It depends.
Chance of a bet paying out? Value them the same.
Amount of information you gained, where you value transferring that learning to other questions, designs, etc? 90% --> 100% is way better.
In a domain where you know you have plenty of uncertainty? 90% --> 100% is a huge red flag that something just went very wrong. ;)
You're basically right, in that it requires much stronger evidence to move you from 45% -> 55% credence than to move you from 90% -> 99.9%.
It is helpful to think in terms of likelihood ratios. To go from 90% -> 99.9% credence requires observing evidence with a likelihood ratio of (0.999 / 0.001) / (0.9 / 0.1) = 111, which is about 6.8 bits of evidence. [edit: fixed flipped / wrong math]
To go from 45% -> 55% credence, you just need a likelihood ratio of = 1.5, or about 0.6 bits of evidence.
(Getting to 100% credence via Bayesian updating requires +inf bits of evidence; remember that 100% isn't really a probability.)
Re getting to 100% probability of a outcome, that's actually surprisingly easy to do sometimes, especially in infinite sets like the real numbers. It's not trivial, but you can get these outcomes sometimes.
If you have some utility function that depends on the amount of money you have, then the improvement from a bet that offers a 45% chance of winning a prize to one that offers a 55% chance is identical to the improvement from a bet that offers a 90% chance to one offering a 100% chance.
Note that this holds only when you have no "intermediate choices".
Suppose you are pretty short of cash at the moment. And you might be getting a prize tomorrow. You have a chance to buy a fancy meal now. If you buy the fancy meal, and then don't get the prize, you will really be struggling to pay off your bills. So it only makes sense to buy the fancy meal if you are >95% sure that you are getting the prize.
In this setup, it does make sense to value the extra certainty.
This is all assuming you don't terminally value certainty in and of itself. You terminally value something else. (If not, then you risk being money pumped where you pay to learn info, even though you can't use that info for anything)
But even if certainty isn't a terminal goal, it can be an instrumental goal.
The framing effect thing is about the chance of winning some prize. Why would you want certainty, what you want is the prize.
45->55% is a 22% relative gain, while 90->100% is only an 11% gain.
On the other hand, 45->55% is a reduction in error by 18%, while 90->100% is a 100% reduction in errors.
Which framing is best depends on the use case. Preferring one naively over the other is definitely an error. :)
In Never Split The Difference: Negotiating As If Your Life Depended On It, Chris Voss discusses cognitive biases:
Through decades of research with Tversky, Kahneman proved that humans all suffer from Cognitive Bias, that is unconscious—and irrational—brain processes that literally distort the way we see the world. Kahneman and Tversky discovered more than 150 of them.
There’s the Framing Effect, which demonstrates that people respond differently to the same choice depending on how it is framed (people place greater value on moving from 90 percent to 100 percent—high probability to certainty—than from 45 percent to 55 percent, even though they’re both ten percentage points) (p. 12).
Isn’t it rational to value 90% → 100% more than 45% → 55%?
Even going from 90% to 95% means you are wrong half as often — instead of — whereas going from 45% to 55% only removes about 20% of your errors.
Is my thinking and math correct? If not, how am I wrong?
Assuming I’m right, I would also really appreciate a better way to explain this.