I would *really* appreciate any help from lesswrong readers in helping me understand something really basic about the standard money pump argument for transitivity of preferences.
So clearly there can be situations, like in a game of Rock Scissors Paper (or games featuring non-transitive dice, like ‘Efron’s dice’) where faced with pairwise choices it seems rational to have non-transitive preferences. And it could be that these non-transitive games/situations pay out money (or utility or whatever) if you make the right choice.
But so then if these kinds of non-transitive games/situations are paying out money (or utility or whatever) I don’t quite see how the standard money pump considerations apply? Sure, I might pay some amount to have Rock over Scissors, and some small amount to have Scissors over paper, and some small amount to have Paper over Rock, etc. But if each time I am also *gaining* at least as much money by making these choices, then I am not being turned into a money pump.
So this seems like a really simple counter-example- i.e. far too simple!! – where having non-transitive preferences seems rational and also financially advantageous What am I missing? I realise I almost certainly have some totally basic and stupid misunderstanding of what the money pump argument is supposed to show.
Rock paper scissors isn’t an example of nontransitive preferences. Consider Alice playing the game against Bob. It is not the case that Alice prefers playing rock to playing scissors, and playing scissors to playing paper, and playing paper to playing rock. Why on Earth would she have preferences like that? Instead, she prefers to choose among rock, paper and scissors with certain probabilities that maximize her chance of winning against Bob.
Yes I phrased my point totally badly and unclearly.
Forget Rock Scissors paper—suppose team A loses to team B, B loses to C and C loses to A. Now you have the choice to bet on team A or team B to win/lose $1 - you choose B. Then you have the choice between B and C—you choose C. Then you have the choice between C and A—you choose A. And so on. Here I might pay anything less than $1 in order to choose my preferred option each time. If we just look at what I am prepared to pay in order to make my pairwise choices then it seems I have become a money pump. But of course once we factor in my winning $1 each time then I am being perfectly sensible.
So my question is just – how come this totally obvious point is not a counter-example to the money pump argument that preferences ought always to be transitive? For there seem to be situations where having cyclical preferences can pay out?
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.
I would *really* appreciate any help from lesswrong readers in helping me understand something really basic about the standard money pump argument for transitivity of preferences.
So clearly there can be situations, like in a game of Rock Scissors Paper (or games featuring non-transitive dice, like ‘Efron’s dice’) where faced with pairwise choices it seems rational to have non-transitive preferences. And it could be that these non-transitive games/situations pay out money (or utility or whatever) if you make the right choice.
But so then if these kinds of non-transitive games/situations are paying out money (or utility or whatever) I don’t quite see how the standard money pump considerations apply? Sure, I might pay some amount to have Rock over Scissors, and some small amount to have Scissors over paper, and some small amount to have Paper over Rock, etc. But if each time I am also *gaining* at least as much money by making these choices, then I am not being turned into a money pump.
So this seems like a really simple counter-example- i.e. far too simple!! – where having non-transitive preferences seems rational and also financially advantageous What am I missing? I realise I almost certainly have some totally basic and stupid misunderstanding of what the money pump argument is supposed to show.
Rock paper scissors isn’t an example of nontransitive preferences. Consider Alice playing the game against Bob. It is not the case that Alice prefers playing rock to playing scissors, and playing scissors to playing paper, and playing paper to playing rock. Why on Earth would she have preferences like that? Instead, she prefers to choose among rock, paper and scissors with certain probabilities that maximize her chance of winning against Bob.
Yes I phrased my point totally badly and unclearly.
Forget Rock Scissors paper—suppose team A loses to team B, B loses to C and C loses to A. Now you have the choice to bet on team A or team B to win/lose $1 - you choose B. Then you have the choice between B and C—you choose C. Then you have the choice between C and A—you choose A. And so on. Here I might pay anything less than $1 in order to choose my preferred option each time. If we just look at what I am prepared to pay in order to make my pairwise choices then it seems I have become a money pump. But of course once we factor in my winning $1 each time then I am being perfectly sensible.
So my question is just – how come this totally obvious point is not a counter-example to the money pump argument that preferences ought always to be transitive? For there seem to be situations where having cyclical preferences can pay out?
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.