# Competitive equilibria in a multi-revenue-stream economy

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**Context:** what happens to SP mining if Filecoin supports staking/lending/other
**Goal:** express mathematically the dynamics of acceptable revenue sharing
**Approach:** coupled first-order non-linear ODEs

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This note explores an incarnation of the network in a future that supports additional streams of revenue for Filecoin beyond storage mining. In this scenario, it's plausible proof-of-spacetime and network security may weaken, if said streams are more economically favourable and canabalize SP mining, so it's worth considering how to model this.

Consider the following revenue streams: staking $S(t)$, lending $L(t)$, as well as storage mining $M(t)$. If the total economic value at any time is $V(t)$, then the sum of revenue streams for participants is the simplex

\begin{align*}
1 & =\frac{S(t)}{V(t)}+\frac{L(t)}{V(t)}+\frac{M(t)}{V(t)}\,.\\
& =s(t)+l(t)+m(t)
\end{align*}
where $s$, $l$ and $m$ are the fraction of incentives attributable to each source. From hereon $s=s(t)$, etc.

The dynamics of this system can be modeled as set of coupled first-order non-linear differential equations:

\begin{align*}
\dot{s} & =r_{\text{s}}\,s\left(1-\frac{s+\alpha_{\text{sl}}l+\alpha_{\text{sm}}m}{K_{\text{s}}}\right)\\
\dot{l} & =r_{\text{l}}\,l\left(1-\frac{l+\alpha_{\text{ls}}s+\alpha_{\text{lm}}m}{K_{\text{l}}}\right)\\
\dot{m} & =r_{\text{m}}\,m\left(1-\frac{m+\alpha_{\text{ms}}s+\alpha_{\text{ml}}l}{K_{\text{m}}}\right)
\end{align*}

where $r_{i}$, $K_i$ and $\alpha_{ij}$ are constant coefficients.

These expressions are an example of replicator equations in evolutionary game theory and competitive Lotka-Volterra equations in population ecology.

Modeling Filecoin revenue dynamics in this paradigm could be useful for three reasons:
1. They give a straightforward interpretation of non-linear interacting dynamics in terms of only a few coefficients, and the coefficients are easily explainable:
* $r_{i}$ is the inherent growth rate of revenue stream $i$
* $\alpha_{ij}$ is the effect of revenue stream $j$ on $i$, which for $\alpha_{ij}>0$ is competitive (harmful), and $\alpha_{ij}<0$ is enhancing
* $K_{i}$ is the carrying capacity of revenue stream $i$: the maximum proportion of revenue the system supports being attributable to this stream (potentially always just 1, but a target could be 0.1 for lending for example)
2. The coefficients in the coupled set of equations can straightforwardly be worked out from real data when available, for exampling using standard ODE sovlers to generate sample trajectories for statistical inference
3. The equations can easily be simulated

### Example

Consider simulating the dynamics for a plausible scenario.

Currently all economic value is derived from SP mining, so $m=1$. Imagine lending ($l$) and staking ($s$) are introduced, and we wish to target $K_{s}=0.15$ and $K_{l}=0.25$ (15\% of Filecoin revenue streams being attributable to staking and 25\% lending). Then what do the revenue dynamics look like for different growth and interaction levels for the introduced staking and lending?

If the inherent growth rate of staking is an increase of $r_{\text{s}}=0.5$ times per month, and lending increase at $r_{\text{l}}=1.5$ times per month, and we assume competition between lending and staking is symmetric as $\alpha_{\text{ls}}=\alpha_{\text{sl}}=0.01$, and let lending and staking both be *supportive* of mining revenue, $\alpha_{\text{lm}}=\alpha_{\text{sm}}=-0.01$ (perhaps locked supply alters circulation dynamics and makes mining more attractive). In this scenario, we see the following dynamics:


### Next

What would really be valuable is connecting the macroeconomic coefficients in the model here, to protocol level parameters. And intermediate dynamics aside, how does one fix the end-point equilibria i.e. carrying capacities at protocol level. On these points ideas most welcome!:)