Editing REP with attenuation



[[Reputation|REP tokens]] can be programmed to have a finite lifetime, where they lose their power after a finite amount of time. REP tokens can also be programmed to diminish in value in much more complicated ways than merely programming an expiration date. We can instead design the [[DAO|DAOs]] in DGF can also program their REP tokens to shrink (or grow) in power as time goes on. In this page, we use [[Reputation tokenomics|REP tokenomics]] formulas to give valuation formulas for REP tokens with attenuation.

== Valuation ==
To program a REP token with attenuation, 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>]]