I think the resulting odds won’t reflect the probability of anything, because they depend a lot on whether Alice or Bob is more risk-tolerant (=rich).
If one of them is willing to tolerate risk equal to the value of Judy’s time to hear out the argument, then you are fine. If the total willingness to risk of people who believe “Judy will believe X on reflection” is lower than the value of Judy’s time, then I think you are basically inevitably stuck unless Judy is willing to risk her own attention. If she is willing to risk her own attention, then she can just give people a budget of “minutes” to spend making wagers, as discussed in the post, and as long as the budget is large enough relative to the size of the disagreement it seems like you are OK.
Also, it seems to me that your scheme works best for yes/no questions. For anything more complicated, Alice and Bob can cooperate to mislead Judy, which is especially scary in case of AIs. I’m not sure how to fix that problem: it seems to require a way for a non-expert to check the work of a malicious expert, not just adjudicate between two experts.
The scheme works if one of the experts advocates for the truth. If there are two options, and both players want to manipulate Judy into believing “yes,” then you are similarly in trouble. I agree that if there are more options than experts then it becomes less likely that “by chance” someone wants to advocate for the right answer. But I think in general you are banking on there being some density of experts who want to argue for the truth because it is the truth.
If the total willingness to risk of people who believe “Judy will believe X on reflection” is lower than the value of Alice’s time