These are decisions in different situations. Transitivity of preference is about a single situation. There should be three possible actions A, B and C that can be performed in a single situation, with B preferred to A and C preferred to B. Transitivity of preference says that C is then preferred to A in that same situation. Betting on a fight of B vs. A is not a situation where you could also bet on C, and would prefer to bet on C over betting on B.
Also—if we have a set of 3 non-transitive dice, and I just want to roll the highest number possible, then I can prefer A to B, B to C and C to A, where all 3 dice are available to roll in the same situation.
If I get paid depending on how high a number I roll, then this would seem to prevent me from becoming a money pump over the long term.
Thanks very much for your reply Vladimir. But are you sure that is correct?
I have never seen that kind of restriction to a single choice-situation mentioned before when transitivity is presented. E.g. there is nothing like that, as far as I can see, in Peterson’s Decision theory textbook, nor in Bonano’s presentation of transitivity in his online Textbook ‘Decision Making’. All the statements of transitivity I have read just require that if a is preferred to b in a pairwise comparison, and b is preferred to c in a pairwise comparison, then a is also preferred to c in a pairwise comparison. There is no further clause requiring that a, b, and c are all simultaneously available in a single situation.
These are decisions in different situations. Transitivity of preference is about a single situation. There should be three possible actions A, B and C that can be performed in a single situation, with B preferred to A and C preferred to B. Transitivity of preference says that C is then preferred to A in that same situation. Betting on a fight of B vs. A is not a situation where you could also bet on C, and would prefer to bet on C over betting on B.
Also—if we have a set of 3 non-transitive dice, and I just want to roll the highest number possible, then I can prefer A to B, B to C and C to A, where all 3 dice are available to roll in the same situation.
If I get paid depending on how high a number I roll, then this would seem to prevent me from becoming a money pump over the long term.
Thanks very much for your reply Vladimir. But are you sure that is correct?
I have never seen that kind of restriction to a single choice-situation mentioned before when transitivity is presented. E.g. there is nothing like that, as far as I can see, in Peterson’s Decision theory textbook, nor in Bonano’s presentation of transitivity in his online Textbook ‘Decision Making’. All the statements of transitivity I have read just require that if a is preferred to b in a pairwise comparison, and b is preferred to c in a pairwise comparison, then a is also preferred to c in a pairwise comparison. There is no further clause requiring that a, b, and c are all simultaneously available in a single situation.