Tl;dr: I think we can vaguely guess a 1.5-2x risk reduction from a third shot, maybe pushing 2-2.5x if you get it close to 6 months after the first; for those with less immunity after two shots, it might be more like expected 3-6x, though it’s unclear if one can discern who these are. If you do get a third shot, you should definitely try to get a different kind, as others have said, though unclear if you want to sacrifice efficacy for this.
First, to establish an expected effect size, how good are the first two shots? The first Pfizer/Moderna takes you to about 80% protection, or a 4/5 risk reduction. The second takes you to 95% efficacy, a 4x risk reduction. To compare, Delta variant takes Pfizer from 95% back to 80, so is a 4x risk increase. So most likely, the third vaccine is not going to be as efficient as the second vaccine. So we expect less than a 4x effect; realistically, maybe I expect something like 2x at the outset since secondary boosters are common in immunology and third boosters aren't. I wouldn't be that surprised at seeing 3x or 1.25x though. (Note that this is all about transmission, for a shot of the same type as your first two; probably slightly stronger effect than this in mortality, and mixed vaccinations I'll get to at the end).
There's no released data, so we have to get creative fast. The best empirical evidence I know of comes from this study on how much vaccinations boost your neutralizing antibody (NAb) titers after already having COVID. (NAbs are by no means all of your immunity, but they’ve been shown to correlate surprisingly well with vaccine effectiveness. Hilariously, there are nonetheless papers claiming we have some evidence that NAbs can CAUSE infections to be worse—while this may be true, all I can do is plead for them to make a better case that this isn’t bog-standard Treatments Correlated with Harm.)
Anyways, this study shows limited effect from a second Pfizer vaccine dose if the patient had already contracted COVID! It’s useful because very roughly speaking, getting COVID gives similar titers as a vaccine dose—and COVID plus one dose is similar to two doses, so looking at COVID plus two doses is a good approximation for a third dose. This gives us some limited evidence that a third dose won’t be that effective.
Notably though, it did significantly increase the titers of the least-protected! This seems like a decent reason to get a third vaccine if you are vulnerable. To put very rough numbers on it, my guess is that for the vulnerable it would cut risk by a factor of 3, whereas the group as a whole only gets a factor of about 1.5.
- Calculation: Getting a second dose cuts risk by 4 as we’ve already said; on this graph the first to second shot gives a jump in titers by about 3 logs, whereas it looks like the vulnerable have their titers go up about 2.5 logs from the third shot, so I’ll give this a ~3-fold reduction. The group as a whole only goes up about 1 log on the third-dose-equivalent, which is maybe a factor of 1.5 protection. Both of these figures match well with our original bound of finding less than a 4x average reduction, and quite well with my prior of about 2x!
As a second creative line of evidence, could we figure out anything from the claims from Pfizer that their third dose increases neutralizing antibody (NAb) titers by by 5-10x?
Is 5-10x a lot? Unfortunately, a major hurdle is that every vaccine appears to measure NAbs using different assays, with no translation to efficacy. I really hope I’m mistaken here, because otherwise it seems like every report of increasing titers by X-fold is almost completely meaningless, e.g. this otherwise-nice comparison. But I don’t think I am.
I tried going through other Pfizer studies to see if maybe they at least used a single assay across the board. One Pfizer study says a factor of 1.5 difference in unnamed geometric titers for a strain meant roughly nothing. Another shows about a 4x factor in log2PRNT50 difference between Delta and wild-type, which corresponds to about a 4x decreased efficacy as we know from elsewhere. But this can’t be the same as the above-stated titer, because that would mean a 5-10x reduced risk from a third shot, which is way higher than our bounds. It’s possible that if we escape the log though, this becomes a 16x titer difference for 4x risk reduction, which is at least in the ballpark and would imply a 2x risk reduction from the third shot. I think this is on shaky ground though and I don’t place much stock at all in it, even if it happens to pop out the magic number.
A very important thing here that a few people mentioned: combating waning vaccine efficacy is important. Many studies show this effect, but I’ll cite my favorite that I linked earlier. It predicts about a 2x risk increase (or effectiveness decrease) over 6 months. Another study shows variable waning from natural infection that roughly accords with this, at least insofar as NAb titers from this or similar papers can be interpreted.
There are two questions here: first, would a third vaccine reverse this waning effect? There aren’t data to confirm this yet, but surely the answer is yes—this is how other vaccine boosters work, and all theory points to this being the case. The second question is whether the third vaccine’s natural effect, as described early to be ~~1.5x normally or ~~3x for the vulnerable, would fully stack on top of a waning-reversal. I think the answer is probably no, but with partial effect, closer to fully stacking than not. Part of this answer comes from the idea that the longer you wait between doses, the more effective they are. In total then, I expect a booster after 3 months to give you about 2x risk reduction, and 4x for the vulnerable.
Last, I completely agree with the other people bringing up that getting a different third dose seems better than the same (though the same shot seems still worth it for many, as above). I wish I had gotten my second dose of a different kind—this is significantly because of “original antigenic sin”, the tendency to get antibody lock-in toward whatever strains you experience early. I expect different vaccines as boosters will help somewhat with this problem, though I really can hardly speculate quantitatively about this—all I’ll say is I would be unsurprised to see that taking this strategy with the right vaccine might get you 4x instead of 2x from a third shot. (However, you’d need both an effective vaccine and for it to be substantially different than the first one or two—and I don’t know if Moderna/Pfizer fit that bill. Perhaps I can research this in the future, or how much effectiveness to sacrifice for difference, though it sounds hard and by default I won’t since I don’t think it’s all that decision-relevant.)
Two last thoughts: if Elizabeth is right in the comments about a possible low limit for mRNA vaccines, that’s a major downside. If it weren’t for that, it would almost surely be worth it to get a 3rd shot for most people by the calculus, since Delta is growing so fast and has so little morbidity for the vaccinated that there isn’t much incentive to keep it out of young and healthy circles. (Cf about 4 days of expected lost life from Delta while vaccinated, and getting a third shot would save you 2 expected days at a cost of two days lightly sick. But if there's much cost in the vein of what Elizabeth mentions, that washes out.)