Randomly choose 0.001% of the books.

If you think that's a bad plan, it's been the norm for the history of books.

Outside of the thought experiment, banning books is different than choosing to not preserve them or keep them in a collection.

Removing a book that would otherwise fit the criteria of preservation just because it covers a "politicized" topic is different than a book becoming low value, getting superseded by newer editions, or no longer being worth preserving by that particular institution.

Maybe this is outside of the thought experiment but I would focus on digitization. Text compresses very well and you can fit 100Gb on a CD sized disc with an estimated 50+ years lifespan (m-disc). So you could easily fit over 30 million text only books on a single 100 disc spindle which is the size of 3 small physical books. Add some redundancy and it might be 25 million books. Books with images would be slightly less compressible but you could still fit 100s of thousands on a single spindle with redundancy. Get yourself a small bar sized wine fridge to control humidity and you could probably fit every book every made in there.

This all assumes you want to preserve the content of the books and not the books themselves. You obviously can't digitize every aspect of a physical book like the ornate artwork on the spine etc. in which case I would focus my preserving efforts on those books and digitize everything else.

General Criteria:

Present and potential relevance to community needs
Suitability of physical form for library use
Suitability of subject and style for intended audience
Cost
Importance as a document of the times
Relation to the existing collection and to other materials on the subject
Attention by critics and reviewers
Potential user appeal
Requests by library patrons

Content Criteria:

Authority
Comprehensiveness and depth of treatment
Skill, competence, and purpose of the author
Reputation and significance of the author
Objectivity
Consideration of the work as a whole
Clarity
Currency
Technical quality
Representation of diverse points of view
Representation of important movements, genres, or trends
Vitality and originality
Artistic presentation and/or experimentation
Sustained interest
Relevance and use of the information
Effective characterization
Authenticity of history or social setting

Stolen entirely form here . Seems like a very good starting point to me, as I would expect from a Libraries Association.

I'd say I'd prioritize the ones that have the most impact and usefulness on society. That means:

• all those self-help motivational books gotta go
• the books written by politicians for the sake of campaigns have no place here
• most school & college textbooks come and go all the time (fick you, Pearson) so I wouldn't include them, but there are all time classics that are worth saving
• all the religious scriptures (e.g. holy bible, quran, tripettaka) gotta stay, no matter how bullshit you and I might think
• classical philosophy stays, probably some modern ones as well (but I don't know where to draw the line)
• books about facts (i.e. science & history) documents stays, but not all of them; this is a hard one, some rigorous verification
• technical guidebooks (e.g. construction, gardening, electronic, software, machinery, chemistry, metalurgy) gotta stay, but needs rigorous verification, also technology changes all the time, so most of the outdated ones might need to go (saving some for historical purpose)
• fiction is the hardest, but generally the all time classic (1984, anyone?) as well as the really popular ones (sadly includes Twilight) stays

Whew... This is way more work than I initially anticipated.

I would only allow books about, or involving, cheese.

Books that help you to think critically are important. Especially when it comes to religion and government. I think “1984” is a good example of this. I’d add “Fahrenheit 451” and “Handmaid’s Tale” to that cache as well.

Fan fiction could pretty much all be tossed without review and no one would know.

Value.

They obviously have value if people want to read them (they're popular). Or they could have artistic value. Or document something and it's important to keep them around because the info inside might be important later on.

You can discard any books like Excel 2006 or Windows 8 beginners guide. I'd say they don't have any value anymore. Or like bad cooking books that no one reads anyways.

Also a library might not be an archive at the same time. So you could focus on which books actually have some use for the patrons and judge by if they're being used/read.

And I'd like to add: Selecting books and tidying up to make space for new popular books... And banning books are two very different things. Banning books for grown-ups isn't a good idea. Never, and under no circumstances. Unless it's 1933 and you're the nazis.

If it's magic, why can't I make it bigger?

Cutting right wing "we can be ð cool kids too!" type media would probably eliminate a full half of ð regular churn of incoming writing.