Game theory, Aumann, Posterior Probability

Question

I have a small technical question related to the paper 'Agreeing to Disagree' (Aumann, 1976) by the Nobel laureate in Economics Aumann.

What I do not understand at all in what follows is why the posteriors for the event $A$ will be $\frac23$ and $\frac13$ respectively.

The crucial point is the following: "Suppose that the agents $1$ and $2$ have a uniform prior on the parameter of a coin, and let $A$ be the event that the coin will come up $H$ (heads) on the next toss. Suppose that each person is permitted to make one previous toss, and that these tosses come up Hand $T$ (tails) respectively.

If each one's information consists precisely of the outcome of his toss, then the posteriors for $A$ will be $\frac23$ and $\frac13$ respectively. If each one then informs the other one of his posterior, then they will both conclude that the previous tosses came up once $H$ and once $T$, so that both posteriors will be revised to $\frac12$".

Thank you very much in advance!

Answer 1

I think he suppressed some assumptions. Assume that each person has a uniform distribution on [0,1] for a prior on the probability of heads,r, f(r)=1/2. Assume one person sees heads. Then the posterior distribution pdf is $f(r|heads)=2r$. (see https://en.wikipedia.org/wiki/Checking_whether_a_coin_is_fair for more calculations). Then the expected value of r, given the posterior is $\int_0^1r\times 2r dr=\int_0^12r^2$ which becomes 2/3. The number wouldn't work out, if, say a person were almost certain that the probability of heads was 1/1000.