How Did Water Solve the 1800-Year-Old Talmudic Bankruptcy Problem?

0:06 Welcome to another Mathologer video.

0:07 Today’s topic is a very old and very famous problem.

0:11 It is the bankruptcy problem from the 1800-year-old Babylonian Talmud,

0:16 the central text of Rabbinic

0:18 Judaism :)In particular,

0:20 I’ll tell you about the ingenious and beautiful solution of this

0:24 problem by Nobel prize winner Robert Aumann and

0:27 his colleague Michael Maschler—as well as Marek

0:30 Kaminski’s surprisingly visual,

0:33 water-powered interpretation using communicating vessels.Here

0:37 is the problem:A man dies.

0:39 Okay that was obviously a big problem for the man,

0:43 but that’s not our mathematical problem.

0:45 The mathematical problem is that when he died,

0:48 the man owed three creditors the amounts of 100, 200 and 300 dollars, making for

0:54 a total debt of 600 dollars.

0:56 Yes, probably they were shekels or dinars or something,

1:00 but let’s go with dollars :)Now,

1:03 how much of the man’s estate should each creditor receive?

1:08 Well, obviously, if the man had more than 600 dollars,

1:11 then everybody gets exactly what they are owed and

1:14 walks away happy.But what if the man’s estate is worth less than 600 dollars?

1:19 What happens then?

1:20 Sad faces, obviously :)but how sad exactly?

1:23 Now you may have an “obvious” answer in mind,

1:27 but save that for now:) Let’s start with the Talmud,

1:31 which provides advice for three

1:33 different scenarios.

1:34 First scenario, suppose the estate is worth 100 dollars.

1:38 In this scenario the Talmud directs that everybody

1:42 is supposed to receive the same amount:

1:44 100 divided by 3,

1:46 that’s 33 and a third.

1:47 Okay, that’s fair in some sense, although the guys owed more money may not

1:52 be so happy with this solution.

1:54 If you are owed more, wouldn’t you expect to get more?

1:57 Anyway, scenario two, what if the estate is worth 200 dollars?Easy right?

2:02 By the same logic as before,

2:05 just divide 200 by 3, and that’s 66 and two thirds for everybody.But that’s NOT

2:13 what the Talmud says.

2:14 Instead, the advised split is:Creditor 1 gets 50 and the other two creditors

2:20 get 75 each.

2:21 Why?

2:21 Who knows:) Well, maybe the third scenario will help us make sense of it

2:26 all.In the third scenario there is 300 dollars to split,

2:32 and the split is 50, 100, 150.

2:36 Everybody gets exactly half of what they are owed.

2:39 That’s the way you were thinking from the beginning,

2:42 right?

2:42 This proportional split is what most of us

2:45 would think to do and what most modern

2:47 bankruptcy laws would demand.

2:49 So what is going on with the Talmud?

2:52 What is the general principle

2:54 underlying these three scenarios?Definitely not clear, right?

2:57 How do you infer from these three

3:00 examples what you are supposed to do if the

3:03 estate is worth 150 dollars or 400 dollars or

3:05 any other amount?

3:06 And what if we are dealing with more or fewer creditors?

3:10 As I said, this mystery

3:12 has fascinated scholars for two millennia.

3:14 Let’s solve it together!

3:16 Will be fun:) Let me start by

3:19 telling you about this communicating vessels business

3:24 :)Did they show you this experiment

3:29 in school?

3:29 Pour water into a set of containers connected at the base.

3:33 Once the water settles,

3:35 it will level out to the same height in all the containers,

3:38 regardless of their shape or size.

3:40 There, and there.This principle is incredibly powerful—and

3:44 one of its coolest applications is

3:48 the water level.If you connect two points with a water-filled tube,

3:53 the water will naturally settle

3:55 at the same height on both ends.

3:57 This makes it an excellent tool for levelling in construction,

4:00 landscaping, or any situation requiring precision—even around obstacles!

4:05 I’ve actually used this method in a few DIY projects myself.

4:10 Practical magic :)Anyway, back to work:) Have

4:13 a look at this special set of 2d communicating vessels.

4:17 Three rectangles of the same height

4:20 have areas 100, 200 and 300.

4:22 Connect them at the bottom with tiny tubes.We’ll be ignoring the area

4:26 of these tubes.

4:27 Now we can use this setup as a water-powered analog computer.

4:31 We can use the

4:33 apparatus to proportionally allocate shares.

4:35 Here the estate corresponds to the area of the

4:39 2d water that we are pouring into the

4:43 vessels.And proportional distribution means that the shares of

4:46 the estate that our three creditors receive

4:50 will always be proportional to the debts.

4:53 Let’s first have a look at the extreme cases.Nothing:)

4:58 an estate of 0 dollars corresponds to three shares

5:01 of 0 dollars each.Makes sense.

5:03 An estate of 600 dollars corresponds to everybody getting all their

5:08 money back.

5:09 Okay.Now what about an estate of 100

5:12 dollars as in the Talmud’s first example.Well,

5:16 100 is one sixth of the total debt of 600 dollars and,

5:20 yep, every creditor receives exactly one sixth of

5:22 what they are owed.Okay what about an estate of

5:26 300 dollars as in the Talmud’s third example.Tick

5:30 again 300 is half of 600 and everybody really

5:32 gets back half of what they are owed.

5:35 Remember this is the same distribution as in the third Talmud scenario.

5:39 However, this is the only one of the

5:41 three scenarios where a proportional distribution of shares is advised.

5:46 Why did the Talmud give

5:48 different rules for smaller estates?

5:51 Let’s investigate further.Actually, at this point let’s

5:55 also cap off the rectangles at the top.

5:58 We’re locking them up so that nobody can ever get more

6:02 money than what they are owed.There all capped off.

6:06 Now before we continue, and so that we’ll

6:11 all be able to appreciate how powerful these

6:14 sorts of water-powered analog computers are at capturing

6:18 complicated bankruptcy setups,

6:19 here is the whole American bankruptcy law at a glance.

6:24 The American bankruptcy law features a hierarchy of

6:27 debts that have to be satisfied in order.

6:30 Government tax is first.

6:31 Nobody gets anything until all tax that is owed has been paid.

6:36 No surprises there:) Then there are the so-called secured claims.

6:41 These are satisfied in a proportional

6:43 way.Trustee expenses are next in line.And finally

6:48 unsecured claims.Such a simple one-glance way to

6:52 visualise a very complicated law, don’t you think?

6:55 Alright we are all impressed,

6:57 aren’t we?

6:59 Now,

6:59 let’s ponder a very different set of containers.With these three containers of

7:08 areas 100, 200, and 300 we can divide estates into equal shares,

7:13 at least to the extent possible.

7:15 For example, an estate of 100gets divided into three equal shares,

7:20 just like in the Talmud.

7:22 An estate of 200also results in equal shares.

7:25 And, 300.Equal shares again.

7:27 In fact, all estates between 0 and

7:30 300 result in equal shares.

7:32 But from here on whatever else happens, the share of the first

7:37 creditor 1 is capped at 100,

7:40 the amount owed.and so on.Now this is not exactly what’s happening in

7:47 the Talmud but there are definitely a number of striking similarities.

7:52 Have another look.So for

7:53 100 we get exactly the same distribution as in

7:57 the Talmud.Everybody gets the same share of the

8:01 estate.

8:01 Next, let’s jump above the first cap.Okay,

8:04 not the same as in the Talmud but the same sort of

8:08 stepping up with creditor 1 maxed out and

8:11 the other two getting the same higher share.

8:14 Also, do you notice something else?

8:16 Yes, all the numbers on the left are exactly double those

8:20 on the right.

8:21 200 times 2 is 400, 50 times 2 is 100, and so on.

8:26 Okay, last example, let’s fill

8:28 up all the way to the top again.Every number on the right gets doubled again.

8:33 300 times 2 is 600.

8:35 50 times 2 is 100, and so on.

8:38 Very interesting, isn't it?Now, doesn’t this doubling business

8:42 suggest a simple way to modify our containers

8:45 to capture exactly what’s happening in the Talmud.

8:49 Can you see it?

8:50 No?

8:50 Well, how about to compensate for the doubling, we simply halve the heights

8:55 of the containers?Let’s check.Works.Works again.Perfect.

8:59 Okay, so there is this very

9:04 simple way of interpolating between the

9:07 three cases considered in the Talmud.But,

9:09 of course, this is just one among infinitely many different sets

9:13 of containers that will give you the Talmud’s

9:16 three distributions on the right.

9:18 Here is another such set.Very unlikely that this monster set is

9:23 what the ancient rabbis had in mind but

9:26 it’s important to realise that just because this

9:28 simple set of containers (click)is a perfect fit,

9:31 does not guarantee that it really is what we are

9:35 looking for.

9:35 Also, currently we have not predicted yet what

9:39 is supposed to happen for estates greater

9:42 than 300, as well as different numbers of creditors and debt amounts.

9:48 Most importantly, we have no clue yet as to the exact

9:52 fairness logic behind this set-up— how exactly would

9:55 the ancient rabbis justify this way of sharing?

9:58 Well, let’s first try to guess what’s most likely

10:01 going to happen beyond estates worth 300.

10:04 Because of the halving, at the moment the three containers

10:09 are each only half the size they are supposed to be.

10:12 Well, there are lots of ways to complete our

10:15 three containers.

10:16 Here is one of them.

10:18 By just doubling up everything, now all the containers

10:21 have the right sizewith the thin

10:24 vertical connections having negligible area again.Well,

10:30 as I will explain,

10:31 what is now believed to be the correct

10:34 solution to the Talmud problem is this one

10:36 here.

10:37 Neat, isn’t it?

10:38 Very pretty, symmetrical and easy to generalise to any number of creditors and

10:43 debt sizes.

10:44 And just to straightaway give you a quick

10:47 taste for why this configuration may be a

10:49 desirable way of arranging a bankruptcy system, have a look at this.

10:54 What does it mean for those

10:57 three grey rectangles to be the same?

10:59 Well, what this means is that the three creditors suffer the

11:03 same loss.

11:04 Right?

11:05 Everybody gets all their money back except for 25 dollars each.So the Talmudic

11:12 system appears to be a compromise between the aims

11:15 of balancing the gains and the losses of

11:17 the creditors.

11:18 And, right in the middle we get a proportional distribution.

11:22 Very satisfying, don’t you think?

11:24 Very satisfying, yes, but how can we be sure that good looks are not deceiving,

11:29 that this is really the bankruptcy system that the Talmud is talking about?

11:34 Well, the best way to

11:35 make sure is to check out what the Talmud has

11:38 to say about other situations where some sort of fair

11:41 sharing is required and how those other scenarios square

11:46 up with what we are considering here :)And

11:49 that’s exactly what mathematicians Aumann and Maschler

11:52 did as part of their analysis.

11:55 Also, these two researchers specialise in a branch of

11:58 mathematics called game theory and what they also

12:01 succeeded in proving is that the Talmudic solution is,

12:04 in a certain natural sense,

12:06 a unique and optimal system of distributing shares.

12:09 We’ll talk about all this in the rest of

12:12 the video.This is what a typical page in

12:17 a modern edition of the Talmud looks like.

12:22 Very interesting isn't it?

12:24 In fact, this is the page that features our bankruptcy problem.

12:27 The oldest part of the

12:29 Talmud is the text right in the middle.Other parts

12:32 have been added by later generations of scholars

12:35 in chronological order from the inside out.

12:38 Like the rings in a tree :)Anyway, in a different part

12:42 of the Talmud we find a discussion of another

12:45 very interesting legal case that can provide us with

12:47 vital insights into the Talmudic logic behind

12:50 the resolution of the bankruptcy problem.Let’s check

12:53 it out.

12:54 As in the case of the bankruptcy problem,

12:57 this legal case starts with a man dying.One

12:59 extra detail.

13:00 He dies childless.

13:02 Now his widowis required to marry her deceased

13:06 husband’s brother.This brother already has two sons

13:10 from his first wife.Eight months later the

13:14 new wife gives birth to another sonand it’s not clear

13:18 whether this baby is the son of her first

13:21 or her second husband.

13:23 Now to make the story even cheerier the second brother dies too.A

13:29 lot of death but that’s not the end:) The two brother’s father,

13:33 the three little ones’ grandad

13:34 is still alive at this stage..But guess what,

13:38 before long he dies too.Very tragic:) Anyway,

13:42 now the grandad’s estate has to be split up among the three children.

13:46 But how?

13:47 Well, if it was certain that the youngest child was

13:51 the second brother’s son,then all three children would

13:53 inherit an equal share of grandad’s estate.If,

13:56 on the other hand, it was certain that the youngest

13:59 is the first brother’s son.Then the youngest would

14:02 get 1/2and the other two children would

14:05 get 1/4th each.Can you see why?

14:07 Right, basically we are sorting out two different family lines,

14:11 entitled to 1/2 each.Kind of makes sense so far.

14:15 But we’re supposing that we don’t know who the

14:18 father of the youngest child is.

14:20 So how should the estate be split up?

14:23 Very interesting problem, right?

14:24 And it’s really amazing how two millennia ago

14:27 people would really tackle this tricky problem

14:29 and would come up with the, in some sense,

14:32 optimal solution.Anyway, here is how this problem is

14:35 resolved in the Talmud.

14:36 Essentially, there are two parties fighting over the estate, the red party

14:40 and the green party.

14:42 The red party’s maximum claim is for 1/2 of the estate.and the green party’s

14:47 maximum claim is for 2/3rd of the estate.Let’s draw

14:51 a diagram.The whole estate is the gray bit in

14:54 the middle.

14:55 Now,

14:55 the Talmud argues that the part of the estate that is claimed by both parties is

15:01 this overlap.On the other hand,

15:03 the gray bit at the top is only claimed by redand the one

15:07 at the bottom is only claimed by green.

15:09 Now the Talmud simply splits the contested part in half.

15:12 And eventually, this results into this split into red and green.

15:17 and a little bit of simple algebra

15:20 acrobatics gives this :)Very interesting and definitely makes sense.

15:24 This resolution is called

15:26 splitting according to the Contested Garment rule.

15:29 It is named after another property battle in the

15:32 Talmud, over a contested garment.In the case of the contested garment,

15:37 one party claims the

15:39 entire garment while the other party claims half.

15:41 We arrive at the final split by the same method.

15:45 First identify the contested part.The top is

15:48 only claimed by red.Split the contested part

15:51 in two equal parts.

15:52 and we arrive at the final split.Cool.

15:55 Now, on the face of it, this garment

15:59 splitting does not seem to have anything

16:01 to do with our original bankruptcy problem.

16:04 However, on closer inspection it’s there staring at us.

16:09 Let me explain.Let’s focus on two payouts, just two

16:15 parties fighting over the loot.Okay, two parties, that’s promising.

16:19 Now let’s think of the total

16:21 payout 50+100 dollars that’s 150 dollars as the

16:25 estate that these two parties are fighting over.

16:28 And let’s translate the debts into claims.

16:31 Again, the total estate is 50+ 100= 150 and 100 is

16:36 2/3rds of that.Now 200 is actually more than

16:39 the whole estate.But in terms of a claim,

16:42 it simply

16:43 amounts to claiming all.Now let’s split according to the contested garment rule.

16:47 What do you expect

16:48 to get?

16:50 Well, let’s see.

16:51 Here is the contested part.The bottom is not contestedSplit the

16:56 contested part in two.There that’s the split.Now remember,

17:00 the total estate was 150.

17:03 And 1/3 of 150

17:05 is 50and 2/3 of 150 is 100.And so we

17:10 arrive back at our original red and green shares.

17:15 Whaaat??

17:16 :)In other words:

17:17 If the two creditors use the Contested Garment Rule to split the amount they

17:21 were jointly awarded, each will get the amount they were actually awarded.

17:26 What a nice surprise,

17:28 don’t you agree?

17:28 And this seems to not be a coincidence because the same turns out to be

17:33 true no matter how we pick two creditors

17:37 and two related payouts from our table.There are

17:40 nine cases in total, all consistent with the contested garment rule.

17:45 We just considered the first case.

17:47 Here is the second, Let’s check again.

17:51 Estate is 50+75, that’s 125.

17:53 100 is 4/5th of that.200 is greater than 125.But

17:57 that just means that green claims all again.Now autopilot

18:02 garment splitting.Now remember, the total estate was 125.

18:09 And 2/5 of 125 is 50and 3/5 is 75.Checks

18:17 out again!

18:17 And as I said, this works for all nine possible combinations in our table.

18:22 We just had a close look at the first two.

18:25 Here is the third combination,the fourth one,and so

18:34 on.Wonderful,

18:34 what all this seems to say is that if faced with a bankruptcy problem,

18:38 we should aim for a split that is consistent with the contested garment rule?

18:45 Okay, but how to do

18:47 this in practise?

18:48 What’s the algorithms to find such a split?Well, if we are just dealing with two

18:54 creditors that’s easy, just apply the contested Garment rule.

18:57 But what about more creditors?

18:58 Is there always a consistent solution?

19:01 And if there is a consistent solution,

19:04 is it unique?

19:05 And if there is a unique consistent solution, how do we find it?

19:09 Well, to answer all

19:10 these questions our special communicating vessels come to

19:17 the rescue.Let’s say we have two arbitrary

19:22 claims.Let’s picture this scenario using a

19:26 system of communicating vessels like before.Then,

19:30 as before,

19:31 the amount of water we pour in corresponds to the estate.Now, let’s perform

19:36 the corresponding contested garment split on the right.

19:40 Okay, there, that’s the estate.and here are

19:43 the claims.We want to convince ourselves that on the right,

19:47 the split of the estate according to

19:50 the contested garment rule is exactly the water split,

19:53 no matter the size of the estate we are

19:56 dealing with.

19:57 Let’s first check this in this simple case.Here both claims are larger than

20:03 the estate in the middle and so both claim all the estate.

20:06 And this means that the whole estate is

20:08 contested.In turn,

20:09 this means that the estate will get split equally between red and green

20:15 exactly like the water does.Great :)Let’s make the estate larger.

20:20 That just means pouring in

20:23 more water.As before both parties claim all and so

20:27 as before the whole estate is contested.And so we

20:30 get another split into two equal halves.But notice that

20:34 we are dealing with a borderline case here,

20:37 both in the water splitting view on the left

20:40 and the garment splitting views on the right.

20:44 Upping the estate a little bitwill take us above the

20:47 lower red part on the leftand for the first time

20:51 will make the red claim smaller than the whole

20:56 estate.Now the contested part is exactly as large

21:00 as the small red claim.And so halfway down the

21:05 red is also where the garment cut will be.Great

21:13 :)Now the argument for why we get the same split on

21:16 the left and right stays the same for a while.

21:19 ThereThereThereAt this point we’ve arrived at the second borderline case.

21:25 Raising the estate slash

21:28 water level furthertakes us into the top part of

21:31 the red container on the leftand on the right,

21:35 both claims are now less than the whole estate.

21:38 How can we see in this case that the water

21:41 also gets us the correct garment split on the right?Well,

21:45 at the top of the two containers on

21:48 the leftare two rectangles of the same height.

21:51 For the moment let’s attach copies of these rectangles

21:54 to the claims on the rightThe red in the middle

21:58 is now equal to the full red claimand the green

22:02 in the middle is now equal to the full

22:05 green claim.And so things are nicely lined up along

22:09 here.This means that when we now simulataneously move

22:13 the two claims back to where they belong,

22:17 it is clear that the long horizontal line on right

22:20 will be right in the middle of the yellow

22:23 overlap.

22:23 And so, again, we can see that the water splits the estate correctly.

22:31 Great :)Okay so where

22:33 does this all leave us in terms of

22:36 solving an arbitrary bankruptcy problem a la Talmud?

22:39 Well, it’s all under control now:) Say we are dealing with

22:43 a couple of debts.Split each of them in two

22:45 equal partsand turn everything in sight into a

22:49 set of our special containers.Pour in the

22:52 amount of water corresponding to the estate in question.Obviously,

22:55 this split of the estate is

22:57 consistent with the contested garment rule in the sense

23:00 that focussing on any two of the creditors,

23:02 like this or like thatyou always see in front

23:05 of you a split conforming to the contested

23:08 garment rule.

23:09 Fantastic :)Also, turns out this overall split of the estate is the

23:16 ONLY possible split that is consistent with the contested garment rule.

23:21 That’s very important, right?

23:23 Because if there were different consistent splits, then we’d be uncertain

23:28 again which one’s the one that the Talmud is talking about.

23:32 Luckily that’s not the case,

23:33 and so our communicating vessels solution must be

23:36 the solution that the Talmud is talking about.

23:38 Great :)Okay, but how can we be sure that there

23:42 cannot be two different consistent splits of an

23:45 estate?

23:45 I am sure that quite a few of you won’t be

23:48 able to sleep tonight if I don’t tell you:) Well,

23:51 here is a quick water-powered proof:Let’s say I

23:55 tell you that we are dealing with a

23:58 distribution of some estate among our creditors

24:01 that is consistent with the contested Garment

24:04 rule.

24:04 And I tell you that the largest creditor receives this much.

24:08 Then, because we are dealing

24:10 with a consistent distribution,

24:12 we can use our special containers to figure out what the shares

24:16 of all other creditors are.

24:18 Right?

24:19 There,That creditor on the left must get this much.And

24:23 that one theremust receive this much,and so on.And

24:27 so it is clear that any consistent

24:30 solution is the one given by our special set-up.

24:34 Also, it’s clear that the larger the estate,

24:37 the higher the water level.

24:38 And this implies that for every possible estate there is a

24:42 unique split of the estate that is consistent with the contested garment rule.

24:46 Nifty proof, don’t you think?Well, and that’s pretty much it for today.

24:57 Just a few more remarks.Maybe,

25:01 after watching this video, you got the impression that the bankruptcy problem

25:05 isn’t all that hard and shouldn’t have

25:07 remained unsolved for nearly two millennia.

25:09 However, keep in mind that what I’ve presented here is a clean,

25:13 visually optimized, Mathologer-style path

25:15 to the solution—one that owes much of its beauty

25:18 and simplicity to hindsight and the ingenious idea

25:21 of representing things in terms of communicating

25:25 vessels :)Without the communicating vessels idea

25:27 in our repertoire,

25:28 grappling with and discussing the bankruptcy problem quickly becomes very messy.

25:33 If you explore the relevant academic papers,

25:35 you’ll often find them to be a dense thicket

25:38 of case distinctions and inequalities :)Also,

25:42 of course Jewish courts have been solving bankruptcy

25:46 problems for thousands of years just fine

25:49 using a combination of textual analysis,

25:51 moral reasoning, and pragmatic compromise.

25:54 While their solutions may not always align perfectly with

25:58 the mathematically optimal solution presented in this video,

26:00 they were likely just as effective—or

26:02 even better suited—to the specific circumstances of the

26:07 cases in question.I never gave much thought

26:10 to bankruptcy problems before this video,

26:12 nor did I question the idea that proportional distribution

26:15 is the best solution.

26:17 But now, I actually see the Talmudic approach as superior in many real-life

26:22 situations :)It values each creditor equally

26:26 while considering individual claims under

26:29 reasonable constraints,

26:30 whereas proportional allocation treats every dollar as equal,

26:34 regardless of who holds it.

26:36 Especially, in a David vs.

26:38 Goliath creditor scenario, with poor

26:40 David being owed one dollar and the bank a million dollars,

26:45 the Talmudic system seems far more fair

26:47 to me :)Overall, with all the containers split evenly,

26:52 the Talmudic solution reminds me of the

26:55 classic dilemma: is a half-filled glass half full or half empty?

26:59 Aumann and Maschler explain the

27:01 Talmudic approach this way:

27:03 they suggest that half of the total claim, representing half a container,

27:07 serves as a psychological threshold between two perspectives.

27:11 If a creditor receives less than

27:13 half of their claim,they focus on the loss—seeing the

27:17 situation as a complete loss with only a small

27:20 portion salvaged.

27:21 If a creditor receives more than half,they focus on what they recovered,

27:26 perceiving it as a full repayment with a minor shortfall.

27:30 And so when the estate is less than

27:32 half of the total claims, all creditors are in a total loss scenario.

27:38 The fairest approach is to

27:40 divide the available amount equally, based on half their claims.

27:44 On the other hand,

27:45 when the estate exceeds half of the total claims,

27:49 all creditors experience a partial loss,

27:52 so the remaining funds are distributed to balance out their losses,

27:55 again using half their claims

27:57 as a reference.

27:58 Makes sense, right?

27:59 Also, notice how the horizontal flip symmetry of the Talmudic

28:03 solution reflects the idea that the Talmud

28:06 treats loss and gain as equally important.Finally,

28:10 I also mentioned that our solution has an interesting game-theoretic aspect.

28:16 Imagine locking all creditors in a room and asking

28:19 them to agree on how to divide the estate.

28:21 There’s a catch: if they can’t agree, no one gets anything.

28:25 Plus, if any creditor is offered 100% of their claim,

28:28 they must accept and leave.

28:30 This setup mirrors real-world bargaining scenarios often studied

28:33 in economics.Aumann and Maschler analysed

28:36 the Talmudic bankruptcy problem through

28:39 this lens.

28:40 They showed that, under reasonable assumptions,

28:42 the creditors would naturally reach

28:44 an agreement that aligns perfectly with the Talmud’s proposed solution.

28:48 In other words, the Talmud’s method isn’t just an

28:52 arbitrary rule—it’s exactly what rational creditors

28:55 would agree upon if given enough

28:58 time and fair bargaining conditions.Mathematically,

29:01 this is connected to a concept in game theory

29:05 called the nucleolus of the bankruptcy game.

29:08 The nucleolus is a way of dividing up the

29:12 estate ensuring that no creditor feels disproportionately disadvantaged

29:16 compared to others.

29:17 It minimises the largest dissatisfaction among all creditors, making

29:21 it the fairest possible outcome under cooperative bargaining rules.

29:25 For the hardcore ones among you

29:28 I’ve included some extra material about all this after the credits,

29:32 at the very end of this video.

29:35 Check it out if you dare :)In any case,

29:39 isn’t it really fascinating that the Talmudic authors

29:41 figured out this optimal bargaining solution more

29:44 than 2000 years before game theorists

29:47 formally defined it!

29:48 Pretty remarkable, right?

29:50 :)Finally I’d like to thank Tamas Fleiner for

29:54 telling me about the bankruptcy problem and the

29:57 ingenious water-powered way of making sense of it,

30:01 and for all his help in sorting out the details:)?

30:16 And that’s it for today.

30:23 Until next time :)You are still here?

31:55 Very good!

31:56 Alright, and so to really finish off for today, let me tell

32:01 you in what sense the Talmudic solution is

32:03 also a mathematically optimal solution to a bankruptcy

32:06 problem.

32:07 Say the estate in our original 3-creditor

32:11 bankruptcy problem is 400 dollars.Of course,

32:14 in theory,

32:15 there are lots and lots of different ways in which we can divvy up the estate

32:18 among the creditors.

32:19 We want to identify the so-called nucleolus of the bankruptcy problem,

32:25 a distribution that in a precise sense maximises the

32:29 level of happiness of all the creditors with

32:32 what they receive.

32:33 For this we need a way to measure happiness.Here is one way of doing this:

32:39 Let’s first list all the different ways in which

32:42 the creditors can team up against everybody else

32:46 to maximise their common share,

32:48 as it’s often done when mathematically modelling these sorts

32:52 of problems.

32:53 In our example,

32:54 there are six such coalitions.Underneath those coalitions, let’s note

32:59 down their worst case outcomes in dollars.To

33:02 explain where those numbers come from,

33:04 let’s first focus on the coalition that just consists of

33:09 creditor 1.Why is creditor 1’s worst case outcome

33:12 0 dollars.

33:13 Well, what’s the worst thing that can happen from this creditor’s perspective?

33:17 Of course,

33:18 that everybody else gets as much of the money as they can possibly get:) Well,

33:23 everybody else, that’s creditors 2 and 3 together,

33:27 they are owed 200+300= 500 dollars.

33:29 And, so, if all 400 dollars first goes to them, then creditor 1 gets nothing.

33:35 And so the worst case outcome for creditor 1 is 0 dollars.

33:39 Another example.

33:40 Let’s consider the coalition

33:41 consisting of creditor 1 and creditor 3.Worst case outcome?

33:45 Who is everybody else in this case?

33:47 Well, that’s just creditor 2.

33:49 If creditor 2 who is owed 200 maxes out first, then there is only 400-200,

33:56 that’s 200 dollars left for creditor 1 and 3 together.

34:01 Okay, next step.Let’s note down how

34:04 much each possible coalition gets if we distribute

34:08 the estate as prescribed by the Talmud.

34:10 Okay, so just pour 400 into our system of containers.And

34:16 so this is what the different teams get.Just as

34:21 in any possible distribution whatsoever,

34:22 all the Talmud numbers are greater or equal to the

34:26 worst case numbers.

34:27 Clear, right?

34:28 Worst case means real shares cannot be less.Now to roughly measure

34:33 the happiness of a coalition,

34:34 we simply subtract their worst case number from their Talmud number.

34:38 Okay50-0 that’s 50125-0 is 125and so on.It’s

34:43 definitely a very rough measure of happiness

34:48 but at the same time makes at least some sense, right?

34:50 Anyway, according to this way of measuring

34:51 the happiness of the individual coalitions with the Talmud’s allocation,

34:51 the coalition consisting of creditor 1 alone is the least happy with

34:51 a happiness score of 50 and the teams consisting

34:51 just of creditor 2 and just of creditor 3 are the happiest,

34:52 both with a happiness score of 125.Okay,

34:52 now how are we going to use these

34:55 happiness scores to figure out which distributions

34:58 optimise happiness overall?

34:59 For that we first order the happiness scores from small to large,

35:05 from sadest to happiest.Getting there.

35:08 Now, let’s imagine populating an infinite spreadsheet with

35:12 the happiness scores of all the infinitely many

35:16 possible distributions of the estate among the

35:19 three creditors.There that’s our spreadsheet.

35:21 Just for fun, let’s just highlight one more

35:24 row in this spreadsheet,

35:26 say the proportional distribution one.Okay last step, let’s sort our

35:31 monster spreadsheet.

35:31 First column B highest to lowest, then column C highest to lowest,

35:37 and so on.

35:38 Basically what we are doing here is putting things into lexicographic order.

35:43 And so the larger the least happiness score of a distribution,

35:46 the higher up this distribution will

35:48 end up in the table.

35:50 And, among the distributions with the same least happiness score, the ranking

35:53 is determined by the second least happiness score,

35:56 and so on.Now the remarkable thing, first observed

36:00 and proved by Aumann and Maschler, is that, after sorting,

36:03 the Talmud ends up at the very top of the

36:06 table.

36:06 And so in this very precise sense

36:09 the Talmud’s distribution maximises the minimal

36:12 happiness among all creditors,

36:15 or equivalently minimises the largest dissatisfaction among all

36:21 creditors, making it the fairest possible outcome.

36:24 Definitely a very neat result, don’t you think?

36:28 :)What about the proof?

36:30 As Tamas pointed out to me, also surprisingly intuitive if we use our

36:35 visual tricks of thinking about these problems.

36:39 Let me finish by sketching Tamas and Balázs

36:43 Sziklai’s proof.

36:43 Baláz is one of Tamas’s former PhD students.

36:47 I link to the relevant paper in the

36:50 description of this video.

36:52 Start with the simplest case of two creditors.

36:57 It’s best to consider this

36:59 case using the contested garment view.There, usual story.

37:03 Estate in the middle and the two claims on

37:08 either side.

37:08 Worst case for red is that green gets

37:11 everything and red only gets the uncontested red

37:15 part.Similarly the worst case for green is that

37:20 green only get the uncontested green part.This

37:23 also means that every possible split of the estate

37:25 in this scenario has to occur in the

37:27 contested part.But then the happiness score of red for

37:33 one of these distributions is the area of the

37:38 rectangle above the barand the happiness score of

37:42 green is the area of this rectangle.But now

37:45 for the split to be optimal in the lexicographic order,

37:49 we need the minimum of those two happiness

37:52 scores to be maximal.

37:53 And obviously that’s the case exactly when the two happiness scores are

37:56 equal.And, as you remember,

37:57 this is exactly the case when we are dealing with the Talmud solution.

38:01 Neat how everything falls into place, don’t you think?

38:04 Now, if we think about this 2-creditor

38:06 setup in the communicating vessel view,

38:11 things looks like this.Any other distribution of the

38:17 estate fitted in these containers would result

38:20 in different water levels.On the other hand,

38:22 by having water flow from the container with the

38:25 higher water level into that with the lower

38:27 water level,

38:27 as the water flows…the ranking of the different distributions we come across

38:33 will rise in our spreadsheet until it reaches the optimal solution.

38:38 Something similar turns out to be true in general.

38:41 For example, in our 3-creditor example, fill our special containers

38:44 with a possible distribution of the estate.Then the

38:47 water levels will be at different levels,

38:49 some above and some below the common water level

38:53 of the Talmud solution.Now pick two containers,

38:56 one with the water level above and one below, say these two.Then,

39:00 as in the case of the 2-creditor setup,

39:04 let water flow from the high level container to

39:07 the low-level container until at least one of

39:10 the water levels coincides with the common water level of the Talmud solution.

39:15 Then it turns out

39:17 that the new distribution of the estate will have

39:19 a higher ranking in the spreadsheet than the one

39:22 we started with.Now just repeat this levelling operation a

39:24 couple of times and you arrive at the

39:27 Talmud solution.

39:28 Since at every levelling step the ranking goes up

39:31 or at least does not go down this

39:34 proves that the Talmud solution is at the very top of the spreadsheet.

39:40 What an amazingly slick proof,

39:42 don’t you think?Challenge for the keen among you:

39:44 fill in the details of the proof that I skipped in

39:46 the comments.

39:47 Alright, and this is really all for today.

39:51 Also make sure to leave a comment saying

39:54 that you made it all the way to the very end

39:56 :)Why is that?Since we are not dealing with the Talmud

39:57 distribution, the water levels are not the same everywhere.

39:57 Then we can prove that by letting some

39:58 water from a high level tank into a low level one,

39:58 the ranking of our distribution with improve.

39:58 And if n is the number of creditors,

39:58 we can also prove that by performing this levelling out maneuvre at

39:59 most n times we can guarantee to reach the Talmud distribution.

39:59 This then finishes the proof that

39:59 the Talmud distribution is optimal.What, you are still here?

40:00 Okay, you are asking for it.

40:00 Here are the gory details of the proof.

40:00 What’s the happiness score of a team in terms of what

40:00 we see in front of us?

40:01 For example, what’s the happiness score of the coalition consisting of

40:01 creditors 1 and 2?The answer is very simple:

40:01 The happiness score is the smaller of two numbers:

40:02 the first number is the total amount of water

40:02 in the containers of creditors 1 and 2and the

40:02 second number is the total amount of air

40:02 in the containers of the competition.In this case,

40:03 there is less air and the amount of air is 75.

40:03 And so the happiness score for our coalition is

40:03 75.

40:03 Challenge for you.

40:03 Fill in the proof in the comments that this really always works.Alright,

40:04 And a communicating vessels based proof of

40:04 this surprising fact is also fairly easy.

40:04 Here is a sketch.

40:04 Let’s take any distribution of the estate and plus

40:05 the little tubes at the bottom of our

40:05 comunicating vessels and fill the creditor 1 2

40:05 3 containers with the new distribution of the

40:05 estate.

40:05 For example,

40:06 here is what you get in the case of the proportional distribution.Note

40:06 that with this non-Talmud distribution the water level is all over the place.