Forum reference mechanisms

The reference mechanisms which support the judicial governance process of reweighting the Forum under DGF are leaching and donating. Combined with an incinerator function, these primitives are complete, in the sense that using only those functions, any possible Forum WDAG can be re-weighted in any way desired.

Primitives

Leaching, donating, and incinerating are primitives of all possible reference mechanisms. This means any complicated type of revaluation mechanism can be achieved by combining these three primitives in series.

Leaching

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Leaching reference code

Main page: Leaching reference code

Donating

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Donating reference code

Main page: Donating reference code

Incinerator

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Incinerator code

Main page: Incinerator code

Combining primitives

Leach-donate

A post may simultaneously leach and donate to another post. It is important to note that the order in which the references are executed matters. By default, a post with both leaching and donating references executes leaching first and donating second, though this can be reversed by fiat.

If there are several leaching references, it is also important to note that the order in which the leaching references are executed matters--switching the order of execution may change the outcome. This non-commutativity arises because the order of references in time is different. For example, suppose post3 leaches from an older post1 first, then leaches from a newer post2 which has already leached from post1, then you get a lesser result for post3 than when the order is reversed.

Chains of references

Donation and leaching are not perfectly symmetric. By default, a chain of posts which all donate will recalculate the values of the posts down the chain when a new donating post is added to the chain. However, a leaching post only affects the value of the post it references--value changes from leaching do not propagate down the chain.

For example, suppose postK at time $j$ has value ${\textstyle R_{K,j}}$. If post1 with initial value ${\textstyle R_{2,0}}$ donates 0.5 to post0, and post2 donates 0.7 to post1, then at time 2 we have $R_{1,2}=R_{1,1}+0.7R_{2,2}$ and ${\textstyle R_{0,2}=R_{0,0}+0.5R_{1,1}+(0.7)(0.5)R_{2,2}}$. So at time 2 the value from post2 flows down through post1 to post0.

However, a post which leaches does not propagate down the chain. It only leaches value from the post it references, which does not affect any ancestors of the object post. For instance, continue the example in the previous paragraph, adding post3 which leaches with strength -0.4. Then we get ${\textstyle R_{2,3}=(1-0.7)R_{2,2}-0.4R_{3,3}}$ and $R_{3,3}=R_{3,3}+0.4R_{2,2}$ assuming the initial value of post3 is smaller than $0.4R_{2,2}$. The other post0 and post1 do not change value with the addition of post3.

If the DAO wishes to leach value from several posts, it must make separate leaching references to each one.

Governance examples

Transferring value from one post to another

The figure displayed illustrates a scheme for redistributing power from Post1 to Post2.

In this case we can imagine governance concluding that Post1 was bad and Post2 was good. Then governance makes Post3 with power 10 REP and validates it. Then in the displayed example the Forum WDAG has been revaluated, i.e., re-weighted. This is the primitive that allows governance to review previous judgements and correct mistakes made in the past. The incinerator is a complication that allows this re-weighting action to keep the total graph weight constant.

This isn't the only way to achieve re-weighting. Every approach has its own advantages and disadvantages. This approach encourages stability because it is particularly inefficient. E.g., it costs 10 to move 10, and 10 REP is destroyed. So conscious effort is required to reweight the Forum to correct past mistakes. This effort requires the sacrifice of 10 REP on the part of those who make the change (the 10 REP from Post3 was governance value).

However note that in this case, the 10 REP "sacrifice" will not burden any particular member to achieve the revaluation, if it is done in concert with the governance tax. In this case the governance tax is free since the total amount of REP in the DAO before and after the revaluation is unchanged, because the excess governance tax is incinerated.

Leaching from a leaching post

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Completeness

A set of reference primitives is complete if any finite WDAG can be changed into any other finite WDAG through a finite series of new posts. (Equality of WDAGs here is measured by weights. A WDAG which includes a post with 0 REP value is equal to a WDAG that deletes that post.)

The single mechanism of simultaneous donation and leaching is complete when combined with the incinerator mechanism, as we now demonstrate.

Proof of completeness:

Given post1 with weight W1 we can reweight it to any other weight W by creating post2 which donates to post1 if W1 < W or leaches from post1 if W1 > W. If leaching is required, post2 simultaneously donates to the incinerator all the value it has. Thus post2 ends with 0 REP value.

This is not the end of the proof, since other posts in the sub-DAG of referenced ancestors are now changed, as well as the set of descendants affected by leaching from post1. We need to balance all the changes so that the leaching and donating processes affect the entire DAG correctly. We will need to solve a system of linear equations in order to find the right reweighting.

To organize the process, first note that any DAG can be given a total order. Since our Forum is a finite WDAG, we can define a Layer 0 consisting of all the roots of the DAG, i.e., those posts with no ancestors, i.e., those posts which have no references going out, but only references possibly coming in. Layer 1 is the set of all posts which reference out only to Layer 0 posts. Layer 2 posts are those which only have references out to Layers 0 or 1. Inductively, Layer ${\textstyle i}$ posts are those which only reference posts in Layer ${\textstyle k}$ for any ${\textstyle k<i}$.

We now work inductively on layers. The posts in Layer 0 may be reweighted without difficulty by leaching or donating. The posts in Layer 1 can be reweighted, but any change in such a post must also reweight any post from Layer 0 that is referenced. To do this we solve the following linear equation:

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End proof.

Alternate primitives

Absolute redistribution bounded by the value of a post

Another complete set of primitives exist which are more restrictive. For example, an even more restrictive governance protocol would only allow the sum of absolute values of redistribution to be less than the value of a post. In this case, for instance, a DAO would use Post3 to only leach 10 and incinerate it. Then a Post4 is required to donate 10 to Post2. With this restriction it would take 20 REP and two posts to achieve the same redistribution that 10 REP completes with one post as demonstrated in the previously displayed figure.

This same limitation could be enforced under the previous hard protocol primitives by instituting a soft protocol requiring it in all posts. Therefore we consider the previous primitives as a better default for DAO governance.

Governmental fiat

Consider another system that a DAO may choose is much less inefficient, but also less stable. We can alternatively program primitives that give governance complete control by fiat. Specifically, a DAO may choose to make a primitive which allows any redistribution desired without any cost in REP. In this case, as long as an arbitrary redistribution was validated it could be achieved if the DAO allowed that primitive.

In general, this is much more dangerous than the previous examples of primitives, because one validated post could put all the value of the entire WDAG into any new account the poster wishes. However, it would be extremely efficient if the DAO wishes to move quickly and trusts its governance.