**Keeping Ownership Decentralized**

Money represents a future commodity ownership. However, the only way of keeping this ownership rightful, hence decentralized, is to price commodities in metarepresented money. Any otherwise priced future ownership will not remain rightfully decentralized.

Still, what is metarepresented money?

**Direct Commodity Exchange**

Let there be two owners *A* and *B* of commodities *x* and *y*, respectively, of whom *A* wants *y* and *B* wants *x*. Without any money — whether metarepresented or not — the only way for both people to obtain their desired commodities is directly from each other:

A-->y|B-->xx_____ |yy_____ |x

Otherwise, *A* and *B* must *delegate* their commodity ownership to someone who then *redistributes* it between them. However, such a centralized solution would at least partially contradict the same ownership, by at least partially taking it away from its rightful controllers. Hence, only a decentralized solution can preserve all commodity ownership legitimizing this exchange, by *A* and *B* exchanging *x* and *y* directly.

**Individual Multiequivalence**

Still, direct commodity exchange poses two problems:

- Let there be now (as follows) three owners
*A*,*B*, and*C*of one unit of commodity*x*, one of*y*, and two units of*y*, respectively. Additionally, let*A*want the most units of*y*, while*B*and*C*want at least one of*x*each. Then, the available unit of*x*will be worth one and a half units of*y*. So either*A*loses value to*B*or*C*to*A*— since the exchangeable quantities of*x*and*y*are not worth the same:*A*-->*y*|*B*-->*x*|*C*-->*x**x*(1.5*y*) |*y*_____ | 2*y* - Let (as follows)
*A*,*B*, and*C*own a single unit respectively of*x*,*y*, and*z*. Additionally, let*A*want*y*,*B*want*z*, and*C*want*x*. Then, direct exchange could not give any of those three owners their desired commodity — as none of them has the same commodity wanted by who owns their wanted one. Moneyless exchange now can only happen if one of their commodities becomes a simultaneous equivalent of the other two, at least for whom neither wants nor has it. So it becomes a*multiequivalent*, whether the other two owners also know of that multiequivalence or not. For example,*A*could give*x*in exchange for*z*just to then give*z*for*y*, this way making*z*a multiequivalent (as asterisked):*A*-->*y*|*B*-->*z*|*C*-->*x**x*_____ |*y*_____ |*z***z**____ |*y*_____ |*x**y*_____ |*z*_____ |*x*

Likewise, this individually handled multiequivalence poses a new pair of problems:

- It allows for conflicting indirect exchanges. In the same example, any two or even all three owners could simultaneously try to handle it. For instance, while
*A*would give*x*in exchange for*z*(then*z*for*y*),*B*could rather try to give*y*for the same*x*(then*x*for*z*). To avoid this conflict,*A*,*B*, and*C*must delegate now their individual choice of handling multiequivalence to a public authority — whether to their consensual one or even to other people’s. However, such a centralized solution would again at least partially contradict their commodity ownership, by at least partially taking it away from them. - In addition to allowing the exchangeable quantities of two commodities not to be equivalent, its indirectness increases the likelihood of that mismatch, by requiring additional direct exchanges. Let the same owners
*A*,*B*, and*C*of a single unit respectively of*x*,*y*, and*z*want the most units respectively of*y*,*z*, and*x*. Additionally, let a fourth owner*D*of two units of*z*want at least one of*x*. Then, the available units of*x*and*y*will each be worth one and a half units of*z*. Finally, again let*z*be an individual multiequivalent. Now, either*A*loses value to*C*or*D*to*A*, then respectively*B*to*A*and*A*to*B*— since the exchangeable quantities of*x*,*y*, and*z*are not worth the same.

**Social Multiequivalence (Money)**

Fortunately, all those problems have the same and only resolution of a single multiequivalent *m* becoming *social*, or *money*. Then, commodity owners can either give (sell) their commodities in exchange for *m* or give *m* for (buy) the commodities they want. For example, again let *A*, *B*, and *C* own commodities *x*, *y*, and *z*, respectively. Still assuming *A* wants *y*, *B* wants *z*, and *C* wants *x*, if now they only exchange their commodities for that *m* social multiequivalent — initially owned just by *A* — then:

A-->y|B-->z|C-->xx,m__ |y_____ |zx,y__ |m_____ |zx,y__ |z_____ |my,m__ |z_____ |x

With social (rather than individual) multiequivalence:

- There are only two exchanges (either a buy or a sell) for each commodity, regardless of who owns or wants which commodities.
- All commodity owners exchange a common (social) multiequivalent, which eventually returns to its original owner.

Finally, with a social multiequivalent (money) divisible into small and similar enough units, any two commodities can always be equivalent, even if their exchangeable quantities are not. For example, let commodities *x* and *y* be worth three and two units of a social multiequivalent *m*, respectively — *x*(3*m*) and *y*(2*m*). Then, let their owners *A* of *x* and *B* of *y* be also the owners respectively of two and three units of that money — *A* of 2*m* and *B* of 3*m*. If *A* and *B* want *y* and *x*, respectively, but only exchange their commodities for *m* units — *x* for 3*m* and *y* for 2*m* — then:

A-->y_ |B-->xx(3m), 2m|y(2m), 3my(2m), 3m|x(3m), 2m

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