Editing Executive governance (section)

=== Attenuation function & policing ===
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 <math>L:\Bbb R \rightarrow \Bbb R</math> which means that a token minted at time <math>0</math> will have potency <math>L(t)</math> at time <math>t</math>. So <math>L(t)</math> is a multiplier for the value of any token minted at time <math>0</math>. This means that one token minted at time <math>0</math> pays out the fraction <math>L(t)/R(t)</math> of the fees the DAO earns at time <math>t</math> instead of the usual fraction <math>1/R(t)</math>. Typically <math>L</math> will naturally be constrained <math>0 \leq L(t) \leq 1</math>, 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 <math>L</math>.

Here as before, <math>R(t)</math> represents the total amount of token power that exists in the DAO at time <math>t</math>. This is different from <math>R^T(t)</math> which is defined to be the total number of tokens ever minted (ignoring attenuation). As before <math>R^T(t)</math> satisfies[[Validation Pool#Vote|<math display="block">R^T(t)=R^T_0+\int_{0}^t m*f' (s)ds </math>]]<nowiki/>where [[Validation Pool#Vote|<math display="inline">R^T(0)=:R^T_0 </math>]] which gives[[Validation Pool#Vote|<math display="block">\frac{dR^T}{dt}(t)= m*f' (t). </math>]]This time, however, we have [[Validation Pool#Vote|<math display="block">R(t)=\int_{-\infty}^t L(t-s)\frac{dR^T}{dt}(s)ds. </math>]]''<small>Equation 13</small>''

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

Assuming a single token was minted at time <math>t=0</math> the fees it earns is given by [[Validation Pool#Vote|<math display="block">f_0^{1,L}(t)=\int_0^t f'(s)\frac{L(s)}{R(s)}ds</math>]]Then as before, the present value of a single token with attenuation (but without policing) has present value [[Validation Pool#Vote|<math display="block">PVf_0^{1,L}=\int_0^\infty e^{-rt} \frac{d}{dt} f_0^{1,L} (t)dt</math>]]


Finally, we add automated policing with parameter [[Validation Pool#Vote|<math display="inline">p_3</math>]] and the analogous quantity of active tokens under attenuation <math>L</math> is now denoted by [[Validation Pool#Vote|<math display="inline">P_0^{1,L}</math>]]. It is easier to keep track of all policing tokens earned from one token [[Validation Pool#Vote|<math display="inline">P_0^{1,U}=P_0^1</math>]] unadjusted by <math>L</math> which evolve according to the DDE[[Validation Pool#Vote|<math display="block">\frac{dP_0^{1,U}(t)}{dt}  =p_3mf'(t)\frac{P_0^{1,L}(t)}{R(t)}  .</math><math display="block">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</math>]][[Validation Pool#Vote|<math display="block">P_0^1(0)=1</math>]]This is a DDE since <math>\frac{dP_0^{1,U}}{dt}(t)</math> depends on the past of <math>P_0^{1,U}(t)</math> for <math>s<t</math>.


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.