openBarter

We give here a detailed information on how openBarter finds exchange cycles from the set of bids of participants that minimises the collective effort, and how a draft agreement is obtained as a compromise between these unilateral commitments.

Price and ω

We know that price using a monetary medium can be written as follows:

price=^{quantity of money}/_{quantity of commodities}

that buyer and seller agree when:

price_buyer=price_seller

and that one price is defined for each commodity.

When measurable commodities are exchanged without monetary medium, we note ω the ratio between received and provided quantities. We have hence:

ω=price_seller=1/price_buyer

And the agreement is obtained when:

ω_seller=1/ω_buyer

ω is also defined for a couple of commodities, one requested and the other provided.

ω observed for executed agreements of the market are defined for a pair (unordered couple) of commodities.

This notion of ω equivalent to a seller price is useful to avoid the confusion between the buyer’s and seller’s points of view of the same bid, and also to understand how openBarter works.

Lowest collective effort

We show here how openBarter matches bids producing the lowest collective effort.

On a market using a monetary medium

The market place proposes to the buyer the lowest price and to the seller for the highest price. A detailed expression of the rules of the market place can written as follows:

• When only price_seller is defined, the buyer searches for the minimum price_seller.
• When only price_buyer is defined, the seller searches for the maximum price_buyer.
• When both price_buyer and price_seller are defined, the market place matches buyer and seller having the minimum ratio price_seller/price_buyer that is the minimum price_seller for the buyer and the maximum price_buyer for the seller.

This last rule can be applied only when there is a single buyer (resp. seller) for a set of sellers (resp. buyers). We see that the ratio price_seller/price_buyer, a non dimensional number, measures the common effort the buyer and seller must provide to find a compromise; an effort they will have to share to cooperate. It is usually greater than 1; when buyer should increase it’s price, and seller should decrease it’s price.

We call this ratio the collective effort required to find a compromise and required for cooperation. It is a real effort when it is ≥ 1, and the most usual situation.

Using ω

If we note:

ω_buyer=1/price_buyer and ω_seller=price_seller

the rules of the market place can be simplified as follows:

• when one ω is unknown, the market place proposes to the participant the minimum ω,
• when both are known; the market place proposes an exchange between bids whose ω’s product, noted Ω, is minimum.

with the same restriction on the last rule.

The product Ω is the collective effort that the market place minimizes.

When the market allows relations between more than two parners

When the commodity one bid provides is what an other requests, they can be chained and the global ω is simply the product of ω’s of the bids we also note Ω.
Openbarter extends the search criteria to exchange cycles larger than two partners using the following rules:

• When a participants looks for a potential agreement with other participants, openBarter searches for the chain of partners having the lowest Ω.
• When all bids of participants defined their ω’s, the proposed agreement is the one minimizing Ω.

This last rule can be applied if all possible matches are found and satisfied before a new bid is submitted.

It is exactly the way openBarter matches bids to minimize the collective effort.

Agreement as a result of a compromize

The compromize found between bids will depend on the knowledge of the author of the new bid and ω of bids will be adjusted. ω_agreement (ω_a) will be obtained from ω_bid (ω_b), and a draft agrement will be submitted for approval to partners.

The compromise will be such as the product of ω_a of the cycle will be 1, this condition being necessary for the draft to be formed.

The participant doesn’t care about other bids

It’s not the correct attitude, but cannot be excluded. openBarter then finds a compromise by adjusting ω’s of bids so that the effort is equally shared between partners.

The draft agreement is then submitted to partners for approval.

The participant cares about published bids

Since ω is define for a couple of commodities, the participant asks for ω proposed by potential partners by defining the commodity he requests and the commodity he provides.

openBarter gives him the chain of bids with the best Ω_b (product of ω_b), potential partners and quantity he can obtain and should provide using this chain, defined according to available quantities of bids of the chain. This proposal is made supposing that its ω_a is such as Ω_b*ω_a=1.

He can then change the quantity he wants to provide, that should be less than what the chain can provide, and the ω_a he wants.

From this new ω_a, openBarter computes the effort required by partners and shares it equally between authors of bids of the chain to adjust ω_a’s on the chain.

The participant can change these two parameters until he is satisfied by compromise. Then, he asks for the draft agreement formation.

To make it, openBarter computes a ω_b for a new bid in order than it produces the wanted ω_a. This bid is submitted using this value ω_b and the defined quantity provided as when the participant doesn’t care about the market. The draft created is then submitted to it’s partners.

Agreement execution

A draft agreement is executed by transferring the ownership of commodities from one to an other for each bid met by the draft.

Quantities reserved for the agreement are transferred back to their corresponding bids when the agreement is cancelled.

The agreement is executed when all partners approved it. It is cancelled when one of them refused it. And if a participant accepted it, but wants to use the reserved commodities for other purpose, he can also refuse.