The Allais Paradox is indeed quite puzzling. Here are my thoughts:
0. Some commenters simply dismiss Bayesian reasoning. This doesn't solve the problem, it just strips us of any mathematical way to analyze the problem. On the other hand, the fact that the inconsistent choice seems ok does mean that the Bayesian way is missing something. Simply dismissing the inconsistent choice doesn't solve the problem either.
1. If I understand correctly, you argue that situation 1 can be turned into situation 2 by randomization. In other words, if you sell me situation 1, I can sell somebody else (named X) situation 2 by throwing some dies and using your offer. More specifically, I throw a 100-sided die. If it's > 34, X looses. Otherwise, I play X's option with you. However, this can't be reversed. Given only situation 2, I can't sell situation 1, assuming I have only $0 initial capital.
Hence, it seems that assuming invertibility of situations (I can both buy and sell them) and unlimited money buffers for that purpose are important for the demanded consistency.
The Allais Paradox is indeed quite puzzling. Here are my thoughts:
0. Some commenters simply dismiss Bayesian reasoning. This doesn't solve the problem, it just strips us of any mathematical way to analyze the problem. On the other hand, the fact that the inconsistent choice seems ok does mean that the Bayesian way is missing something. Simply dismissing the inconsistent choice doesn't solve the problem either.
1. If I understand correctly, you argue that situation 1 can be turned into situation 2 by randomization. In other words, if you sell me situation 1, I can sell somebody else (named X) situation 2 by throwing some dies and using your offer. More specifically, I throw a 100-sided die. If it's > 34, X looses. Otherwise, I play X's option with you. However, this can't be reversed. Given only situation 2, I can't sell situation 1, assuming I have only $0 initial capital.
Hence, it seems that assuming invertibility of situations (I can both buy and sell them) and unlimited money buffers for that purpose are important for the demanded consistency.