I'm reading a poker theory book that has a chart with optimal calling frequency vs different bet sizes in a toy game where you know you have 50% equity. (Play Optimal Poker, Ch2, part 8).

It's saying that if someone overbets 100x pot (with a bluffing frequency approaching 50%), that it's correct to call 1/101 times. How can this be the case? If you have 50% equity, wouldn't it correct to call any overbet every time, since the odds can never be worse than 1:1?

50% equity must be BEFORE betting starts. (If we wanted to calculate things with different starting equities, it would be more difficult).

For example: If I am sitting with 10 nut combos and 10 air combos BEFORE betting, then your medium strenght hand has 50% equity vs my entire range if we checked down.

But when I bet 1x pot, I do that with 10 nut combos + 5 air combos, so your equity is now 33.3%.
33.3% is the same as your pot odds, so you are indifferent between calling and folding.

If I bluffed more often, you would aways call.
If I bluffed less often, you would always fold.

You should call 50%, because my bluff needs to work 50% of the time to be break-even.
If you called less often, I would bluff any two cards profitably.
If you called more often, I would stop bluffing and just value bet.

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If I bet 100x pot, your pot odds are going to be 100/201 = 49.75%.
This means I should use 49.75% bluffs = 9.9 bluff combos for every 10 nut combos.

You should call 1/101 of the time, because my bluff needs to work 100/101 of the time.
If you called less often, I would bluff any two cards profitably.
If you called more often, I would stop bluffing and just value bet.

Your equity is approaching 50%, not exactly 50%. Actually assuming there are no blocker effects, your EV in calling should be 0, same as folding. But if we call less than 1/101 times, our opponent can auto profit vs us by bluffing 50% of the time or more. If we call more often than that, our opponent should never bluff us.

In practice it is more of a situation of ebbs and flows. We don't always need to be perfect in figuring our if our opponent is underbluffing or overbluffing to decide to call or not when it's close. We call sometimes to prevent being exploited. When it's close we might use some randomization method to call sometimes.

In reality in those situations our opponent is probably either underbluffing or overbluffing, so the right answer is probably either call or fold, but in practice we might not know exactly where our opponent is with their bluffing frequency. We do know that if we overfold, a good opponent can eventually exploit us by overbluffing, and if we overcall, a good opponent can exploit us by eventually underbluffing. So it makes sense to try to defend at a frequency around MDF vs good opponents who might exploit us if we don't when we don't have a clear exploitative approach to a spot on whether our opponent is imbalanced.

I guess what makes it confusing then is I just have these 2 conflicting modes of thinking that clash when thinking about the toy game:

1. "What is my equity and am I getting good enough pot odds to call"
vs
2. "How do I call in a way that makes his bluffs indifferent".

Do I just have to discard the first mode of thinking when thinking about GTO?

If its a toy game where villain is perfectly polarized, then he will be bluffing at a frequency such that your equity is equal to that bluff frequency... so your calls are 0 ev in (1.). This assumes no blockers/perfectly polarized, etc.

You definitely should always think about the first mode of thinking as this is the main thing you'll be thinking about when playing real games.

Either you misunderstood or it's badly explained, if a play is higher EV you make that play. You only play at a frequency when both plays are the same EV