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

Welcome to another Mathologer video. Today’s topic is a very old and very famous problem. It is the bankruptcy problem from the 1800-year-old Babylonian Talmud, the central text of Rabbinic Judaism:)In particular, I’ll tell you about the ingenious and beautiful solution of this problem by Nobel prize winner Robert Aumann and his colleague Michael Maschler—as well as Marek Kaminski’s surprisingly visual, water-powered interpretation using communicating vessels. Here is the problem: A man dies. Okay that was obviously a big problem for the man, but that’s not our mathematical problem. The mathematical problem is that when he died, the man owed three creditors the amounts of 100, 200 and 300 dollars, making for a total debt of 600 dollars. Yes, probably they were shekels or dinars or something, but let’s go with dollars:)Now, how much of the man’s estate should each creditor receive? Well, obviously, if the man had more than 600 dollars, then everybody gets exactly what they are owed and walks away happy. But what if the man’s estate is worth less than 600 dollars? What happens then? Sad faces, obviously:)but how sad exactly? Now you may have an “obvious” answer in mind, but save that for now:) Let’s start with the Talmud, which provides advice for three different scenarios. First scenario, suppose the estate is worth 100 dollars. In this scenario the Talmud directs that everybody is supposed to receive the same amount: 100 divided by 3, that’s 33 and a third. Okay, that’s fair in some sense, although the guys owed more money may not be so happy with this solution. If you are owed more, wouldn’t you expect to get more?:) Anyway, scenario two, what if the estate is worth 200 dollars? Easy right? By the same logic as before, just divide 200 by 3, and that’s 66 and two thirds for everybody. But that’s NOT what the Talmud says. Instead, the advised split is: Creditor 1 gets 50 and the other two creditors get 75 each. Why? Who knows:) Well, maybe the third scenario will help us make sense of it all. In the third scenario there is 300 dollars to split, and the split is 50, 100, 150. Everybody gets exactly half of what they are owed. That’s the way you were thinking from the beginning, right?:) This proportional split is what most of us would think to do and what most modern bankruptcy laws would demand. So what is going on with the Talmud? What is the general principle underlying these three scenarios? Definitely not clear, right? How do you infer from these three examples what you are supposed to do if the estate is worth 150 dollars or 400 dollars or any other amount? And what if we are dealing with more or fewer creditors? As I said, this mystery has fascinated scholars for two millennia. Let’s solve it together! Will be fun:) Let me start by telling you about this communicating vessels business:)Did they show you this experiment in school? Pour water into a set of containers connected at the base. Once the water settles, it will level out to the same height in all the containers, regardless of their shape or size. There, and there. This principle is incredibly powerful—and one of its coolest applications is the water level. If you connect two points with a water-filled tube, the water will naturally settle at the same height on both ends. This makes it an excellent tool for levelling in construction, landscaping, or any situation requiring precision—even around obstacles! I’ve actually used this method in a few DIY projects myself. Practical magic:)Anyway, back to work:) Have a look at this special set of 2d communicating vessels. Three rectangles of the same height have areas 100, 200 and 300. Connect them at the bottom with tiny tubes. We’ll be ignoring the area of these tubes. Now we can use this setup as a water-powered analog computer. We can use the apparatus to proportionally allocate shares. Here the estate corresponds to the area of the 2d water that we are pouring into the vessels. And proportional distribution means that the shares of the estate that our three creditors receive will always be proportional to the debts. Let’s first have a look at the extreme cases. Nothing:) an estate of 0 dollars corresponds to three shares

of 0 dollars each. Makes sense. An estate of 600 dollars corresponds to everybody getting all their money back. Okay. Now what about an estate of 100 dollars as in the Talmud’s first example. Well, 100 is one sixth of the total debt of 600 dollars and, yep, every creditor receives exactly one sixth of what they are owed. Okay what about an estate of 300 dollars as in the Talmud’s third example. Tick again 300 is half of 600 and everybody really gets back half of what they are owed. Remember this is the same distribution as in the third Talmud scenario. However, this is the only one of the three scenarios where a proportional distribution of shares is advised. Why did the Talmud give different rules for smaller estates? Let’s investigate further. Actually, at this point let’s also cap off the rectangles at the top. We’re locking them up so that nobody can ever get more money than what they are owed. There all capped off. Now before we continue, and so that we’ll all be able to appreciate how powerful these sorts of water-powered analog computers are at capturing complicated bankruptcy setups, here is the whole American bankruptcy law at a glance. The American bankruptcy law features a hierarchy of debts that have to be satisfied in order. Government tax is first. Nobody gets anything until all tax that is owed has been paid. No surprises there:) Then there are the so-called secured claims. These are satisfied in a proportional way. Trustee expenses are next in line. And finally unsecured claims. Such a simple one-glance way to visualise a very complicated law, don’t you think? Alright we are all impressed, aren’t we?:) Now, let’s ponder a very different set of containers. With these three containers of areas 100, 200, and 300 we can divide estates into equal shares, at least to the extent possible. For example, an estate of 100gets divided into three equal shares, just like in the Talmud. An estate of 200also results in equal shares. And, 300. Equal shares again. In fact, all estates between 0 and 300 result in equal shares. But from here on whatever else happens, the share of the first creditor 1 is capped at 100, the amount owed. and so on. Now this is not exactly what’s happening in the Talmud but there are definitely a number of striking similarities. Have another look. So for 100 we get exactly the same distribution as in the Talmud. Everybody gets the same share of the estate. Next, let’s jump above the first cap. Okay, not the same as in the Talmud but the same sort of stepping up with creditor 1 maxed out and the other two getting the same higher share. Also, do you notice something else? Yes, all the numbers on the left are exactly double those on the right. 200 times 2 is 400, 50 times 2 is 100, and so on. Okay, last example, let’s fill up all the way to the top again. Every number on the right gets doubled again. 300 times 2 is 600. 50 times 2 is 100, and so on. Very interesting, isn't it? Now, doesn’t this doubling business suggest a simple way to modify our containers to capture exactly what’s happening in the Talmud. Can you see it? No? Well, how about to compensate for the doubling, we simply halve the heights of the containers? Let’s check. Works. Works again. Perfect. Okay, so there is this very simple way of interpolating between the three cases considered in the Talmud. But, of course, this is just one among infinitely many different sets of containers that will give you the Talmud’s three distributions on the right. Here is another such set. Very unlikely that this monster set is what the ancient rabbis had in mind but it’s important to realise that just because this simple set of containers (click)is a perfect fit, does not guarantee that it really is what we are looking for. Also, currently we have not predicted yet what is supposed to happen for estates greater than 300, as well as different numbers of creditors and debt amounts. Most importantly, we have no clue yet as to the exact fairness logic behind this set-up — how exactly would the ancient rabbis justify this way of sharing? Well, let’s first try to guess what’s most likely

going to happen beyond estates worth 300. Because of the halving, at the moment the three containers are each only half the size they are supposed to be. Well, there are lots of ways to complete our three containers. Here is one of them. By just doubling up everything, now all the containers have the right sizewith the thin vertical connections having negligible area again. Well, as I will explain, what is now believed to be the correct solution to the Talmud problem is this one here. Neat, isn’t it? Very pretty, symmetrical and easy to generalise to any number of creditors and debt sizes. And just to straightaway give you a quick taste for why this configuration may be a desirable way of arranging a bankruptcy system, have a look at this. What does it mean for those three grey rectangles to be the same? Well, what this means is that the three creditors suffer the same loss. Right? Everybody gets all their money back except for 25 dollars each. So the Talmudic system appears to be a compromise between the aims of balancing the gains and the losses of the creditors. And, right in the middle we get a proportional distribution. Very satisfying, don’t you think? Very satisfying, yes, but how can we be sure that good looks are not deceiving, that this is really the bankruptcy system that the Talmud is talking about? Well, the best way to make sure is to check out what the Talmud has to say about other situations where some sort of fair sharing is required and how those other scenarios square up with what we are considering here:)And that’s exactly what mathematicians Aumann and Maschler did as part of their analysis. Also, these two researchers specialise in a branch of mathematics called game theory and what they also succeeded in proving is that the Talmudic solution is, in a certain natural sense, a unique and optimal system of distributing shares. We’ll talk about all this in the rest of the video. This is what a typical page in a modern edition of the Talmud looks like. Very interesting isn't it? In fact, this is the page that features our bankruptcy problem. The oldest part of the Talmud is the text right in the middle. Other parts have been added by later generations of scholars in chronological order from the inside out. Like the rings in a tree:)Anyway, in a different part of the Talmud we find a discussion of another very interesting legal case that can provide us with vital insights into the Talmudic logic behind the resolution of the bankruptcy problem. Let’s check it out. As in the case of the bankruptcy problem, this legal case starts with a man dying. One extra detail. He dies childless. Now his widowis required to marry her deceased husband’s brother. This brother already has two sons from his first wife. Eight months later the new wife gives birth to another sonand it’s not clear whether this baby is the son of her first or her second husband. Now to make the story even cheerier the second brother dies too. A lot of death but that’s not the end:) The two brother’s father, the three little ones’ grandad is still alive at this stage... But guess what, before long he dies too. Very tragic:) Anyway, now the grandad’s estate has to be split up among the three children. But how? Well, if it was certain that the youngest child was the second brother’s son, then all three children would inherit an equal share of grandad’s estate. If, on the other hand, it was certain that the youngest is the first brother’s son. Then the youngest would get 1/2and the other two children would get 1/4th each. Can you see why? Right, basically we are sorting out two different family lines, entitled to 1/2 each. Kind of makes sense so far. But we’re supposing that we don’t know who the father of the youngest child is. So how should the estate be split up? Very interesting problem, right? And it’s really amazing how two millennia ago people would really tackle this tricky problem and would come up with the, in some sense, optimal solution. Anyway, here is how this problem is resolved in the Talmud. Essentially, there are two parties fighting over the estate, the red party and the green party. The red party’s maximum claim is for 1/2 of the estate. and the green party’s maximum claim is for 2/3rd of the estate. Let’s draw a diagram. The whole estate is the gray bit in the middle. Now, the Talmud argues that the part of the estate that is claimed by both parties is

this overlap. On the other hand, the gray bit at the top is only claimed by redand the one at the bottom is only claimed by green. Now the Talmud simply splits the contested part in half. And eventually, this results into this split into red and green. and a little bit of simple algebra acrobatics gives this:)Very interesting and definitely makes sense. This resolution is called splitting according to the Contested Garment rule. It is named after another property battle in the Talmud, over a contested garment. In the case of the contested garment, one party claims the entire garment while the other party claims half. We arrive at the final split by the same method. First identify the contested part. The top is only claimed by red. Split the contested part in two equal parts. and we arrive at the final split. Cool. Now, on the face of it, this garment splitting does not seem to have anything to do with our original bankruptcy problem. However, on closer inspection it’s there staring at us. Let me explain. Let’s focus on two payouts, just two parties fighting over the loot. Okay, two parties, that’s promising. Now let’s think of the total payout 50+100 dollars that’s 150 dollars as the estate that these two parties are fighting over. And let’s translate the debts into claims. Again, the total estate is 50 + 100 = 150 and 100 is 2/3rds of that. Now 200 is actually more than the whole estate. But in terms of a claim, it simply amounts to claiming all. Now let’s split according to the contested garment rule. What do you expect to get? Well, let’s see. Here is the contested part. The bottom is not contestedSplit the contested part in two. There that’s the split. Now remember, the total estate was 150. And 1/3 of 150 is 50and 2/3 of 150 is 100. And so we arrive back at our original red and green shares. Whaaat?:)In other words: If the two creditors use the Contested Garment Rule to split the amount they were jointly awarded, each will get actually awarded. What a nice surprise, don’t you agree? And this seems to not be a coincidence because the same turns out to be true no matter how we pick two creditors and two related payouts from our table. There are nine cases in total, all consistent with the contested garment rule. We just considered the first case. Here is the second, Let’s check again. Estate is 50+75, that’s 125. 100 is 4/5th of that. 200 is greater than 125. But that just means that green claims all again. Now autopilot garment splitting. Now remember, the total estate was 125. And 2/5 of 125 is 50and 3/5 is 75. Checks out again! And as I said, this works for all nine possible combinations in our table. We just had a close look at the first two. Here is the third combination, the fourth one, and so on. Wonderful, what all this seems to say is that if faced with a bankruptcy problem, we should aim for a split that is consistent with the contested garment rule? Okay, but how to do this in practise? What’s the algorithms to find such a split? Well, if we are just dealing with two creditors that’s easy, just apply the contested Garment rule. But what about more creditors? Is there always a consistent solution? And if there is a consistent solution, is it unique? unique consistent solution, how do we find it? Well, to answer all these questions our special communicating vessels come to the rescue. Let’s say we have two arbitrary claims. Let’s picture this scenario using a system of communicating vessels like before. Then, as before, the amount of water we pour in corresponds to the estate. Now, let’s perform the corresponding contested garment split on the right. Okay, there, that’s the estate. and here are the claims. We want to convince ourselves that on the right, the split of the estate according to the contested garment rule is exactly the water split, no matter the size of the estate we are dealing with. Let’s first check this in this simple case. Here both claims are larger than

the estate in the middle and so both claim all the estate. And this means that the whole estate is contested. In turn, estate will get split equally between red and green exactly like the water does. Great:)Let’s make the estate larger. That just means pouring in more water. As before both parties claim all and so as before the whole estate is contested. And so we get another split into two equal halves. But notice that we are dealing with a borderline case here, both in the water splitting view on the left and the garment splitting views on the right. Upping the estate a little bitwill take us above the lower red part on the leftand for the first time will make the red claim smaller than the whole estate. Now the contested part is exactly as large as the small red claim. And so halfway down the red is also where the garment cut will be. Great:)Now the argument for why we get the same split on the left and right stays the same for a while. ThereThereThereAt this point we’ve arrived at the second borderline case. Raising the estate slash water level furthertakes us into the top part of the red container on the leftand on the right, both claims are now less than the whole estate. How can we see in this case that the water also gets us the correct garment split on the right? Well, at the top of the two containers on the leftare two rectangles of the same height. For the moment let’s attach copies of these rectangles to the claims on the rightThe red in the middle is now equal to the full red claimand the green green claim. And so things are nicely lined up along here. This means that when we now simulataneously move the two claims back to where they belong, it is clear that the long horizontal line on right will be right in the middle of the yellow overlap. And so, again, we can see that the water splits the estate correctly. Great:)Okay so where does this all leave us in terms of solving an arbitrary bankruptcy problem a la Talmud? Well, it’s all under control now:) Say we are dealing with a couple of debts. Split each of them in two equal partsand turn everything in sight into a set of our special containers. Pour in the amount of water corresponding to the estate in question. Obviously, this split of the estate is consistent with the contested garment rule in the sense that focussing on any two of the creditors, like this or like thatyou always see in front of you a split conforming to the contested garment rule. Fantastic:)Also, turns out this overall split of the estate is the ONLY possible split that is consistent with the contested garment rule. That’s very important, right?:) Because if there were different consistent splits, then we’d be uncertain again which one’s the one that the Talmud is talking about. Luckily that’s not the case, and so our communicating vessels solution must be the solution that the Talmud is talking about. Great:)Okay, but how can we be sure that there cannot be two different consistent splits of an estate? I am sure that quite a few of you won’t be able to sleep tonight if I don’t tell you:) Well, here is a quick water-powered proof: Let’s say I tell you that we are dealing with a distribution of some estate among our creditors that is consistent with the contested Garment rule. And I tell you that the largest creditor receives this much. Then, because we are dealing with a consistent distribution, we can use our special containers to figure out what the shares of all other creditors are. Right? There, That creditor on the left must get this much. And that one theremust receive this much, and so on. And so it is clear that any consistent solution is the one given by our special set-up. Also, it’s clear that the larger the estate, the higher the water level. And this implies that for every possible estate there is a unique split of the estate that is consistent with the contested garment rule. Nifty proof, don’t you think? Well, and that’s pretty much it for today. Just a few more remarks. Maybe

after watching this video, you got the impression that the bankruptcy problem isn’t all that hard and shouldn’t have remained unsolved for nearly two millennia. However, keep in mind that what I’ve presented here is a clean, visually optimized, Mathologer-style path to the solution—one that owes much of its beauty and simplicity to hindsight and the ingenious idea of representing things in terms of communicating vessels:)Without the communicating vessels idea in our repertoire, grappling with and discussing the bankruptcy problem quickly becomes very messy. If you explore the relevant academic papers, you’ll often find them to be a dense thicket of case distinctions and inequalities:)Also, of course Jewish courts have been solving bankruptcy problems for thousands of years just fine using a combination of textual analysis, moral reasoning, and pragmatic compromise. While their solutions may not always align perfectly with the mathematically optimal solution presented in this video, they were likely just as effective—or even better suited—to the specific circumstances of the cases in question. I never gave much thought to bankruptcy problems before this video, nor did I question the idea that proportional distribution is the best solution. But now, I actually see the Talmudic approach as superior in many real-life situations:)It values each creditor equally while considering individual claims under reasonable constraints, whereas proportional allocation treats every dollar as equal, regardless of who holds it. Especially, in a David vs. Goliath creditor scenario, with poor David being owed one dollar and the bank a million dollars, the Talmudic system seems far more fair to me:)Overall, with all the containers split evenly, the Talmudic solution reminds me of the classic dilemma: is a half-filled glass half full or half empty?:) Aumann and Maschler explain the Talmudic approach this way: they suggest that half of the total claim, representing half a container, serves as a psychological threshold between two perspectives. If a creditor receives less than half of their claim, they focus on the loss—seeing the situation as a complete loss with only a small portion salvaged. If a creditor receives more than half, they focus on what they recovered, perceiving it as a full repayment with a minor shortfall. And so when the estate is less than half of the total claims, all creditors are in a total loss scenario. The fairest approach is to divide the available amount equally, based on half their claims. On the other hand, when the estate exceeds half of the total claims, all creditors experience a partial loss, so the remaining funds are distributed to balance out their losses, again using half their claims as a reference. Makes sense, right? Also, notice how the horizontal flip symmetry of the Talmudic solution reflects the idea that the Talmud treats loss and gain as equally important. Finally, I also mentioned that our solution has an interesting game-theoretic aspect. Imagine locking all creditors in a room and asking them to agree on how to divide the estate. There’s a catch: if they can’t agree, no one gets anything. Plus, if any creditor is offered 100% of their claim, they must accept and leave. This setup mirrors real-world bargaining scenarios often studied in economics. Aumann and Maschler analysed the Talmudic bankruptcy problem through this lens. They showed that, under reasonable assumptions, the creditors would naturally reach an agreement that aligns perfectly with the Talmud’s proposed solution. In other words, the Talmud’s method isn’t just an arbitrary rule—it’s exactly what rational creditors would agree upon if given enough time and fair bargaining conditions. Mathematically, this is connected to a concept in game theory called the nucleolus of the bankruptcy game. The nucleolus is a way of dividing up the estate ensuring that no creditor feels disproportionately disadvantaged compared to others. It minimises the largest dissatisfaction among all creditors, making it the fairest possible outcome under cooperative bargaining rules. For the hardcore ones among you I’ve included some extra material about all this after the credits, at the very end of this video. Check it out if you dare:)In any case, isn’t it really fascinating that the Talmudic authors figured out this optimal bargaining solution more than 2000 years before game theorists formally defined it! Pretty remarkable, right?:)Finally I’d like to thank Tamas Fleiner for telling me about the bankruptcy problem and the ingenious water-powered way of making sense of it

and for all his help in sorting out the details:)? And that’s it for today. Until next time:)You are still here? Very good! Alright, and so to really finish off for today, let me tell you in what sense the Talmudic solution is also a mathematically optimal solution to a bankruptcy problem. Say the estate in our original 3-creditor bankruptcy problem is 400 dollars. Of course, in theory, there are lots and lots of different ways in which we can divvy up the estate among the creditors. We want to identify the so-called nucleolus of the bankruptcy problem, a distribution that in a precise sense maximises the level of happiness of all the creditors with what they receive. For this we need a way to measure happiness. Here is one way of doing this: Let’s first list all the different ways in which the creditors can team up against everybody else to maximise their common share, as it’s often done when mathematically modelling these sorts of problems. In our example, there are six such coalitions. Underneath those coalitions, let’s note down their worst case outcomes in dollars. To explain where those numbers come from, let’s first focus on the coalition that just consists of creditor 1. Why is creditor 1’s worst case outcome 0 dollars. Well, what’s the worst thing that can happen from this creditor’s perspective? Of course, that everybody else gets as much of the money as they can possibly get:) Well, everybody else, that’s creditors 2 and 3 together, they are owed 200+300 = 500 dollars. And, so, if all 400 dollars first goes to them, then creditor 1 gets nothing. And so the worst case outcome for creditor 1 is 0 dollars. Another example. Let’s consider the coalition consisting of creditor 1 and creditor 3. Worst case outcome? Who is everybody else in this case? Well, that’s just creditor 2. If creditor 2 who is owed 200 maxes out first, then there is only 400-200, that’s 200 dollars left for creditor 1 and 3 together. Okay, next step. Let’s note down how much each possible coalition gets if we distribute the estate as prescribed by the Talmud. Okay, so just pour 400 into our system of containers. And so this is what the different teams get. Just as in any possible distribution whatsoever, all the Talmud numbers are greater or equal to the worst case numbers. Clear, right? Worst case means real shares cannot be less. Now to roughly measure the happiness of a coalition, we simply subtract their worst case number from their Talmud number. Okay50-0 that’s 50125-0 is 125and so on. It’s definitely a very rough measure of happiness but at the same time makes at least some sense, right? Anyway, according to this way of measuring the happiness of the individual coalitions with the Talmud’s allocation, the coalition consisting of creditor 1 alone is the least happy with a happiness score of 50 and the teams consisting just of creditor 2 and just of creditor 3 are the happiest, both with a happiness score of 125. Okay, now how are we going to use these happiness scores to figure out which distributions optimise happiness overall? For that we first order the happiness scores from small to large

from sadest to happiest. Getting there. Now, let’s imagine populating an infinite spreadsheet with the happiness scores of all the infinitely many possible distributions of the estate among the three creditors. There that’s our spreadsheet. Just for fun, let’s just highlight one more row in this spreadsheet, say the proportional distribution one. Okay last step, let’s sort our monster spreadsheet. First column B highest to lowest, then column C highest to lowest, and so on. Basically what we are doing here is putting things into lexicographic order. And so the larger the least happiness score of a distribution, the higher up this distribution will end up in the table. And, among the distributions with the same least happiness score, the ranking is determined by the second least happiness score, and so on. Now the remarkable thing, first observed and proved by Aumann and Maschler, is that, after sorting, the Talmud ends up at the very top of the table. And so in this very precise sense the Talmud’s distribution maximises the minimal happiness among all creditors, or equivalently minimises the largest dissatisfaction among all creditors, making it the fairest possible outcome. Definitely a very neat result, don’t you think?:)What about the proof? As Tamas pointed out to me, also surprisingly intuitive if we use our visual tricks of thinking about these problems. Let me finish by sketching Tamas and Balázs Sziklai’s proof. Baláz is one of Tamas’s former PhD students. I link to the relevant paper in the description of this video. Start with the simplest case of two creditors. It’s best to consider this case using the contested garment view. There, usual story. Estate in the middle and the two claims on either side. Worst case for red is that green gets everything and red only gets the uncontested red part. Similarly the worst case for green is that green only get the uncontested green part. This also means that every possible split of the estate in this scenario has to occur in the contested part. But then the happiness score of red for one of these distributions is the area of the rectangle above the barand the happiness score of green is the area of this rectangle. But now for the split to be optimal in the lexicographic order, we need the minimum of those two happiness scores to be maximal. And obviously that’s the case exactly when the two happiness scores are equal. And, as you remember, this is exactly the case when we are dealing with the Talmud solution. Neat how everything falls into place, don’t you think? Now, if we think about this 2-creditor setup in the communicating vessel view, things looks like this. Any other distribution of the estate fitted in these containers would result in different water levels. On the other hand, by having water flow from the container with the higher water level into that with the lower water level, as the water flows…the ranking of the different distributions we come across will rise in our spreadsheet until it reaches the optimal solution. Something similar turns out to be true in general. For example, in our 3-creditor example, fill our special containers with a possible distribution of the estate. Then the water levels will be at different levels, some above and some below the common water level of the Talmud solution. Now pick two containers, one with the water level above and one below, say these two. Then, as in the case of the 2-creditor setup, let water flow from the high level container to the low-level container until at least one of the water levels coincides with the common water level of the Talmud solution. Then it turns out that the new distribution of the estate will have a higher ranking in the spreadsheet than the one we started with. Now just repeat this levelling operation a couple of times and you arrive at the Talmud solution. Since at every levelling step the ranking goes up or at least does not go down this proves that the Talmud solution is at the very top of the spreadsheet. What an amazingly slick proof, don’t you think? Challenge for the keen among you: fill in the details of the proof that I skipped in the comments. Alright, and this is really all for today. Also make sure to leave a comment saying that you made it all the way to the very end:)Why is that? Since we are not dealing with the Talmud distribution, the water levels are not the same everywhere. Then we can prove that by letting some water from a high level tank into a low level one, the ranking of our distribution with improve. And if n is the number of creditors, we can also prove that by performing this levelling out maneuvre at most n times we can guarantee to reach the Talmud distribution. This then finishes the proof that the Talmud distribution is optimal. What, you are still here? Okay, you are asking for it.

Here are the gory details of the proof. What’s the happiness score of a team in terms of what we see in front of us? For example, what’s the happiness score of the coalition consisting of creditors 1 and 2? The answer is very simple: The happiness score is the smaller of two numbers: the first number is the total amount of water in the containers of creditors 1 and 2and the second number is the total amount of air in the containers of the competition. In this case, there is less air and the amount of air is 75. And so the happiness score for our coalition is 75. Challenge for you. Fill in the proof in the comments that this really always works. Alright, And a communicating vessels based proof of this surprising fact is also fairly easy. Here is a sketch. Let’s take any distribution of the estate and plus the little tubes at the bottom of our comunicating vessels and fill the creditor 1 2 3 containers with the new distribution of the estate. For example, here is what you get in the case of the proportional distribution. Note that with this non-Talmud distribution the water level is all over the place.