Here is what the AI researcher Douglas Hofstadter wrote on the topic. The answer to your question is in the realization that reasoning is not subjective.

“You might feel that each person is completely unique and therefore that no one can be relied on as a predictor of how other people will act, especially in an intensely dilemmatic situation. There is more to the story, however.

Any number of ideal rational thinkers faced with the same situation and undergoing similar throes of reasoning agony will necessarily come up with the identical answer eventually, so long as reasoning alone is the ultimate justification for their conclusion. Otherwise reasoning would be subjective, not objective as arithmetic is. A conclusion reached by reasoning would be a matter of preference, not of necessity. Now some people may believe this of reasoning, but rational thinkers understand that a valid argument must be universally compelling, otherwise it is simply not a valid argument.

If you’ll grant this, then you are 90% of the way. All you need ask now is which world is better for the individual rational thinker: (one with thinkers all cooperating or all defecting)
...
Since I am typical, cooperating must be preferred by all rational thinkers. So I’ll cooperate.” Another way of stating it, making it sound weirder, is this: “If I choose cooperation, then everyone will choose cooperation.”
...
“We live in a world filled with opposing belief systems so similar as to be nearly interchangeable, yet whose adherents are blind to that symmetry. This description applies not only to myriad small, conflicts in the world but also to the colossal... Yet the recognition of symmetry - in short, the sanity - has not yet come. In fact, the insanity seems only to grow, rather than be supplanted by sanity. What has an intelligent species like our own done to get itself into this horrible dilemma? What can it do to get itself out? Are we all helpless as we watch this spectacle unfold, or does the answer lie, for each one of us, in recognition of our own typicality, and in small steps taken on an individual level toward sanity?”
...
“To many people, this sounds like a belief in voodoo or sympathetic magic, a vision of a universe permeated by tenuous threads of synchronicity, conveying thoughts from mind to mind like pneumatic tubes carrying messages across Paris, and making people resonate to a secret harmony. Nothing could be further from the truth. This solution depends in no way on telepathy or bizarre forms of causality. It’s just that the statement “I’ll choose C and then everyone will”, though entirely correct, is somewhat misleadingly phrased. It involves the word “choice”, which is incompatible with the compelling quality of logic. Schoolchildren do not choose what 507 divided by 13 is; they figure it out. Analogously, my letter really did not allow choice; it demanded reasoning. Thus, a better way to phrase the “voodoo” statement would be this: “If reasoning guides me to say C, then, as I am no different from anyone else as far as rational thinking is concerned, it will guide everyone to say C.””