Principles for a free, powerful and stable monetary system for the digital era

• develop mathematical descriptions of the 2 rather complex aspects of how to make it work:
• Credit : to specify who owes to who, that is, who is expected to be responsible for providing the real value of money, and therefore, which transactions are possible while still ensuring their values)
• Value stabilization : to define a meaningful expression of the agreement about how much real value a monetary unit is supposed to be worth (how much real value will the ones be supposed to give back to the others in the future), while the exact future market prices of specific things are yet unknown.
• then implementing these theoretical solutions in the form of software.

This page is a draft; sorry I did not clean up everything. But it brings essential information on the subject that you can't find elsewhere.
(in red or italic are excerpts of received messages, that I reply to)

Simple expression of the nature of money, by an example

Theory of credit: some mathematical definitions

Let C be a map from P×P with values in the set D⁺ of positive quantities of money (without use of any diagonal element; we may write C(x,x)=0 or any positive value without this having any effect).
(This is a first, simple version of the concept, to introduce it; the useful possibility of admitting negative values of C will be discussed later)

The meaning of this map is that C(x,y) represents the amount of credit that x granted to y. It is a contract: each individual x freely chose the amount of credit C(x,y) which he granted to every y, (in exchange of a possible remuneration, like an interest rate corresponding to the risk taken - but let's forget it for now, as the theory is already difficult enough without it).

(In fact there will also be deadlines with possible progressive decreases before deadline, so that it is C(x,y,t) where t represents future times, and this C(x,y,t) comes down to 0 for large t; but this time dimension will not be discussed here, as the main mathematical structure that needs to be developed is about instantaneous situations disregarding future.)

In practice, for all x there will be only a few y such that C(x,y) is not zero. (For some, "financial people" whose job is to make many investments, there can be a little more than a few, but...)

Notation. Let R={r : P → D | ∑_x∈P r(x) = 0}.

Definition. We define the set of allowed distributions of balances, as
L={r∈R|∀ A⊂P, ∑_x∈A r(x) ≤ ∑_{x∈A, y∉A} C(x,y)}

Theorem. For any distribution of balances r ∈R, the condition r ∈ L of allowance, is equivalent to the existence of a corresponding antisymmetric loans matrix, M : P×P → D, such that

• ∀ x,y∈P, M(x,y) = - M(y,x) ≤ C(x,y)
• ∀ x∈P, r(x)= ∑_y∈P M(x,y).

For any positive quantity of money u and any two distinct individual x and y, we call payment of u from x to y the vector v in the space R, whose only two nonzero components are v(x)=-u and v(y)=u. The individual x is called the receiver or seller, and the individual y is called the donor or payer. Given the distribution of balances r at a time t, and a submitted payment v, this payment will pass through if and only if r +v belongs to L (is also an allowed distribution of balances). If it passes through, then the distribution of balances at the next time t' will be r' = r + v.

Theorem. We have equivalence :  (L has nonempty interior in R) ⇔ (the graph G defined by C with P as set of vertices and {{ x,y }|C(x, y)> 0 } as set of (non-oriented) edges, is connected).

(Math definition reminder: an interior point of a set is a point of the set that is not at its boundary, for example a cube has boundary points, those of faces; and interior points hidden inside the cube)

(The condition for some {x,y} to be an edge, can also be written : C(x,y) + C(y,x) >0)

Let us prove both above theorems. 

Note that both definitions of L, that will be shown equivalent, are visibly definitions of convex sets.

Note that the role of the map M (distribution of loans) is to prove the fact that a given distribution of balances is indeed "possible" in a computationally reasonable way (as the first definition was in terms of the set of all subsets of the population, which is exponentially big). Indeed, if it exists, then the truth of the system of inequalities of "possible distribution of payments" can be verified very easily. A little more subtle is to prove the converse: that for any distribution of balances satisfying those inequalities, there exists some corresponding distribution of loans.

So the method to find whether a payment will be possible, will be to find a way to modify M correspondingly. In fact, it is enough to look for the variation of M among the positive linear combinations of paths from the payer to the receiver (and moreover, where two paths passing by the same edge have the same orientation).

Before continuing the proof of the first theorem, let us prove the second one.

If the graph is not connected, the fact this set has empty interior is obvious : it is contained in the subset of distribution of balances for which the sum of balances of the nodes of a given connected component, is zero.

If it is connected, then we can take randomly (or at the middle) a value of each M(x,y) for an edge { x,y } inside the interval where it can vary.

This gives some possible distribution of balances, which is in the interior of the set because any sufficiently small payment between two points is possible by any path connected them. (and the same for small combinations of payments, by convexity).

So the L defined in terms of loans has nonempty interior, and as we know it is contained in the L defined by inequalities, so both have nonempty interior.

Finally, to show that they are equal it is enough to show that their boundaries (or faces) are equal.

We reach a face in terms of loans if no payment (by paths) from some x to some y is possible. Then, consider the set A of people x' so that a small enough payment from x to x' is possible. So, nobody inside A can make any payment to someone outside A. This A is the set for which the inequality becomes equality, in the system of inequalities defining L. So this is a face of L.

This ends the proofs of above theorems.

Reading your message yesterday I noticed an interesting parameter that you mentioned and that is not including in my formalism: "Each neighbor node individually decides how much funds it can commit either directly or by taking loan from its neighbors and then re-lending it."

Then I thought about it : how can one grant credit 1000 to the one and 1500 to the other, but no more than 2000 in total ? It's no problem, it can easily be built on top of my formalism. Here is how: let every individual x be represented in my formalism not by one node but by two nodes, say x₀ and x₁. Only x₀ can receive credit from other people, but credits can be granted by x₁ and for this, the limit is expressed by the quantity of credit from x₀ to x₁.

Now an important condition for the honesty of all operations is that if a node of the individual x (or a group that this node belongs to) is bankrupted, then all nodes of x must be liable for it : "one cannot escape paying back debts on the one side and spend money on the other side". How can this be expressed: the possibility for a node to run bankrupt and the individual be responsible for it, is when this node received some credit by another person and this credit expires. To make all nodes of the individual be liable for it, is that they all grant infinite credit to this node.

But if 2 nodes grant infinite credit to each other, then they are the same node. So there must be only one node of a given individual that can receive credit from other people, and all other nodes of the same person must grant infinite credit to it.

So x₁ must grant infinite credit to x₀ but this is worth mentioning only if there is a chance that M(x₁,x₀) be positive. Since there is no credit from other people to x₁, this can happen only if:

(x₁ has positive balance) AND ((x₀ has negative balance) OR (x₀ granted credit to another person)).

Possibly we don't need to care about this, and for example simply make automatic credit from x₁ to x₀ corresponding the balance of x₁.

Now, there is here one information too much: among the three parameters

b₀ = balance of x₀

b₁ = balance of x₁

C = credit from x₀ to x₁

only two are meaningful, namely

b = b0 + b1 (one's balance)

C'=C+b1 (effective total credit granted to others by x₁, amount that one takes the risk to lose by the credits from x₁)

or, if you prefer, b and C" where C" = C'-b = C-b0 (it is the effective total loan that one can receive from others, the debt one will have if all the people one granted credit to make the worst bankruptcies)

Now, we can wonder: in current payment operations, will we involve x₀ or x₁ as actor ? both ways can be considered.

  If Alice own 100kg to Bob and Bob own 100kg to Carol it does not automatically mean that Bob can just cancel his debt and just make Alice own 100kg to Carol Alice usually wouldn't mind but Carol might. Carol does not know Alice so she might not substitute Bob's promise to repay with Alice's promise to pay.

Precisely, this situation is as follows: Carol granted credit 100 to Bob Bob granted credit 100 to Alice These credits are being used as loans. Carol is not interested to know that this loan is not personally used by Bob but used to lend to Alice. This is Bob's business, not Carols. Carol is only interested to consider that Bob is a responsible person that would not grant foolish credit and run away from his responsibilities in case it would be lost. So, if Alice fails to pay back, then Carol expects Bob to pay back by himself anyway.

Financial operations

- Granting credit to a given person: this has an amount and an expiring date. Or, instead of an expiring date, it can have a schedule of successive expiring dates for divisions of this amount, as if it was a superposition of different credits with different expiring dates.

- Delaying the expiring date : a person who granted a credit can modify the expiring date to a later date.

- Instant payment: it was already mentioned (modification of distribution of balances at 2 elements by searching for a possible sum of paths between the nodes whose total amount is the one required; it can pass through or not)

- Therefore, when a credit expires, the computer will try to let it disappear. The trivial way is if, in this credit, the loan is not positive. Else the computer can try to reduce the situation to this case, by searching for a better distribution of loans for the same distribution of balances: this procedure that is identical to the instant payment procedure ("paying back the debt") with amount the value of the loan, considered in the absence of this expiring credit. If it passes through, it's OK. Else, the computer will do this paying back with the maximum possible value, and reduce the amount of the credit accordingly. This expired credit with remaining value, that cannot disappear because it is not paid back yet, is called bankrupted credit. When there is a bankrupted credit from x to y, the graph is divided into two subsets: the bankrupt set B (containing y) and the rest (containing x). It is characterized by the fact that the loan from every node x outside B to a node y in B, has its maximum value C(x,y), and that at least one of them is bankrupt. Therefore, no payment is possible by a node in B to a node outside B, and every next payment or new credit (or unwarranted donation as below) from a node outside B to a node in B will be used to try again to pay back the bankrupted credit.

- Unwarranted donation from x to y: decided by x, it is the same as simultaneously making instant payment from x to y and having y grant credit to x with the same amount. Unlike instant payment, this operation needs no verification of whether it passes through because it is trivial: this payment is done by the very loan from y to x allowed by the created credit.

Before expiring time, this has exactly the same effect on affordability of all payments, as if it was a simple credit from x to y. The differences are: before expiring, the displayed values of balances on the screen; then the fact that at the expiring time, the value is got by y instead of coming back to x. - If a person is dead:

If he had negative balance, its value is a loss that is shared among those who granted him credit

If he had positive balance and chose heirs, there will be payments to them with values shares of this amount

Anyway, all credits he made should be divided and converted into credits by:

- heirs if balance was positive, preserving expiring times

- those who granted him credit, but expiring times will be the min of both expiring times of credits made into one.

I did not think very exactly about all this, can you find yourself some exact formulas ?

There may be some fundamental indeterminacy of the result. Considering how people can foresee this problem and express priorities among the possibilities of what can happen, can be useful development.

- Commitments : see corresponding thread

Another possibility, that we will discuss later, is to involve the notion of "financial power delegation". See the general concept of power delegation in http://spoirier.lautre.net/trick.html

This trust is a trust even stronger than credit. It is to allow someone to grant credit in one's name. If A grants B financial power, then B can use this power to grant credit to C in A's name. If C does not pay back (go positive again), then A will carry the loss, not B.

Back to our island. Alice grants Bob financial power to grant A's 100kg of bread to Carol that she does not know. Bob does that. Carol does not pay. Alice is at loss. Guess what happens to Bob's financial power? He looses it.

Possible but not necessary. There may be a justification why Bob really did the best and is not responsible for the result. For example there may have been a flood or a fire that destroyed Carol's working installation, that could not have been foreseen. This is important because Alice had to consider well before granting Bob financial power. She had to know and trust him very much, to agree this way that she would carry the loss herself in case the credit failed. If she takes precisely this risk to carry the responsibility instead of Bob, it should not be for holding Bob responsible once the failure actually happened. It is not so likely that she was wrong doing so, else she wouldn't have made this choice. She thought he was better skilled than her for choosing who to grant credit to. But failure is always possible. If she does not trust him so much, then she will not grant him financial power, but simple credit, so that he will be liable for it. But he may refuse to grant credit himself because it is risky for him. If he does not grant credit himself, and Alice does not herself grant credit to others, then the group (Alice+Bob) may be not granting enough credit to be able to sell things and same money. So this combination (Alice ready to take the risk chosen by Bob) may be helpful.

Alice will not trust his judgment anymore. What happens to Bob's trust to Carol (she presumably promised him to repay loan). She looses Bob's credit.

 When a credit was not honored, It is too late for losing it. On the contrary, it is because the credit is expired, that we can discover that it is not honored.

forced auto-repayment feature is something that both nodes should agree beforehand.

I think it will be the default agreement, except for the case of the credit made by an Unwarranted donation (that even should let a margin between the credit reduction done and the one that would be possible.

If a person is dead then his account is either inherited by his heir(s) or abandoned (no heirs). In a first case problem solved - his rights and obligation keep living. In second case it keeps living acting on previously issued and accepted loans as normal. Loans expire and money go in and out of his account. When they all expire we have final balance

Loans go in an out, but balance remains constant.

, which is zeroed. Destroyed. If it is positive we have other persons' obligations that nobody can claim anymore so they are useless.

The problem is that this is unfair to let other people earn some money because their creditor died. They do not deserve it. They are behaving like heirs, without having been chosen. I think the system should oblige people to choose heirs.

If it is negative we have obligations held by his creditors that cannot be fulfilled so they are useless. And they are destroyed. This is the opposite process of credit(money) creation.

The problem is that it is unfair too. There should be some agreement to decide how the loss is shared by creditors. But I have another idea: that the person can have done Unwarranted donation, which should not expire as long as the person has debt, and the underlying credit of this donation should be the one that is first carrying the loss when the person dies.

We can see that the concept of heritage is very close to the one of Unwarranted donation. If instead of saying he chooses a heir someone makes a big unwarranted donation, it is more or less the same.

I once had another idea for expressing measures of trust, but it may be a bit complicated. I would say: "I grant him credit 100 only for only if his debt goes no more than 150. If his debt goes down lower (balance x < -150), then I will have this priority of refunding: my loan will be refunded and my credit to him will be over as soon as his balances goes up to x+150".

This way my responsibility is limited in case he would abuse of too many credits from different people: if 10 people grant him credit but with the same condition as mine, he cannot get too endebted.

But, of course, for practical purposes at least initially we will have to settle for some simple model. Credit limit is probably sufficient for now. So let's continue with it.

Ok, so to simplify, assume that all creditors agree for the same endebtment of a person A. But here we should include not only the negative balance of A but also the loans made by A, in other words, the sum of all positive loans made by creditors to A. This is because else A would have an incentive to cheat taking more money by lending it to someone else in the trick, which may even be a virtual user made and controlled by A. Then I just remark that this model seems identical with the one we considered before and formalised by representing a user by 2 nodes, except that the credit limit cannot be modified by A without the agreement of the creditors, and (to make this meaningful in terms of the only meaningful parameters) the balance of the node that receives credits is not allowed to go negative.

Then, could we express also by a adding more nodes, the possibility that one creditor would allow for a bigger debt than the others allowed ?

Philosophical aspects

"That still leaves us with with one difference - issuer. Value of two promises even denominated in the same units and in the same amount will be different to different people. Each one will factor his private measure of trust."

The problem is that the trust to someone cannot correctly reflect the evaluation of the real risk and amount of loss due to the fact that this person will not pay back. This real risk does not just depend on how honest or how good manager is this person, but is highly dependent on the context of all other credits and loans around the people involved. It is a very complicated problem.

For example you can imagine that a person receives credit from 2 people who have a different evaluation of the risk. So when the person goes to negative balance, this can be expressed by different possibilities: either the loan is from the one, or from the other.

In my formalism, this makes no difference: we cannot afford computationnally to optimise the distribution of loans corresponding to a given distribution of balances depending on how to reduce the fear of the people (lending preferably from the people who trust more), it would be too complicated, and rather impossible to decentralise; and anyway, in my formalism again it would be useless as the only thing I consider important is to decide whether a payment passes trough or not, using some existing credit no matter their values.

So, in computer, the debt is randomly and meaninglessly split as loans from different people. If the negative balance remains, who will carry the loss ? The one who trusted more or the one who trusted less ? Obviously, if we want to do something fair, we will have to recompute how it is divided into loans, no matter how the computer considered the division previously.

Then, how do you express your different values of promises ? By the fact I will accept as well giving something to someone, in exchange of a bigger promise from someone I trust less, than from someone I trust more ?

Is it a price for the credit I grant, or for the actual loan ? Why ask for and get a price for credit if it happened that this credit will in fact never be used for loans ?

But, I know what credit I grant, but, I just explain that I cannot know what are my actual loans, which anyway will constantly change from the fact that many times, two people which have nothing to do with me nor with this person are using my credit to this person as an edge in the middle of the route of their payment, as well as the computer could have found another route.

If it is not this way, then which way ?

"Trust is not a binary value (yes or no). There are different levels of trust. It makes sense to say I trust both Alice and Bob, but I trust Alice more, isn't it? In this case boolean logic does not work."

Boolean logic will work though the fact that if you really feel that youtrust more Alice than Bob, then you should not grant a credit to Bob at all. Just let closer relative to Bobs grant him credit if they feel so.

In other case, if Bob really needs a credit from you and you still trust him enough to grant him credit, so well you do it, but I see no way to make a practical use of this fact (that your trust to him is weaker than your trust to someone else) in the financial system. Bob pays you to grant him credit ? well. You want to sell this to someone else (to buy the agreement that someone else grants credit to Bob in order to have your own credit expire now) ? But, this operation may not pass trough, and even if it does, Bob may not agree with this : maybe he cannot afford something from the other's credit that he could have afforded with your credit. In this case, the other may be in trouble too (as he also reaches the affordability limit) but if Bob wanted to make a payment of the type Big-or-nothing, it will be Nothing, therefore that person will not be at affordability limit.

"I define trust in this context as a measure of subjective probability that a party will fulfill its obligations. So, 100% means that I take your promise at a face value. 50% - I have doubts, so you will have to promise me to pay twice as match so offset my doubts"

Yeah but it is not fair. Suppose that person is really honest. He will lose money just because other people don't believe it.

Suppose he really intends to pay back, but only the same value and not double value. Is this dishonest ?

If it was a matter of chance that does not depend on his will, like insurance for accident, then I would have no problem with it.

Well OK, I can consider to be paid for the merit of trusting someone that others do not trust, so that one's trust makes possible an investment that would not have been possible otherwise.

But I would say it is a price for granting a credit, I would not express it in terms of %. Of what would it be % ? Consider for example how it can depend on expiring date.

Cancelling interest rates

In the context that I described, the question of whether granting a credit to a person is effectively useful to him to let him make the payments he needs, is very doubtful and dependent on context. Therefore I suggest that we consider making no interest rate for it.

So granting credit will bring no benefit to the person. Never mind, it will still happen for two reasons: one reason is that one needs to grant some credit for being able to receive payment. The second reason is that credit is an act of personal trust, that should in most cases be only granted to relatives, and so one will like to offer this service to their relatives for free.

This was for nominal interest rates and interest for risk. Now, for the global stability of the system there will need to be a determination of the global real interest rate by market. This need not be expressed in the form of the nominal interest rate, as it should rather take the form of a mechanism of evolution of the level of prices for the goods:

Real interest = Nominal interest - inflation

Nominal interest = 0

Therefore, inflation = - Real interest

The condition of market stability is a determination of the stable real interest rate, therefore of the negative inflation. This will be a mechanism that deals with the price of things (that will slowly decrease), not the money itself.

This problem will be considered separately later.

Commitments

Definition of a commitment

A commitment between two people is a quantity of money (balance) put "between them" in the sense that it reserves the possibility that this money can come to either of them in the future.

That x commits to y for a quantity of money c (so puts a quantity of balance c between them), is formalised by letting C(y,x)= - c (if without this commitment there was C(y,x)=0). Indeed, it is a theorem that the system of inequalities that we wrote before over all subsets of the population with the map C, but now letting C have some positive values (credits) and some negative values (commitments), is equivalent to :

- Any system of attributions which will let the money of any commitment on either of the two people between which it is committed, will give a possible distribution of balances.

- Or in other words, letting C=(C+) - C- where C+ and C- are maps with positive values, or even merely satisfying the condition that for all x, y, C+(x,y)+ C+(y,x) and C-(x,y)+C-(x,y) are positive, the condition on the distribution of balances b is equivalent to: For any antisymmetric map g such that for all, x,y, g(x,y) is lower or equal to C-(x,y) (or in other words, between -C-(y,x) and C-(x,y)), the distribution of balances b(x) + sum over y of g(x,y), is a possible distribution of balances.

You notice that the condition for positive credits was formulated as : there exists a loans map limited by the credits; The condition for a mixed positive and negative credits, is formulated as : for all loans map limited by the negative credits there exists a loans map limited by the positive credits, such that the sum of these 2 maps gives this distribution of balances.

In a commitment, so when a money is put between 2 people, then it can come to the one only by the agreement of the other.

It differs from unwarranted donation, in that the two steps (donation and expiration) are done in the reverse order.

It is for every transaction that takes time to be completed and there is a problem of trust between the partners. For example, you want to order something. So you commit, then the seller sends you the object, and when you received it and checked that it is OK you complete the payment. This way everyone is sure to not be tricked by the other. If the one does not do what he should (let the money to the other) then anyway he can't get it for himself.

Later we will discuss a way to force the money to go to the right hand in case of a disagreement.

Commitment breaking and disagreements

"Ok, I think I see where are you going with this I just couldn't exactly follow complex interactions between your four friends. I had to start drawing diagrams and still I am lost."

The diagram is as follows:

Claude -> Jacques <- Luke -> Paul

Lending diagram is between ...... ...... <- ..... .... and ...... -> ...... .... -> ...

Possible breaks of chain (between those who agree with Claude and those who agree with Paul) are:

- Between Claude and Jacques (both Jacques and Luke agree with Paul) : Jacques will give back to Luke and not to Claude, so Claude is upset against Jacques that he granted credit to;

- Between Jacques and Luke (Jacques agrees with Claude and Luke with Paul): Jacques will give back to Claude and not to Luke, so Luke is upset against Jacques;

- Between Luke and Paul (both Jacques and Luke agree with Claude - possibility not considered in the "story") : Paul will not give anything to Luke, but Luke agrees that Jacques will not pay him back anything because he can rightly pay his debt to Claude instead. So Luke lost his balance, but is upset against Paul for this, not against Jacques.

In other words, the world is anyway divided into 2 subsets: the supporters of the one, and of the others. These two worlds disagree, and break (or question) the credits between them. The subset with total positive balance loses its total balance (from its own viewpoint), up to the amount of the disagreement (the commitment), and this loss is carried among its members by the authors of credits to members of the other subset. If the total balance of the one from its own viewpoint is lower than amount of disagreement, then the other part considers too, from its own viewpoint, that it has a lost, that is carried by its authors of credits to the first part. This would mean the question of which of both disagreeing groups wins, is determined by the question of which has positive and negative balance : the one with negative balance needs not pay back. However, to the same question we can give a different answer: that, as one of the groups is small and the other big (in most cases one group will be only a few people, the rest of the world in the opposite position), the big group wins and the small loses. Why ? Because the big group can cope without the small group, whereas the small cannot cope without the big. The ability for a group to work without the other, gives it the power to win in conflict. Indeed, consider that the small group has negative balance. So they can choose to not pay back, however they will be cut off from the rest of the world (and we can even consider to cut their accounts to no more use the system). It's up to them whether they stay and pay back, or leave and not pay back. But they may be in trouble if they leave and not pay back though in a sense they "won".

Paradoxes;restricting possibilities for practical reasons

The concept of commitment as I defined, with the associated equivalent conditions as formulated, is OK when there is only one commitment. These equivalent conditions can also be reformulated in terms of the existence of a path or a sum of paths in the credits graph between the 2 people, where each edge in the path has its interval of possible lendings, reduced by the amount of the commitment that is seen as following this path. This path would be the path of payment if the money was being paid from the one to the other, but this path remains as long time as the commitment remains. The only problem would be that other changes that happen meanwhile (other payments between other people) may require to change these paths, replace a fraction of an existing path by another path. Indeed, this path of commitment is the expression of the fact that we want to keep 2 possibilities for the distribution of balances, and these possibilities require 2 distributions of loans for being cheched, and this path or sum of paths, expresses the difference between these 2 distributions of loans. Now, problems come when we want to have several commitments present simultaneously. Indeed, having N commitments means that the image of some N-dimensional parallelepiped is contained in the set of possible distributions of balances, so that the verification would require to check that the 2^N vertices of this parallelepiped are satisfied by some distributions of loans. This is not computationnally reasonable.

Instead, we can be satisfied with a more restrictive, sufficient condition that ensures all is secure though it will not allow some possibilities that could theoretically have been allowed according to previous definitions.

This sufficient condition will be the following : that every commitment is ensured by a path or a sum of paths along which the interval of possible loans is restricted; and what is restrictive here is that the sum of paths that ensures a given commitment, will be asked to be independent of the choice of the distribution of balances that the other commitments ask to keep the possibility of.

Geometrically, the previous condition was that the image (in the set of distributions of balances) of the parallelepiped of commitments, was contained in the image of the parallelepiped of (positive) credits (that define possible distributions of loans).

Now, the restrictive condition says that there is a LINEAR map from the parallelepiped of commitments, into to the parallelepiped of credits, that fits.

Requiring this map to be linear, is what restricts the possibilities, while making it reasonably computable.

This indeed restricts the possibilities, it is not an equivalent condition.

Here is the typical example of situation that was permitted by previous condition, but no more by the new one:

Take a graph of 4 nodes (users) presented like the 4 vertices of a square. Let the 4 edges be credits (with some arbitrary orientation, so in an edge {x,y}, we may have C(x,y)=1 and C(y,x)=0) , and the 2 diagonals be commitments (negative credits), all of the same amount. What ever the arbitrary orientations of credits, the set of possible balances in this configuration is a point with previous condition, and the empty set with the new condition.

Indeed, what we have here is a square of commitments mapped to the image of a 4-cube in the 3-space. I don't remember the exact reasoning, but I remember that somehow the problem was translated into the fact that the square was mapped identically to the square that is the projected image of a tetrahedron, while it is not possible to enter linearly the square into the tetrahedron in this way.

With this same example, I have studied according to the choices of orientations of the credits, what would happen if both commitments break into conflicts. I found that in many cases the result is more or less clear, but in some cases it leads to paradoxes: the 2 breakings of the commitments do not commute. Even in one case I found that the resulting balance of one node differs by 2 units depending on which commitment breaks before the other.

This can be considered as a good reason to restrict the condition into the more computable new one : to avoid such paradoxical situations.

Computation method

Principle for a computational construction

Let's now start the construction of a possible computation method.

First, we recall what was the problem: we have a large graph of credits in a database, with, say, between 10,000 and 100,000 nodes, with the data of the balances of every node, and as a proof that this distribution of balances is allowed we have an example of a possible data of loans at each credit. And the problem is that, for every payment or reduction of credit, we would have to search for a sum of paths between two nodes to find out some new loans that prove that this operation is allowed. This operation would be lengthy if the graph is big. Maybe not too lenghy just like this, perhaps, but... the situation is made more complicated, and this time much harder, by the presence of commitments that might eat too much credits if represented by long unfortunate paths in the graph, and that we would need to sometimes modify to arrange other payments. And we would need a way to anticipate the risks of unaffordability of possible future payments.

For these reasons, I think it will be useful to mathematically reformulate and slightly modify the problem in the following way, that will give rise to a somehow completely different computational method that will make most payments computationally immediate (only a few operations instead of a whole path finding !).

First, let us define a new computational approach to this very above problem. Later we will introduce a slight modification of the problem, that will allow for a more completely arranged practical computational method that will forget classical pathfinding algorithms.

The idea is to introduce inside the database, as an intermediate tool of calculation, a virtual partition of the big set of all nodes into a set of a smaller number of big nodes, each representing a group of elementary nodes. The question is how to make this partition not an arbitrary useless partition, but a "good" one. Roughly, I think most nodes will be in groups of about 10 to 100, though unfortunately a number of remaining ones will stay more or less isolated.

Now I will describe the partitioning method. Note that this method will be dependent on the distribution of loans, which is more or less arbitrary. So if we revise the representation of the situation by loans, the corresponding partition will be modified. But anyway, as situations evolve, the partition need not be every time recomputed. It has some margin of tolerance, until it reaches some limits and asks for a revision. Moreover, as concerns the precise fact of revising the loans for the same "real" other data (credits and balances), it happens that it will not so much affect the partition of nodes as defined from these loans.

In the first step of the construction, we will assume that there is no commitment. We will introduce commitments later, in the more favorable situation that will be given thanks to the slight modification of the problem that we will explain then.

To understand the partitioning process, you should first have some notions of percolation theory, though our problem has its specificites that differ from usual percolations problems. http://en.wikipedia.org/wiki/Percolation_theory

The idea is as follows: for a given distribution of loans M(x,y), so, antisymmetrical and lower than C(x,y), for every positive value of a parameter u, we define the binary relation T_u on the set of nodes, by: T_u (x,y) if and only if (M(x,y)+u)<C(x,y).

Now, still for a given value of u, we define a partition of the set of nodes, by the equivalence relation: "x and y are in the same class if in the relation T_u there is a path from x to y and also a path from y to x".

In other words, the condition is that x can pay to y the quantity u through a SINGLE PATH and conversely.

You can also understand this saying that it is the equivalence relation defined by the smallest preorder containing T_u (the equivalence relation by which you quotient a preorder to obtain an order).

What we defined here is not yet exactly the "good" partition that we were looking for, because it has some big defects, in particular that it depends on u, and for most interesting u there is too much discrepancy between the sizes of the parts, some are too big and most are too small. But we will work from this series of partitions that depend on u, to obtain the more useful partition we need.

This work is the following: start with the largest values of u, and reduce it progressively. First, most nodes are more or less isolated, and they progressively gather into parts. When you first obtain a group of more than a hundred nodes, then you take it from the graph, it will be the Main Group. Or there can be a few Main Groups like this, if several big parts appear for the same value of u and not immediately melting into one when you reduce u a little. They are big groups, with big value of u. Then, with the rest of the graph, you reduce the value of u until you find other large groups, and the more you do this reducing u, the smaller groups you admit to separate from the rest. At the end you are left with a number of unfortunate isolated nodes.

Finally, this builds a partition of the nodes made of, rougly, large parts with associated large values of u, and small parts with associated small values of u.

What I described here was the initialisation process of the database. In practice, once initialised, the partition should be locally revised from time to time according to the needs so to more or less follow what it would become if we recomputed it all, though we need not make such a full recomputation of the whole partition.

Potential function

Here is an idea of a trick that may be useful to guide the calculation. On any edge of the graph, so that has an interval [-C(y,x),C(x,y)] of possible values of the loan, arbitrarily define some convex map on this interval, a "potential energy" function. For a fixed data of balances, the sum of the energies of all loans, is a convex map with domain the set of all possible loans that correspond to these balances. It has a minimum value, and a unique map of loans that gives this value if the potential maps are strictly convex (else a convex set).

Now, let's define a "difference of pressure" between 2 nodes for a given distribution of balances: it is the derivative of the energy with respect to a payment from one node to the other. Except in some pathological situations (if the derivative of the potential on an interval is not continuous from -infty to + infty and the distribution of balances is at some corner), at the minimum of the energy, this derivative is well-defined and independent of the path, and equal to the derivative of the "minimum energy" map on the set of distributions of balances.

Now, the use of this tool is that, given of choice of several possible payment paths found for a needed payment, it is "better" to choose the payment that gives the lowest potential, in order to stay close to the minimum energy of the system. The advantages of staying near the minimum energy, are that

1) Many loans remain "well inside" their intervals, and thus available for small payment paths

2) The difference of pressure between two nodes as measured through one or a few payment paths, is "not too false". The data of pressures can be used to detect a risk that the balances approach the border of affordability, and warn the corresponding users that they may be in trouble and need to pay back or search for new credit for their next payments to pass through.

Introducing cash to simplify computations

As announced at the beginning, it is possible to bring big simplification to the computational problem thanks to a small variation of the economic problem.

This change is the following.

We bring the assumption that, either all nodes, or at least a large set of nodes that fills the graph, granted credit to one same node. So this node that receives credits from all or many people is very "powerful". Too much power ? When the general power system of the project will be implemented, such "powerful" nodes will naturally be safely created.

This power should be limited by the condition that the sum of all loans to this central node, must have a limit value like 1% of the sum of all positive balances in the system. So it "masters" only 1% of the value of the money in circulation. There are 3 possible "meanings" of this sum of loans to a "power" node, inside the general power system of my project.

It may be for a negative balance of the power node: "public debt" or "parts in the value of the hosting company".

Or it may be a "banking power", that gathers this 1% of the savings of the population, for granting credits to a number of companies while mutualising the risks.

Anyway, in the real world, power exists and it is not necessarily bad.

There exists and will always exist some center of authority. Today's states have heavy debts. At least to protect the environment, there must be a green tax, that should be paid by polluters to "nobody" (public money).

To accept to have their financial account to some host, the people must anyway trust the owner of this host.

In a future development, we can consider the coexistence of several power nodes between which every user would have the choice. But in a first implementation, to make things reasonably simple, I think it is better to keep only one power node.

So, it is a little change in the economic problem (only 1% difference), but a dramatical change in the computational possibilities.

The loan from another node to the power node, will be called "cash". If you have cash, then you can pay to anyone with it, with no need to search for any payment path in the graph. Your payment path has only one intermediate : the power node.

If you don't have enough cash, then you just need to borrow it from all the people closely connected to you that may have, then pay with this cash.

Now, remember: I told you that the population should be virtually partitioned into groups, for a matter of calculation. In each such group, you don't need to bother sharing the cash between all the members.

You just need to keep one common reserve of cash in each group. To operate a payment from a member x of a group to a member y of another group, x often only needs to borrow the cash from the reserve of his group, to pay it to y that will let it to the reserve of its own group.

Then, since every group is defined by a condition of affordability of payments inside the group from any member to any other up to a certain amount, you don't always need to look for a payment path between a node and the depositor of the cash reserve: you know in advance that such a path exists. Thus, you can allow individual members of the group to have negative cash, meaning that they virtually borrowed this cash from the depositor of the cash reserve of the group, and the update of internal loans in the group to concretise this borrowing, will be operated later (once a number of payment will have been done). So: inside each group, the total sum of cash must be positive (defining the cash reserve of the group), and the total sum of negative cash should not exceed the internal payment affordability of the group.

As concerns commitments, there are 2 possibilities : either there is a chance to find a short commitment path, in particular a commitment path that is internal to the group.

Or, the commitment path can be the short path through the power node. It just blocks an amount of cash in the system.

A previous draft on theory of credit, more "philosophical" (but still very mathematical)

0. Introduction

1. Political and technological foundations

2. The theory of credit

The prototype

The "classical interpretation" of the prototype model

Credit delegation

Time dimension

Credit under condition of limitation of debt

Reservations of individual cash

First reasons to drop the conditions on the majoritarian parts A.

A mathematical reformulation of the main problem 

Possible occurence of close communities

The problem of adapting the distribution of cash to the needs

Degrees of trust, priorities of refunding

Retroactive elasticity of liabilities to release cash with minimal risk (second reason to drop the conditions on majoritarian parts A)

The problem of closed communities, again

3. Payments and pledges

The notion of credit can often serve as a sufficient substitute for the trust transitivity. Not for all cases but for many of them, namely when all information is known (with no risk of hidden cheating of opportunities that the other will never know or things like this, and where the maximum risk taken is known) (I have four French names, what short English names would you take ? Luc=Luke - ordered alphabetically): 

The underlying mathematical problem is described here.

The second part of the problem: defining the value of money based on term markets

Miscellaneous comments

Why a protocol like Ripple is not needed

If there is no direct relation between two hosts, one intermediate will be sufficient, so that it can easily be done too without protocol.

This way, in my project, the main work will be on the centralized knowledge intrahost routing, which will be quite a task already.

Here are the characteristics of my conception for computation method:

- The programming will be quite a task, but once done, it will work very quickly: most payments or other financial operations will only take very few mathematical operations to be made (less than 10 mathematical operations, even for a set of 10,000 users), though of course in some cases, some tougher data restructuration will have to be done by more complicated parts of the program. You may be surprised by this fact as it is usually said that path finding takes at least a number of operation about the square root of the number of nodes, but... you will see

- It makes it reasonable to include in this program a detection of the risk of payment unaffordability : if some relatively small payment between any unknown users would be impossible, this will be automatically detected BEFORE the payment is requested. This way, users can get warning : "You are in a group that is poor or received small credit, so you have the risk to not afford payment", or on the contrary "You are in a group that is rich and granted small credit for it, so you should try to grant more credit to someone to be able to receive payment".

So, why I don't need a protocol:

A site with thousands of users, has by its many users, direct connections with hundreds of peer sites.
Therefore any two sites, if not directly connected, will surely have in common a number of sites with which they are directly connected.
This will make pathfinding very easy.