Executive governance

Executive governance in DGF consists of automated policing of peer actions. The primary mechanism for executive governance is the Work smart contract (WSC), which sets the protocols for successful participation in the profit-making aspect of the DAO. The secondary executive governance mechanism is the Validation Pool, which is used to validate the completion of every WSC, using binding votes (voting against the majority results in loss of tokens) on the acceptability of all actions that affect the DAO.

Executive governance in a DAO is the active, direct control of ownership in the organization. Executive governance consists of policing who is allowed inside the group, how much power insiders have, and how outsiders can interact with the group. Executive governance is therefore the accounting of the lists of owners of the various types of tokens that have power to engage the DAO’s smart contracts, whether those owners are using their tokens according to previously established protocols, and whether outsiders who engage with the DAO are engaging properly. Inasmuch as a DAO is truly decentralized, this policing control must be ultimately democratic. Therefore, some sort of token-weighted voting is necessary. Inasmuch as executive governance is the process of executing the regulations that the organization has already agreed to follow, executive governance should be automated as much as possible. In a DAO this means algorithmic enforcement using smart contracts run on a decentralized computing platform.

Policing motivation

The most basic and strongest direct incentive a DAO member has for participating in the policing process is that they are rewarded with new REP tokens when they participate in the Validation Pool. Based on the policing parameter $c_{3}$ (which has default value $c_{3}=1/2$) and the rate of total fees $f'$ the DAO earns, we can determine how valuable it is to participate in executive governance.

An indirect motivation for an individual to police is that it keeps the DAO, and therefore the individual's REP tokens, secure and valuable. A higher policing parameter motivates more participation, but it also makes it more difficult and expensive to achieve a 51% attack arbitrage.

However, a higher policing parameter makes finite lifetime REP tokens last longer, as new REP is automatically minted with fresh tenure. This leads to

Such motivations need to be weighed carefully when a DAO decides how to set policing parameter. The decision is clarified with a rigorous valuation of policing tokens.

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 for executive governance, that is, to automatedly police all actions in the DAO by participating in each Validation Pool. This will be typical when the reward for policing is sufficient to motivate members to participate in policing. This means the reward is more than the price to run the machines needed to participate.

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.

Policing valuation

In this section we use tokenomics formulas to 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.

Infinite Lifetime REP

Whenever a validation pool is opened with a fee of $f$ in fungible currency units, then $mf$ reputation tokens are minted, and $c_{3}mf$ tokens are staked in the poster’s name and $(1-c_{3})mf$ tokens are given to the bench for policing. Here $0\leq c_{3}\leq 1$ gives the work-to-policing reward ratio of $c_{3}/(1-c_{3})$. Default is $c_{3}=1/2$ when work and policing are equal. When $c_{3}<1/2$ policing is augmented. When $c_{3}>1/2$ 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 $c_{3}\approx 1$ encouraging new members to join and earn REP through work, and $c_{3}\approx 0$ 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 $f'$ determines the present value of a policing token. The total number of tokens in a DAO is again given by

$$R(t)=\int _{-\infty }^{t}m*f'(s)ds=R_{0}+\int _{0}^{t}m*f'(s)ds$$

where $R_{0}=R(0)$. Assuming a single token was minted at time $t=0$ the fees it earns is again given by

$$f_{0}^{1}(t)=\int _{0}^{t}{\frac {f'(s)}{R(s)}}ds.$$

The present value at time $t_{0}=0$ when 1 token was minted is

$$PVf_{0}^{1}=\int _{0}^{\infty }e^{-rt}{\frac {d}{dt}}f_{0}^{1}(t)dt=\int _{0}^{\infty }e^{-rt}{\frac {f'(t)}{R(t)}}dt.$$

But the new REP tokens the single token earns by policing is governed by the parameter $c_{3}$. For each fee that enters the DAO, the first REP token earns the fraction $p_{3}1/R^{p}(t)$ of the $mf'$ newly minted REP tokens, where the policing parameter is simply $p_{3}:=1-c_{3}$ and $R^{p}$ is the number of tokens that participate in the policing actions. So $R^{p}\leq R$. For simplicity, we will conservatively assume $R^{p}=R$ which is reasonable if $p_{3}$ 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 token that 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 ${\textstyle P_{0}^{1}(t)}$ which satisfies the initial condition ${\textstyle P_{0}^{1}(0)=1}$. So ${\textstyle P_{0}^{1}(t)}$ grows according to the ODE

$${\frac {dP_{0}^{1}(t)}{dt}}=p_{3}mf'(t){\frac {P_{0}^{1}(t)}{R(t)}}.$$

Equation 10

which is separable and can be explicitly solved as

$$\int {\frac {dP}{P}}=p_{3}m\int {\frac {f'(t)}{R(t)}}.$$

assuming $p_{3}$ and $m$ are constant. Then

$$P_{0}^{1}(t)=exp{\Biggl (}p_{3}m\int _{0}^{t}{\frac {f'(t)}{R(t)}}{\Biggr )}.$$

Equation 11

1 token which is actively policing then produces ${\textstyle P_{0}^{1}}$ coins which together have the reputational salary ${\textstyle f_{0}^{1,P}}$ given by the formula

$$f_{0}^{1,P}(t)=\int _{0}^{t}f'(s){\frac {P_{0}^{1}(s)}{R(s)}}ds.$$

1 token which is actively policing then has present value

$$PVf_{0}^{1,P}=\int _{0}^{\infty }e^{-rt}{\frac {d}{dt}}f_{0}^{1,P}(t)dt$$

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 $f'$.

Constant fees

For constant fees $f'(t):=f_{0}'$ we get

$$R(t)=R_{0}+\int _{0}^{t}m*f'(s)ds=R_{0}+mf_{0}'t$$

and so Equation 11 gives

$$P_{0}^{1}(t)=exp{\Biggl (}p_{3}m\int _{0}^{t}{\frac {f_{0}'}{R_{0}+mf_{0}'s}}{\Biggr )}ds=exp{\Biggl (}p_{3}{\biggl (}ln(R_{0}+mf_{0}'s)-ln(R_{0}){\biggr )}{\Biggr )}={\biggl (}1+{\frac {mf_{0}'}{R_{0}}}t{\biggr )}^{p_{3}}.$$

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

$$P_{0}^{R_{0}}(t)=R_{0}{\biggl (}1+{\frac {mf_{0}'}{R_{0}}}t{\biggr )}^{p_{3}}.$$

gives the formula for the collection of all policing tokens ${\textstyle P_{0}^{R_{0}}}$ generated from the original tokens ${\textstyle R_{0}}$ from time ${\textstyle t=0}$. The policing tokens from a single token  grow without bound, however the relative power that all original tokens ${\textstyle R_{0}}$ maintain under automated policing is

$${\frac {P_{0}^{R_{0}}(t)}{R(t)}}={\frac {R_{0}{\biggl (}1+{\frac {mf_{0}'}{R_{0}}}t{\biggr )}^{p_{3}}}{R_{0}+mf_{0}t}}={\biggl (}1+{\frac {mf_{0}'}{R_{0}}}t{\biggr )}^{p_{3}-1}\rightarrow 0$$

as $t\rightarrow \infty$ anytime $p_{3}<1$. 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.

We also can calculate the income stream ${\textstyle f_{0}^{1,P}(t)}$ of 1 token with policing ${\textstyle P_{0}^{1}(t)}$ and its present value under the assumption of constant fees:

$$f_{0}^{1,P}(t)=\int _{0}^{t}f'(s){\frac {P_{0}^{1}(s)}{R(s)}}ds=\int _{0}^{t}f_{0}'{\frac {{\biggl (}1+{\frac {mf_{0}'}{R_{0}}}s{\biggr )}^{p_{3}}}{R_{0}+mf_{0}'s}}ds={\frac {f_{0}'}{R_{0}}}\int _{0}^{t}{\biggl (}1+{\frac {mf_{0}'}{R_{0}}}s{\biggr )}^{p_{3}-1}ds={\frac {1}{mp_{3}}}{\Biggl (}{\biggl (}1+{\frac {mf_{0}'}{R_{0}}}t{\biggr )}^{p_{3}}-1{\Biggr )}.$$

and

$$PVf_{0}^{1,P}=\int _{0}^{\infty }e^{-rt}{\frac {d}{dt}}f_{0}^{1,P}(t)dt={\frac {f_{0}'}{R_{0}}}\int _{0}^{\infty }e^{-rt}{\biggl (}1+{\frac {mf_{0}'}{R_{0}}}t{\biggr )}^{p_{3}-1}dt$$

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.

Exponential fees

Now we assume, as above, the fees follow an exponential rate ${\textstyle f'(t):=f_{0}'e^{ct}}$

This gives

$$R(t)=R_{0}+\int _{0}^{t}m*f'(s)ds=R_{0}+mf_{0}'{\frac {e^{ct}-1}{c}}$$

Using Equation 11 we get

$$P_{0}^{1}(t)=exp{\Biggl (}p_{3}m\int _{0}^{t}{\frac {f'(t)}{R(t)}}{\Biggr )}={\Biggl (}1+{\frac {mf_{0}'}{R_{0}}}{\frac {e^{ct}-1}{c}}{\Biggr )}^{p_{3}}$$

The single policing token gives the reputational salary

$$f_{0}^{1,P}(t)=\int _{0}^{t}f'(s){\frac {P_{0}^{1}(s)}{R(s)}}ds={\frac {1}{mp_{3}}}{\Biggl (}{\biggl (}1+{\frac {mf_{0}'}{R_{0}}}{\frac {e^{ct}-1}{c}}{\biggr )}^{p_{3}}-1{\Biggr )}.$$

Notice policing makes a qualitative difference under exponential fees, since a passive token had salary ${\textstyle f_{0}^{1}(t)\sim {\frac {1}{m}}ct}$ asymptotically linear, whereas policing gives ${\textstyle f_{0}^{1,P}(t)\sim e^{cp_{3}t}}$ asymptotically exponential salary.

This salary’s present value is

$$PVf_{0}^{1,P}=\int _{0}^{\infty }e^{-rt}{\frac {d}{dt}}f_{0}^{1,P}(t)dt$$

which is not expressible using elementary functions, but is given by the hypergeometric function $2F_{1}$. Notice ${\textstyle PVf_{0}^{1,P}}$ is finite when $cp_{3}<r$ and infinite otherwise. Therefore, we see that when the policing ratio $p_{3}$ and the exponential growth rate $c$ overcome the interest rate $r$ 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 $R_{0}$ original tokens, even if they all participate reliably in policing will be diluted in relative power according to

$${\frac {R_{0}P_{0}^{1}(t)}{R(t)}}={\frac {R_{0}{\Biggl (}1+{\frac {mf_{0}'}{R_{0}}}{\frac {e^{ct}-1}{c}}{\Biggr )}^{p_{3}}}{R_{0}+mf_{0}'{\frac {e^{ct}-1}{c}}}}={\Biggl (}1+{\frac {mf_{0}'}{R_{0}}}{\frac {e^{ct}-1}{c}}{\Biggr )}^{p_{3}-1}\rightarrow 0$$

as $t\rightarrow \infty$ 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.

Finite Lifetime Automated Policing

For the case of REP tokens with finite lifetime, the ODE of Equation 7 becomes a delay differential equation (DDE)

$${\frac {dP_{0}^{1}}{dt}}(t)=p_{3}mf'(t){\frac {P_{0}^{1}(t)}{R(t)}}-{\frac {dP_{0}^{1}}{dt}}(t-L).$$

Equation 12

with initial condition

$${\frac {dP_{0}^{1}}{dt}}(t)={\begin{cases}0&{\text{if}}\quad -L\leq t<0\\1&{\text{if }}\quad t=0.\end{cases}}$$

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

$$R(t)=\int _{t-L}^{t}m*f'(s)ds$$

Again, $p_{3}:=1-c_{3}$ is the policing parameter, and the tokens that result from this policing process from one REP token is denoted ${\textstyle P_{0}^{1}(t)}$ with ${\textstyle P_{0}^{1}(0)=1}$ which have the reputational salary ${\textstyle f_{0}^{1,P}}$ given by the formula

$$f_{0}^{1,P}(t)=\int _{0}^{t}f'(s){\frac {P_{0}^{1}(s)}{R(s)}}ds$$

which has present value

$$PVf_{0}^{1,P}=\int _{0}^{\infty }e^{-rt}{\frac {d}{dt}}f_{0}^{1,P}(t)dt$$

To explain why the policing token holdings ${\textstyle P_{0}^{1}}$ evolve according to the DDE of Equation 12, we focus on the two terms on the right-hand side. The first term ${\textstyle p_{3}mf'(t){\frac {P_{0}^{1}(t)}{R(t)}}}$ is the same as Equation 10, coming from the fact that the rate ${\textstyle dP_{0}^{1}/dt}$ of newly generated policing tokens is proportional to the newly minted tokens as a fraction of the total active policing tokens ${\textstyle R(t)}$. These new tokens are minted in proportion to fees ${\textstyle f'}$ and the minting ratio ${\textstyle m}$ and the amount policing is rewarded ${\textstyle p_{3}}$. The second term ${\textstyle -{\frac {dP_{0}^{1}}{dt}}(t-L)}$ comes from the fact that tokens ${\textstyle L}$ units of time old are expiring at the same rate they were added ${\textstyle L}$ units in the past.

The term ${\textstyle -{\frac {dP_{0}^{1}}{dt}}(t-L)}$ makes Equation 12 a delay differential equation, because the way ${\textstyle P_{0}^{1}}$ changes at time ${\textstyle t}$ depends on the moment  ${\textstyle L}$ units before the present. Such DDEs can be solved numerically for any practical choice of parameters ${\textstyle p_{3}}$, ${\textstyle m}$, and ${\textstyle f'}$.

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 $L:{\mathbb {R}}\rightarrow {\mathbb {R}}$ which means that a token minted at time $0$ will have potency $L(t)$ at time $t$. So $L(t)$ is a multiplier for the value of any token minted at time $0$. This means that one token minted at time $0$ pays out the fraction $L(t)/R(t)$ of the fees the DAO earns at time $t$ instead of the usual fraction $1/R(t)$. Typically $L$ will naturally be constrained $0\leq L(t)\leq 1$, 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 $L$.

Here as before, $R(t)$ represents the total amount of token power that exists in the DAO at time $t$. This is different from $R^{T}(t)$ which is defined to be the total number of tokens ever minted (ignoring attenuation). As before $R^{T}(t)$ satisfies

$$R^{T}(t)=R_{0}^{T}+\int _{0}^{t}m*f'(s)ds$$

where ${\textstyle R^{T}(0)=:R_{0}^{T}}$ which gives

$${\frac {dR^{T}}{dt}}(t)=m*f'(t).$$

This time, however, we have

$$R(t)=\int _{-\infty }^{t}L(t-s){\frac {dR^{T}}{dt}}(s)ds.$$

Equation 13

To justify the form of $R(t)$ in Equation 13, consider how at time $t$ the term $L(t-s)$ attenuates the tokens that have aged $t-s$ units of time. The integral in Equation 13 sums all the tokens ever minted, with the term $dR^{T}/dt$ over all times $s$ before time $t$ modified by how much they have attenuated in their current age $L(t-s)$.

Assuming a single token was minted at time $t=0$ the fees it earns is given by

$$f_{0}^{1,L}(t)=\int _{0}^{t}f'(s){\frac {L(s)}{R(s)}}ds$$

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

$$PVf_{0}^{1,L}=\int _{0}^{\infty }e^{-rt}{\frac {d}{dt}}f_{0}^{1,L}(t)dt$$

Finally, we add automated policing with parameter ${\textstyle p_{3}}$ and the analogous quantity of active tokens under attenuation $L$ is now denoted by ${\textstyle P_{0}^{1,L}}$. It is easier to keep track of all policing tokens earned from one token ${\textstyle P_{0}^{1,U}=P_{0}^{1}}$ unadjusted by $L$ which evolve according to the DDE

$${\frac {dP_{0}^{1,U}(t)}{dt}}=p_{3}mf'(t){\frac {P_{0}^{1,L}(t)}{R(t)}}.$$

$$P_{0}^{1,L}(t)=L(t)P_{0}^{1}(0)+\int _{0}^{t}L(t-s){\frac {dP_{0}^{1,U}}{ds}}(s)ds$$

$$P_{0}^{1}(0)=1$$

This is a DDE since ${\frac {dP_{0}^{1,U}}{dt}}(t)$ depends on the past of $P_{0}^{1,U}(t)$ for $s<t$.

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 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 be profitable. If it is on the edge of profitability, 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 subsequently generated policing REP will persist for all time, 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.