There were two things I wanted to do with the following
post. First, I wanted to get people’s feedback about a tricky decision problem.
Secondly, I wanted to tell a story, just for the sake of adding some story-telling
to the site. If you want to skip the story, and have the example given in a
more traditionally philosophical format, feel free to jump to the end of the
story, which is clearly marked with a horizontal line. There are many allusions
to various themes from economics, literature, philosophy and theology in the
story. Most of them are obscure, to sy the least. In some cases, you might reasonably
think that the most salient property the alluser and allusee have in common
is that they are members of a particular pair set. Despite that, I’m not going
to explain all of them, but there are some footnotes explaining some,
particuarly where I directly borrow ideas without clear citation in the story.
Walking out of the bar, Leopold and Stephen turned
right onto Upper Ormond Quay. It was a long walk to the park, especially in
the snow, but they seemed resigned to it. Stephen stopped at a cart selling
old books and picked up a tattered journal.
—This is the most bizarre theory of the afterlife
I’ve ever read, said Stephen. It claims that some of the people that are sentenced
to purgatory are occasionally moved around between heaven and hell because of
games they play with gods and angels. And it says that the values of days in
heaven and hell are equal, opposite and constant.
—Opposite relative to what, asked Leopold?
—Relative to time in purgatory, said Stephen.
So two days in heaven followed by two days in hell is exactly as good as four
days in purgatory. What I can’t believe is the constancy. It seems to me that
as you got used to heaven it would be less pleasurable than when you first arrive,
but as your pain tolerance diminished, hell would just get worse and worse.
—That’s a very depressing view of the afterlife.
I remember Pascal argued eloquently for the constant marginal utility of extra
days in heaven.
—Are you sure? I recall Jevons arguing finely that
everything has a constant marginal utility.
—Jevons? Don’t believe everything you read by Australians.
—No, I think he made a very plausible case.
—LOTTERY TICKETS!
—What? asked Stephen, turning around to see a small
boy holding out a small book.
—Lottery tickets, said the boy. First prize a week
in paradise.
—We’ve just been talking about paradise, haven’t
we Leopold? Maybe we should buy and if we win we can go and see if the seventh
day is as good as the first.
—I think the seventh day is normally better than
the first, said Leopold.
—Where, pray tell, is paradise boy? asked Stephen.
—Margate, said the boy.
—Paradise is in England? snorted Stephen. I doubt
even God can see through the clouds that hover over England.
At that Stephen and Leopold turned around to walk
off. But they never made it, as at that moment a runaway horse came tearing
up the path, trampling over Stephen and Leopold, killing them both.
Instants later, they were in Purgatory, with instructions
ringing in their ears that that was where they were to stay the afterlife. Stephen
thought it was a fair cop, but Leopold was understandably aggrieved at the injustice
of it all.
They were rather surprised to learn that the theories
of heaven and hell Stephen had recited just before their deaths were entirely
true. It took them a while to discover this, and it was mainly through Leopold’s
work that they learned it was true. At first, Leopold accepted every deal offered
by a passing angel. One day in hell followed by six in heaven. Sixteen days
in hell, with a day in heaven after the seventh. Four days in heaven for free.
The last deal from the angel who felt he must make up to Leopold for exploiting
his curiosity in the previous deal. Leopold took everything that came by, and
concluded that the value of days in heaven and hell really was equal, opposite
and constant.
Stephen disparaged all these journeys. Heaven isn’t
all it was cracked up to be. Life is a journey, and once you’ve arrived in heaven
you’ve got nowhere to go, so you may as well be dead. The chase, he’d say, is
always better than the kill. Leopold noted that Stephen disliked being reminded
of this little morsel of wisdom after another failed sexual adventure. But eventually
Stephen agreed with the common wisdom that the value of time in heaven and hell
is equal opposite and constant.
Purgatory was a dull and dreary place. A failed seaside
resort, perhaps, but even then there was something missing. So Leopold became
heavily involved with the gambling scene in Purgatory. He found it hard to follow
all the terminology sometimes. He didn’t understand how a shilling was a thousand
pounds, except when it was a shilling, which it almost never was because a shilling
didn’t buy you anything nowadays. So a shilling was a thousand pounds. And there
were some new currencies to learn. A hundred pounds made a cent, and a hundred
cents made a roam. And roams were very important, because you could trade a
roam for a day in heaven, and if you went into debt you had to buy back roams
from the bankruptcy court at one roam per day spent in hell. It was all very
painful, but it meant all debts got paid, which was nice. No one knew how this
currency came into being, nor how the bank could transfer roams for days in
heaven, or the bankruptcy court repay debts in this way, but when pushed most
experts turned out to be Boussetians.
—Oh look! There’s Buck, said Stephen. Should we
go and see what he’s up to?
—I suppose it can’t hurt, said Leopold, thinking
mostly of how he could trick Buck out of some roams.
When they wandered over, Buck was talking with an
angel called Daniele. Or at least that’s what Buck said the name was, though
Leopold and Stephen were suspicious, for the angel didn’t look Italian. Daniele
was explaining a particular deal Buck was being offered.
—A rock is an envelope with a number written inside
it. I figure out what number to write in it by flipping a coin repeatedly until
it lands tails.
—A fair coin? asked Buck suspiciously.
—Yes, it’s all perfectly fair, said Daniele. If
it lands tails the first time, I write 1, if it lands tails the second time
I write 2, if it lands tails the third time I write 4, the fourth time I write
8, and so on.
—And if it never lands tails? asked Buck, seeming
to recall something from his past.
—I tear up the envelope and start again, said Daniele.
—Doesn’t that take a while? asked Stephen.
—I flip the coins pretty fast, said Daniele. So,
that’s a standard rock. A tenrock is just like a normal rock, except I multiply
the numbers by ten. And for a hundredrock I multiply the numbers by one hundred.
And that, Buck my lad, is just what I have here. Two shiny hundred rocks, one
green and one gold.
—It looks more yellow than gold to me, said Buck,
still not trusting Daniele.
—Well, it’s faded a bit over time. So let’s call
it yellow. Here’s the deal. If you agree as many days in hell as is written
in the yellow envelope, you can spend as many days in heaven as is written in
the green envelope. What do you think?
—I don’t like that deal, said Buck, I don’t trust
those rocks.
—Hmmm, said Daniele meaningfully. What about this
deal then? If you agree to spend one billion days in hell, you can spend the
number of days in heaven written in the yellow envelope.
—NO! screamed Buck.
—That sounded fairly definite to me, said Daniele,
and started to drift off.
Leopold and Stephen, intrigued by the last exchange,
followed in pursuit.
—Why did you offer him that last deal? asked Stephen.
—I’m allowed to do anything I like to your poor
souls as long overall you’re better off having me around than not having me
around. So now in the future I can make up some rules for our Buck to break,
and then send him to hell for a decade or two for breaking them.
Stephen didn’t follow this at all, but he couldn’t
help but be impressed by the majesty of the apparent smile on Daniele’s face.
Meanwhile, Leopold was busy calculating what Buck
had been offered. How much and how many and how much and how many and how much
and how many and, oh this is going to take a while. Stephen always thought how
much and how many and so on was a foolish way to choose how to bet, but he had
to admit that Leopold had been doing fairly well since he started using it.
So if it’s going to take so long, is it worth even doing? Perhaps. How much
and how many, that’s 100 and a half, that’s 50, and how much and how many, that’s
200 and a quarter, that’s another 50, and 400 and an eight, that’s another 50,
and oh these rocks could be worth quite a bit. But how many fifties does it
take to make a billion?
—So, here’s four tenrocks, two green and two yellow.
Will you each take one of each colour? asked Daniele.
Stephen agreed, and so did Leopold.
—I’ll offer you each the same deal I offered Buck.
Well, perhaps with a twist. For you Leopold I’ll make your yellow envelope a
tenrock minus ten.
And, snapping his fingers, he did just that.
—What’s a tenrock minus ten? asked Leopold.
—It’s just like a tenrock, except I take ten off
the answer. So if your envelope previously said forty, it now says thirty. And
for you Stephen, I think I’ll make that yellow envelope a tenrock minus twenty.
Leopold started to splutter. Daniele ignored him
and clicked his fingers.
—What if my yellow rock only said ten, asked Stephen?
Now it says minus ten, but I can’t spend minus ten days in hell.
—If it’s negative they’ll be extra days in heaven,
said Daniele.
—Why does he get twenty off and I only get ten off?
asked Leopold who managed to regain his voice.
Ten roam was more than a man could make in a year
of good gambling, and here Stephen was being given it on a whim.
—I like Catholics, said Daniele. Besides, you’re
doing better than ugly Buck.
Nobody noticed that Leopold rudely resumed spluttering
as loudly as ever. Stephen started twirling his envelope around aimlessly while
trying to think about the deal.
—Oh disaster, said Daniele. Stephen’s green rock
seems to be a window envelope. I do hope you haven’t seen the number inside
there before you made the decision.
Stephen had stopped twirling the green envelope,
and was now staring at it in shocked disbelief. Daniele, in a rare feat of interpretative
genius, took that to be a negative answer, and pressed on for an answer.
—What’ll it be boys? Green days and yellow days
for a while, or no?
—I’m in, said Leopold, though I think it’s rotten
how you gave him a better deal than I.
Stephen went to speak, but was clearly struggling
to produce words.
—I, I, I suppose I must, I have to, that is, I can’t,
I’m sure it’s a very generous offer, well almost sure that is, there’s the rub.
Almost. I decline, he finally said. But you won’t be sending me to hell will
you?
—Much as I’d like to, I think someone might think
I’m overstepping my role if I did. So I’ll take Leopold and my leave now.
—Let’s see, how long are you going to be in heaven
for, said Daniele when they were safely out of earshot. It’s heaven first you
know, I hope you don’t mind.
—I think I’d rather not know how long I’m there
for, said Leopold, I’d rather it be a surprise.
—Suit yourself, said Daniele, but I have to book
rooms for you upstairs and down, so let’s see those envelopes.
He tore open the first, and took out a telephone.
—Central bookings? Yes, I’d like to reserve a place
for Leopold for
Leopold tried to put his fingers in his ears, but
found that he now lacked fingers, and ears.
—eighty days. That’s pretty good isn’t it, Daniele
said turning to Leopold.
It was the worst moment of Leopold’s life.
—Now let’s see what you’ve got at the other end.
He tore open the other envelope.
—Zero, so no hell for you this time. A lucky break
all in all I’d say. Do you want to know what Stephen turned down?
—Yes, said Leopold, only just having regained his
composure.
—The green envelope says 20,480, this one goes to
eleven, and, let me see what’s in this yellow envelope, minus 10, so make that
20,490 days. Still, I think he made the right choice, don’t you Leopold?
—20,490 days, that’s practically a lifetime.
—Terrific thinking Leopold, said Daniele. Now try
looking at the envelopes from Stephen’s perspective.
—Oh I see, said Leopold after a pause. Yes, I think
he did the right thing. And so did I I think. Well, all’s well that ends well.
When are you coming back to Purgatory?
Daniele didn’t answer, and Leopold was rushed into
his Heavenly rooms.
Who
made the right decisions: Buck, Leopold or Stephen?
So here’s a summary of the decisions the characters
faced, for those who wanted to skip the story.
First, some stipulations. The default setting for
the agents is that they will spend eternity in Purgatory. They can, from time
to time, rise to Heaven or fall to Hell. The value of time spent in these places
is, as Stephen puts it, equal, opposite and constant. So five days in Heaven
and five days in Hell, in any order, is just as worthwhile for an agent as ten
days spent at the default condition, being in Purgatory. The constant marginal
utility of time out of Purgatory means that a 50% chance of ten days in Heaven
plus a 100% chance of 5 days in Hell is worth literally nothing to our agents.
A rock is a special kind of unknown value object.
It is an envelope with a number inside it. A green envelope means that the bearer
gets to spend that many days in heaven. A yellow envelope means that the bearer
must spend that many days in hell. The number is chosen by flipping a fair coin
repeatedly until it turns up tails. (It’s a St
Petersburg process, for those who like thinking of things that way.) If
it turns up tails the first time, 1 is written in the envelope, 2 if tails the
second time, 4 if tails the third time, 8 the fourth time, and so on. A tenrock
is a rock where all those numbers are multiplied by ten. A hundred rock is a
rock where the numbers are multiplied by 100. A tenrock minus x is a rock where
the numbers are multiplied by ten and then x is subtracted from the result.
If this leads to a negative number then the bearer is sent to the opposite place.
(So a green envelope saying minus fifteen means fifteen days in Hell, and a
yellow envelope saying minus ten, as turns up towards the end of the story,
means ten days in heaven.)
The first deal offered is that the angel ‘Daniele’
offers Buck a green hundredrock and a yellow hundredrock. Buck declines the
offer. Daniele then changes the deal. Buck is now offered a green hundredrock
in exchange also accepting a billion days in Hell. Buck declines this offer
as well, and we are never told what is in his green hundredrock.
The second deal is that Daniele offers Leopold a
green tenrock in exchange for also accepting a yellow tenrock minus ten. Leopold
accepts the deal, and ends up spending eighty days in Heaven, and none in Hell.
The third deal is that Daniele offers Stephen a green
tenrock in exchange for also accepting a yellow tenrock minus twenty. Before
he can accept or decline, Stephen sees that his green envelope contains the
number 20,480. This doesn’t make Daniele withdraw the offer, but Stephen declines
it. It turns out Stephen’s yellow envelope says minus 10, so he turned down
20,490 days in Heaven.
So,
which of the three characters do you think made the right choice? Click
on that link to answer.
Occasional Footnotes.
This is the most bizarre theory of the afterlife
I’ve ever read, said Stephen. Stephen is reading Frank Arntzenius and David
McCarthy’s "The Two Envelope Paradox and Infinite Expectations" (Analysis
1997) as theology.
I recall Jevons arguing finely. Stephen has
been reading, or perhaps misreading, this
paper, especially sections 8 and 9.
For you Leopold I’ll make your yellow envelope
a tenrock minus ten. I’m not sure where I learned of the decision problem
Daniele sets Leopold here. I think I learned it during a course on infinity
that Graham Oppy
ran when I was at Monash. But I’m really not sure. So this is a disclaimer of
originality.