Editing Executive governance




== Policing Valuation ==

= 1   =
It is assumed most members will not leave their reputation tokens unused, merely passively earning the reputation salary. At a minimum, most members can be expected to also use their REP to executively govern, that is, to automatedly police all actions in the DAO. This will be typical when the reward for policing is sufficient to motivate members to participate in policing.

In this section we find the present value of automated policing, based on the work-to-policing ratio set in the validation pool. This will help guide DAO governance decisions for how to decide this parameter’s value to motivate more or less policing.

'''Proper Policing''' means continually using the most recent version of the canonical front end, which runs the automated algorithms that vote in accordance with governmentally established soft protocols. We distinguish Proper Policing from general policing, because in a decentralized system Byzantine behaviors are possible. Anyone can choose to directly vote in any way, e.g., against the accepted standards of the DAO.

An example of policing is given in the first Bitcoin node software. Running a full node means your computer will store the entire information of every transaction on the blockchain and run the full software package for participating in blockchain consensus. This software package runs automatically and deploys the following protocol: Listen for new users’ proposed transactions ('''TXs'''), send those TXs to the network (this is the '''gossip network'''), collect all known TXs for the last 10 minutes into a '''block''', calculate hashes to try to win the hash lottery for PoW ('''hash mining'''), submit your block for approval if you win the lottery, or check the other submitted block(s) are made according to protocol if you lose. The vast majority of participants in the Bitcoin blockchain skip most of these steps. They instead choose to run an edited version of a full node which limits their computing resources to the task of has mining so they can maximize their chances of winning the bitcoin reward. Policing, in this context, means the other steps in the process, which secure the network—participating in the gossip network and especially checking that the proposed blocks are formed correctly according to protocol.

How much is policing rewarded? In the Bitcoin network, there is no immediate incentive for policing. In more sophisticated PoS (proof of stake) consensus algorithms, such as Ethereum’s LMD-GHOST, policing rewards are built into the protocol. When a DAO implements a reputation mechanism, we need to account for policing rewards explicitly, so we can optimize the incentive. The formulas given previously show how much a token is worth for passively collecting the reputation-weighted salary. But in this section, we need to determine how much more a token gains by policing. Specifically, how much extra REP is gained automatically by participating in the non-contentious, binding validation pools that should result in unanimous decisions? If this value is not more than the price of running the machines necessary to execute the full version, then most users will eventually opt to execute simplified versions, which is a security threat.

== 1.1     Infinite Lifetime Automated Policing ==
[??consider ignoring the  naming convention and simply name the policing parameter  or .  ??]

Whenever a validation pool is opened with a fee of  in fungible currency units, then  reputation tokens are minted, and  tokens are staked in the poster’s name and  tokens are given to the bench for policing.[1] Here   gives the '''work-to-policing reward ratio''' of  Default is  when work and policing are equal. When  policing is augmented. When  work is encouraged. The term “policing” may give the wrong impression, since automated policing may have a greater effect on how much entrenched power is protected, with  encouraging new members to join and earn REP through work, and  favoring older members who run the automated policing algorithms to protect their power. These claims will become more specific and obvious from the mathematical analysis that follows. In the end, these calculations will dictate how to set the policing reward parameter to motivate optimal policing and also whether to encourage new members to participate or to protect established members’ power.

As in the previous calculations, the rate of fees  determines the present value of a policing token. The total number of tokens in a DAO is again given by

Assuming a single token was minted at time  the fees it earns is again given by

But the new REP tokens the single token earns by policing is governed by the parameter . For each fee that enters the DAO, the first REP token earns the fraction  of the  newly minted REP tokens, where the policing parameter is simply  and  is the number of tokens that participate in the policing actions at time . So . For simplicity, we will conservatively assume  which is reasonable if  is large enough to justify participation. However, this means our estimates will undervalue reliably active tokens, since all DAO members’ continual participation is not likely.

Now each of the tokens the first token earns by policing will also earn further policing tokens at the same rate, ''ad infinitum''. The tokens that result from this policing process from a single initial REP token is denoted by  which satisfies the initial condition . So  grows according to the ODE

Equation 10

which is separable and can be explicitly solved as

assuming  and  are constant. Then

Equation 11

1 token which is actively policing then produces  coins which together have the reputational salary  given by the formula 

1 token which is actively policing then has present value 

similar to the present value of a passive token.

To improve our intuition for these formulas and gain some mathematical perspective about how important policing is, next we make simplifying assumptions, such as constant or exponential fee rates .

=== 1.1.1      Constant fees ===
For constant fees  we get   

and so Equation 8 gives 

If (as we are assuming) all members police all the time, then

gives the formula for the collection of all policing tokens  generated from the original tokens from time . The policing tokens from a single token  grow without bound, however the relative power that all original tokens  maintain under automated policing is

anytime . This proves policing alone cannot maintain your relative power in the DAO, nor the power of the original cohort . The inflationary minting mechanism will dilute your power if you don’t continue to contribute to the DAO with further work. So the entire starting DAO members will eventually be usurped if they don’t add more original contributions beyond automated policing.

?? The new tokens that are minted through work fees not including policing is  Those tokens and all of their policing rewards are 

??Can we understand this more clearly? What is the differential equation that governs how these new tokens are growing?

??


We also can calculate the income stream of 1 token with policing  and its present value under the assumption of constant fees:


and 

This last expression is finite, but not expressible using elementary functions. This is common for any expression that derives from a complicated continuous process—most elementary functions lack elementary antiderivatives. Technically it is an incomplete gamma function. When necessary, we can make tables of values for any relevant parameters we happen to be using in order to gain intuition for how it behaves.

=== 1.1.2      Exponential fees ===
Now we assume, as above, the fees follow an exponential rate

This gives

Using Equation 8 we get 

The single policing token gives the reputational salary

Notice policing makes a qualitative difference under exponential fees, since a passive token had salary  asymptotically linear, whereas policing gives  asymptotically exponential salary.

This salary’s present value is 

which is not expressible using elementary functions, but is given by the hypergeometric function . Notice  is finite when  and infinite otherwise. Therefore, we see that when the policing ratio  and the exponential growth rate  overcome the interest rate  we can expect explosive returns, which is a threat to the stability of the DAO since it is likely a hype cycle would form, generating unreasonable expectations of future earnings.

Even though exponentially growing fees gives an exponentially growing salary, notice that the  original tokens, even if they all participate reliably in policing will be diluted in relative power according to

so the DAO will predictably decentralize, even if the founders continually police the newcomers. The only way founders can maintain majority power is by bring the majority of new fees to the DAO by continually performing the majority of productive work.

== 1.2     Finite Lifetime Automated Policing ==
For the case of REP tokens with finite lifetime, the ODE of Equation 7 becomes a delay differential equation (DDE)

Equation 12

with initial condition

As before,  represents the total number of tokens in a DAO with finite lifetime  and is given by

Again,  is the policing parameter, and the tokens that result from this policing process from one REP token is denoted  with  which have the reputational salary  given by the formula 

which has present value 

To explain why the policing token holdings  evolve according to the DDE of Equation 9, we focus on the two terms on the right-hand side. The first term  is the same as Equation 7, coming from the fact that the rate  of newly generated policing tokens is proportional to the newly minted tokens as a fraction of the total active policing tokens . These new tokens are minted in proportion to fees  and the minting ratio and the amount policing is rewarded . The second term  comes from the fact that tokens  units of time old are expiring at the same rate they were added  units in the past.

The factor  makes Equation 9 a delay differential equation[2], because the way  changes at time  depends on the moment  units before the present. Such DDEs can be solved numerically for any practical choice of parameters and .

== 1.3     Attenuation Function ==
In this section we detail the formulas for a more complex tool that DAOs have at their disposal. The tokens can be programmed to diminish in value in much more complicated ways than merely programming an expiration date. We can instead design the DAO so that the tokens’ power shrinks (or grows) in value as time goes on. To do this we specify an attenuation function  which means that a token minted at time  will have potency  at time . So  is a multiplier for the value of any token minted at time . This means that one token minted at time  pays out the fraction  of the fees the DAO earns at time  instead of the usual fraction . Typically  will naturally be constrained , but the formulas are valid for more general functions. In particular, these formulas can be used by a DAO’s governing body to account for the reputation tokenomics consequences from diminishing or enhancing older power according to any chosen design, which amounts to choosing a time formula for .

Here as before,  represents the total amount of token power that exists at time . This is different from  which is the total number of tokens ever minted (ignoring attenuation). As before  satisfies


where  but now we have


Equation 13

where, of course,


To justify the form of  in Equation 10, consider how at time  the term  attenuates the tokens that have aged  units of time. The integral in Equation 10 sums all the tokens ever minted, with the term  over all times  before time  modified by how much they have attenuated in their current age .

Assuming a single token was minted at time  the fees it earns is given by

Then as before, the present value of a single token with attenuation (but without policing) has present value



Finally, we add automated policing with parameter  and the analogous quantity of active tokens under attenuation  is now denoted by . It is easier to keep track of all policing tokens earned from one token  unadjusted by  which evolve according to the DDE



This is a DDE since  depends on the past of  for .


Using an attenuation function with non-zero derivative will improve the non-fungibility of the REP tokens. Technically two tokens minted at the exact same moment will be fungible if you ignore the possibility of slashing from review. However, similar to financial derivatives, any token minted at a different time is non-fungible with any token minted at any other time. This makes the market for REP tokens much shallower, which would result in REP tokens being more closely tied to its owner, decreasing the need for any KYC protocols.

== 1.4     Policing Conclusion ==
The policing parameter determines the reward for policing, which motivates members to participate in the automated function or not. If it is high enough, then we can expect everyone in the DAO to participate, because it will profit them. If it is on the edge, some will drop off, which will give greater reward to those who remain.

Secondly, if the policing parameter is high, the ability to pay for REP to effect a 51% attack arbitrage is diminished, since the higher the parameter, the more costly the attack.

Thirdly, the higher the policing parameter, the more old REP remains in the DAO, even with finite lifetime for REP, because policing creates new REP from old REP. Even though policing REP diminishes in time, some policing REP will persist for all time as it is continually minted even though individual REP tokens have a finite lifetime.

This leads to the problems identified with infinite life-time tokens, albeit in a diminished form. Older members have some advantage over younger members. However, in this respect it is more justified. Older members only have the minor advantage as long as they continue policing. If they ever turn off their machines, then all their REP eventually will be extinguished.




Now that we’ve justified the basic formulas for the present value of REP tokens, we can evaluate more complicated scenarios. The next section introduces the construction of new types of bonds that can be minted thanks to the versatility of smart contracts, whose value is amplified by the transparency of the typical DAO. The final section uses the present value formulas to evaluate the consequences of DAO governance parameter adjustments.
----[1][1] ''N.b.'': There are many different ways this can be achieved with different particular hard protocols. For our present purpose, these parameters represent averages to analyze general scenarios. For example, the present analysis assumes  REP is shared with the group per fee , but different protocol choices may mean that sometimes tokens are burned when a post is downvoted, so in that case  means the average ratio of tokens that are distributed to the DAO per fee .  ???Delete footnote?

[2] Technically this is a '''neutral delay differential equation''' (NDDE), since the delay comes from the function’s past derivative, not directly from the function itself.