Conference rusure::math

I'm not sure I agree with your sentence "both are then satisfied".

I split up the beer.  You then choose the larger one.  I'm unhappy.

I know, you want to know *why* I created a larger one.  Well, I didn't mean to.
It's just hard to split beer evenly.

Perhaps we should say that after I split up the beer, you have to even them
up and then I get to choose first.

In other words, I'm reopening the original 2-person question.

/Eric

The usual formulation of these problems assumes that the cost of doing the split
and the cost of analyzing the split are both zero.  

In the real world, the first assumption is often true, because you can do and
redo the split on paper.  However, the second assumption may be false in some
cases.  In a large financial deal, millions of dollars may be spent deciding the
value of a package of assets.

When the second assumption is false, the same procedure works, but the result is
only approximate.  Person A does a preliminary analysis, then splits, then
analyzes the split, then resplits, continuing until the difference is smaller
than the cost of continuing.  Person B analyzes the split until the cost of
further analysis is less than the perceived or more than the expected
difference.  Then B chooses.  A can be satisfied that B got about the same as A,
plus or minus the value of the last analysis that A did.

When the first assumption is false, I don't see a clean solution.  

If the cost of splitting the loot is high, you can substitute splitting
probability, which can usually be presumed to be cheap.  For example, suppose
that there is a single item whose cost of splitting is infinite, or equal to the
item's value: a vase, perhaps.  Suppose further that transfer payments of
present or future value between the participants are somehow forbidden.  It's
clear that (by symmetry) the solution that yields the best value must be to give
each participant a 1/n chance of getting the entire item.  (This assumes they
are not so jealous that they would prefer a certainty of nobody getting it to
the 1/n chance of their getting it!)

Similarly, suppose two people are to split $2.01 in cash.  We may reasonably
assume that utility is nearly enough linear in money on the scale of pennies.
So giving each a 1/2 chance of the extra penny splits the value of that penny,
whereas physically splitting it would cost nearly the entire value of the penny.

...reminds me of the story of wise King Solomon and the 2 mothers.  Anyone
else remember that one ?

Of course!  Solomon had to divide 2 mothers and a kid with one of his courtiers.
Dividing the mothers was easy, but the kid was a problem, so he ended by calling
for a sword.  But one of the mothers pointed out that the kid had eaten three
pomegranate seeds, which the swordsman would have to divide evenly too.  This
seemed overly difficult, so the king wisely decided the kid would have to spend
six months of the year with one mother, and six with the other.

Or something like that.

    Version I heard was that 2 mothers both claimed the kid and took the
    case to court for custody rights. Solomon noted that the situation and
    the evidence were copmletely symmetrical between the two mums so
    mathematical esthetics demanded a symmetrical solution. He called for
    the sword, and one of the mums said "Aaaaargh! No! Let her have the kid
    rather than that." So Solomon figured she was the real mum and gave her
    the child.
    
    There's a three player non-zero sum game in there somewhere.
    
    Andy.
    
    BTW, I was born in 1955.7 what odds on that coincidence?

    I was born in 1955.10.  Another coincidence!

Clearly, if Solomon had known modern mathematics, he would have let one mom
divide, and the other choose...

More seriously, I recall reading in my childhood a set of stories about some
famous judge (the term "judge" isn't quite accurate but will do; my recollection
is that the stories came from Chinese culture) who was put in exactly the same
situation as Solomon, and came up with the same solution.  Unfortunately, the
false mom was cleverer in this story (or already knew the trick) and both moms
immediately clamored to give the kid to the other rather than kill the child. 
So he had to concoct another method, which I don't recall, to tell which was the
real mother.

Can anyone else remember reading a set of stories like this, and supply more
details?

Anyone remember a puzzle about how do you divide 17 horses amongst 3 sons ?  The answer
was something like borrow a horse from the neighbor so you have a divisible number,
divide them up, then such-and-such is conveniently now true so you return the horse
to the neighbor and you're done ?

What's the part I'm forgetting ?

/Eric

    Anyone remember a puzzle about how do you divide 17 horses amongst 3
    sons ?  The answer was something like borrow a horse from the neighbor
    so you have a divisible number, divide them up, then such-and-such is
    conveniently now true so you return the horse to the neighbor and
    you're done ?

> What's the part I'm forgetting ?
    
    The part about keeping your entries to 80 columns or less :-)
    
    (Actually I remember the puzzle too, but no more than what is cited above)
    
      John

    
    Wasn't it that the sons should get 1/2, 1/3 and 1/9 of the horses?
    

  
01234567890123456789012345678901234567890123456789012345678901234567890123456789


Yes, that was it, thanks.  I'll try to keep this reply to 80 columns.  I'm
on decwindows notes so I must have dragged the window size to something
large before.

Yes, the problem was he had to divide his 17 horses amoungts his 3 sons,
in the amounts 1/2 1/3 1/9.  So he borrowed a neighbor's horse to make 18.
He gave out 9 horses, 6 horses, and 2 horses, then returned the neighbor's
horse !



/Eric

I heard this as an inheritance problem, in which the will specified the
fractions.  The three sons were quarreling when a dervish wandered by with an
extra horse and the solution.  

I've heard this told so that the dervish makes a profit on the deal.  I can't
remember the math.

I've also heard a variation with coconuts and a monkey.  Was that in this
conference?

    I can tell you the coconut and monkey one, and someone else can tell
    you what note it's in (and whether it's in this conf or the brain
    bogglers one)
    
    Anyway, 5 sailors on an island collect coconuts with intention to then
    sleep and divide up the lot in the morning.
    
    Each of the 5 sailors wakes up in the middle of the night, and not
    trusting his buddies, he takes his 1/5 of the coconuts and hides them,
    except that there's always 1 coconut left over, which each sailor
    gives to a monkey.
    
    In the morning, they divide the remaining coconuts, and of course
    there's again 1 left over for the monkey.
    
    How many did they start with ?
    
    /Eric

    It's note 1094 where they discuss "cocoanuts" and monkeys.

        In 1094 the monkey didn't get another coconut in the
        morning.
        
>It's note 1094 where they discuss "cocoanuts" and monkeys.
        
        The dictionary does give "cocoanuts" as an alternative
        spelling of "coconut" so that is okay.  But some titles
        in this notes conference are so much fun.  I'll never
        forget how I searched and searched before finding the
        topic on how freqently a calendar month has two full
        moons in it:
        
        	Notes> dir/title="you saw me standing alone"
        
        :-)
        
        Dan
        

>        topic on how freqently a calendar month has two full
>        moons in it:
>        
>        	Notes> dir/title="you saw me standing alone"
    
    Obviously a deficiency in NOTES. Clearly we need two type of titles:
    a "display" title and a "reference" title.
    
    Anyhow, methinks any nuts'n'monkeys puzzles oughta go in 1094.

Would a month with 2 full moons tend to follow one with 2 new moons ?

/Eric

> Would a month with 2 full moons tend to follow one with 2 new moons ?
    
    No. With 2 new moons in a month the full moon is around the 15th of
    that month. Given a period averaging 29.53053 days between full moons,
    the full moon the following month would be around the 13th-15th.
    Exactly the opposite of what you want, i.e., the full moon on the 1st.
    
      John

	I love this note. It's like concept association football.

	The infant bisection problem cropped up in Cervantes' Don Quixote
as well, didn't it?

Andrew.