yesterday i ran a poll with a coin flip trolley problem.

unfortunately, the clear majority answer leads to EVERYONE dying with 100% probability!!

what happened??


here's the poll. i took @ole_b_peters' excellent coin flip betting scenario and recast it as a trolley problem

it's an example of a problem where the expected value and the long-run average are NOT the same! this blew my mind when i first heard it


the expected value calculation is simple

50% chance of gaining 50% (×150%)
50% chance of losing 40% (×60%)
→ 0.5×1.5 + 0.5×0.6 = 1.05

so on average, the expected GAIN is 5% per coin flip

so how can that possibly lead to LOSING 100% with 100% probability??

to factor out the risk aversion, this follow-up poll flips the coin infinitely many times. that way, you DEFINITELY get the long run average, instead of having to worry about getting an unlucky run.

unfortunately, in the long run it always goes to 0


consider what happens when you flip 1 heads and 1 tails.

you go up 50% but down 40%.
150% × 60% = 90%

so with an even number of heads and tails, you LOSE 10% every two rounds

ALL of the positive expected value comes from EXTREMELY rare streaks of many extra heads flips

how rare?

well, exceptionally rare. exponentially rare. in the limit of infinite flips, so rare that the probability actually goes to zero.

if you're uncomfortable with the infinity, then flip for "only" some reasonable finite time – say one flip per second for a week

you could reset and try that over & over for the life of the universe and never find a single positive week-long run

if you have 20 minutes, you can watch this video where ole peters walks you through the problem step by step.

(it's intended for a general audience, so you don't need any crazy math prerequisites to watch it!)


hopefully this is a useful example for the discussion on "risk-neutral utilitarianism"

we often motivate bayes with "dutch book" arguments

this is kinda a dutch book for risk-neutral utilitarianism. you'd keep flipping the +EV coin forever, until everyone's dead :(

wait i forgot the punchline!!!!

this is an example of a "non-ergodic" system: even as N→∞, individual runs DON'T converge to the expected value.

that is,,,,,


the end.
thank u for reading :)

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