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.
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?