REP with attenuation: Difference between revisions
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[[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. | [[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 [[DAO Governance Framework|DGF]] can also program their REP tokens to shrink (or grow) in power as time goes on. This is called '''attenuation'''. In this page, we use [[Reputation tokenomics|REP tokenomics]] formulas to give valuation formulas for REP tokens with attenuation. | ||
== Valuation == | == 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>. | 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> of the [[Reputation#REP Salary Mechanism|reputation salary]]. 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>'' | 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>'' | ||
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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>. | 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>]] | 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>]] | ||
== Applications == | == Applications == | ||
* [[Executive governance#Attenuation function & policing|REP with atttenuation & policing]] | * [[Executive governance#Attenuation function & policing|REP with atttenuation & policing]] | ||
* [[Reputation]] | |||
** [[Reputation tokenomics]] | |||
Latest revision as of 15:09, 18 June 2023
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. This is called attenuation. In this page, we use REP tokenomics formulas to give valuation formulas for REP tokens with attenuation.
Valuation[edit | edit source]
To program a REP token with attenuation, 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 of the reputation salary. 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 in the DAO at time . This is different from which is defined to be the total number of tokens ever minted (ignoring attenuation). As before satisfies
To justify the form of in Equation 13, consider how at time the term attenuates the tokens that have aged units of time. The integral in Equation 13 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
