Spent some time with this as recreation from AI Class. Now see by drawing a probability diagram the 2/3 chance of winning for swappers is obvious:
Set swap-probability x to 1 and there is a 2/3 chance of winning the car.
With swap-probability 1/2 and there is a 1/2 chance of winning.
p.s. for posterity: the only comprehensive description of Bayes Ball I found at the time was in these slides nos. 21, 22, 23.