# Worldcoin: Thoughts about Monetary policy

- become a distributor to lower lockups
- random unlock
-

## Relevant Links

- [NEAR Token Supply and Distribution](https://near.org/blog/near-token-supply-and-distribution/)
- [Algorand Token Supply](https://algorand.foundation/algo-dynamics)
<!--- [How Secure is Cardano?](https://medium.com/@undersearcher/how-secure-is-cardano-5f1e076be968)-->
- [Economics in the Ethereum Whitepaper](https://ethereum.org/en/whitepaper/#currency-and-issuance)
- [Balaji: Quantifying Decentralization](https://news.earn.com/quantifying-decentralization-e39db233c28e)

## Dashboard to play with the Numbers
Link: http://54.197.116.0:8501/
Password: worldcoin2020

<!-- ## Questions for Sam
1) How much quality insurance will worldcoin do for distributors/signups?
2) Ownership of the device: Buying, renting, "staking"?
3) Assuming we have over 100mio in cash: Use of funds? -->

<!-- ## Coin Distribution
- Team
- Foundation
- What roles does the foundation have: Staking, giving out grants, funding internal development
- Investors:
- Seed Round (Sam): 3M
- Series A: 25M
- ICO: 250M
- Distributors
- Users



**Thoughts about ICO valuations**
1) Telegram raised 1.7 bn @ 4 bn val
2) Filecoin: 257 Mil ICO @ 850 mil valuation.
3) Ethererum raised 18 mn at 40 mil valuation. The interesting thing was, they framed it at 1% of the BTC market cap at the time.

**Design Principles**
1) No party should be "overly powerful" (Not more than 5 percent) at any point in time -> Read Balanji
2) Gini Coefficient at the end of distribution should be significantly better than Bitcoins which is 0.88.

Both results in the idea that we should think about **Milestone/User** oriented vesting. Active users is kind of the only measure we care about, so why not align everyone, also the investors on it. -->
<!-- ## Coin Distribution

* **Core Contributors:** The founding team
* **Series A Financing:** 5% will be soldin a Series A financing with a goal of 35 million USD, Sam already holds x percent?
* **ICO Financing:** In Q1 of 2021, prior to the global launch of Worldcoin a ICO with a target sum of 500 mio USD will take place.
* **Foundation (Ecosystem and Distributor Rewards)**: The Worldcoin Foundation will control 15% of the Network to drive growth. Ten percent are used to give out the Distributor incentives for every signup. Five percent to promote projects and fund the ecosystem. The funds are controlled by the foundation counsel consisting of founders, investors and influential personalities. -->

## Signup and Distributors Rewards
### Signup reward function
The signup reward functions decreases with a powerlaw. This creates the desired urgency. On the other hand, assuming that the price of worldcoin rises with some kind of powerlaw with the number of active users, such function still results in a reasonable high fiat signup reward at the end of the claim period.

The used function that seems to have desired quantities is:

$$R(n) = \frac{A}{(n + n_{offset})^{\gamma}}$$

where A integration constant, $n_{offset} = 300$ and $\gamma = 0.55$ ($\gamma_{BTC} = 0.8$)


Still not completley happy with this function, but we are getting there! Probably it is ok if more goes to the first 100m users, and after a billion the reward falls below 1 wld really fast, because if we reach this growth, the worldcoin value will be very high anyway.

### Distributor reward function
* We reserve a defined share of the currency pool to pay distributors.
* In the optimal case Worldcoin has a team that is defining the Distributor reward in required complexity to maximize growth in the network and motivate Distributors. The extrem case would be the Uber model, where a very complex function changes the price to limit and increase driver density in realtime. We probably want something more predefined and the ability to change it over time steps. For example every three months we can update the "function" with a governance vote, also the function is always public.
* To define the slice to pay distributors, at first it sounds like a good idea to define a fiat equivalent (hourly rate * (targeted users/singups per hour). However, this calculation get's intractable if you assume fluctuation of multiple 100 percent for WLD in fiat. However, fixing it to 6.5% seems like a legitimate estimate. This is exactly 10% of the user share. At an achievable signup rate (without virality) of 10 users/hour the distributor would get paid one user singup bonus per hour. Therefore, if the user rewared is greater than the average wage - which it should be anyways - everything will work. Our estimates are much higher. Educated guess is >10x average wage (see BTC scaling).

## Lockups and Vesting
* Users are vested over time and get a immediate unlock of 10 percent. The vesting time is 1 year.
> **TODO** some users should get really rich and buy crazy things. Anyone has ideas for that?
* Investors and all other shareholders are vested by "users". So they keep their relative share and it get's bigger as more users are signed up. Therefore we promise not a fixed number of coins, but a relative fraction of the network.
* In addition to that investors and team get a immediate unlock of 10 percent at launch (How does ICO trading go? Secondaries?), so investments can get derisked.
* **Arguments for relative investor unlocking:**
* We can make sure that no party get's too powerful in any point in time.
* Liquidity of the market increases with signed up users
* You can make sure to be sufficiently decentralized fast
* Investors like coinbase, are incentivized to increase growth with marketing etc

> **TODO**\
> After you guys agreed that that's a good idea i will spend some time to make the plot below and all it's assumptions clear and understandable. What happens here is basically assuming a growth scenario as result of a linear rollout of orb ownership to 50k orbs. Then let the users vest over 1 year and invevstors and team vest relative to the users until 1bn signups. Then all investor and team coins get unlocked assuming that the network will be sufficiently decentral by then.


**What happens if we sign up less users?**
At end auf launch period we have two options:
* We burn all not claimed coins, by that every party of the network has the percentage share they agreed on. However we run risk that significantly less than 10bn coins are in circulation.
* We distribute the unclaimed coins by percentage to every shareholder over a period of five years, by which we inflate the whole system. We certainly need to do this over time otherwise we flood the market and destroy the price.
- My personal best guess would be to burn them, it's the least complex version and the only counterarguemtn is that we don't end up 10b coins.
> **TODO**
> * Research if cliff needed for user coins due to regulatory reasons
> * Get opinion on user vesting time & immediate unlock

### Fees
Fees are absolute and are determined market driven. The Blockchain is not infinitely scalable, a free market setting those fees is the optimal solution.
More important than always having a constant fee is that the fee of a transaction can be well predicted by the user. This means that the user can be sure that a transaction will get included within a certain time span. Many blockchains currently fail to provide those guarantees and transaction fees set inefficiently.

<!-- **Thesis:
Fees should be relative
**Counterargument:**
You can’t know the financial value of a transaction by looking at it. It could be encoded in arbitrary ways. Imagine a token on top of our transactions. Who sets the price of this token in comparison worldcoin? (Also obvious: private transaction will make knowing the financial value impossible)
The only thing one can estimate, is the computational load and burden on bandwidth caused by a transaction. However, this is fully decoupled from economic value.
Side note: We live in a non-infinitely scalable world. Miners need to prioritise transactions by some criterium and this is the fee. If one used relative fees, small transactions would get highly discriminated.

**Thesis:**
Supply soak up is a "mechanical" way to increase the price of worldcoin
**Counterargument:**
* Digital gold implies low velocity. See bitcoin as an example: [24h trading volume in bitcoin is approx 0.5% of market cap.](https://messari.io/screener) Of those 0.5% we take away 0.5% in fees (we also need to pay validators from this amount).
$$(1 - 0.005^2)^{365} \approx 0.99$$ Which means we take away less then 1 percent of circulating supply per year through fees.
* Also we look for a major value increase of the network of a factor 100 or 1000, we can never achieve this solely with fees, no matter how high the throughput, we would need to burn 99% with fees. We need network effects such as Metcalfe's law to kick in. -->

## Current Proposal in the Whitepaper


The problem with the current proposal is:
1) The resulting Gini coefficient is even higher than bitcoins
2) A power law, rather than a exponential function seems to be better. Exponential decay drops to fast.

<!-- #### How did we get to powerlaws

We analyze bitcoin to find out
* how network value / market cap $M(n)$ grows with the number of users $n$
* how the price $C(n)$ and the circulating supply $N(n)$ correlate with the number of users $n$
* find a function for added supply per user $N'(n)$

We assume the **market cap** can be modeled as a powerlaw:

$$M(n)\propto n^{\beta}$$

We find that those assumptions fit the data very well. We find $\beta=1.8$. The market cap can be split up in the **circulating supply** and the **coin price** $M=N*C$. We assume that those two quantities can be modeled as powerlaws as well:

$$
N(n) \propto n^{\gamma} \\
C(n) \propto n^{\chi}
$$

where $\beta = \gamma + \chi$. We again find that those assumptions fit the data very well. We find:

$$
\gamma = 0.2 \\
\chi = 1.6
$$

We can now calculate the added supply per user $N'(n)$:

$$N'(n) = R_T(n) \propto \frac{1}{n^{0.8}} $$

There is no such thing as signup rewards in other cryptocurrencies and therefore, the powerlaw we fitted to the added supply per user is not added because of the user but would have been added to the system anyways through mining whether users would have joined or not.

What we can conclude is that at a given growth of $n$ the network seems to **tolerate** a certain **inflation** while the price $C(n)$ is still increasing and therefore one can make it plausible that the **benefit** of the added users **outweighs** the inflation through virtual added supply per user and leads to a net deflation.

Therefore, paying people for becoming active users with a similar reward. This should be tolerated by the network while still increasing the coin price. Since we are aiming for a fair launch we slightly adjust the signup reward function

$$R_T(n) = \frac{A}{(n_{initial} + n)^{\beta}}$$

Our adjustments are:
* The additional $n_{initial}$ in the denominator makes up for a less steep decline in the beginning (i.e. coin reward does not drop by a factor of 10 between the first and the 10th signup).
* decrease the exponent $\beta$ in order to achieve a fair distribution

#### New Proposal


Where $n_{initial} = 300$ and $\beta = 0.5$. And $A$ as integration constant, so we reach the defined number of coins in total.
-->

<!-- ### Bitcoin -->

<!-- ## Not related questions
- What happens to the money we raise? Include projections to the investor memo, what would we do with a billion dollars. Look maybe in Ton projections.
- Balance Sheet fund?! Marketmaking?
- How much money do we want to raise, how many coins do we want to sell for this? Play with Valuations? Resulting coin price...
- Vesting by users still important! Need to think about it. -->

<!-- ## Todos

* Research Gini Coefficient of: Gold, Dollar, Euro, BTC,... -> Table
* How much maximum b
* Fees?
* Anyone can accelearte coin unlocking?
* -->

<!-- ## Boundary Conditions given by the Blockchain

### Validation Costs

Requirements and Parameters:

Blocktime: ~10s
Blocksize: ~100KB
Bandwidth (monthly): SecondsPerMonth * Blocksize / Blocktime = ~25GB (+ overhead)
CPU: ~2 VCPU 
Storage: ~256GB SSD

Total monthly cost on AWS per validator (worst case): ~100$

If we aim for 10K validators, this results in about 1M WLD per month (at $1/WLD) for network security / decentralization.

### Fixed or inflationary supply

The overall question is whether this security can be ensured over long term solely by transaction fees of users. This is a fundamental question, as it was shown that [Bitcoin might become unstable](https://www.cs.princeton.edu/~arvindn/publications/mining_CCS.pdf) without inflationary mining rewards and Ethereum most prominently does not employ a hard cap on supply.

Assuming the average fee will be around 5ct (50 mWLD) per transaction ([comparable to what one expects in Rollups](https://docs.fuel.sh/v1.0.0/Concepts/Fundamentals/Fees.html)), about 20M transactions per month are needed to pay for the above mentioned security costs. This yields roughly 7.7 Txn/s as an needed average load, which equals the maximum throughput of the bitcoin network.

### Staking lockup -->

- Fixed token supply Y/N
- Estimated sum of all fees
- Estimated share of coins locked up by staking
> * Add link for not fixed supply. Make argument here, why it shouldn't be capped.

-->

<!-- -------------
# Monetary thoughts (Sunday Brainstorm)

## Goals
* Signup reward at any point in time should be $>x$ where $x$ $\in$ $[10,100]$ \$ such that it is worth at the time of signup for users to show up
* coin price should go up say $10.000$$\times$ after $10^9$ users to generate hype
* At each point in time the user should know that now is the best point in time to sign up and waiting would only decrease the signup reward
* Maximize growth
* understand which reward functions can accomplish this behaviour
* since it is a dynamic system: understand where it breaks down

## Definitions

Empirically the market capitalization is given by the function:

\begin{equation}
M(n) = \alpha \cdot n ^ \beta
\end{equation}

in units of \$. Where $n$ is the number of signups up to this date. The unit is \$. Where common examples include:

\begin{align}
M_{BTC}(n) &= 0.22 \cdot n^{1.79} \\
M_{ETH}(n) &= 4.3 \cdot n^{1.1}
\end{align}

The absolute value each user brings to the network is given by the derivative:
$$\delta_U = M(n)' = \alpha \cdot \beta \cdot n ^{\beta - 1}$$

in units of $\frac{dollar}{user}$. The market capitalization is also given by the total number of coins to date and the price of the coin and can therefore we written as follows:

\begin{align}
M(n) &= N(n) \cdot C(n) \\
&= (\gamma \cdot n ^{\xi}) \cdot (\lambda \cdot n ^{\eta}) \\
&= \alpha \cdot n ^{\beta}
\end{align}

where $N$ is the total number of coins in $[WLD]$ and $C$ is the price of Worldcoin in $[\frac{dollars}{WLD}]$. This yields the constraints:

\begin{align}
\gamma \cdot \lambda &= \alpha \\
\xi + \eta &= \beta
\end{align}

The total number of coins is approximately give by the total signup reward function $R_T$:

\begin{equation}
N(n) = \int_0^n R_T(n) \ dn
\end{equation}

where the reward $R_T$ in $[\frac{WLD}{user}]$ needs to be split between distributor and the user:

\begin{equation}
R_T(n) = R_D(n) + R_U(n)
\end{equation}

Another concept is the inflation generated by each user defined by:

\begin{equation}
I(n) = \frac{R_T(n)}{\int_1^{n-1} R_T(n)}
\end{equation}

## Case study for established currencies
Above we assumed that the following quantities can be described by powerlaws:
* $M(n) = N(n) \cdot C(n) = n^\beta$
* $N(n)=\gamma \cdot n ^{\xi}$
* $C(n) = \frac{M(n)}{N(n)} \propto n^{\beta-\xi}=n^{\eta}$

To validate this assumption we fit Bitcoin and Ethereum and find that this assumption can be made. We found the following values:

Bitcoin:
* $\beta_{BTC}=1.79$
* $\xi_{BTC}=0.2$
* $\eta_{BTC}=1.59$

Ethereum:
* $\beta_{ETH}=1.11$
* $\xi_{ETH}=0.06$
* $\eta_{ETH}=1.05$

We are looking for the signup reward function $R_T$ for Worldcoin which determines the reward of each signup and therefore the total supply of Worldcoin $N(n)$. While no other cryptocurrency has such a function we can however find analogies. For any currency we can determine the correlation between the number of users and the number of coins added to the total system. This is the derivative of the total coin supply depending on the number of users $N'(n)=R_{T,equivalent}(n)$. The major difference to Worldcoin is however, that users mine the coins through proof of work while at Worldcoin they mine them through their identity and get paid for the value they add to the network simply by using it and threfore make it more useful. With the numbers of Bitcoin we get:

$$R_{T,equivalent}(n) \propto n^{\eta -1}=\frac{1}{n^{0.8}}$$

Our assumption is that this amount of inflation is tolerated by the system and still allows it to grow in a somewhat similar way to Bitcoin. Interestingly, when we couple this reward function with the current coin price $C(n)$ in [\$] we get that the instantaneous dollar equivalent at the time of signup increases:

$$R_T(n) \cdot C(n) \propto n^{0.6} $$

Note: Users who sign up in the beginning would still get more Worldcoin but the perceived value at the time of signup - i.e. if the user would exchange for fiat instantaneously (which is not possible due to vesting) - the reward would increase.

TODO: calculate exact numbers: what would it mean in absolute values for each user. What value do we start with, what do we end up with.

Quick check:

BTC_marketcap / BTC_activeaddresses = 200.000 pro active address
BTC_marketcap / BTC_totaladdresses = 4.000 pro active address

## Conclusion

We have shown on the example of two established currencies, that the dynamics of crypto currencies can be described by powerlaws. The empirically obtained values of the exponents and coefficients can be used as bounds for the equations, which we expect to govern the dynamics of world coin. Furthermore, one can see fruitful analogies between the increase of bitcoin by mining and the increase circulating world coin by giving away world coin to new users. In the case of world coin, the growing number of active users leads to system effects which increase the fiat value of each world coin.

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