One of the things I've found useful when doing that comparison is figuring out which existential risks will actually make Earth less livable than other planets. If, say, the worst-case CO2-based climate change scenario comes to pass, Earth will still be a better place to live than Mars, and easier to terraform to human-optimal than Mars will be.
Again, when considering something like nuclear war, I believe a post-apocalyptic Earth would be better than an untouched Mars.
Against other existential risks- like UFAI- I don't think fleeing helps. There's probably one or two where it does- but even for stuff like gamma ray bursts or asteroid strikes the back of the envelope calculations I've done in the past have suggested fleeing Earth is a bad plan.
While Earth would be easier to terraform due to available resources and global conditions already closer to something inhabitable, it would not be safer, as mistakes in the terraforming process are not going to be as catastrophic when you try to terraform a backup, uninhabited planet.
Toying with complex, poorly understood processes at a time when we wouldn't even have our current resources, manpower on a ravaged Earth whose environment might just be one wrong step from becoming much worse, could destroy a majority of what remains of humanity, the economy and valuable resources, making it impossible for us to ever recover.
(I am however assuming we were talking about global terraforming of the whole planet, not making minute changes to local spots)
The old L5 Society would be rather disappointed with your focus on planets. Incremental construction of space habitats would be significantly easier than whole-planet terraforming, and putting the eggs in multiple baskets in multiple orbits is more survivable than any handful-of-planets scenario.
Quite possibly- My impression is that space habitats tend to be more fragile and expensive than planets for a number of reasons, and so when you compare equal-cost options a lot of benefits of space habitats evaporate (with the caveat that any estimate of costs is going to be very, very wide).
Space habitats more expensive than terraforming Mars? You're having to import large quantities of volatiles in either case, Mars needs more imported per person supported (since you'll be using lots of them in the upper atmosphere where they only provide radiation shielding and pressure, instead of smaller quantities to directly support biologically active processes), Mars outgasses the hydrogen you import thanks to photodisassociation of water molecules, and Mars has half the available solar power of a 1 AU orbit (and while you can possibly power a habitat with nuclear power, good luck using anything but solar for a terraforming project).
Now, if you're dealing with habitats, yes, Mars gives you the advantage that by reasonably good placement you can have all the CHONSP you need for your habitat right at your doorstep, and an atmosphere that unmodified already shields reasonable amounts of radiation on its own, et cetera. Mars habitats will be cheaper than orbitals (or sealed habitats anywhere but Earth). But you were talking terraforming. Orbitals vastly outperform terraforming Mars in cost effectiveness per person supported (and Mars outperforms any other solar system body other than Earth).
Now, fragility. Yes, orbitals are relatively fragile. But they are also more numerous. If an orbital faces an independent one-in-a-hundred chance of a local catastrophic total failure leading to death of all inhabitants in a decade, and Earth faces a one-in-a-trillion chance of a local catastrophic total failure leading to death of all inhabitants in a decade, then ten orbitals are a hundred million times less fragile collectively than Earth despite being ten billion times more fragile individually. When you consider the raw number of orbitals you can make for the cost of terraforming Mars, orbitals as a group vastly outclass terraforming Mars in their ability to avoid existential risk, over all classes of risk where a terraformed Mars would actually reduce existential risk.
Space habitats more expensive than terraforming Mars?
No, more expensive than Mars habitats, for the reasons you mentioned. When I said "and easier to terraform," I meant that as a bonus, not as a requirement. Closed habitats seem like a reasonable norm for small societies in hostile conditions.
independent one-in-a-hundred chance
Sure; but there must also be dependent chances (like gamma ray bursts or ubiquitous design faults or so on). It seems difficult to have the baseline dependent fragility of orbital habitats at lower than the baseline dependent fragility of Earth. I would rather have 10 Cheyenne Mountain-style habitats a bit below the Earth's surface than 10 orbital habitats- and, again, at equal cost expect to have a lot more ones on Earth than off it.
Ten Cheyenne Mountains are fine—as long as nothing happens to the Earth that would stop the people surviving in them from resuming agriculture on the Earth's surface (whether or not under glass). I'd like there to be humans elsewhere in the solar system that are already growing their own food as a hedge against any of the things, already thought of or not, that can take out a planet but not a solar system.
I'm not aware of any systematic investigations. One thing to keep in mind is there's a big distinction between say going to Mars and going to planets in another solar system. Going to a whole other system reduces more existential risks but requires a lot more investment.
It might make more sense to think of this in terms of specific risks and whether they would be avoided simply by having a backup population. Obvious examples in that category are asteroids and nanotech disasters. The first however is more easily dealt with by careful observation and moving the asteroids out of the way. The second is much harder to quantify but many people who actually study nanotech consider it to be unlikely.
Colonies do however also protect against the unknown unknowns. But for obvious reasons it is very difficult to estimate how much they do so.
I'm trying to find out how short-term or long-term your thinking is. Moving to Mars seems very fragile, depending on constant input from planet earth. The challenges of moving to another star system where you could have a self-sustaining life are immense. Neither option is available in your lifetime. I think quantitative estimates of the survival of sapience on earth are pretty much useless -- the uncertainties of individual estimates are way too high. As a young man in 1981 I debated moving from the US to Australia as a hedge against nuclear war, a much more modest proposition. I decided not to, partly because I could be a more effective activist against nuclear war in a superpower that was my native culture. So if you go down this path, you could think of your utility in terms of preventing the destruction of sapience on earth.
Can anyone recommend some resources, or provide actual examples, of any relevant evolutionary-biology-style equations which could describe what factors would most likely predominate to allow for the survival of sapience, given such factors as the ease which any given individual can acquire the means of killing large numbers of people, the scale of those means, the willingness o people to use those means, the ability of small groups to travel away from other groups, and so on?
Or, put another way - is there any way I can quantitatively check my intuition that a valuable way to avoid certain existential risks is to flee Earth?