Links to Dahl's "Man from the South"
The
questions are repeated in this post.
Preliminary answers
Q1. What does American Boy think is the
probability of his lighter lighting during any single attempt?
Most people accept a bet when they
think the odds are at least 5050. Since Man from the South and
American Boy have had time to negotiate the terms and think about it,
both think they can win it. So as neutral observers we can consider
the probability that the American Boy will win the bet to be 0.5.
So what does this tell us about what he
thinks is the probability of the lighter lighting on any one attempt?
Let this probability be p.
The probability that he will win the bet, i.e. that it will light N
times out of N trials, is p^{N}.
We just decided that this is ½, for N=10. So,
p^{10} = ½
I was
driving and couldn't very well use my laptop (my phone is not a very
smart phone), leave alone the log tables, slide rule or abacus in my
back seat. And I can't do powers of 1/10 in my head, not in general.
But I can calculate powers of numbers near 1! So let's calculate
q=1p instead, which is bound to be small since the boy is so
sure of success.
½ = (1q)^{N}
~ 1qN, so for N= 10, q=0.5/10
= 0.05.
Which
gives us our first answer: p = 95%, which we know is a bit of
an overestimate. (The exact answer is 93.3%.)
But this
is like focusing on conversion ratio and not on costperaction.
Q4. Were there to have been a 9^{th}
attempt, and were American Boy to have failed in it, what would he
have lost?
A: His left pinky, those were the terms
of the bet. So American Boy is staking his pinky every time! All but
the last time, if he wins, all he wins is the right to stay in the
game, if he loses, chopchop (and the right to play!)
So this is kinda sorta like conversion
attribution: every bidrequest we (RFI) win and then every impression
we serve only gives us the right to stay in the game, in the sense
that if we don't serve the impression we certainly won't get
attribution credit.
Q2. How many fingers to a Cadillac?
On the face of it, it would seem to be
one finger to a Cadillac, C = F, since if he loses, he loses a
finger, and if he wins, he wins a Cadillac. But as we saw above, the
American Boy stakes his one finger 10 times and thus faces 10
opportunities to lose his finger vs. one opportunity to win the
Cadillac. So,
10F = C
After the eighth successful attempt,
Man from the South's wife surprises them and ends the game. She sends
American Boy away empty handed.
Q3. How should he have been
compensated, if at all?
Aha! “How to distribute the spoils in
an interrupted game?” The very question that my colleague Jack
pointed out was the leading cause of the rise of probability theory!
The way I think of this is in terms of
“vesting”. Each time they play and American Boy wins, he wins a
10^{th} of a Caddy, but, BUT, he only gets to keep his
winnings (the entire Cadillac) if
he keeps his head (figuratively) and finger (literally) for 10
trials. American Boy can't get cold feet and walk out after say 5
trials and demand half a Cadillac – so really he can
get cold feet and walk out but he can't drive off. So there are two
possibilities. Under one, the Man from the South gets cold feet and
decides not to play anymore. In this case he forfeits his Cadillac,
which was held in escrow by the narrator. Under the other
possibility, the “authorities” intervene. Since neither party has
broken their terms, in this case I think American Boy gets to keep
his “unvested” winnings, which would be 8/10 of a Cadillac. Now
since the Cadillac wasn't the possession of the Man to begin with,
but belonged to the “authorities”, … this isn't a math problem
anymore. (By the way, who intervened and broke up Cardano and
Pascal's poker game?)
Legal opinions?
Aji, Joanne?
Q1 (Reevaluated). What does American
Boy think is the probability of his lighter lighting during any
single attempt?
Most people accept a bet when they
think it is a game with at least a zero sum in their favor. Since Man
from the South and American Boy have had time to negotiate the terms
and think about it, both think they can win it. So as neutral
observers we can consider the game to be a zerosum game.
What does it mean for this to be a
zerosum game? Winnings * probability of winning – losses *
probability of losing = 0! Which yields:
p = 1/(W/L + 1). (Check: if W are high,
p is low; if W are nearly 0, p is nearly 1 and if W=L, p =1/2.)
In our case the winnings are the
Cadillac C, the losses are the finger F and the probability P of
winning the entire game is P = p^{N}, where p
is the probability of the lighter lighting in a single trial.
Combining things we have
C*P – F*(1P) = 1,
or P = 1/(C/F + 1)
Putting in C/F = 10,
p^{10} = 1/11
1/11 = (1q)^{N}
~ 1qN, so for N= 10, q=1/11
= 0.1.
Which
gives us 78.7%.
Really?
Would you play that game with a lighter which only lights less than
80% of the time? I think that the Man from the South has fuddled the
American Boy into undervaluing his finger, by making him think he is
wagering a finger vs. a Cadillac, whereas really he is wagering a
finger against a 10^{th} of a Cadillac.
Some
preliminaries
If you aren't interested in the nuts and bolts, skip them, but this is so anyone can check my work.
Notation: Sum[i,
0, Infinity] f(i) is to be interpreted as the sum of the function or
series f(i) over the index i from i=0 to i= Infinity. Then,
Sum[n,0,Infinity]
p^{n} = 1/(1p)
Sum[n,0,N]
p^{n} = (1p^{N+1})/(1p)
and
Sum[n,1,N]
p^{n} = (1p^{N})*p/(1p)
Look at
the following table of outcomes of consecutive tosses and the overall
probabilities
W ←
1 → L
1: p
(1p) (End)
2: p^{2}
p(1p) (End)
3:
p^{3} p^{2}(1p) (End)
…
So after
n trials, the probability of winning all is p^{n} and the
probability of losing any is
(1p)* Sum[i,0,n1] p^{i} = (1p) * (1p^{n})/(1p) =
1p^{n} = 1 prob(Winning). Which is good since it indicates
I can still sum correctly.
Back to
the problem, to get a handle on
what if anything AB deserves
when the game is stopped.
From
making the last, Nth, trial a zerosum game, we know that
p^{N}
= 1/(C/F + 1). We've assumed that AB wins 1/N th of a Cadillac
(virtually) when his lighter lights. So assuming the first trial is
also a 0sum game, we have:
p*C/N =
(1p) *F, which resolves as
p =
(C/NF + 1) ^(1).
Can
these two equations be solved simultaneously for both C/F and p?
Yes, but
the solution is i) independent of N and ii) meaningless:
We have
(1+C/NF)^N = (1 + C/F), but the RHS is simply the first two terms in
the binomial expansion for the LHS, so the equality holds only when
C/F = 0 and p =1.
So one
of our assumptions above is wrong.
Let's try another tack.
Assume
that at the nth trial, AB wins some unknown portion of the Cadillac
a(n)*C. (When Ari and I were talking about this last week, Ari
guessed, “Wouldn't it be some quadratic or increasing portion that
he wins?” Ari's motivation was to take into account the wearing
out of the flint, the gas running out, the thumb getting tired, AB
getting nervous etc. Dahl, spends an entire paragraph describing the
care and attention to detail taken by AB, after each light, he blows
on the lighter, closes the lighter, waits a few seconds perhaps for
gas pressure to build up again, reopens it and then flicks it once.
) What we know is that
Sum[n,1,N]
a(n) = 1, over the course of the entire game, if he survives, he wins
the entire Cadillac. So at every trial, AB stands to lose not just
his finger and the right to play, but also the “won but not vested”
portion of the Cadillac, and he stands to win some portion of the
remaining.
At the
last, Nth trial: W : L
p
: (1p)
a(N)C
: F + (1a(N))C
Using
the 0sum equation, we get:
a(N) =
(1 +F/C)(1p).
Great,
so now we have ONE equation and 3 unknowns: a(N), F/C and p. But we
also know that if the game is 0Sum over all: p^N * C = F*(1p^N),
which
yields:
0 < p
= (1+C/F)^(1/N) < 1. So now at least we have two equations for
three unknowns, and the solution for p is valid. This doesn't
guarantee that 0
If we
knew C/F, we could solve the problem. However, clearly, C=F is no
longer valid. C = 10F could be used for AB's assumption. We also have
another source of information: The Man from the South's wife explains
that he has lost eleven cars and taken fortyseven fingers. Assuming
that he considers these equivalent, we have 11C = 47F
so,

The fraction C/F is the ratio of the value of one Cadillac to the value of one Finger. 

But all
this hasn't answered the question of
what AB deserves when the game
is interrupted.
Also, we've assumed that the entire game is 0sum and
that the last trial is 0sum. Can't we make use of the assumption
that
every trial is also 0 sum
to see if we can figure out the
intermediate nonvested winnings?
Recall
that at the nth trial, AB stakes his previous winnings and his finger
for a chance to win a(n)*C portion of the Cadillac. So the 0Sum
equation for the nth trial is:
a(n)*C =
(1/p 1) *(F + C* Sum[i, 0, n1] a(i)). The resulting recursion
relation is for a geometric series!
a(n+1) =
(1/p)* a(n),
whose
solution is
a(n) =
a(0)/p^n.
Note
immediately that a(0) != 0, so AB has to stake (even if only
virtually) something more than just his finger.
We find a(0)
by using
the fact that the total portion of the Cadillac gained over 10 trials
is 1:
1 =
Sum[i,1,N] a(i) = a(0) * Sum[i,1,N] (1/p)^i = a(0) * (1/p^N
1)/(1p), or
a(0) =
(1p)/(1/p^N – 1)
For the
first trial (note that this is independent information since so far
we have used the recursion relation and established a(0) using the
“normalization”, but we haven't yet used any
“initial
conditions”):
a(1)*C =
(1/p – 1) * ( F + a(0)*C), which
yields
C/F
= ((1/p)^{N}
1)/p, which is a different relationship between C/F, N and p than we
had previously. I am not sure I can invert this to yield p(C/F), but
it can certainly be numerically solved.
Let
us also calculate the virtual winnings after each trial:
W(n)
= C*Sum[i,1,n]a(i) = C*a(0)*(1/p^n 1)/(1p), which simplifies to
W(n)
= ((1/p)^{n}
 1)/((1/p)^{N}
 1)
So
given p we could calculate C/F (or vice versa), a(0), a(n) and W(n)
Working
on the “wife's numbers”, we see that the American Boy
should be compensated with 66% of the Cadillac when the wife
interrupts the game after the 8^{th} trial.
How
do the winnings increase as the trials proceed:

Winnings in units of "Cadillacs" 

which
look like

Levenfeld curve 

Conclusion
and final answers:
Q1. What does American Boy think is the
probability of his lighter lighting during any single attempt?
About 80%.
Q2. How many fingers to a Cadillac?
In American Boy's valuation based on
his behaviour, 10 fingers to a Cadillac.
After the eighth successful attempt,
Man from the South's wife surprises them and ends the game. She sends
American Boy away empty handed.
Q3. How should he have been
compensated, if at all?
With 60 or 66% of a Cadillac. I would
go with 66%, which is based on the Man from the South's experienced
equivalence between Cadillacs and fingers.
Q4. Were there to have been a 9^{th}
attempt, and were American Boy to have failed in it, what would he
have lost?
Ah, his finger, of course, and, his
virtual stake, which is 3.9 % of a Cadillac. How could he have lost
something he never had to begin with? Well, for the bet to proceed,
AB would have had to ask the narrator to spot him 4% of a Cadillac,
or its cash equivalent, or its (OUCH) finger equivalent, which is
17% (3.9% * 4.3 F/C).
If AB's lighter fails during the game,
he loses 1.17 fingers since he would have to sell 0.17 fingers to the
Man from the South to pay off the debt to the narrator. If AB's
lighter doesn't fail during the game, he simply returns the cash or
Cadillac equivalent from whoever he borrowed it, and is ahead one
Cadillac.
Why did I ever think of approaching the
problem this way, with a “virtual stake”? In particle physics,
one can borrow virtual particles from the vaccuum in order to
simplify calculations. It is all halal as long as the virtual
particles don't violate any conservation laws for quantum numbers and
the massenergy of the particles exists for a short enough duration
of time that Heisenberg's Uncertainty principle is not violated. The
really interesting thing is that these virtual particles have real
effects: A pair of uncharged conducting plates will attract
each other because a virtual charged particle – antiparticle pair
will come into existence from the vacuum for a brief time, and the
effective dipole and its images will cause the plates to experience
an attractive force. Don't believe me, look up the Casimir Effect.
What happens if you grab those
particles and forcibly separate them from each other and prevent them
from annihilating each other as any decent particleantiparticle
pair should do? You end up creating a Black HoleWhite Hole pair,
which you can then use for superluminal transportation and as a
timemachine! (Okay, I just made that up, but is it really any
crappier than “The Secret” or Deepak Chopra?)
Back to reality: The Man from the
South's wife explains that he has lost eleven cars and taken
fortyseven fingers.
Q5. What does Man from the South think
is the probability that American Boy's lighter will light during a
single attempt?
86%.
Q6. How many fingers to a Cadillac does
Man from the South figure?
Th ratio of the values is C/F = 47/11.
Q7. Do your answers to Q3 and Q4
change?
Yes.
Added on 16th Nov. 2012
What was Fermat and Pascal's approach? Instead of looking backwards,
they looked forward and calculated the probability (on the condition
of the current circumstances) of winning or losing the game and
divided the spoils accordingly. So if AB has a probability of p
of lighting the lighter and has already done so 8 times, the
probability that he will then do so 10 times is simply p^2
and the probability that he will lose is (1p^2). According to
this approach, AB wins p^2 of the Cadillac and loses 1
p^2 of his pinky. I think this is close to Jon's suggestion, who
strongly felt that since the game hadn't finished AB would have to
lose part of his finger in exchange for part of the Cadillac.