Editing Executive governance (section)

=== Finite lifetime automated policing ===
For the case of REP tokens with finite lifetime, the ODE of Equation 7 becomes a delay differential equation (DDE)[[Validation Pool#Vote|<math display="block">\frac{dP_0^1}{dt}(t)  =p_3mf'(t)\frac{P_0^1(t)}{R(t)} - \frac{dP_0^1}{dt}(t-L) .</math>]]''<small>Equation 12</small>''

with initial condition [[Validation Pool#Vote|<math display="block">\frac{dP_0^1}{dt}(t)  = \begin{cases} 0 & \text{if} \quad -L\leq t<0 \\ 1 & \text{if } \quad t=0.  \end{cases}</math>]]As before, <math>R</math>  represents the total number of tokens in a DAO with finite lifetime <math>L</math> and is given by [[Validation Pool#Vote|<math display="block">R(t)=\int_{t-L}^t m*f' (s)ds </math>]]

Again, [[Validation Pool#Vote|<math>p_3:=1-c_3</math>]] is the policing parameter, and the tokens that result from this policing process from one REP token is denoted [[Validation Pool#Vote|<math display="inline">P_0^1(t)</math>]] with [[Validation Pool#Vote|<math display="inline">P_0^1(0)=1</math>]] which have the reputational salary [[Validation Pool#Vote|<math display="inline">f_0^{1,P}</math>]] given by the formula[[Validation Pool#Vote|<math display="block">f_0^{1,P}(t)=\int_0^t f'(s)\frac{P_0^1(s)}{R(s)}ds</math>]]<nowiki/>which has present value[[Validation Pool#Vote|<math display="block">PVf_0^{1,P}=\int_0^\infty e^{-rt} \frac{d}{dt} f_0^{1,P} (t)dt</math>]] 

To explain why the policing token holdings [[Validation Pool#Vote|<math display="inline">P_0^1</math>]] evolve according to the DDE of Equation 12, we focus on the two terms on the right-hand side. The first term [[Validation Pool#Vote|<math display="inline">p_3mf'(t)\frac{P_0^1(t)}{R(t)}</math>]] is the same as Equation 10, coming from the fact that the rate [[Validation Pool#Vote|<math display="inline">dP_0^1/dt</math>]] of newly generated policing tokens is proportional to the newly minted tokens as a fraction of the total active policing tokens [[Validation Pool#Vote|<math display="inline">R(t)</math>]]. These new tokens are minted in proportion to fees [[Validation Pool#Vote|<math display="inline">f'</math>]] and the minting ratio [[Validation Pool#Vote|<math display="inline">m</math>]] and the amount policing is rewarded [[Validation Pool#Vote|<math display="inline">p_3</math>]]. The second term [[Validation Pool#Vote|<math display="inline"> - \frac{dP_0^1}{dt}(t-L)</math>]] comes from the fact that tokens [[Validation Pool#Vote|<math display="inline">L</math>]] units of time old are expiring at the same rate they were added [[Validation Pool#Vote|<math display="inline">L</math>]] units in the past. 

The term [[Validation Pool#Vote|<math display="inline"> - \frac{dP_0^1}{dt}(t-L)</math>]] makes Equation 12 a delay differential equation<ref>Technically this is a '''neutral delay differential equation''' (NDDE), since the delay comes from the function’s past derivative, not directly from the function itself.</ref>, because the way [[Validation Pool#Vote|<math display="inline">P_0^1</math>]] changes at time [[Validation Pool#Vote|<math display="inline">t</math>]] depends on the moment  [[Validation Pool#Vote|<math display="inline">L</math>]] units before the present. Such DDEs can be solved numerically for any practical choice of parameters [[Validation Pool#Vote|<math display="inline">p_3</math>]], [[Validation Pool#Vote|<math display="inline">m</math>]], and [[Validation Pool#Vote|<math display="inline">f'</math>]].