I am a structural philosopher, and voting systems is one of my favorite subject matters.
Let me first say what I do not like, before saying what I do like about Nigeria's form of democracy.
1/ I dislike empowered presidents. I do not mind presidents when they don't have all that much power.
I can show you in the following image why:
Very obviously, the elite ended up getting more when there is an empowered president in top. Particularly, columns #3 and #5 tell the story loud and clear.
With that single empowered person, the entire game can get yanked into a specific direction, and that will of course benefit those that have access to the president.
2/ I dislike single seat elections. I do not mind multi-seat districts, but only when there are at least 12 seats up for grabs at the same time in that district. The lower the number of seats (and 1 seat per district is therefore the very worst there could ever be), the more the elite can control the outcome.
See Column #1 in that same image, and you can see it is more all over the place than columns #4 and #5.
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The reason a single empowered president and elections in districts for one seat (or only a few seat) is bad news is that after twenty or more years a large group will realize that they are never represented at all.
Someone else said this better than I can:
"Two very different ideas are usually confounded under the name democracy. The pure idea of democracy, according to its definition, is the government of the whole people by the whole people, equally represented. Democracy as commonly conceived and hitherto practiced is the government of the whole people by a mere majority of the people, exclusively represented. The former is synonymous with the equality of all citizens; the latter, strangely confounded with it, is a government of privilege, in favor of the numerical majority, who alone possess practically any voice in the State. This is the inevitable consequence of the manner in which the votes are now taken, to the complete disfranchisement of minorities."
—John Stuart Mill, Representative Government, 1861
So, a system that is unfair can still be called a democracy because voting is involved. Yet a real democracy is based on full representation, and the happiest nations in the world happen to have full representation. Meaning, that all voters can point at the person or party they picked themselves, sitting in a seat.
In winner-take-all, that tends to be about 60% of the voters (sometimes more, sometimes less). And there is then a reason to feel disgruntled because the government picked a voting system of divide-and-conquer (majority gets all, the minority of voters gets nothing).
The powers that be like it that way because they don't want to listen to everyone because that will cost them some of their money. Rather, they want everyone to speak via two or three mouth pieces (parties) and deal with them that way. Meanwhile, these parties don't have to listen all too carefully to all the voters because all these parties need is to serve the majority, not all the people.
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The good part is that, because Nigeria is a large nation, it is important to have a stable democracy. It will not help Nigeria to have twenty parties all pulling in their own political directions.
So, at the National level, it is actually good to have three, four, perhaps five parties, but not twenty. Limiting democracy (i.e. not using proportional voting, or mix it up with winner-take-all like they do in Germany and New Zealand) is good for stability.
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So, that is good about the current setup. It establishes a stable nation, but it can get improved.
I would still try to minimize the power the president has.
Plus, I would make sure that all elections at the 36 State levels and at the local levels make use of the clean voting system, and do not use winner-take-all. Three levels of winner-take-all is too much. It makes people unhappy in the long run.
Particularly when the Nigerian nation is secured with having three, four, or five parties (and not more), then the rest of the governmental organizations should offer the greatest political freedom possible at their State and local levels.
That way, people can feel free in their own location, ruled by their own people and not pushed into submission by the winner-take-all system. They will then understand that at the national level, Nigeria cannot be fully-free, but delivers the next best thing of representation in somewhat limited measures.
Good luck with Nigeria. I hope Nigeria will embrace the real form of democracy for all but the National level, and yet have a decent but smart kind of democracy at the national level.