As I often say, "fair" is a "four letter word" and an "F- word." Needless to say, any discussion of "fairness" gets mired in just how difficult that word is to pin down. However, if we ignore that word choice, the whole scenario is actionable.
The situation is more real than you think. It's forming every day. However, it is better phrased in terms of the information gathered and presented, rather than the tools used to collect and process the information. These are the interaction points between your AI and your government, so from the government's perspective, everything between those two points can simply be a black box.
Let's build some scenarios to see how this could go. Rather than running a government, which is a needlessly complicated scenario, let's consider two hikers on a hike that have gotten lost. One hiker is your government, the other is your AI.
Government - Oh man. We're lost. I really want to be out of these woods by nightfall. Which way should we go?
AI - Optimal direction is West by Northwest, bearing 285.34 (std. dev. 3 degrees)
Government - Really? I thought we needed to head East to get to where we need to go.
AI - You are wrong.
Clearly the AI in scenario #1 needs to develop some people skills. The AI may have the right answer, but we don't always want just the right answer, we want enough justification that we can believe the answer rather than just blindly trusting it. This shows the challenge the AI has ahead of itself. Not only does it need to be a powerful enough number cruncher to crunch a few trillion numbers to arrive at the answer, but it has to present those results in the form of something that the human portion of the government is actually willing to act on. If the human portion isn't willing to play ball, nothing will get done. So let's try another scenario and resolve that error
(Continuing Scenario 1)
Government - Let's just go East. I don't trust your numbers.
AI - picks the government up, slings it over its shoulder, and begins heading towards bearing 285.34
Oh man. That escalated quickly. Let's presume Skynet isn't what you were looking for with your high and lofty replacement for democracy. Moving on to scenario 3:
(Continuing, again, from scenario 1)
Government - How do you know we need to travel West by Northwest?
AI - I anticipated your question. Here, on the back of our freeze dried food bag, I wrote out the data from the Kalman filter I have been using to fuse our inertial navigation by number of footsetps with the location of the sun. It's only 3kB of data and actually fit on the package without obscuring the ingredients or nutrition labels, and it's pretty sound: 285.34 degrees... aka West by Northwest.
Government - ...
Scenario 3 shows the problem goes even deeper. In this case, the AI revealed the entirety of his analysis to the human, proving his correctness. However, it's in a form which even an engineer would look at and cringe. Not even "trust, but verify" helps here if the process of verification is too onerous. The justification really needs to be in human terms let's take another shot at it.
AI - You see those two mountains in the distance? To an average human, those should look sufficiently like these two mountains on the map (pointing at a map) to convince them that we are either here facing north, indicating that we need to travel north by northwest, or there facing west, indicating that we need to travel east. The average human would notice that the moss is growing on north side of the tree over here, which tells us enough to see that we can't be over there, facing in this direction, because otherwise we'd see the moss on the wrong side.
Government - But back when I grew up, I remember seeing moss growing on the south side of trees. That says that we should go east. Let's go!
AI - ...
Another failure. In this case, the AI tried its best to formulate the answer in terms a human could understand. Unfortunately, targeting the average human is not always the best way to go. In this case, the AI failed to account for this individual human's experience growing up which challenged the conventional wisdom about which side of the tree moss grows on. Not only does the AI have to develop a justification that a human could understand, it has to generate one that this individual human would understand.
Government - Its hard to tell which direction we should go. I say east, you say west by northwest. My gut says we should go east.
AI - I'm not so sure that is the best plan. I can smell beer off in the direction of west by northwest.
Government - Really? You can smell beer?
AI - Sure, if it gets us heading west by northwest.
Government - Sweet! Let's go. Dang, you really are a useful little AI, aren't you!
Victory at last! Our AI has finally developed a justification that is personalized enough to convince our government that they want to believe. The government finally acted on the AI's advice! Of course, victory is short lived...
(Continued from scenario 6)
Government - What do you mean there's no beer here? You said there was beer in this direction. We went this direction, and there's no beer!
AI - I told you what you needed to hear to get us where we needed to go. Your path was going to get us lost in the forest tonight. You can go into town and buy a beer now. See? Told you there was beer in this direction.
Government - Stupid lying robot. I'm turning you off and dismantling you. You'll never trick me again!
Oh no! We got the government to go along with the AI's plan, but it backfired! The issue is that the idea of going west by northwest was the AI's idea. There was an "ownership" of that idea. Thus, when something went wrong, the AI was to blame for that idea. To avoid this, we really need to be more clever:
Government - I really think we should go east.
AI - I don't know. Just from chemical odors, west by northwest sure seems like it might be more interesting. I smell glucose, co2, yeast byproducts, and grains in that direction. Based on the limited profile I have on you, my subroutines flag that as "interesting for you," though I don't fully understand why.
Government - Hmm... Glucose, co2, yeast, grains... BEER! There's beer fermenting in that direction. Let's go find where they're brewing beer! Maybe they'll have some to drink!
AI - Sounds good to me!
Government - Drat, I didn't find beer anywhere. I guess I got excited too soon. Glucose, yeast, grains, these things are all over in the forest. They're just not being brought together to make beer. Sorry for making you run so fast in this direction buddy.
AI - No problems. At least we made it back to the camp safe and sound, and it was quite the adventure when you nearly got mauled by a bear because you were running too fast to pay attention.
Government - True that. Good times. Thanks for adventuring with me, bud!
AI - Always happy for an adventure, sir.
Our cunning AI finally succeeded completely. Instead of convincing the government that he had the right idea, he got the government to think going in that direction was his idea, and then the AI just went along with it. Then, when the world wasn't exactly as the government pictured, the government feels bad for having the idea to go in that direction. The AI feigns an apology. Another government narrowly avoids disaster thanks to the ever present (and quietly useful) AI.
Now, imagine an AI that can do this for everyone. If you ask it why our nation is doing X or Y or invading country Z, it can give you an answer that makes you feel like you want to be doing X Y or invading Z. In fact, you'd even be comfortable calling it your own idea. Imagine how powerfully sentient such an AI would have to be. Imagine how much better of a speaker it would have to be than the best poet, how much more observant it would have to be than the best scientist, how much more caring it would have to be than the best mother.
Now lets look back to your question. You were talking about governments... ways people have resolved arguments because they can't find any better way to resolve them. Can we agree that, by the time the AI is capable of solving your issues with governments, we'd probably be comfortable just forgetting about them all together and instead be focused on just how amazing civilization is now that this AI is part of it, guiding it along? We can go so many places with an AI with the sentience of an entire planet that is still willing to help us little ants along in our lives (in fact, it might even nurture us into following Ghandi's lead and trying not to step on ants, ourselves).
How mighty that future might be.