Making the best future money system, requires to

- 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)

Jacques said to Luke: "lend me a coat, I need some and will return you the value later". Luke thus granted a coat to Jacques, and announced it.

Paul manufactured boots, but needed an electric razor, which Claude was selling.

Claude said to Luke: "I trust my friend Jacques, and I am ready to advance the value of this object to him. I thus will send a razor to Paul, good reception guaranteed. From now on it is Paul who will owe you boots, for since you trust him you know that he will provide them to you".

Luke reported to Jacques: "from now on, it is towards Claude that you will have a debt".

All agreed.

Later, Paul received a pack from Claude, opened it, but did not have satisfaction: the razor was defective and could not function.

He sent a complaint to Claude, who did not do anything of it.

He said to Jacques then: "the pack from Claude was not good. Thus, do not return this value to him, give it back rather to Luke, to whom I do not owe anything more in this situation".

He also forwarded his complaint to Luke, who supported this request.

But Jacques refused, not to betray his friend Claude. He answered that Claude had made his sending and is not responsible for bad operations from Paul who broke his razor, and so that he will rather return something to Claude.

Paul said to Luke: "Luke, you trusted me, you know that I do not lie when I say that the pack arrived defective. Jacques is a dishonest person, he did not want to recognize the defective character of the pack of Claude. You were the one trusting Jacques, therefore you take the responsibility for the consequences of his wrong judgements. I do not know you, I do not owe you anything."

Luke called Jacques: "you betrayed my trust in you by your irresponsible judgement. As you refused to return the value of this coat to me after I lent it to you, I will not get it back. "

Jacques repented, and said to Claude: "Paul is right, I will return to Luke his due".

Claude, seeing he would get no profit from this business under such conditions, sent a new pack to Paul, in good condition that one. Paul announced it, and all did everything as they said appropriate for this transaction.

(In practice, choosing a monetary unit, it is represented as D=ℝ, the set of all real number; I just did not write ℝ because its elements are quantity like in physics (like lengths, times, masses and so on), that is, it has additions but the unit 1 plays no role. Just like the same distance can be measured in meters, feet or miles, in the same way, the same money can be expressed in dollars, euros, cents and so on provided that the exchange rates are considered fixed. )

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

Mcorrespondingly. 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 *x*_{0} and *x*_{1}.
Only *x*_{0} can receive credit from other people,
but credits can be granted by *x*_{1} and for this,
the limit is expressed by the quantity of credit from *x*_{0}
to *x*_{1}.

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 *x*_{1} must grant infinite credit to *x*_{0}
but this is worth mentioning only if there is a chance that M(*x*_{1},*x*_{0})
be positive. Since there is no credit from other people to *x*_{1},
this can happen only if:

(*x*_{1} has positive balance) AND ((*x*_{0}
has negative balance) OR (*x*_{0} granted credit to
another person)).

Possibly we don't need to care about this, and for example simply
make automatic credit from *x*_{1} to *x*_{0}
corresponding the balance of *x*_{1}.

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

*b*_{0} = balance of *x*_{0}

*b*_{1} = balance of *x*_{1}

*C* = credit from *x*_{0} to *x*_{1}

only two are meaningful, namely

b = b0 + b1 (one's balance)

C'=C+b1 (effective total credit granted to others by *x*_{1},
amount that one takes the risk to lose by the credits from *x*_{1})

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 *x*_{0} or *x*_{1} 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.

- 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 ?

"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.

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.

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.

"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".

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.

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.

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.

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.

But whereas this virtualisation represents a danger to the current system and is made inevitable by technological progress and the natural movement of markets towards reductions of the transaction costs, it brings new opportunities to set up a finally coherent monetary system based on new principles.

But this can happen only if these new principles are understood and implemented. This understanding means first to distinguish the necessary concepts and features of money systems which come from the real needs of markets and their stability, and which we should keep, from unnecessary human constructions. These latter meaningless complex conventions may come from the irrelevant materiality of the exchange means, all mistakes and ways to deal with them, and the methods developed by banks and governments to keep their domination as financial actors and use it for their profit.

Second, this understanding means to innovate in the formulation of the necessary features, and find the right formalism to express them in their purest and most accurate form. This does not mean it should be always simple, but that its complexity should precisely reflect the complexity needs of the economic system it is concerned with.

The object of this article is to express what features, an ideal money system should have to optimize its stability and usefulness for its users and the society. Thus, it would make it the most competitive currency in an economy open to a free choice of currencies, assuming people's rationality. In a way, this competitiveness can be understood as coming from the fact such a system would be in itself a kind of supermarket of currencies.

In fact, we shall not develop here all details of such an ideal system, but only some of them while other problems are either just listed or given the idea of the the work that remains to do. However, it should be already enough to give a rather clear idea of how it is really possible to make such a better money system than the classical ones, and to give the means for other people to complete this work.

More precisely, the complex computations at the base of the new money systems are roughly divided into two main problems that can be studied as independent theories. Namely, they are

- the theory of credit that deals with responsibilities and limits of loans, to process the respect of money's value;

- the theory of value standards and exchange markets, that defines the agreement on how much the money is supposed to be worth.

The interferences between these problems are likely to be scarce, but may still happen and require specific studies, especially if the too unforeseeable character of the economic events (or of history in general) proves to put these two stabilization mechanisms at hard test.

Economists accustomed to the traditional monetary theories in which the regulation of money creation by public authorities is seen as the instrument to stabilize the value of the currency, might be surprised to see here these two problems considered quite independently of each other, with money creation nearly abandoned in the hands of the individual free choice and responsibility.

(Note: the author of this article was trained as a mathematician and physician and not as an economist, which may explain the sometimes strange style used).

Instead of a bank, there would be one, or several, complementary and/or redundant data bases. Actually, to avoid any loss of data, one can of course duplicate and store them on independent computers. Another situation to consider is when the data are not centralized but each computer is monitoring the arrangement of the only data it contains, and it only communicates the necessary information outside, to make possible the relations between individuals registered at different data bases. Of course that might reduce the possibilities of arrangements compared to the case where all data would be centralized, unless sufficently enough data is exchanged.

I did not study this question of how such a network of distinct data bases could work. So, to simplify the problem from a money theoretical point of view, in all the following we will assume all the data is stored in the same web server.

This web server that proceeds all the operations can be maintained by people that are by no means personnally involved in the financial transactions proceeded there, as the only people responsible for these operations are those who request them. The only question the administrators of those web servers are concerned with is to ensure that these servers reliably run the right software, as users expect. In practice, this implies the software must be open source, and the maintainers must declare what software they use. If people don't like it, they will choose to interact instead with another such web server on the market that runs a prefered software.

Each individual, therefore, can make the financial transactions and take the responsabilities he wants, by electronically signing the contracts and the cheques and sending them into the data base.

The problem we will focus on is not a search for such an absolute anonimity of transactions that even the roots (system administrators) could not track them, which would require special cryptographic tools as developed by DigiCash or similar projects. Indeed, as the trustworthiness of the administrators is one of the main qualities required from them for other reasons, the point of questioning their access to the data and their respect for confidentiality of information when due, is hard to make. The existence of such a possiblity to track transactions can also serve as a good way to oppose corruption and money laundering.

Instead, all the payments, contracts of credit and other transactions will be documents simply electronically signed by the partners. They happen online, sent by secure connections to the central data base web server (to check that they go through). They are recorded and will be consultable at least by the system administrators.

Up to now we just detailed a suitable material context in which the new monetary system can take place. Now we shall enter the core of the problem.

We shall first define a prototype, a reference concept that is relatively simple but coherent, and already represents a big step forward compared to the existing monetary systems. Then we will enumerate a probably nonexhaustive list of its insufficiencies and tracks of developments which would have to be formalized more precisely to obtain a complete system. Many of these developments are related to the details that should be added to the contracts of credits mentionned in the prototype model. Some of the main necessary developments will be expressed and commented in details, thus showing the style of the required solution. This will make it clear what work must still be done on this part of the problem until the development of an effective system. The point is not only to formalize a good solution to each one of these insufficiencies taken separately, but also to assemble these various corrections from the initial prototype to work together.

But, this prototype still remains valid as a set of basic concepts giving the language in which the final solution will be expressed.

Let D be a vectorial line, whose elements are the quantities of money.

Let

The meaning of this map is that C(x, y) represents the amount of credit that x grants 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. In practice, for all x there will be only a few y such that C(x,y) is not zero.

Notation. Let R be the set of the maps r from P to D such as the sum of all the r(x) for x in P is equal to zero.

Definition. A

We denote by L the set of possible ditributions of balances.

We can note that this condition is also equivalent to the same condition restricted to the parts A of P which are connected (for the graph obtained by keeping only the edges { x,y } where x and y are in A) and whose complementary set is also connected. Indeed, if it is true for the connected subgraphs, its satisfaction for the non-connected ones comes by summing this condition over all its connected components. In the same way by symmetry the condition on the subgraphs whose complementary set is not connected can be deduced from the conditions on the others.

To any given time t is associated an element r_t of L, called the distribution of balances at the time t: the quantity r_t(x) is the

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 p of R whose only two nonzero components are p(x)=-u and p(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_t at a time t, and a submitted payment v, this payment will pass through if and only if r+v is also a possible distribution of balances. If it passes through, then the distribution of balances at the next time t' will be r_t'=r_+v.

The map C can evolve in time as follows: each x can freely decide to increase the credit C(x, y) which he grants to y, but he can decrease it only if two conditions are satisfied:

1) It is allowed by the credit contract, which defined a schedule of delays that limited the rythm of credit reductions in future times, yet anyway allowing for a total cancellation of the credit after a sufficiently long delay.

2) The new set L defined from the new credit map still contains the value of the distribution of balances at that time. (This distribution of balances is not modified).

If the first condition is satisfied but the second one is not, then we can request a lesser credit reduction to the maximal extent still possible, and make a system of automatic requests to try again to complete this reduction at regular future intervals (we say that y is ruined). If this appears hopeless, then we can operate a

Theorem and Definition. Suppose that C(x,y)>0 while no reduction of C(x,y) is possible still letting r_t in L, in other words, r_t \in L but there exists a subset A of P such that x is not in A and y is in A, and that

the sum of the r(y') for y' in A equals minus the sum of C(x', y') for y' in A and x' in the complementary set of A.

Then for any positive quantity of money u</= C(x,y), a replacement of C(x,y) by C(x,y) - u (all other values of C staying unchanged), together with a payment of u from x to y, gives a new distribution of balances that is possible with this new credit map. It is called the bankrupt credit reduction of C(x,y) by u.

This quantity u is the amount of money that x lost in y's bankruptcy, by this bankruptcy operation.

The above theorem can be proved independently, or seen as an obvious consequence of the "classical explanation" of the above constructions, that we shall present now.

This construction can be seen as the classical interpretation of the concepts we defined, in terms of the "hidden variable" M which stands for the description of the "underlying reality" behind the effective variable r: at every time, the value of M(x, y) stands for "the quantity of money that x actually lent to y". Its relations with C express the fact that y cannot actually borrow from x more than the credit that x granted to him.

The value of this variable remains undetermined, except when some of its components are produced into concrete reality by events of bankruptcy. Indeed, the further developments we shall present in the next paragraphs imply that the values of this variable as "observed" during bankruptcies, cannot in fact be understood as having any determined real value preexisting the occurence of these events, for they will depend on the way these bankruptcies take place in practice.

From now on we will suppose that G is connected.

Now we shall see some various problems requiring changes from the above prototype to some more sophisticated models.

This list of problems to take into account for a theory of credit does not claim to be necessarily exhaustive.

In a way, we can say that is is normal for the risk to exist, for the same reason that credits exist with anyway, rigorously speaking, no absolute guarantee to be honored. But more specifically, we can consider such possible discrepancies in schedules in the case of credit delegation: if x delegates to y the right to give credits, but y grants this way longer term credits than the delay when x needs his money back, what can we do ?

I currently did not develop this question. It may require major developments.

The next questions on the theory of the credit we shall see from now on will complement each other in a remarkable way to form an interestingly consistent whole.

This point of the limitation of debt as clause of contract of credit would deserve to be developed more.

A consequence of this, that we should care about to know if we agree, is this one: for example let each of two individuals a and b grant a credit to c of an amount of 2000 units with a limitation of debt of 3000. In the same way, c and d grant credits of 1500 each one to e with limitation at 2000. Finally c,d,e all go bankrupt whereas c had a debt of 2800 and e a debt of 2000. Thus each of a and b lose 1900, which is in agreement with the contracts limiting the responsibility to 2000 units each, since c was not involved in debt himself beyond 3000. Are we sure to agree with that?

Let m be a map from P to D such that for any x, (- sum over y of C(y, x)) < m(x) <r(x). By this we define a neighbourhood V(m) of r in R, as

V(m) = { r' in R such as for any x, r' (x) is equal to or higher than m(x) }.

The size (diameter) of these V(m) vanishes when the sum of the m(x) approaches 0 (in other words when every m(x) approaches r(x)). Thus, there exists such an m, so that V(m) is a neighbourhood of r included in L.

If such is the case, then we can consider to use this V(m) instead of L in the role of a set of possible distributions of balances, i.e. any payment which would lead outside this set V(m) will be blocked. More generally, even if V(m) is not included in L, we can consider to use V(m) this way if the circumstances would hardly leave any practical risk that a series of payments occurs, that keep in V(m) but lead outside L. So:

(Here we call "cash" an account figure and do not distinguish between "virtual" and "concrete" money, because the time is coming when the virtual reality, made of information, will become more real than the so-called "physical objects", that we are just used to see as real conditioned by the viewpoint of our common direct perceptions, which are anyway but a poor glance of the actual realities from atoms to the plans of our destiny.)

This restriction of the set of possible distributions of balances to such a set V(m), could seem a disadvantage in the sense that it prohibits certain payments which would have been possible with L.

In return, this brings the advantage that the risk that a payment is blocked can only come from the case the paid sum exceeds the reserve of cash of the payer, which he can clearly foresee; instead the situation where that depends on the last events of the payments carried out by other people who are not his business: "Sorry, we cannot accept your payment because the people to whom you granted credits have just used them by borrowing your money, while people who granted credit to your credit did not receive yet their wages and the other credits which they granted also already have been just used".

The condition that V(m) is included in L, is expressed by:

For each connected subgraph A of P, A \not = P, \sum_A m(x)+ \sum_{x\in A,y\notin A}C(y,x)\geq 0.

The problem to be solved is as follows:

Take a credits function C whose graph is connected, with a big set P of vertices connected each one to a small number of others in a nearly random way, and an interior element r of L. Then, can we find such a neighbourhood V(m) of R included in L, but also which offers enough narrow money for the needs of the market, that is, an amount proportional to the population (so that each person can have an average quantity of cash that remains significant in a growing population), so that the resulting restriction of the possible payments is not dramatic ? Although that can seem surprising at first, we will show that it is possible. Of course, then if one wants to effectively implement the system, it will be necessary to study in more details the set of the possible maps m and its properties to facilitate the variations of m, to see how to adapt it to the needs and to optimize it to limit the wasting of its resources (as when a reservation of cash for one requires the inhibition of many times more cash for all the others). I did not study this problem.

We will thus be satisfied to show in an approximate way the presence of broad possibilities.

If one does not know oneself enough reliable investors, one can make use of the system of delegation of credits mentioned above, where delegates take care to redistribute the assets of the various savers through the wide world down to the diverse investors. No restricted group can require to be selected in particular to receive credits from the rest of the world, but any restricted group is free to spread and grant its credits towards the rest of the world, and for this, has a broad choice.

As a conclusion, among the above conditions expressing the inclusion of V(m) in L, we need not care about those for the majoritarian parts A. Indeed, the above decisions resulting from an evolution of r towards any element r' in general, and thus any element r' of V(m) (even an extreme one) in particular, will modify C so that the corresponding conditions for r' to be in the new L will come true.

Note that once the conditions on the big subgraphs A are dropped this way, the remaining conditions on the small subraphs remain equivalently summed up by those on the connected small subgraphs, because the condition on a nonconnected sugraph is a consequence of those on its connected components which are smaller and thus kept under consideration. This argument would not apply to the connectivity condition of the complements of the subgraphs.

We shall see more reasons to drop the large parts A later.

Consider the weighting of the edges of the graph defined by: for all individuals x and y,

B({x, y }) := C(x, y)+C(y, x).

The search for the map m is equivalent to the search for the map f defined by

f(x)=m(x) + sum of C(y, x).

The desired conditions are expressed by: on the one hand, the sum of the m(x) is negative, that is to say

sum of the f(x) = sum of the B(e) - (narrow money)

In addition, for each connected subgraph with set of vertices A, the condition

sum on A of the m(x) > - sum of C(y, x) for x in A and y out of A

is equivalent through adding on both sides the sum of C(y, x) for all x in A and all y, with

sum on A of the f(x) > sum of C(y, x) for all x and y in A = sum of B(e) for all the edges e of A.

Note that for A=P or nearly, this conditions becomes impossible since narrow money must be positive. Note also that, according to this condition applied to all A={x} for x in P, all f(x) must be positive.

Finally, the heart of the mathematical problem is expressed as follows:

Suppose we have given a finite but large and (more or less) random connected graph (where each vertice is connected to a small number of edges and for each relatively small subgraph, the field of order of its number of edges does not exceed that of the number of edges connecting it to its complementary), together with a weighting B(e) of the edges e by strictly positive quantities.

Find a method to search for the "minimal" weightings f of the vertices by positive quantities, such that the sum of the weights f(x) over all vertices x be strictly lower than that of the weights B(e) of the edges, and for each relatively small subgraph, the sum of the weights of its vertices be no less than the one of its edges.

("minimal" = in the mathematical sense for the pointwise order, in other words, such that any f' that fits such that for all x, f'(x) is less than f(x), is identical to f)

(Compared to a population of a million, "small" = form 3 to ten, and "relatively small" = from 1 up to some size somewhere between a few tens and a few thousands (which should give rougly equivalent results.)

(Maybe we should not look for the minimal values in fact, to facilitate the variations of this choice during time).

These conditions are equivalent to the same conditions restricted to the connected subgraphs.

So let us show that these conditions on small subgraphs is compatible with a significant amount of narrow money. According to the way this problem was defined and with the help of the "classical interpretation", there exists a decomposition of the wheight of every edges as a sum of two positive quantities attached to the ahlf-edges, such that the value of m at each vertex is lower than the sum of the quantities attached to its half-edges.

Here comes the following essential assumption, which is a consequence of the rather random character of the graph structure:

Precisely, this proportion of the number of exterior edges to the number of vertices does not become small for any subgraph, even those of a significant size, as long as this size is small compared to the whole population.

In other words, there is no community of people that nearly forms a close world that does not receive significan trust from the rest of the world. Instead, we assume that we are in an opened world, with multiple relations which cross the various geographical borders permanently, of communities or anything which one could consider, so that any not majoritary set of people will have as many relations outside as inside.

Thus will be satisfied the required condition.

So, consider a community A of people who form a relatively closed world, that the rest of the world would hardly know and trust with some exceptions. The reason to respect the corresponding condition, that is, to restrict the cash they can have, is this: imagine that these people trade with each other, behaving as having each one a good deal of cash, but at some time by misfortune they all decide to use this money to make purchases outside of their community. This means the community is collectively involved in debt beyond the credits which are granted to them from the outside, so the outside world won't accept it.

How can such a thing happen ?

For example, this community A of people who spend their cash could be the set of the extravagant people to which it would be too risky to grant credits. If such were the case, it can be good in any case to observe the rule, namely not to let to these irresponsible people an amount of cash corresponding to the amount of credits which they granted one another. But however, this story is rather incredible. Before all these people spend their credits which they agreed the ones to the others, there are chances that they experience themselves the effect of their own mistake, as a number of them note that these excessive credits towards people too involved in debt paralyse their transactions...

A more serious and interesting situation would be the case of a poorer social class, or, in the case of an international system including inhabitants of rich and poor countries, the set of members living in poor countries.

Else, vis-a-vis the presence of a closed community that may miss credit from outside to validate its internal cash, one could still consider a way of officializing this fact by recognizing the presence of several distinct currencies in circulation: those inside the community on the one hand, those recognized by the outside on the other hand. Vis-a-vis such a situation, it would be necessary to develop specific rules of operation of the relations and convertibilities between these currencies, rules which remain to be developed (but a possible outline is described below).

Then, a difficulty which will arise is to manage the modifications of m to follow the needs, and try to deter people from asking for reservations of more cash than possible to respect our condition : will one tax the cash requested like an interest rate ? Will the cost of holding cash for the individual depend on the circumstances and how ? Would the one who would ask for more cash buy this right as on a market from others ready to reduce their own cash for this ? Anyway, we should try to avoid the trap of tricks where people make business of reserving more cash than they really need, just for the possible profit of selling back this reservation when it "becomes expensive".

One can notice that the previous concept of credit under condition of limitation of debt completes already a work of limitation in the direction of the concept sought here. As all the problem that we raised consists in deciding up to what point the cash of the ones should be limited to preserve the one of the others, we notice that this individual and natural rule of limitation of debt (and thus of cash) already puts a restriction to the field of possibilities to explore. This limit to the cash of the ones contributes to freeing the cash of the the others, which could possibly allow it to reach this same limit, without obliging us to raise any more questions. Or maybe not, but the range of remaining questions to raise may be simpler.

But, fault of knowing anyone for this in particular, one wishes to be able not to choose and leave the rest of the world, as a whole, assume this responsibility, relying on the global functioning mode of the economic system to carry out that. There would be thus a certain share of the sum of cash in circulation freed from any risk (as long as the rest does not globally collapse), and thus an increased risk for the rest of the world, beyond its theoretical total affordability. Never mind, it is enough for this to avoid defining the division of the risks in a rigourously local and individual way, so that even if a certain number of bankruptcies happen, their responsibilities will be taken among those who took the risks, without affecting the unrisky currency itself, as long as there remains sufficiently many healthy other companies to ensure that.

In fact, this question is but the natural extension of the problem of the cash itself we already considered, somehow the same problem interpreted under a slightly different prospect. To speak about a cash without risk is a redundancy. Because, even in the previous case where those who carry cash take risks, the risks in question are not those of the cash itself, because these risks keep their positions while the cash circulates. Thus, considering what difference exists between successive configurations shows what really moved: it is a nonrisky cash.

This problem of riskless cash slightly differs from the previous question on large communities A and reasons to drop the corresponding inequalities in the condition (V(m) included in L), as a difference between the potential and real characters of the property: after extending the set V(m) to the case when it is not included in L but r which evolves in V(m) stays in L, we now extend the consideration to the case when r which evolves in V(m), may even happen to leave L.

So, the set L we started with is replaced here by a larger set defined by the same conditions on the debts of groups, but only the ones of minoritarian groups.

In this context, here are two possible means to facilitate the increase of narrow money able to effectively leave the risk attached, in other words to facilitate the possible concentration of the risks on some part of the population beyond the total affordability of this part. This means that for at least a certain number of individuals, the average amount of assumed risk, defined as being the abstract sum on the credits which one granted, of the (average ?) amount of money which one would lose if just the recipient of this credit could not pay back anything more after a given time (only the account of this recipient of credit cannot go up any more above the negative amount it has at that time), will be higher than the total amount of money that one agrees (can afford) to lose.

A means to do this is to explicitly formulate a convention such as: the share of risk taken by an individual on a contract of credit which he granted in parallel to other creditors of the same debtor, will be a function of how lucky he will be on the credits which he will have granted to other debtors (and of the chance of the other creditors of these debtors). If he has 2000 units on its account and grants 3 credits with a responsibility limited to 1000 for each one, if the 3 businesses succeed he will keep his 2000; if one goes bankrupt and the 2 others succeed he will keep for example only 1100 (or even 1000 if the other creditors have other bad lucks); if two go bankrupt he will keep 300 or 400; if all the three go bankrupt he will keep zero.

Another means, which can be seen as just a traditional method that we recall for use in this new context, is to (assess/guarantee) by some different circuit of credit the more risky share and the less risky share of risk associated with a credit. The more risky share of risk is taken by some people more close to contractor and involved in his business. The less risky share of risk is taken by some more collective and anonymous financial circuit, something like a work of bank or of insurance, able by its size to collect a large number of small risks that are not likely to crash all together, and to assume them on the basis of relatively weak financial assets compared to the sum of all quantities of money risked. These anonymous financial circuits are likely to be those handled by the delegates of credit power that we mentionned.

So consider again the risk to have a big group of people that grant themselves credits that would locally (individually) allow them cash inside the group, but whose sum on the whole of the group would too much exceed the credits granted to the group by the outside world. But what would really forbid us to just ignore the problem and tell everybody that they really own their locally appeared cash ? The only risk, as we already explained, occurs when the whole group happens to spend this cash towards the rest of the world, therefore having no more the total of credits necessary to cover this debt. But, what if we continued to ignore this and allowed them however to come to this debt ? Ah, but we need to specify: a debt unto whom ? Unto people refusing to take the risk to grant credit to particular individuals in the group, but still agreeing to take the risk to grant it to the whole group, so that finally the only real risk they take corresponds to the case of a general bankruptcy of the group. Therefore, unto people who decide to carry the cash of minimal risk created by the internal system of this community.

As a conclusion, let us sum up the problem of what should be computed to face this matter of closed communities.

The delcate problem which remains is that of the limitation of the debt of the groups of average size (minority but not so small).

The first question is to find the most significant violations of the conditions (sum of m(x)+ sum of C(y,x)>0) on groups with average size while it is always kept true for those of small size.

Then, evaluate the effective risk that such a gap, which does not have consequences first, comes to happen, namely the risk that this group of average size gets involved in debt beyond the condition. If there is such a risk, consider the possibility to define a specific currency for the group.

Then, detect when it happens that this community gets involved in this debt, announce this event, and study the problem of whether it has any sense to fear that this excessive debt ends by a mass bankruptcy of this group.

The larger the group is without being one same organization or activity, the weaker the risk of a general bankruptcy is in probability and proportion but large in total value, and the more likely the quantity of received credits is also to be small in proportion but large in total value. If all is well random, the quantity of received credits increases more quickly than the size of the bankruptcy that might happen. Still it is necessary to check well that nothing is faked, to pay attention to all that and to seek to identify and describe the groups at risk.

Then, if such a situation occurs, launch a currency specifically representing the debt of this group, which makes it possible for everyone to speculate on the reality of this risk, and to thus provides a tool of market regulation of this problem.

When two people undertake a transaction, they may need to require that one (or both) commit financially by blocking some amount of cash until the transaction is completed. For example one can put a guarantee which will be returned except in the event of a problem. More precisely, the contract of transaction may require that the guarantee will be returned back unless its author gives it away or the other part makes a complaint against him. For example, sending a message can require the deposit of a small guarantee which will be returned except if the receiver makes an antispam complaint. Similarly, if a transaction involves a risk if the other part is not trustworthy, the commitment will be expressed this way, with a money blocked then automatically given back if there is no complaint.

Another situation is the online sale where the purchaser must pay in advance, but the salesman will be able to really receive the money once the buyer received the thing and is satisfied; and if the buyer is not satisfied he must make a complaint to not pay.

I started a project aimed to handle a system of complaints: see the description at http://spoirier.lautre.net/trustedforum.html.

[Some corrections have been made up to this point; below is a mere draft]

Thus in general we may need to let an amount of money blocked between oneself and someone else, so that no one will be able to take it without the agreement of the other. As long as it is not freed, the sum of the cash held by the individuals is diminished by the same amount. It can be freed only by one in favour of the other, or maybe by a common agreement in favour of a third to be determined, or some partition between these three possibilities, if the contract of transaction allows for it. If it is not freed, its treatment can go through a climbing (?) of complaints (see the above link) or a complaint addressed to a third or a group of third (a court) recognized by both. Then it can be end by a mutual agreement as a donation for charity or any work of public utility recognized by both (see the system of budgetary power of public utility in the liberal theory of the power).

If we had to express this notion of money put between two people in terms of limitation of debt for groups instead of cash, it would be like this: the total debt of each group cannot exceed the total credit received by the group from the outside, where the money put between two people of the group belongs to the group, whereas the one between two people at least one of whom is outside the group, is seen as not belonging to the group.

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.

http://spoirier.lautre.net

Anyway, a lot of more theoretical research would be needed to produce a really satisfying practical solution.

Mathematicians and theoretical physicists are IMHO the right people to do this work: please, dear thinkers of the abstract idealities, despite appearances (the expression what you are currently being paid for), be aware that this is one of the most urgent and useful things that such people could do for mankind in this century, much more than most of what they are currently working on (even more than the Millenium Prize Problems all together) !

The starting problem is that the basic concept of "perfect" market with equilibrium prices, is only a static concept, not a dynamical one. In the static concept, the perfect distribution of prices is defined by a system of many equations in many variables, with a unique solution. This unique solution depends on all functions (behaviours) of all people, none of the values of the solution can be determined independently of any other function.

In "real life", the problem is that things happen along time. With time, causality is only one-way: no future event can ever influence a past one. So the condition of total interdependence that was necessary to ensure reaching market stability, is a priori not satisfied.

So the question is : what new structure should we develop, to reach a sort of maket stability that, while anyway it can never be perfect because of the above problem, will still be as close as possible to perfection ?

The first step of the answer is to introduce a second time dimension: the virtual time.

So we have two dimensions of time: the real time t, and the virtual time t'.

What is interesting is the domain (t'>t).

So, at every real time t, the virtual time axis (t' varying along [t, +oo]) is the static dimension in which we are figuring expected future events. Since it is a static dimension where everything remains possible, the pursuit of market equilibrium can be operated there. Not exactly of course, since the future necessarily has uncertainties, but something approaching.

But, this concept is not sufficient, there are special phenomena and troubles that need to be handled.

Now, what's next, in the stability problem. In the definition of stability, there is a neutral free variable: the real value of the monetary unit. Because the monetary unit is a pure convention, that is meaningless in itself, its only meaning is to serve as intermediate figure to calculate the ratio between different real values on the market.

This concept is very similar to the one of gauge theory in physics: what is meaningful is the difference of electic potential between 2 points, while the value of the electric potential at one point is meaningless.

The laws of physics speak about static equilibrium defined in terms of the differences of potential, while the concept of potential at one point is a phantom concept that is discarded (just like the fact that movement is relative). Concerning the monetary stability problem, a trouble comes here: while at a fixed real time t, the stability of market in the picture along the virtual future time t' can be searched for, and defined up to a neutral "meaningless" factor for the monetary unit, the risk that the conventional unit used at a real given time t1 will not be consistent with the conventional unit used at another real time t2. In other words, what is a meaningless factor from the viewpoint of a fixed real time (well, meaningless only if all accounts are initially zero at this real time and considering to vary only in the future), can be subject to variations when the real time passes, which are meaningful.

These variations have to be "controlled", to satisfy the market consistency. If the value of the monetary unit was let "free", it would be subject to arbitrary variations along real time that respect no rule, no meaning, no concept, nothing. Just a wind, pure absurdity: what you borrow to invest in a business, you cannot know how much you would have to really pay back later. No sort of market stabilisation would ever be possible in such conditions.

Now here is the next step of the concept.

Fix a real time t, and consider for it, how the stabilisation process of the market in the picture extended in the virtual (future) time, happens.

As said before concerning the stabilisation in a static market, all is interdependent. But more precisely, all REAL things are interdependent. So, there is an exception: the conventional unit of money, that is not a real thing (well, if we forget the important fact that at t'=t, people have nonzero balances and are very concerned about it; but for whatever value of this monetary unit revised at t'=t, the conditions of stability of the market along the virtual

future admit a theoretical solution, and this is these solutions that we will be speaking about now).

So, this "unreal" or "arbitrary" thing of one-dimensional set of possibilities of revision of the real value of monetary unit, contrary to what has a unique solution determined by market stability, would be a priori not causally determined and therefore have a priori no reason to obey a precise order of causality, either mutual interdependence or unidirectional causality.

However, we can distinguish in this picture a "natural process" of unidirectional causality that runs backwards in the virtual time dimension. And this "natural process" is something fundamental that the final solution of the problem will have to use.

Here is this backwards causality relation: the present (t'=t) real value of the monetary unit, is naturally determined by market forces to conform to the only present value that would be consistent with the market stability condition with respect to its expected future values (t'>t), which are "supposed" to be dictated by unknown exterior forces.

Indeed, when you are selling something to someone, the price that you choose to agree on is determined by what you think that this money you

are getting will be worth in the future. You are not, at this time for this agreement, interested with the numerical value of this money, but

only in the future real value that you are expecting to get through this amount of money you are now receiving.

So, the present operations are naturally uniquely determined to be what conforms to the context of the expected future.

This is the backwards causality determination along the virtual time dimension.

Now, what is the structure that needs to be implemented to ensure monetary stability ?

It is an "artificial" retroaction process whose object is to correct, by convention, the picture of the "official" (contractual, through term markets) future expectations, according to the real time observation of its (backwards causal) effects on the present market prices, in order to make these present market prices conform to the past agreements that had been done in this same way.

That's it.

Very theoretical problem, and that will still require a big amount of work to develop into the necessary practical implementable details.

This is why I do not envision the realisation of a monetary system in the short term.

What kind of concepts I dream about as progress in this research: to define a sort of "superfluid markets" including things like buying and selling uncertainties in the ratio between given values.

Indeed, the problem of making a money system is a mathematically complex problem, and it is clear for me that no solution can be really good and sustainable while getting rid of the central banks'control, unless it involves some huge pack of mathematical formalization. I know there also exists in the world some simple technologies that can be simply developed by anyone without important mathematical work. Unfortunately, it happens that the money problem is not in this situation. So I consider that the existing online or "alternative" currencies (LETS, time money, Ripple, Bitcoin...), have no future, as, no matter how many million people would use them, they cannot be the right systems and can never be the right basis to make one, because their basic concepts are fundamentally flawed : what's the point of super- securing a system that is not the right one ???? All their claims and interests are missing the very whole point, which is

Ripple is (among the projects I know) the one that is closest to mine as it presents some similarity to mine in its mathematical concept.

(Other projects like LETS and Time Moneys, ignoring the mathematics and economics of all this, are just sorts of cults promoting some stupid fanatical and paranoid litterature, that is worth nothing: such systems can never work (even if their supporters can self-persuade they do). They just ignore that money is a hard technology, and successfully making a good technology is mainly a matter of science and mathematics, not a matter of litterature, fanatism and paranoia.)

Though my ideas start from rather similar principles from an economical point of view as the Ripple project, the implementation method I envision would be completely different, and, I think, much better, so I think the Ripple project will become obsolete when mine will work. I told it to Ryan, but he still goes on in his project.

As one part among others of my infoliberalism concepts, the implementation method I envision would consist in a later additional functionality on top of the infrastructure of my Trust-forum project, that can first become successful for its other uses without any monetary function yet.

Then, on the technical implementation of the money functions would involve many users per web server, grouped so that many connections will be between users of the same server. So, each database would contain many (thousands of) users'accounts in averagle. In this way, payments will generally involve only 2 servers, sometimes 3 (where the third intermediate one would usually be easy to choose using the data of both servers making a transaction), and no special protocol between servers would be needed.

Getting rid of this fuss of making a special protocol for money, we can focus the work on more important problems: make the computation most often immediate after a mathematical reformulation of a slightly modified version of the problem that simplifies the calculations by getting rid of the path searching trouble in most operations (long path searching will not even be needed inside a single database !!), involving a small percentage of "cash"; complete the system with other economic functions to solve conflicts, make different sorts of payments adapted to the needs, handle bankruptcies and provide monetary stability.

The first part, is the theory of credit. It is developed below in 2 drafts I wrote at different times. A first draft that expresses some general ideas, and a second draft, more recently written, focuses on what to implement concretely.

The second part, is a theory of defining the value of money based on term markets, that is necessary to stabilize the system. I did not develop it yet.

If you are interested to work on its implementation as a component of my trust-forum project, please contact me (trustforum at gmail.com).

What I envision is a network of thousands of hosts, where each hosts contains thousands of nodes. (for a network of 10 million people, we can say for example 2,000 hosts of 5,000 users each). The advantage is that for every two users A and B of the network, there is good probability of having a credit relation between some user of A's host and some user of B's host, so that decentralised interhost routing will be trivial.

Please read about the Global Login System that is included in my project:

http://spoirier.lautre.net/trustedforum.html

This system is already implemented.

This way, there will be no significant economical incentive to be hosted to one place rather than another, so that connected monetary accounts can be often grouped to the same hosts to facilitate payments.

Moreover, I suggest that one's financial account can be hosted to a different place than one's main account, and accessing financial account will be in 3 steps: log in to main account, use GLS to financial site, then use other password for using financial account. This way will be more secure.

And it will be easy to arrange to group many connected nodes into the same host.

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.

The whole purpose of my project is to improve the determination of trust so that there will be trust. Not trust in anyone, but a determination of who is reliable, so that he can be chosen, and therefore the chosen ones will be reliable.

There will need no big organisation to handle a machine with thousands of accounts. Most of the work will have been done by programmers and contained in the free software. The human remaining "controlling" work can have been programmed to be done by some users as determined from database (trust declarations by other users), independently of who is materially hosting the machine.

And thanks to the rest of my infoliberalism project, it will be easy to find trusted people for hosting account. His only role will be to install the software correctly. This requires not so much competence, and a more technical competence, contrary to political or bank competence that are too melted with corruption.

So it will not be too difficult to find some innocent hand to do it.

Ultimately, in a future when computers will be very cheap and technology will be good, it will be easy and sufficient that a few innocent people, even not rich, install the right softwares in which all the data will be stored secure and private, so that all privacy problems will be solved.

Also, the economic and mathematical problems of finance have many subtleties, special functions and tricks that are necessary to understand well to start with, before making a protocol that otherwise will be useless and unable to carry them.

So again I consider the fuss of finding a protocol before having a good economical, political and mathematical vision of the goal, is a complete waste of time.

The organisation of data into more centralised databases that I propose, will contribute to the privacy

of user information.

All the expected qualities of the Ripple protocol become much easier to ensure when the data are in some central databases maintained by trusted people rather than dispersed all over the web.

If the network is large enough, and time will pass, it will be easy and economical to host data at trusted people.

If it is an important question, then it will be cared about.

(French version)

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