Trying to make this more intuitive: consider a prediction market which is currently priced at x, where each share will pay out $1 if it resolves as True.
If you think it’s underpriced because your probability is y, where y>x, then your subjective EV from buying a share is y-x. e.g., If it’s priced at $0.70 and you think p=0.8, your subjective EV from buying a share is $0.10.
If you think it’s overpriced because your probability is z, where z<x, then your subjective EV from selling a share is x-z. e.g., If it’s priced at $0.70 and you think p=0.56, your subjective EV from selling a share is $0.14.
Those two will be equal if x is halfway between y and z, at their arithmetic mean.
So if two people disagree on whether the price should be y or z, then they will have equal EV by setting a price at the arithmetic mean of y & z, and trading some number of prediction market shares at that price. i.e., The fair (equal subjective EV) betting odds are at the arithmetic mean of their probabilities.
Trying to make this more intuitive: consider a prediction market which is currently priced at x, where each share will pay out $1 if it resolves as True.
If you think it’s underpriced because your probability is y, where y>x, then your subjective EV from buying a share is y-x. e.g., If it’s priced at $0.70 and you think p=0.8, your subjective EV from buying a share is $0.10.
If you think it’s overpriced because your probability is z, where z<x, then your subjective EV from selling a share is x-z. e.g., If it’s priced at $0.70 and you think p=0.56, your subjective EV from selling a share is $0.14.
Those two will be equal if x is halfway between y and z, at their arithmetic mean.
So if two people disagree on whether the price should be y or z, then they will have equal EV by setting a price at the arithmetic mean of y & z, and trading some number of prediction market shares at that price. i.e., The fair (equal subjective EV) betting odds are at the arithmetic mean of their probabilities.