REP with attenuation

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 DAOs in DGF can also program their REP tokens to shrink (or grow) in power as time goes on. In this page, we use 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 $L:{\mathbb {R}}\rightarrow {\mathbb {R}}$ 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

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

Applications

• REP with atttenuation & policing