Here’s a more intuitive way to view the first proof (under very slightly different assumptions).
Suppose that Omega offers two lotteries: a St. Petersburg lottery X∞ and a “half chance of a St. Petersburg lottery” 12X0+12X∞. Suppose I draw an outcome from X∞, I see what it is, and then I’m given the option to switch to the other (currently uncertain) lottery.
No matter what finite value I see, it’s very easy to argue that I’m going to want to switch (and I’d want to switch even if I truncated the second St. Petersburg lottery after some finite number of steps). But then by a reasonable dominance or “sure thing” principle I might as well just switch before I even look at the outcome.
But that implies 12X0+12X∞≥X∞, i.e. I’m just as happy with a half chance of a St. Petersburg lottery as a sure thing of a St. Petersburg lottery. And similarly I’m just as happy with an ε chance for any ε>0. That violates Intermediate Mixtures (and is generally just a kind of bizarre preference to have).
If you truncate the second lottery so that the outcome that you got from the first one doesn’t appear then you would not want to switch.
I am a little confused as I read this setup multiple times as “We draw a an outcome ‘jackpot’ from St. Petersburg. Then you get to choose between ‘jackpot’ and ‘heads goat, tails jackpot’”. If goat were just an arbitrary condolence price then it might happen that jackpot<goat and it might happen that jackpot>goat. But goat is not an unconnected outcome but the shittiest thing that St Petersburg can spit out. So jackpot>=goat and I am fine staying with jackpot. And I think the setup intends that the lotteries are drawn “separately”, that you can get a diffrent thing out. But in that case it escapes me how rolling one die would make me update in any direction about disconnected dice even if they have the same makeup. That somebody wins the lottery doesn’t make me update the value of lottery tickets upwards.
To my mind “half St. Petersburg” has a transfinite expectation value which is smaller than St. Petersburg. The main gist how the argument goes forward is imply that transfinite values breaking things sufficiently mean they need to have equal value. If you can’t say > and can’t say < one option is that the things are equal but another option is that they are incoparable or can’t be compared by those methods one wishes to use.
Oh, nice! It seems more irrational to me to violate this “sure thing” principle than the axioms in your post, or at least, this comment makes it clear that you can get Dutch booked and money pumped if you do so. You have a Dutch book, since the strategy forces you to commit to switching to a lottery that’s stochastically dominated by a lottery available at the start that you previously held (assuming X0 has identically 0 payoff). There’s also a money pump here, since Omega can offer you a new St. Petersburg lottery after you see the outcome of your previous lottery and charge you an arbitrarily large finite amount to switch.
Still, this kind of behaviour seems hard to exploit in practice, because someone needs to be able to offer you a finite unbounded lottery with infinite expected value (or something similar, if we aren’t using expected values).
Here’s a more intuitive way to view the first proof (under very slightly different assumptions).
Suppose that Omega offers two lotteries: a St. Petersburg lottery X∞ and a “half chance of a St. Petersburg lottery” 12X0+12X∞. Suppose I draw an outcome from X∞, I see what it is, and then I’m given the option to switch to the other (currently uncertain) lottery.
No matter what finite value I see, it’s very easy to argue that I’m going to want to switch (and I’d want to switch even if I truncated the second St. Petersburg lottery after some finite number of steps). But then by a reasonable dominance or “sure thing” principle I might as well just switch before I even look at the outcome.
But that implies 12X0+12X∞≥X∞, i.e. I’m just as happy with a half chance of a St. Petersburg lottery as a sure thing of a St. Petersburg lottery. And similarly I’m just as happy with an ε chance for any ε>0. That violates Intermediate Mixtures (and is generally just a kind of bizarre preference to have).
If you truncate the second lottery so that the outcome that you got from the first one doesn’t appear then you would not want to switch.
I am a little confused as I read this setup multiple times as “We draw a an outcome ‘jackpot’ from St. Petersburg. Then you get to choose between ‘jackpot’ and ‘heads goat, tails jackpot’”. If goat were just an arbitrary condolence price then it might happen that jackpot<goat and it might happen that jackpot>goat. But goat is not an unconnected outcome but the shittiest thing that St Petersburg can spit out. So jackpot>=goat and I am fine staying with jackpot. And I think the setup intends that the lotteries are drawn “separately”, that you can get a diffrent thing out. But in that case it escapes me how rolling one die would make me update in any direction about disconnected dice even if they have the same makeup. That somebody wins the lottery doesn’t make me update the value of lottery tickets upwards.
To my mind “half St. Petersburg” has a transfinite expectation value which is smaller than St. Petersburg. The main gist how the argument goes forward is imply that transfinite values breaking things sufficiently mean they need to have equal value. If you can’t say > and can’t say < one option is that the things are equal but another option is that they are incoparable or can’t be compared by those methods one wishes to use.
Oh, nice! It seems more irrational to me to violate this “sure thing” principle than the axioms in your post, or at least, this comment makes it clear that you can get Dutch booked and money pumped if you do so. You have a Dutch book, since the strategy forces you to commit to switching to a lottery that’s stochastically dominated by a lottery available at the start that you previously held (assuming X0 has identically 0 payoff). There’s also a money pump here, since Omega can offer you a new St. Petersburg lottery after you see the outcome of your previous lottery and charge you an arbitrarily large finite amount to switch.
Still, this kind of behaviour seems hard to exploit in practice, because someone needs to be able to offer you a finite unbounded lottery with infinite expected value (or something similar, if we aren’t using expected values).