# Public Goods Funding (PGF)
Namada's existence relies on its open-source software stack, research, and ecosystem tooling all of which have relied on standing on the shoulders of other open-source information, readily available to the public through the internet. All of these foundations share the common attribute of being non-rivalrous and non-excludable, which are sufficient charactersitics to define them as public goods.

It is a well known concept in the economics literature that public goods tend to be underfunded due to these qualities. In short, this occurs because of coordination problems between individuals, whereby equilibrium market prices do not reflect the benefit the goods existence brings to the society as a whole, but rather the buyers of the good.

It can be shown that if the *consumers* of the good (i.e buyers and free-loaders respectively), all merged into a single actor that made a single purchasing decision, this collective problem can be solved trivially. In economics, this imaginative all-knowing figure is sometimes referred to as the "social planner". In reality, unfortunately, merging all benefactors into one individual is not feasible, and therefore we must use other tools at our disposal to attempt to reach the same outcomes as the social planner would arrive at.

## Example
Here is a short example to demonstrate this.

Let’s simplify the world where we have some public good free open source software. Let the number of blocks committed towards this project be denoted by $x$. Assume that each additional block of code committed provides the end user with $(100 - x)$ additional “monies” of benefit, and costs the developer $x$ “monies” to commit.

To put this in as obvious a manner as possible we have

$$ D_i = (100 - x) $$
$$ S = x $$

Where $D_i$ denotes "individual demand", also known as "marginal benefit" to the consumer, and $S$ denotes the "supply" or the "marignal cost" to the producer.

When we have individuals coordinate in the free-market, the 50th line of code is provided at a cost of $100 - x = x \Rightarrow x = 50$ monies. In equilibrium we have 50 blocks of code at 50 monies each, costing society a total of 2500 monies for a benefit of (99 + 98 + 97 + ... 50 =) 3725 monies *per user*.


Let's say there are $N>100$ users. Now what would happen if they could all commit to "splitting the bill" before-hand?

This means that each additional line of code now costs them $\frac{x}{N}$ monies rather than $x$.

$$ 100 - x = \frac{x}{N} => x = \frac{100}{1+\frac{1}{N}}$$

As $N \rightarrow \infty$, we start to arrive at a social solution with the maximum number of blocks of code comitted.


What economic tool do we have at our disposal that forces people to "split the bill" amongst all the money holders?

Hint: It starts with "i" and ends with "nflation".

## Proposed solution

One proposed solution to avoid such coordination failures is to subsidise public goods in a way that allows individual incentives to align with that of the social planner. In a dream world, all public goods are well-known and all market failures can be quantified precisely. In the real world, we are not only tasked with finding public goods, but also with quantifying the *total realised + potential* benefit they bring *over an infinite time horizon*. In Namada, this simple task is assigned to a group of democratically elected individuals.

There will be a periodically elected council tasked with voting on projects/goods that should be funded.

The council is responsible for two types of public goods funding.

1. Continuous funding - in order to fund ongoing public goods projects in order to assist their coming to fruition
2. Retroactive funding - in order to fund completed and easily evaluated public goods, in order to incentivise future public goods to be built

Funding public goods is sourced through native asset (NAM) inflation, dictated by the protocol.

At the time of writing, 10% of Namada annual inflation is dedicated towards PGF.

## Retroactive PGF (rPGF)
Retroactive public goods funding has become an increasingly popular method for funding public goods. By retroactive, we mean to say funding goods in a post-fruition fashion. Goods receive a lump-sum subsidy in proportion to their perceived social utility at some point in time in which the perceived utility is a more accurate reflection of the realisable long-term social utility.

rPGF means to answer the question:
"What would society pay a time-traveler NOT to go back in time and destroy project A so that it doesn’t exist today (if society could coordinate)?"

The benefit of this method of PGF is that it is often more clear to evaluate the benefit public goods have brought to society after-the-fact. Because of the nature of public goods being non-excludable, its benefit to society is a function of the number of people that use it, which may not be obvious at first. Especially for public goods that have utility which is multiplicative to the number of people that make use of it (e.g the internet), retroactive funding is an especially useful tool. In this way, the uncertainty of the "potential utility" the good will bring under an "infinite time horizon" is greatly reduced in an important way.

The cons with retroactive PGF are two-fold.

1. Retroactive PGF only allows for ex-post assessment of projects at the time the assessment is being made. This means that any public good that was unable to come to fruition because of a lack of funding or other circumstances, as well as any counterfactual of the good not existing, is ignored.

2. Increased trust assumptions. In order for rPGF to correctly function as an incentive mechanism for the entrepreneur (agent) to take on risk in diverting resources to the project, the agent must be confident that the rPGF assessment mechanism will both work correctly and exist in the future at the time the assessment will be made. This may be able to be offset by some additional reward, but with risk-averse agents, this reward will have to be inefficiently allocated.

### Figuring out the right amount of funding

#### Assumptions
When doing economic modeling, there will inevitably be a number of assumptions. I will attempt to outline them carefully here.

**1. Rational agent**
One of the greatest assumptions we tend to make is that the agent is *rational*. In this sense, we mean an agent that is able to perfectly determine its preferences over any two alternatives in such a way that if the agent were given all the alternatives the universe had to offer, the agent would be able to rank them from "most preferred" to "least preferred".

**2. Risk neutral agent (relaxed later)**
This means that the agent is trying to maximize the reward in expectation. A risk-averse agent will inevitably require more funding.

**3. Risk free interest rate**
We will assume a risk-free interest rate $r$ that remains constant over time. This is a very silly but simplifying assumption. It is arguably silly for the reason that it's hard to imagine that a "risk-free" way of making money even exists, so this "risk-free" is always relative to something, and further that it remains constant is likely impossible. A changing $r$ can easily be incorporated at the cost of uglier looking equations.

**4. Known probability of receiving funding**
The rational agent is able to accurately assess the probablity of receiving funding at time $t=T$ to be $s$.

**5. The public good's benefit to society is great enough**
Additionally, we assume that the ex-post benefit to society $U$ at the time of assessment $T$ is greater than the cost of investing in the project (so that the rPGF council will find it worth funding ex-post). I.e
$$ U_{society}(\text{public good}, t=T) > C(1+r)^T$$

#### Scenario
The agent is considering investing in a public goods project. The agent assesses the probability of the project being successfully funded by the rPGF council at some rate $0<s<1$. Note that the success rate $s$ includes both
1. The inherent risk of the project itself as well as
2. The risk attached to the rPGF council existing and functioning properly at the time of assessment.

The project comes at one-time sunk cost $C$, which includes the opportunity cost to the agent of choosing this endeavour over any other in addition to any other non-monetary cost to the agent.

$$\mathbb{E}[\text{Funding} - \text{Costs}] > 0 \Rightarrow
\mathbb{E}[sF - C(1+r)^T]> 0$$

An agent will be incentivised to invest in their project if funding retroactively directed to the project $F$ is at least as high as their costs.

$$ sF \geq C(1+r)^T \Rightarrow F \geq \frac{C}{s}(1+r)^T $$

If the agent is risk-averse, then this becomes more complicated. A common way to measure risk aversion is to assign the agent a utility function
$$u(c) = \begin{cases}
\frac{c^{1-\eta}}{1-\eta} &\text{if }\eta \geq 0 \text{ and } \eta \neq 1\\
\ln(c) &\text{if } \eta = 1
\end{cases}$$
(known as [isoelastic utility](https://en.wikipedia.org/wiki/Isoelastic_utility)).

In non-nonsense speak, this basically means that the agent is sensitive to uncertainty, and this risk-averseness is captured by $\eta \geq 0 \;\; ; \eta \neq 1$.The bigger the $\eta$, the bigger the aversity to risk. Importantly, this aversion to risk means that the agent must receive a greater amount of funding in order to compensate for her distaste for uncertainty. This additional funding requirement is often referred to as a *risk premium*.

For the case $\eta = 1$, doubling the expected reward increases the utility of the risk-awerse agent only by a additive constant. More concretely, if Alice has the choice between 100 dollars with certainty or flipping a coin to win $x$ dollars, she would need at least $x=10000$ dollars for the coin flip to be worth the trouble.

For the rPGF example, then we have
$$ \mathbb{E}[u(\text{Funding + Endowment - Costs})] > u(\text{Endowment}) \Rightarrow $$

$$ s\frac{(E + F - C(1+r)^T)^{1-\eta}}{1-\eta} + (1-s)\frac{(E - C(1+r)^T)^{1-\eta}}{1-\eta} \geq \frac{E^{1-\eta}}{1-\eta} \\
\Rightarrow
F \geq \big(\frac{E^{1-\eta} - (E - C(1+r)^T)^{1-\eta}(1-s)}{s}\big)^\frac{1}{1-\eta} - (E - C(1+r)^T)
$$

Where $E$ is some initial endowment for the agent (if you start off wealthy, you can take more risks, but that's a topic for another discussion).

We can analyse that the amount of funding required for an agent operating under this utility function to find the investment worthwhile will be greater than a risk neutral agent's preference if and only if $\eta = 0$.


The above image depicts the funding requirements (as a proportion of initial wealth) required for agents with varying risk profiles, assuming the cost of the project is $10 \%$ of their initial wealth and the probability of receiving funding is $20 \%$. As the probability of receiving funding decreases, the required funding increases by a magnitude dictated by their risk aversity.

There is an extensive literature in estimating $\rho$, and estimates vary wildly. [Groom and Maddison](https://link.springer.com/article/10.1007/s10640-018-0242-z) estimated the value to be around 1.5 for the United Kingdom. But of course, this number may easily overestimate risk aversity for certain relevant groups ... 💎🙌🦍🤝💪📈.

:::info
**Ponder**
We assumed that $\eta \geq 0$, but what is the risk profile of someone with $\eta < 0$?
How does this change the amount of required funding?
:::

In a risk averse world, therefore, it is important to establish confidence in agents that rPGF works and that it works well. As $s$ approaches $1$, we begin to move towards a world where retroactive funding achieves all of the benefits of continuous funding, whilst also maintaining its advantage of filtering uncertainty at no additional premium (since if F is large enough this is true regardless). The premium needed whilst $s<1$ will depend on the risk-averseness of the agent, and could be mitigated by the existence of insurance, should a market for that exist.

## Continuous PGF

Continuous PGF is meant to assist public goods coming to fruition.

cPGF aims to answer the question
"What would society pay the PGF *provider* in order to get A in the ***future*** if all the members of society that benefit from it could coordinate?"

Every quarter, the council votes and decides on a group of public good projects that will receive up to 5% of the annual inflation distributed continuously over that quarter. The benefits of continuous funding include but is not limited to:

1. Cheaper. Since the money is received up front, the entrepeneural agent is less concerned with the possibility of not receiving funding despite the success of the project. As long as the agent is risk-averse to some degree, the total funding cost needed will be less

2. Increased likelihood of existing. In an economy where the entrepreneurial agent is unable to take loans at a reasonable price (either because lenders do not have enough faith in the rPGF system or because the agent is unable to provide sufficient collateral), an agent that would be unable to pursue the endeavour is more likely to pursue it.

On the other hand, there are inevitable costs:

1. The council takes on additional risk, since the benefit of the good is less clear without the benefit of hindsight.
2. Unless monitoring is in place, there is a lack of incentive for commitment by the agent, which can lead to uncompleted projects. Monitoring is expensive.
3. Increased uncertainty in evaluating the council's decisions as speculation is harder to assess. This places greater trust assumptions on the council acting faithfully.

## Four goals of PGF
Public goods funding is intended to be directed to the four following categories.

For development related to Namada/Anoma ecosystem:

1. Technical research related to Namada/Anoma, e.g research in cryptography, distributed systems, consensus mechanisms, etc.

2. Engineering related to Namada/Anoma, including but not limited to the user-interface, storage/throughput optimisation, bug-fixes and can be distrubted through e.g developer grants, bug bounties, etc.

3. Social reserach related to Namada/Anoma, more specifically resarch exploring the relationship by the technology and humans. This may be how privacy affects interactions e.g artist grants etc.

For public goods not directly related to Namada/Anoma:

4. External public goods - e.g carbon capture, independent journalism, direct cash transfers, etc.

More detailed information can be found at specs.namada.net/economics/public-goods-funding.html.