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As per above:
"When should Player A announce a double, supposing that she owns the doubling cube? If P(t) < 1a and she announces a double, the opponent will accept the double on the grounds that his expected payoff is greater than S. This is so because his probability of winning, which is 1P(t), exceeds a. As the expected payoff for Player B is minus the expected payoff for Player A, this is undesirable for Player A. Hence, she must wait until P(t) >= 1a, in which case Player B will refuse the double and she wins the stake. This argument shows that necessarily b=1a = 4/5"
I think this is flawed reasoning, specifically the statement that "as the expected payoff for Player B is minus the expected payoff for Player A, this is undesirable for Player A". This misses the distinction that while Player B has to compare the expected payoff from accepting to S, the payout from not accepting, Player A has to compare the expected payoff from doubling to P(t) x S, the expected payoff from not doubling.
This means that in any case where the expected payoff for B from accepting the double is between S and (P(t) x S), it is both correct for A to double and for B to accept the double.
For instance, if P(t) > a such that E(Baccept double) = 0.9S, then A should double as he increases his payoff from <0.8S to 0.9S, but B should accept the 0.9S expected payout because its better than rejecting and forfeiting S.