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Dividing An Estate Among 3 כתובות: A Non-Game Theory Approach
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[Music]
Dr Shalom kelman from Baltimore
he introduced me to the mishnah because
when we met at the recent all that
Baltimore cm and the mishna has a
complicated halukkah a woman or a man
who's paying out three separate subists
to three different wives and the mishna
and igmar they have a complicated
halukkah and Professor Robert Allman
actually wrote
a
Theory or a paper on this Dr Kellman
will explain this a little bit more
but there were still some questions on
this and
Dr Fred Gross introduced a simpler
method of understanding this
mathematical equation there's a little
bit above my pay grade so I'm going to
hand it over to Dr Kellman to explain
but before he takes over I just wanna
introduce Dr Kellman Dr kelman attended
yeshiva's Karen B Avenue in 1973
followed by three years of intensive
studies under rehearsal shakter at Yu
for 14 years he had a weekly learning
session with review hudu cooperman of
meshachma Fame and other swarim Dr
Kellman has published in the areas of
kabulosa aretz and kadushas yushalayim
and regularly speaks at Toro
institutions including REITs and yadav
Herzog and yushalayim he has presented
the Shabbos Baltimore since 2000. so it
was my pleasure for Dr Kellman to take
us through this mathematical equation
thank you so much now by schwed it was a
delight to meet you and uh for all
you've done for the dafiomi and it's my
pleasure to share some exciting material
with you on this upcoming daf which has
been a mystery for almost 2 000 years as
no one has really figured out how to
truly understand the mishna the mishna
we're going to come across like I said
is the mission of the begins
and it says
man is married to three men who mace and
he dies and they have different casubas
this one has different claims so there's
a table here to help us simply uh
understand the question of the mishna
the case of the mishna you have on the
top you have
the claims of each woman a become A1 a
sub 2 a sub 3 and they each have a
different claim on your state the
wealthy one has a 300 First value the
second 200 and the lady a hundred on
this side as you can see you have the
value of the estate so in the value of
the estate there's a hundred
the khaluka it seems very
straightforward you take a hundred each
one gets a third
that makes sense but there's very little
there's divide among the three people
who are proclaiming it
if you look at 300 that's also something
of uh
because it's proportional each one's
claiming let's see 300 to 200 to 100 and
there's only and that would add up to
600 right if Adam or if it was 600
everyone get what they wanted but they
only have 300 so you have half and what
gets half of what they want 150 150.
but the middle of
275 75 and 50
doesn't seem to follow any suburb at all
and this is a mystery and we're seeing
the Gomorrah Memorial has why I'm
struggling getting ukintas how can they
understood in the end the mirror says at
the end of um
I don't understand it understand
and in fact that's how they'll be shown
in
and the Mystery has always been but
what's the what's the mishna and there
are the
goes back there's a famous riff who has
much longer the bigger Rift than you'd
expect on this little piece of Gamora
and he quotes about High Ground who had
an idea maybe it was related to schneid
mocks and the talus but never was able
to implement that idea how was actually
working in this case
and this question I said
baffled Shona
but the answer only came to light in our
time
thanks to Professor Robert Allman almond
is famous in the world today he won the
Nobel Prize in economic Sciences in
2005. there's a famous picture of him
taking the prize with his big yamaka of
the shashem and the area of his
expertise for which he won the prize is
called Game Theory Game Theory is a very
sophisticated mathematical structure and
there are people who understand it I'm
not one of them a lot of smart people
don't understand it's very very
difficult area and the issue is
I'll hit pause before I go this is the
copy of this paper and you all have by
the way hopefully we'll have a PDFs
available of the paper and another paper
you wrote two papers this is the one he
um
was written mainly for mathematical
audience uh it was an economics journal
and as you see it's game theoretic
analysis of bankruptcy problem from the
talmud he he labeled this a bankruptcy
problem because it it's essentially the
same kind of thing you have a and the
state you have to and and doesn't have
enough money so how do you divide if
it's available in the state equitably to
all the the shareholders and the other
set is dedicated to the member you have
his son who was a Tamil who has
killed an action in Lebanon in 1982 and
as he says in other sources it was his
son who brought this to his
attention said Daddy if you know Game
Theory maybe you can apply to this
mishnah and Laura behold he was
successful he was able to do so
and
um the other paper I want to make a
difference you'll have is a paper that
was written more for
non-mathematicians and to understand it
on an intuitive level and the people in
our world of tashiomi so the second
paper is also included it was originally
written in Hebrew then translated into
English published in Bangla and I should
be available to you as well as a PDF
so what were some of Almond's insights
that allowed him to solve this problem
so the first thing you recognized was
indeed that's what I had suggested this
division is based on the principle of
schneid mochs in the Taos
yeah we'll jump ahead a little bit
he also was able to restate rashi's
explanation of how that saluka was done
in a mathematical form and we'll come to
that in a few minutes
another thing he recognized that one can
expand the rule of schnaim of sin to not
from two people to three or four as many
as you need using the same Principle as
we'll explain in a few minutes
the main concept and the main idea of
schneid marks on the talus is that each
claimant shares the same amount of loss
as the other claimant I'll give you an
illustration in just a moment but that
was that's the that's the thing which
we'll come away with all the
participants every claimant when
compared to any other claimant will have
the same loss and that is
that's the fairest way
now that's not always the case you know
in other types of aluca you have
proportional it's not necessary
everybody's going to have the same loss
it's not going to work out that way
is that every week 71 will save the same
amount of loss
another thing almond was able to do was
to suggest that
once he was able to see the mishnah and
say oh this Mr fits the rules of
Muhammad I expect from Game Theory he
was able to develop an algorithm to
actually accomplish this how do you go
about doing it don't have money have
this stuff so what's the how do we do it
in Practical terms he came up with an
algorithm that allowed him to do it
all of this though the entire structure
of what he created was based on his
development of Game Theory
and he acknowledges this in his papers
he says you know once I had Game Theory
I had the key to the garden now I was
able to understand what's going wrong
but and he's able to work back and
figure out algorithms Etc
but how did know about it game theory is
so sophisticated it's very unlikely to
have access to it he struggles with this
he gives some theories how possibly
intuitively they might have Etc but it
still remains a strong question
now comes my chance to introduce Dr Fred
Gross Dr Gross himself was a noted
mathematician he had a PhD in number
Theory but in addition he had Samira
from tar vidas his family came before
the war and he was a major he also he
loved learning and he loved mathematics
and he published extensively in his
field as well known in the field of math
but he also made many fedushim in the
field of combining his mathematical
knowledge with Target insights most of
this material unfortunately is still in
the hernanduscript form and what I'm
sharing with you today is also only a
manuscript I don't have a PDF I can
share with you at this point the family
still wants to keep it until it's really
to be published and then we'll be
sending it out but in addition to the
things he says on this topic he has on
so many other topics he always wanted to
in fact write a textbook with for high
school students showing them how imagine
the mathematics of the of Torah can help
enhance their understanding of math and
vice versa so I was zohra to have a very
nice relationship with Fred going back
several years and we discussed this
topic many times and he sent me a
manuscript that he wrote with his from
his son Arya that I used as the basis of
today's presentation
what was Fred's big insight into this
topic
he also applied the bhava mitzia
division rule that that I had previously
noted I didn't say that point that Alman
called it the contested garment
contested garment based on steinmos and
uh and if all the khalukas have to be
consistent with this contested garment
and
Fred named it the BMV the brother mitzia
revision and what he did was developed
using the government's rule schneid
mohsen using basic algebra and not least
crime in Game Theory
thus it was in the reach above no sun
and all the other tanoim who lived in
the period of time of the Hellenistic
time the Roman Greek periods where basic
algebra was available solving
simultaneous equations which we're going
to do today was accessible to them
they could do quadratic equations they
were very sophisticated in many areas of
mathematics so this is something that
someone like NASA who we know from the
Gomorrah was a very it was she math
genius and astronomy has different
different quotations of NASA we just had
him recently in the daphion
they always being bocaoc and so on but
nothing was known for his mathematical
prowess
and more than that now that we have this
algebraic Prince approach it's much more
accessible to lay audiences like
ourselves and that's what I hope to
share with you today and by doing this I
hope to enhance our learning the dafyami
this Friday
I should emphasize that both Professor
almond and Dr gross's approaches yield
the same results there's no difference
all the numbers came out exactly the
same it's just how about how does one go
about getting there that your approaches
are different
and at the end I will try to show how
they actually
the two approaches actually intersect
beautifully
so let's begin
with the wheels thrust of today what is
the bmd the government see a division
rule
is where
one man says this all belongs to me so I
grabbed I found the Hefner I grabbed
it's mine and the other gentleman says
no we both picked that up at the same
time it belongs to both of us
so the first claimant says it's all mine
the second claimant says no half of it
is mine half of it is yours
what does arashi explain over there he
says a simple explanation which is if I
am the owner if I'm began to claim that
nay who says it's all mine so
I say all of it is mine claim it B is
saying but part of it is yours half of
it is yours I agree and the other half
we're arguing about I say that's mine
and you say it's yours but I disagree
with you so both agree though that half
of it belongs to the first so the time
of day
therefore he gets it without any problem
the contested part the second half is
divided between the two of them and that
leads to
why as he gets half of the half the
other one gets a half of the half so the
total he gets three quarters one gets
one quarter
that's how Rashi explains it
what Alman was able to show
mathematically was that the losses of
the claimants are actually equal
and in order to explain that let me just
explain these few sining equations and
we'll get into more understand a little
deeper so another way of understanding
the government to your rule is that the
losses of the claimants must be shared
there's a loss to everyone you divide it
and each one divide has the same degree
of loss
to understand how you understand what a
loss is a shared loss you look at what
the total claims are
you subtract from the estate what's
what's available
and that is the loss that they have to
share between them
so the shared loss is divided between
them equally
so when you end up with the claim the C
of the claim minus the loss of the
individual that is the Allah that's what
that individual gets so these are the
concepts you're getting here over and
over again
claim minus a loss equals an allotment
now let's do our case for example A1
Claims One Talus a sub 2 claims the half
a tallest So based on our equation the
total claim minus the estate is the
shared loss the total claim is one plus
a half is one and a half
what's the estate value there's only one
Talus what's the difference a half a
half divided between them comes out to a
quarter a half times a half is a quarter
so each one has a loss of a quarter that
is
how do we therefore divide it so the
allotment what each one receives is
their claim the individual claim mind
that's their loss so A1 a in the sub K
subscript A1 a refers to the allotment
that we associate with A1 the big A1 he
has
a claim of one you subtract a quarter
that's his loss he ends up with three
quarters
A2 the other contestant he claimed only
one half and you subtract that which is
that's his claim one half you subtract
his loss which is the same loss as the
other fellow a quarter and you end up
with a quarter so this concept I hope
because clear as can be we're going to
come through it over and over and up to
the Hazara you're more comfortable with
it
so again another way of looking at this
equation is let's move things around
moving loss to the left and allotment to
the right the equivalent is to say a
loss
is a claim minus the allotment
so in this case the loss of the two are
equal the loss is a claim minus
allotment so loss of A1 is 1 minus three
quarters because his claim is one is a
lot must be quarter so his loss is a
quarter and A2 his law claim was only a
half to begin with his allotment was a
quarter and therefore his loss is a
quarter the emphasis again is that they
are the loss here in both it has to be
equal
now what are we going to do how are we
going to outline the approach that we're
going to take now as outlined by Dr
Groves
the general law again the loss of
claimers must be equal
we assume that we're going to prove a
generalization of this rule bmd World
exists in other words we can do it not
just for two individuals we're going to
expand it as three individuals in all of
these cases we'll say that the
individuals themselves A1 A2 and A3
they have claims C1 C2 and C3
and we always put them in order such
that e which represents the entire
estate what's available
is greater or equal to C1 which is the
higher claim which is equal to or
greater than the second claim C2 which
is going to be greater or equal to C3
the lowest claim
so in all these cases we have to assume
that they aren't put in a numerical
order
now what we'll do is assume
one person's array one of the claimers
is receive something so let's assume A3
the one who's the lower value he
receives his allotment a sub three which
is uh I hope it's not clear the
uppercase a is the individual the
lowercase a is his allotment and the
subscript identifies who we're talking
to so the upper uppercase a is three he
gets his allotment of the lowercase A3
that's the value A3
therefore
what's left what's left is what A1 and
A2 they share the remaining A1 Plus A2
there because A3 has been removed so all
we have left is A1 Plus A2
and therefore if the a1a2 have to split
it up
by the by the salukas MS of the bmd rule
their losses must be equal so claim one
minus A1
that's the A1 that's what his claim
minus his allotment his loss equals the
claim of two minus is allotment and
that's his loss and the losses have to
be equal
so I hope this is clear because this is
the key of understanding everything
that's coming afterwards
now take note
you might always assume that the claim
is whatever the guy asks so so for
example in our case in the mishnah the
claim of the first of A1 should be 300.
but that's not always the case because
the claim cannot be greater than what's
available in other words even though he
says his claim is that she'll claim is
C1 which in this case was 300
it can't be 300 if we assume A3 has
taken part of a state away he's been
he's gotten his haluca so we always have
to figure out what is actually the claim
and this is the Hop you have to focus on
so again the claim cannot be greater
than what is available after A3 takes
his allotment leaving A1 and A2
so the way we denote this is say the
claim of A1 is the lesser of either his
actual claim or what's left after A3
gets what he takes which she takes and
leaves behind A1 Plus A2 that would be
left over
this is mathematically this this script
is written as denoted as the minimum of
A1 Plus A2 comma C1 means what is the
lower of these two values either C1
which is his original claim or it can be
less than that because A3 has taken his
part and therefore leaving behind A1 and
A2 which is the lower part
The Point again is that the claim can
never exceed what's available
now that's we having now we have that
now we have the way to achieve what will
give us the solution
how do we solve these equations so let's
assume once AC A3 receives her allotment
A3
we have one equation of known values
because we know C1 and C2 and we have
unknowns we don't know what the
allotment is A1 and A2 so we can write
an equation C1 minus A1 equals C2 minus
A2
however now let's assume we don't know
that we're going to make another
equation based on another assumption
let's assume okay and instead of saying
A3 receives his or her allotment let's
start with A1 let's say one who gets the
hives claim received for his allotment
first we have a second equation because
that's what's left over in whether it's
C2 and a and C2 and have to contend
they're eight I'm sorry A2 and H we have
to contend with each other so what are
their what are their haluca we say the
losses have to be equal so C2 minus A2
the loss now the claim of C2 minus is
allotment is his loss
is equal to the claim of three a three
minus his allotment A3 which is his loss
or her loss so in other words these two
losses have to be equal so we have two
independent linear equations with three
unknowns we don't know the unknowns are
the allotment a1a2a3 but we do know what
the claims are
so they have two equations
this one equation is two equations and
we have a third equation we know A1 A2
and A3 when you add up all the
allotments it has to be what's the
estate was
when we have three equations the three
unknowns we have the solution
we can solve now for A1 A2 and A3
so Fred was able to prove this
algebraically and it's not such a hard
proof you'll see
but the even more astounding was he can
use the same method for four or five or
six claimants because the number of
equations just keeps growing you have
four equations and four unknowns so five
equations and five unknowns and each
that's algebra that is accessible so
people understand algebra
okay now this is the meat of tonight
today's talk and if all you can do is
understand the next three slides four
slides you'll understand the chaluka of
the mishnah and then you'll feel good
about what you accomplished because this
is something that um cannot understand
and now you're going to have the ability
to understand it
so let's begin
with the first case of the mishnah
remember the mishta have the first case
was a value of a hundred the estate's
only worth a hundred
and the mishna said we divide everything
equally
everyone gets 100 divided by three
excuse me
now
let's assume I would say for each one
their claim initially was 300 200 and
100. right
but the estate is only worth a hundred
right so the most anyone can claim is a
hundred in other words even though the
claim C1 is 300 C2 is 200 200 but the
actual claim is only a hundred because
that's what's available
now A1 Plus A2
but there is the allotment of a1a2 can
never exceed 100 why because A3 is
taking apart in other words if you look
at this this A1 A2 and A3 that's each
one's allotment has to equal to 100. if
A3 is getting something then there's
that's less than 100 so A1 and A2 has to
be less than 100.
so assume like we said before A3 his
allotment gets goes first
therefore A1A to each claim A1 and A2
what's ever left
now what's the minimum of the claim of
A1 is that A1 Plus A2 or is its actual
claim what we just said before it has to
be less than his claim of 300 because A1
and A2 has to be less than 100 so we're
absolutely so the answer to this
question has got to be it's a 1 plus A2
it can't be C1 which would have been 300
it's got to be a one and A2 Which is
less than 100. the same argument can be
made with C2 c2's original claim was 200
and here we have a claim of either it's
either the minimum A1 Plus A2 Which is
less than 100 or C2 which is 200 so the
answer is obviously a1a2
so now let's use our equations
the loss of A1 defined by his claim
minus his allotment is the claim is A1
Plus A2 minus this allotment which is A1
so simple it's A2 you subtract A1 minus
A1 you're left with A2
the loss of A2 the same argument the
claim minus the allotment is the loss
his claim is A1 Plus A2 as we pointed
out his claim is his allotment is it
should be a mistake it's a typo it
should be A2
and therefore his value should be A1 he
should get that's that that was his loss
about his loss is A1
so by the rule of the bmd rule that it
should be therefore the losses have to
be equal so A2 is equal to A1
we can do the same argument when A1
receives a lot in the first and it just
compare A2 to A3 because even A3 is
claim was 100 but we know it can't be
100 it's got to be less than 100 because
a in which we move A1 A2 and A3 have to
be less than a hundred so the same
argument applies to if a one and a four
when A1 receives this allotment so you
have basically A1 equals A2 equals A3
which is like three times A1 which is
the same as 100 divided by three so
that's why everybody divides it equal
and it gets 100 divided by three that is
the first case of The Michener
the second case is a little bit more
complicated with an additional hop but
you'll be I hopefully you'll be able to
follow it relatively easily
the mishna's second case was the Estates
worth 200.
all right so for E equals 200 what's the
most A1 and A2 can claim is 200 right
that's all there is so the claim of A1
can't be 300 and A2 can claim a 200 and
remember A3 is only claiming a hundred
now since
A1 Plus A2 plus 3 has to be equal to 200
if you took away A3 A1 and A2 cannot
exceed 200 because if H we get something
it diminishes 200 so A1 and A2 cannot be
greater than 200. and similarly a to an
A3 cannot kinetics at 200. if we took
away A1 from this equation then leaving
A2 and A3 well we know a1s has to be
value something so it's going to
diminish 200. so what's left H1 A3
kinetic c200
so let's begin uh the assessment like we
did before we certainly look what
happens with A3 as assume AC received
his allotment and then we'll do the same
thing assume A1 receipts is alignment so
if A3 A3 but this is a lot more first as
before using the same argument we had
before A1 equals A2
that's the same argument behind the
previous page
but if A1 receives A1 first
right if A1 gets A1 first then what's
left A2 and A3
so A2 and 8th we have to share an A2 and
A3 this A2 plus A3
so let's look at each one of the claims
individually what does a2's claim
a2's claim is we said the minimum
between its actual came C2 200 or A2 and
A3
what we just said A2 and A3 have to be
less than 200 because A1 is there if it
takes this allotment then it has to
diminish 200 so we have to be the A2
plus A3 has to be less than 200. so
therefore the minimum of this expression
is not 200 that's to be less than that
A2 plus 2 has to be less so therefore
the answer is A2 plus A3 so the claim of
A2 is a plus A3
its loss therefore is A2 plus A3 right
minus his allotment A2 so the F gives
you A3 right A2 minus A2 and A3 so you
have A3 left over
but with A3 it's not so passionate a3's
claim is again the same argument we said
before it's either the lesser the
minimum of A2 plus A3 which is left
after A1 takes this piece with A2 with
A3 is left
actual claim
at this point we don't know every chance
we can't we have no way you can't use
these arguments earlier to determine
what it is so we have to leave it as
unknown for now so his loss is therefore
the minimum of that expression
minus whatever you saved X3
now since we said losses between the two
claimants once we've moved day one we
have A2 and A3 their losses have to be
equal so the loss of A2 is A3 the loss
of A3 is this expression and we make
them equal losses are equal so A3 equals
that expression the lesser of these of
either A2 plus 3 versus 100
minus
A3 because that's what this expression
was you said the loss is this expression
minus A3 so these two two this this
equation that's what result is results
this simple manipulation we add A3 to
the side at A3 to this side so you have
2a3 equals what's that minimum of this
this expression
so here's now the the the the lumbar cup
if we assume that the answer to this
what is this value
is A2 plus A3 let's assume the answer is
not 100 but it's A2 plus A3
then we have now the equation resolved
so 2a3 this will be here 2a3 equals A2
plus A3
which then comes out because remember
we're removing 100 we're saying this is
this the Lesser is A2 plus A3 so that
being the case we go back to this
equation 2 A3 equals A2 plus A3 which
then comes out too if you just subtract
A3 from this side and from this side you
have A3 equals A2 A2 equals A3
all right but we know from the first
expression if we assume the A3 receives
as a lot in the first we came to the
conclusion A1 equals A2 and now we'll
come to conclusion A2 equals A3
so if A1 equals A2 so it comes out all
of them all the allotments are equal
like in the previous case A1 will equal
a two equal three which is the same as
three times A1 which is equal equals the
what's the set the state is 200 or
dividing it by three it comes out A1 and
all the others all be the same will be
200 over 3 or 66.6
but if that's the case then A2 plus A3
would be 133 and we said over here right
which is which is sorry 1833 which is
greater than 100 in other words in this
expression we said what is the answer
what is the lesser of these two
expressions so we said let's assume it's
this one A2 plus A3 but if we follow
that we end up with a contradiction
because then it comes out that
A3 is greater than 100. so that
therefore can't be correct
thus our initial assumption that the
minimum of these two would be A2 plus 3
is not valid and therefore we have to
come to the conclusion it must be the
other option which is a hundred is equal
to 100. therefore now going back to this
original equation 2a3 equals 100. 100
there 100 here what A3 equals 50. so
once we now have come to the conclusion
that A3 gets 50 so we know
200 equals A2 plus A3 and we know
therefore once we A3 gets this piece a
50 what's left is 150 A1 and play two
and since we said A1 equals A2 so each
one gets 75.
2A equals 150 and you get 75 so this
answers now the second case we have the
first one getting 50 the the A3 the less
we're getting 50 and then the remaining
150 has to be divided equally because we
came to the conclusion based on what we
said A1 equals A2 and therefore I have
now 75 for the A1 A2 and 50 for A3
let's quick a quick check to see if this
what I just suggested fits what we're
saying all along that the total sums of
the claims minus the estate equals the
total loss and then has to be shared so
let's just let's do a quick a quick
check again assume AC A3 received the
allotment of 50. so
what is the claim of A1 we said it's 300
but it's camping right because what's
left
only uh it's a 200 right so it was 200
if if A3 gets 50 what's left 150 so the
maximum claim of A1 could be 150 same
thing A2 it can't be 200 but it's only
what's left 150. why because you can't
claim more than what is available after
A3 is allotment
so therefore as to ask the numbers the
total claim is 150 150 is 300 right
what is the estate's value after A3
takes his allotment
150 not 200 right because we said HB
gets his 50. so that's left over is 150.
so while do our Quest do our equation
the total claim which we said is 300
it's 150 plus 50 minus the estate which
is not 200 but 150 because A3 got is a
state
is equal to the shared loss so 300 minus
150 is 150 that's the shared loss and
the shared loss has to be divided
between them which is 75 each
so therefore individually the claim of
one or two a1a2 is 150 minus his loss is
75 minus 150 minus 75 and that explains
what each one of them gets an allotment
of 75. 75 75 and A3 getting 50. so it
all works out from both perspectives
that we talked about
and now let's look at commissioner of
the third missionary
A1 and A2 another dimension now is the
allotment the the estate available is
300. so again this is the same arguments
it's a little easier
A1 Plus A2 cannot exceed 300
and A1 Plus n a two plus two you cannot
exceed 300. again for the same argument
over and again since the total is equal
to 300
once we take away A3 then A3 has to
diminish what's available he's getting
something so A1 Plus A2 cannot be 300
has to be less than that and and the
same thing with A3 takes if A1 gets his
allotment then A2 and A3 has to be less
than 300. so let's start we usually
start A3 receives his allotment A3 first
so let's look at the claims what are the
claims and what's the allotment to
determine the loss
so a1's claim is the minimum of his
claim of 300 or A1 Plus A2
well that's clearly the answer is A1
Plus A2 because we already said that A1
Plus A2 has to be less than 300 because
A3 has taken diminished it so this is an
easy one to answer so this expression is
A1 Plus A2 and therefore the loss is the
claim a 1 plus A2 minus his his
assignment or allotment A1 is equal to
A2
so again if A3 receives first so if a
loss of A1 is A2 lowercase A2
if A2 now in the same situation with A3
but receives it first what is the claim
of A2 so we can determine the loss
so in this instance we don't know it's a
minimum of A1 Plus A2 versus 200. well
this could be higher than 200 we don't
know so we'll just leave it as we don't
know and calculate the loss is the
minimum of this a1a2 Plus versus 200
which we don't know minus his actual
allotment A2 so this expression is what
we know about the loss of A2 this is
expression what we know along with the
loss of A1 we equate them so A2 from up
here equals the expression pump here
minimum A1 Plus A2 or 200. right minus
A2 because that's what he said the loss
is its claim minus is allotment or
once we have this expression we just
move the two eight the a sub 2 to the
bar to the other side so add 2 plus 8 to
here and there if you have 2 a sub 2
equals that expression A1 Plus A2
right cannot be less than 200.
right
why since this means A3 is greater than
a hundred
if we look at this again if A2
A1 Plus A2
can we say it has to be it cannot be
less than 200 because if A3 guesses
allotment
well A3 can never get more than 100
because
he can't he's already claiming a hundred
so if A3 is only claiming a hundred it
can't get more than a hundred so
A1 and A2
has to be
it can't be less than a hundred two
hundred because otherwise A3 will be
getting more than a hundred I hope
that's clear I got moment I already lost
myself so again A1 and A2 their claim
together
has to be
cannot has to be greater than 200
because if
a
um
the total claim is 300 or I hate when
this happens but this is the problem
after we do again okay I'm going to
start over from this line start from
over here
so
um A1 so now we have claim we have
decided that this is the equation of
equality between the losses being equal
the A2 from over here that was the claim
the loss of A1
and the loss of A2 is this expression we
trimmed over here it's minimum this
minus A2 we equate them and therefore we
can now move the A2 on this side to that
side and we have 2A 2 equals this now
we're trying to determine what is
the minimum of this expression
so the argument is as follows we can
determine this
it cannot be less this A1 Plus A2 cannot
be less than 200.
we're trying to say that this has got to
be
it can't be less than 200 because if it
was less than 200
then A3
is going to get more than 100. if this
is less than 200
A1 Plus A2 then A3 has got to be more
than a hundred but that can't be because
it's only claiming a hundred so
therefore means let's proved that A1 and
A2 has to be can't be less than 200. so
the answer then must be that in this
expression the minimum is
200.
because this has got this is going to be
more than it has to be more than 200. so
this has got to be the answer is 200. so
if that's the case then we have back to
our equation 2a2 equals 200 and
therefore A2 is equal to 100. so now we
know at least one allotments based on
this reason is equal to a number that we
know a hundred that is for assuming AC
got A3 received his allotment A3 first
what
okay so now let's now look at the other
side uh assuming uh A1 receives his
allotment first and what's left is A2
and A3 to receive their share of A2 plus
A3
now we know from the previous slide we
calculated we've showed that A2 is equal
to 50 100. so now we have a two plus
three is the same as 100 plus A3 yeah
for A2 we're going to try to figure out
the same we did before what is the claim
and what is the loss for the claim of A2
it's the usual it's the minimum of A2
plus A3 the allotment what's left over
after A1 received his allotment versus
his actual came of 200 what is the lower
value now we can one thing we can plug
in is now we know A2 is the same as 100
so this is now equivalent to 100 plus a
sub a sub 3 200.
now but we have some additional
information to help us here since a sub
3 the allotment of three
cannot exceed 100 why because he only
claims a hundred so the minimum of these
thing of 100 plus A3 versus 200 well A3
has to be less than 100 right so what's
going to be the lower value this or that
is going to be 100 plus A3 so this now
becomes the claim 100 plus a sub 3. so
a2's claim is 100 plus A3
and now we can clap we can calculate the
loss because we know we just said the
claim is 100 plus A3
minus his allotment what we calculated
earlier A2 is equal to 100 for the
previous slide so now 100 plus A3 minus
100 comes out loss is A3 so the loss of
A2 is his claim minus his allotment and
is equal to A3
now that's for A2 what about A3 he's the
other remaining Contender against A2
what's A1 is received as a portion for
A3 the same issue the minimum
of 100 plus A3
because it's A2 plus A3 so A2 is 100 A2
plus A3 versus 100. what's the lower
level what's going to be 100 since he
only claims the hundreds it can't be
more than 100 so it can't be something
100 plus something has to be a hundred
so this now comes down to a hundred so
thus a3's loss is his claim which we
said is a hundred minus this allotment
which is A3
and now we can now configure out the
equation of equaling losses we have the
loss A3 which we calculate is 100 minus
A3 that's here
equals the loss of A2 which was A3 so
100 minus A3 equals A3 A3 has to be 50.
right 100 minus 50 is 50. and therefore
if A3 is 50 and A2 is 100 then we know
what's left is a total of 300 so A1 has
to be 150 and thus we have figured out
all the allotments of each of the
individuals on each case of the mishna
now
what frame this if you come away dressed
with this information you'll be able to
say I can explain each one of the Kim in
the mishna and that was probably
something that the tanoim could do like
I said what we did today this kind of
freaking reasoning through what is the
value for each of the situations if you
first take what if A1 gets what he gets
and make the equations for A2 and A3 and
then assume A3 gets what he gets and now
A1 versus A2 and knew the equations for
them you'll arrive in every case at a
very simple sense of equations and it's
easy to solve like we just went through
and again the principle again is that we
assume one person gets what he does and
then we have to reassess okay what is
the claim because it's a change claim is
going to change if someone takes away
some from the from the estate
so you have to decide what is the actual
claim what he says or the Lesser if
there's not enough left in the pot his
claim can't be his high that's the it's
the most important your sowed and based
on that you have to have the two left
two people are left you calculate their
losses you say the the claim minus is
allotment of one equals the claim minus
allotment of the other and they have to
be equal and this will always be the
case and in all cases you know whether
you're trying A2 versus A3 or A1 versus
A2 or A1 versus A3 doesn't make a
difference it always comes out equal
and this is enough to understand the
mishnah and in Practical terms they
didn't have to use all the um they would
do equations because they had they had
numbers to play with they had 300 the
estate is 300 he's claiming this he's
claiming 100 200 300 it had numbers so
it made it easier for them to calculate
but the next level from an algebraic
standpoint would would be to substitute
for numbers
the actual symbols of C
and Fred did that what I'm going to do
next is not to frighten you but just to
show you how taking the algebraic level
to another level will make it easier to
actually program a calculator to do this
for us
so have you did was here he came up with
these nine scenarios because each case
is you take the minimum of a102 and
versus C1 what is it what has could be
it could be all these possibilities
and the same thing can be all these
possibilities and this and these and
then you have you go through all the
possible combinations you end up with
nine combinations and you can just make
What's called the loss equations we know
the loss of this versus equal that the
loss of this equal that and the loss
equation raise points in every scenario
we're going to have two equations and
then you're going to have
a third equation which is the E equals
A1 Plus A2 plus A3 and all the way down
the same equation and when you solve for
these three equations on a
um algebraic method you end up with
these equations
now these equations are easily solved by
the way if you use Mathematica or any
kind of mathematical program you don't
have to use it by hand you can do it you
do it quickly on on the on the on the
Mathematica
once you have this what he did was
recognize that some of those cases one
two three four cases were really special
cases where the claim of two and three
were equal
so he had resolved with basically five
cases five scenarios and in each
scenario by studying he was able to
determine that the oil based on five
different conditions of e depending if e
started with e being very low very high
and so on without going through the
details
once you have this then you're able to
write a nice program
so now that we have the ability to
identify
the E's the Estates in different values
of these inequalities I was able to
write a program which
takes the opportunity to say take in the
value of e and look at what its value is
in these different scenarios if it fits
this one over here then
this is the equations for the allotments
if it's these and those and so on and
that's how this program was written and
once you have it the beauty is you have
a little calculator and you can now play
with this and this will be available
hopefully on the old off website you'll
be able to download this it's called a
CDF it's a type of PDF that you'll be
able to play with and as you see the
estate can be January so right now I can
move the estate and you'll see below how
the values of the very claimants change
so if at 100 until we get to
value of
150 it states 150 continues and then you
see it keeps changing and then the
values
dynamically changed now to understand
this better it may be easier for us to
look at a bar chart I found that's a
little easier for me to follow this so
let me uh do that now
and now you can see
if we go back to the beginning when the
a state is a hundred so just like we
expect the division is equal like the
mishna says everyone gets
a third and that stays that way
as we add more the more and more estate
increases in value we divide the estate
equally among all three claimants
but when the value in our reaches
a hundred and fifty
what happens now is that c has received
half of his claim which was our claim of
100 now she gets 50. so that's half the
claim so now
we just simply divide the rest as the
state gets more money we divide that
among the two remaining C1 and C2
and then when c 2 gets a hundred which
is half of her
of her claim she claims 100 200 now
she's got a hundred to have her claim
then you see the khaluka now just gives
money to as the estate and gets more
money we now divide it only goes now to
C1 to the claim of A1 and here's the
interesting thing when it reaches 350
however then we start dividing it among
both C1 and C2 the higher values and
they share
increasing as the value of the estate
increases they share that value equally
up until about 450 and then C3 starts to
increase and shares in the increased
value of the estate and as I emphasized
earlier in all of these trouble codes
the claim minus the allotment is just
the loss is always going to remain equal
if you cut if you keep C3 alone but if
we move that from the estate and just
compare C1 and C2 their loss will be
equal if you just remove a one like we
said earlier and get it and just compare
C2 to the C3 it's their losses will be
equal and this is what you can
dynamically see as we go further and
then when the estate has lots of money
everybody is increasing until they get
their full requirement which was 600
with everyone getting the 300 the 200
and the 100 equals 600 and everybody
gets the foja Luca
now what's fascinating is when you look
at this again this Dynamic I'll just
repeat it one more time because it's I
want to emphasize these points again we
start the lowest value of 100 they all
share together then when it reaches the
half value of the of the claim of C3
which is 100 which is 50 then we just
add to the values that we divide the
estate only between C1 and C2 until C2
reaches half and then we just get add
add if we want to the C1 until
which is
the heat is 200 when they get to it and
then they still have to share again
the fascinating thing for me is this is
almost ident this is the exact approach
of the algorithm that almond
described
okay now we can look at the Almond
algorithm we'll see how it's identical
it's what we just saw with the bar
charts but you suggest again you order
the credit just from lowest to highest
like we did a 1828 8382 A1 it divided
State equally among all the parties
that's what we're doing from the start
when we have the low values when the
whole that stays rather hydrated by each
one equally and the and you do that
until the lowest creditor receives half
of the claim and that's what we saw in
the bars it went up to 100 of the of the
C3 got up to A3 got up to 50 and then it
stopped moving and now divide the estate
equally among the remaining parties
right except the lowest creditor you
don't give him anything just give the
other A1 and A2 and so the next lowest
creditor was A2 in our case we see it's
half the claim and that's exactly what
happened when he received his half his
claim but his claim was 200 when he
saved 100 then his bar didn't move up
anymore and then we proceed until the
next credit to reach us half of the
original claim
now you work in Reverse start giving the
highest claim more and more money until
his loss actually to be defined before
the difference between the claim and the
award equals the loss of the next
creditor and that's what we'll see in a
minute how that works out it's always
going to be the same loss and continuing
this until you're finished so this is
why it's exciting to see how using
almonds approach is identical to using
Dr gross's approach from a
um
algebraic standpoint
next slide shows another way looking at
this is simply a diagram a graph diagram
as you look again you'll see as we State
increases on the bottom here at the
x-axis it increases until it gets to
50 because we we add money into the
system and these all three are receiving
a1a2h3 receiving the same amounts until
we get to
50 when it hits us 50 A3 the lowest one
that's no longer gets anything but a 1
and A2 continue to divide equally until
A2 divides it gets his 100 and doesn't
get any more as the stake increases in
value because as the state cruises in
value it will be here because this
increases we see that he doesn't get any
extra but rather we give all the money
up to A1
and but
astute Observer will say wait if
anyone's getting all this money how can
the losses be the same
it seems that distance to be intuitive
they're not right how can the loss be
the same if this one's keeping getting
more and more money the answer is how we
defined a loss and that'll be a final
slide to play all together from the
beginning and once we get up to the 300
of the when it gets up to this value of
a 350 then we see A1 and A2 continue to
share until it gets to 450 and then all
of them share at the same rate
to explain how it's possible that this
can keep gaining money by himself and
yet maintain the same loss as A2 let's
go one last slide
remember clay minus allotment equals
loss when the value of the estate was
300 remember we had sine the claim based
on what's available
assume that we A3 got his portion was
50. so the estate even though it's
unlisted on the fars and so it's 300 but
in reality it's 250 is available
so A1 his claim was 300 but what's
available is 250 as we did earlier so
it's 250 and his allotment is 150 100
that's his loss A2 is he his maximum is
200 he can't ask him whether he asked so
it's 200 minus his allotment of 150 so
the loss is remain equal
when we go to the next level when the
estate is 325 remember we always have to
take away the allotment of A3 that's
because we're comparing A1 to A2 so A3
is received his allotment of 50. so
what's the value of the estate is really
275. a1's claim was two seven was three
hundred if you can't ask for more than
there is so he's a kind is 275. he
receives 175 based on the analysis and
the algebra we did earlier and therefore
his loss is a hundred A2 continues to
have the same doesn't change as 100 so
the law stays the same so therefore you
see an opportunity here as A1 is getting
more and more uh his allotment of 175 he
wants 150 went from 150 to 175 yet
because his claim continues to increase
the cane increases this increases and
leaves the loss at 100 which is equal to
the loss of A2 so this is how the A1 can
get more money and yet stay made the
same loss as A2 when we get to three
evaluate with 350 which is beyond
emissions was then the initiative was at
300 and now we're at 350. again the same
process we take away A3 left out the
value of the estate is 300. A3 a1's
claim was only 300 so that's a 300 is
allotment is 200 value and loss is a
hundred and that's the same as A2 it
hasn't changed what if we go beyond that
and it's the last one if we go beyond
that let's say just to make the numbers
easier
um the estate now is up to 372. so again
comparing A1 to A2 A3 is removed that
leaves the 15 removed that's the estate
is 4322 so A1
his name is always 300
and the allotment based on the system is
211 his loss is 18 high and that's the
same a A2 is 200 his his claim is 200
but he's now getting more money now he's
starting to share as we saw the par c2's
graph is starting to move up and he gets
111 and therefore the loss Remains the
Same and this could be you can look at
this or with your calculator and play
with it and the insight into what's
happening okay
fine so thank you very much Dr Kellman
for math Fanatics this is certainly
going to be something which they'll find
very very intriguing for the regular
person
this is something that if they want to
spend the time and go through it they'll
certainly be able to have a much better
understanding of the mishnah