoth-dude realizes that instead of waiting for spear-dude to rebel, he could simply pay off one faction to coerce even more extra payments from another. Ultimately this is shooting himself in the foot - the more negative sum games he plays, the more likely the next tribe over will build a better economy and simply trash his tribe one day. However, just like it's difficult for individual subjects to see that their long-term interest is probably served better by not following mammoth-dude, it is almost impossible (judging from history) for mammoth-dude to see past his own short term gain.
This is basically the only kind of game a president can play - take one faction (for example, the courts) and use it against another. They can't even merchant-prince the way mammoth-dude can, because there are literally thousands of people who know thousands more about how to make a buck, and they're already doing it.
Aside from the odd exception, there is only one thing a president need concern themselves with; making sure they aren't toppled, thus destroying the figurehead bonus.