I’m not sure about the implications, but I just want to register that deciding to roll repeatedly, after each roll for a total of n rolls, is not the same as committing to n rolls at the beginning. The latter is equivalent in expected value to rolling every trial at the same time: the former has a much higher expected value. It is still positive, though.
The cumulative EV of n decisions to roll repeatedly is
A:∑ni=0[u×2i×p]=up∑ni=02i (where u is the initial utility of 10, and p stays constant at 63−163)
whereas the EV of committing to roll up to n times is
B:upn∑ni=02i
Which is much-much lower than A, as you point out.
But then again, for larger values of n, you’re very unlikely to be allowed to roll for n times. The EV of n decisions to roll (A) times the probability of getting to the nth roll (pn−1), is
A×pn−1=(up∑ni=02i)×pn−1=upn∑ni=02i=B
In other words, A collapses to B if you don’t assume any special luck. Which is to say that committing to a strategy has the same EV ex ante as fumbling into the same path unknowingly. This isn’t very surprising, I suppose, but the relevancy is that if you have a revealed tendency to accept one-off St. Petersburg Paradox bets, that tendency has the same expected utility as deliberately committing to accept the same number of SPPs. If the former seems higher, then that’s because your expectancy is wrong.
More generally, this means that it’s important to try to evaluate one-off decisions as clues to what revealeddecision rules you have. When you consider making a one-off decision, and that decision seems better than deliberately committing to using the decision-rules that spawned it, for all the times you expect to be in similar situation, then you are fooling yourself and you should update.
If you can predict that the cumulatively sum of the EV you assign to each one-off decision individually as you go along will be higher, compared to the EV you’d assign ex ante to the same string of decisions, then something has gone wrong in one your one-off predictions and you should update.
I’ve been puzzling over this comment from time to time, and this has been helpfwly clarifying for me, thank you. I’ve long been operating like this, but never entirely grokked why as clearly as now.
“You never make decisions, you only ever decide between strategies.”
Thanks for writing this up!
I’m not sure about the implications, but I just want to register that deciding to roll repeatedly, after each roll for a total of n rolls, is not the same as committing to n rolls at the beginning. The latter is equivalent in expected value to rolling every trial at the same time: the former has a much higher expected value. It is still positive, though.
The cumulative EV of n decisions to roll repeatedly is
A:∑ni=0[u×2i×p]=up∑ni=02i
(where u is the initial utility of 10, and p stays constant at 63−163)
whereas the EV of committing to roll up to n times is
B:upn∑ni=02i
Which is much-much lower than A, as you point out.
But then again, for larger values of n, you’re very unlikely to be allowed to roll for n times. The EV of n decisions to roll (A) times the probability of getting to the nth roll (pn−1), is
A×pn−1=(up∑ni=02i)×pn−1=upn∑ni=02i=B
In other words, A collapses to B if you don’t assume any special luck. Which is to say that committing to a strategy has the same EV ex ante as fumbling into the same path unknowingly. This isn’t very surprising, I suppose, but the relevancy is that if you have a revealed tendency to accept one-off St. Petersburg Paradox bets, that tendency has the same expected utility as deliberately committing to accept the same number of SPPs. If the former seems higher, then that’s because your expectancy is wrong.
More generally, this means that it’s important to try to evaluate one-off decisions as clues to what revealed decision rules you have. When you consider making a one-off decision, and that decision seems better than deliberately committing to using the decision-rules that spawned it, for all the times you expect to be in similar situation, then you are fooling yourself and you should update.
If you can predict that the cumulatively sum of the EV you assign to each one-off decision individually as you go along will be higher, compared to the EV you’d assign ex ante to the same string of decisions, then something has gone wrong in one your one-off predictions and you should update.
I’ve been puzzling over this comment from time to time, and this has been helpfwly clarifying for me, thank you. I’ve long been operating like this, but never entirely grokked why as clearly as now.