Reputation tokenomics

Reputation tokenomics is the economic analysis of the value of REP tokens. REP tokenomics studies the consequences of reputation token design. The fundamental results derive from analyzing the effects of parameter changes in the REP Token Minting Mechanism. Models for REP token evolution depend on assumptions such as the rate of fees $f'$ the DAO earns to share in its REP salary. The analysis follows by determining the income stream of a REP token due to this REP salary, which is equalized over time by calculating its present value.

The results of REP tokenomics help guide governance decisions. The formulas derived from tokenomics give us more precise intuition for how to manipulate the parameters to drive the system in different ways. For instance, a DAO may choose to change the number of tokens minted when fees enter the system (a parameter denoted by $m$). The analysis shows how default inflationary minting of REP encourages decentralization, and to what degree parameter choices strengthen or weaken the effect. Such calculations precisely account for how different types of members (especially older or newer members) benefit from different DAO governance decisions, which clarifies the true moral principles the DAO embodies. This allows us to compare the functioning of a DAO against its marketing, in order to objectively evaluate the group’s values and integrity.

REP Valuation

Basic parameters

REP valuation models are based on the following parameters:

1. $R(t)=$ the total number of REP tokens in the DAO at time $t$.
2. The rate of total fees $f'={df \over dt}$ that the DAO earns. Therefore $f(t)$ denotes the total fees earned from the beginning of the DAO until time $t$.
3. $f_{0}^{1}(t)$ is the cumulative reputational salary collected for one token from start time $t_{0}=0$ when the token was minted until time $t$. This is our function of primary concern. After determining its formula, we are most interested in its present value .
4. $m$ is the minting ratio. This is the proportion of REP tokens that are minted relative to the fees the DAO collects. The default assumption is $m$.
5. $r$ is the base interest rate of the economy. Depending on the context this can alternately denote the inflation rate of the stable coin in which the fees are paid or the CPI. The default assumption is $r=4\%.$
6. $L$ is the lifetime after which a token expires. The default assumption is $L=\infty$. The token can be programmed to maintain full potency until it expires, or dwindle in power according to an attenuation function. In traditional finance, the lifetime is often referred to as its maturity, or expiration, meaning the initial length of a contract upon its inception. The tenor is the length of time remaining in the lifetime of a financial contract, $L-t$.

Fundamental results

The basic results from which all other applications can be derived are summarized in the following theorems, which give the present value of a single REP token when it is minted.

Theorem 1.  (Infinite Life Tokens) 

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

where $R_{0}=R(0)$. The reputational salary of a single token is therefore given by the income stream

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

The present value at time $t_{0}=0$ when a single token is minted in a DAO is

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

Proof.

Theorem 2. (Constant fees)  

Assume the DAO is in the market position of earning fees with a constant rate $f_{0}'$ and the lifetime of a token is infinite, $L=\infty$.

Then the reputational salary of your single REP token is

$$f_{0}^{1}(t)={\frac {1}{m}}ln{\frac {R_{0}+mtf_{0}'}{R_{0}}}$$

and the present value is

$$PVf_{0}^{1}=\int _{0}^{\infty }{\frac {f_{0}'}{R_{0}+mtf_{0}'}}e^{-rt}dt={\frac {1}{m}}exp{\biggl (}{\frac {rR_{0}}{mf_{0}'}}{\biggr )}\int _{\frac {rR_{0}}{mf_{0}'}}^{\infty }{\frac {e^{-s}}{s}}ds$$

Proof.

Theorem 3.  (Finite Life Tokens)

Assume the REP tokens have finite lifetime $L<\infty$. Then the total number of active REP tokens at any time $t$ is

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

Then, assuming a single token was minted at time $t_{0}=0$  the fees it earns is given by

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

(Constant Fees)

Now assume the rate of fees $f_{0}'$ is constant. At any time $t$ after the DAO reaches token number equilibrium $t>L$, there will always be $R(t)=R_{\infty }=mf_{0}'L$ tokens in the system. The income stream of a single token is then

$$f_{0}^{1}(t)={\begin{cases}{\frac {t}{mL}}&{\text{if }}t<L\\{\frac {1}{m}}&{\text{if }}t\geq L\end{cases}}$$

and the present value of 1 token at time $t_{0}=0$ when it is minted is

$$PVf_{0}^{1}={\frac {1}{m}}{\frac {1-e^{-rL}}{rL}}.$$

(Exponential Fees)

Now assume a DAO has exponentially growing fees $f'(t)=f_{0}'e^{ct}$ and lifetime $L<\infty$. After the DAO has been running $L$ units of time, the number of active tokens will grow at a proportional exponential rate. The income stream of a single token is then

$$f_{0}^{1}(t)={\frac {c}{m(1-e^{-cL})}}min\{t,L\}$$

with present value

$$PVf_{0}^{1}={\frac {1}{m}}{\biggl (}{\frac {1-e^{-rL}}{rL}}{\biggr )}{\biggl (}{\frac {cL}{1-e^{-cL}}}{\biggr )}$$

Proof.

Consequences

The formulas cannot directly valuate REP tokens in a simple manner. A token's value is dependent on the rate of fees in the DAO after the token is minted. However, the formulas allow us to analyze the value of tokens by observing the past. And further, by making different assumptions about DAO fee rates, averages of present values can be predicted.

Old tokens with infinite lifetime are more valuable then new tokens

For infinite lifetime REP tokens, the present value of a REP token at minting will change depending on the rate of fees the DAO attracts. In this case, more REP accumulates as time goes on. Typically, a REP token minted earlier is more valuable than a REP token minted later, since a later token shares the REP salary with more owners of REP. One way to combat this inequity, if that is desired, is to limit the lifetime of a token.

Convergence to fair value for finite lifetime tokens

Theorem 3 shows that for finite lifetime REP tokens, the number of tokens that exist in the DAO is not important if the rate of fees stabilizes. Assuming the DAO is healthy, in the sense that it attracts a regular rate of fees, that grow with inflation $r=c$, then the present value of a token minted at equilibrium is $PVf_{0}^{1}=1/m$ which is the same amount of the fees that were brought given to the DAO in order to earn the REP token. Therefore, an expert who earned a REP token does not immediately receive the cash for their work, but they do ultimately receive more than the amount of cash they brought in. The present value of REP tokens (on average) are exactly the same as the amount of fees they attracted with their work. However, the reputation accounted for is also a signal that the owners of REP tokens have done useful work in the past, and their tokens can be slashed at any point up to time $L$.

Applications

• Treasuries
• BONDs
  • iBONDs
  • fciBONDs
  • fREP
• REP Markets
  • graceful exit
  • underwriting
  • generalized chit fund banking
• stable coins
• PoR block production consensus