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Compounded Tokenized Incentives for Initial Token Offering
The technical paper for OpenPad's allocation policy: Compound Hyperbolic
$\text{Can Kocagil}\\ \text{Founding Director of OpenPad}\\ [email protected]$

# Compound Hyperbolic

Compound hyperbolic is a state-of-the-art investment allocation calculation model of OpenPad, considering both time and token assets as input and producing the allocation amount of the users as output.
Let
$A[x], S[x]$
and
$R[x]$
be the allocation, $OPN staked amount, and the generated$OPN reward of the user
$x$
in the form of the token amount for
$x = 1, ..., N$
respectively for a certain time frame between
$t_s$
and
$t_f$
for a specific token sale round. (Say Project XYZ issues a 1,000,000
token for the specific token sale round with a swap rate of
) The final allocation is a function of both
$S[x]$
and
$R[x]$
.
$A[x] = f(S[x], R[x]) \text{ } f: \mathbb{R}\to \mathbb{R}$
​The compound hyperbolic model consists of a two-layer deterministic time-series mathematical process to compute the final allocations of the users, aiming to produce a capital-efficient resource allocation model, long-term time- and asset-based economic incentive schemes to incentive users to stake native launchpad tokens (i.e., OPN), creating a positive-sum incentive alignment between investors, projects owners and OpenPad. # Compound Time Layer Let $R[x] = R(x, \gamma)$ where $\gamma$ is the compound frequency or rebase index, which calculates the reinvestment period for $S[x]$ , and dynamically update the following equations at every $\gamma$ . $S[x]_{t+1} = S[x]_{t} + R[x]_{t}$ which creates an exponential effect based on $\gamma$ . Hence, $R[x]$ can be computed for the time period $t_s$ and $t_f$ as follows. $R[x] = R(x, \gamma) \approx S[x] * (1 + \dfrac{r}{\gamma})^{\gamma}$ where $r$ is the interest rate of the staking pool (vault). Then, let $\delta = \sum_{x=1}^N{S[x]}$ ​and $R = \sum_{x=1}^N{R[x]}$ , representing Total Staked Amount and Total Reward Generated respectively. Then, let's also define $S_p[x]$ and $R_p[x]$ as a staking and reward share of the user $x$ . $S_p[x] = \dfrac{S[x]}{\delta} \\ R_p[x] = \dfrac{R[x]}{R}$ Then, $\varphi[x]$ is the linear aggregator, calculated as a weighted average of $R_p[x]$ and $S_p[x]$ with weights $<\alpha, \beta>$ . $\varphi[x] = \alpha * S_p[x] + \beta * R_p[x]$ where $\alpha < 1, \beta <1$ and $\alpha + \beta = 1$ . $\varphi[x]$ represents the linear aggregation of the staking share and reward share which will be used as an aggregate input for the hyperbolic asset layer. # Hyperbolic Asset Layer After the calculation of $\varphi[x]$ , which represents the weighted average of staking and reward share of the users, a generalized sigmoid curve is adapted to re-design allocations of the users from linear to hyperbolic curves, producing greater multipliers to lower-size capital powers. The final allocation of the user $x$ , $A[x]$ , can be calculated as $A[x] = \sigma(\varphi[x]) \text{ where } \sigma(z) =A+\frac{K-A}{\left(C+Q e^{-B z}\right)^{1 / \nu}}$ $\sigma(z)$ is a generalized sigmoid curve, where $z$ is the input and • $A$ : the lower (left) asymptote; • $K$ : the upper (right) asymptote when $C=1$ . • $B$ : the growth rate; • $\nu > 0$ : affects near which asymptote maximum growth occurs. • $Q$ : is related to the value $\sigma(0)$ • $C$ : typically takes a value of 1 . In the below graph, some sigmoidal curve functions are depicted to illustrate the behavior of the function. More importantly, the range $1> x > 0$ is particularly used in the allocation algorithm, which has mathematical properties as follows. 1. 1. $\sigma(x)$ is always increasing with respect to $x$ . 2. 2. $\dfrac{\partial f(x)}{\partial x}$ is always decreasing. Some sigmoid functions compared. In the drawing all functions are normalized in such a way that their slope at the origin is 1. In the below graph, the red line represents the derivate of the sigmoid curve $\dfrac{\partial f(x)}{\partial x}$ . # The Economic Results: Effects & Analysis ### The Derivation of Plain Language Allocation Policy "The more or longer the stakeOPN, the larger the allocation will be"
The compound hyperbolic allocation policy is a fully-continuous and monotonic increasing function, i.e., always increasing or remaining constant but never decreasing; hence it's strictly increasing. Hence, for our use case range (
$1 > x > 0$
) for every increase in the x-axis, the y-axis is always increasing, creating a plain language allocation policy: "The more or longer the stake $OPN, the larger the allocation will be" ### Low Entry Barriers Entry level defines the lowest capital an investor needs to deposit before investing in the form of staking the native token of the launchpad ($OPN). Compound hyperbolic is a continuous function, i.e., the function always produces
$f(x) > 0$
for
$x>0$
; hence any positive $OPN staking in the auto-compounded staking pool (vault) yields a positive $A[x]$ , enabling any$OPN staker to get an allocation.

### 100% Guaranteed Allocation

It is the binary (either guaranteed or not, there’s no in-between) metric defined by whether the 100% guaranteed allocation is assigned to the investor who already passes entry level in the launchpad. Since compound hyperbolic is a real-valued continuous function, meaning that there are no abrupt changes in value, known as discontinuities, for every $OPN staker, there is a positive $A[x]$ , enabling every$OPN staker to get 100% guaranteed allocation.

### Pre- & Post-IDO incentives of $OPN staking & unstaking “Pre- & Post-IDO incentives of$OPN staking & unstaking” refers to the creation of incentive schemes for investors to stake $OPN longer and unstake$OPN later in both pre- and post-IDO periods, avoiding "stake & unstake" schemes.Since the compound hyperbolic allocation model takes time as an input, investors are incentivized to stake $OPN longer in both periods regardless of what stage they're in (pre- or post-IDO). Moreover, the time has also partially compounded effect on the allocations hence longer-term$OPN stakers allocations are compounded also, creating an exponential positive incentive scheme for investors. Finally, since unstaking-then-re-staking will re-initialize
$R[x]$
component, the allocation of the user will drop significantly with respect to the one who doesn't. This effect is controlled by
$\beta$
parameter.

### The incentives for extra capital contribution

“The incentives for extra capital contribution” refers to the incentive scheme of depositing/allocating extra capital into investment deals on top of existing investments, measured by how much your allocation will be increased with respect to the increase in your staked native launchpad token ($OPN). Best understood under example; e.g., you staked 1000$OPN (
$S[x]=1000$
) and got 2% of the all investment allocation (
$A[x] = 2\%$
). Moreover, then, you decided to invest more and staked your extra 500 $OPN (summing to$1500), and get (2 +
$\pi$
)% allocation. Here,
$\pi$
refers to the change in your allocation in response to the extra staked $OPN. The larger the $\pi$ , the higher the incentives of the investor to deposit more capital. In compound hyperbolic, $\pi > 0 \text{ for every x>0}$ , creating an incentive to deposit more money into investment deals. However, since $\dfrac{\partial A[x]}{\partial x}$ is a monotonic non-increasing function, the rate of increase of $\pi$ is decreasing for every increase in $x$ . ### Fair allocation economics Fair allocation economics refers to the distribution of allocation and multiplier to the investors, aiming to assign resource opportunities to participants equitably and fairly. Equitable distribution of the resources involves allowing the capital resources to be distributed equally to each division of the community rather than accumulating the resources in the hand of a few persons. In allocation economics, there is no 100% accurate metric to understand how fair the distribution is; however, technically, incentive mechanisms should be analyzed to understand how allocation and multiplier might be distributed across investors. It is more statistical than analytic; thus, it’s hard to measure; however, one objective is to avoid too centralized distribution of allocation and multiplier. Critics: Many attempts have been made at defining what a fair distribution of resources in society would look like. Utilitarians have argued for any distribution that maximizes welfare, while libertarians have argued for the legitimacy of any distribution that follows from a just initial acquisition or transfer. Liberal theorists, like John Rawls, have argued that a just distribution maximizes welfare for the worst off — this is called the maximin principle. It entails that inequalities are only allowed in cases where the unequal distribution generates a better outcome for those at the bottom. Compound hyperbolic has inverse quadratic internal disincentive schemes for decreasing the multiplier of investors, creating a financially-disincentived strategy for centralized accumulation of tokens. ### Multiplier Distribution Let $M[x]$ be a multiplier of the $x$ , which is the ratio between allocation and staking pool share, is a metric for capital-efficiency for investors. The larger the multiplier, the better the capital-efficiency of the investor. $M[x] = \dfrac{A[x]}{S_p[x]}$ For example: Bob can have 2M staked$OPN, hence
$S[x] = 2,000,000$
tokens with a total staked amount
$\delta$
of 100M tokens, corresponding 2% shares, i.e.,
$S_p[x]$
of the staking pool. For the allocation share, Bob can get 1M a token allocation hard cap (the maximum amount an investor can deposit) on 25M token sale event, corresponding to
$A[x] = \dfrac{1,000,000}{25,000,000} = 4\%$
shares of the whole investment deal. Hence, the multiplier of the investor is
$\dfrac{A[x]}{S_p[x]} = \dfrac{4\%}{2\%} = 2$
. The larger the multiplier, the more capital efficiency is for the investor. The first-order differential curve of the sigmoid is essential for interpreting the multiplier distribution across investors because the distribution represents the rate change of the allocation with respect to increase in
$S_p[x]$
.
$M[x] \approx \dfrac{\partial A[x]}{\partial x}$
​Hence, we can utilize the graph of the first-order derivate of the allocation to represent the distribution of the multiplier over capital amounts as follows.

# The Hyperbolic Asset Layer Simulation

The allocation of resources is the public finance field in the context of economics, tries to find a set of strategies or per-participant shares under certain preferences and privileges that lead to Pareto efficient outcomes, in which no party's situation can be improved without hurting that of another party. Hence, the state of distribution should be allocative efficient.
Allocative efficiency is a state of the economy in which production is aligned with consumer preferences; in particular, every good or service is produced up to the point where the last unit provides a marginal benefit to consumers equal to the marginal cost of producing.
Resource allocation efficiency includes two aspects:
• At the macro aspect, it is the allocation efficiency of social resources, which is achieved through the economic system arrangements of the entire society [1]
• The micro aspect is the use efficiency of resources, which can be understood as the production efficiency of the organization, which can be improved through innovation and progress within the organization [1]
When a market fails to allocate resources efficiently, there is said to be a market failure. Market failure may occur because of imperfect knowledge, differentiated goods, concentrated market power (e.g., monopoly or oligopoly), or externalities.
With a private-goods market mechanism where people can buy as many voices as they want at the same price per voice, the individual with the strongest preference (or the wealthiest) carries everything. We are OpenPad and here to find allocative efficient and fair ways to distribute capital and voices.

## Fair Division of Resources

Fair division of resources is a challenging and dynamic problem, and an active research area in mathematics, economics, social choice theory, dispute resolution, etc. Moreover, in game theory, the challenge of distributing a set of resources among numerous people who have an entitlement to them so that each person gets their fair share is known as fair division.
Disclaimer: The following code is just a simplified representation of the potential modelling of OpenPad token sales and does not reflect the actual IDOs or similar. Numbers, data, visualizations and everything else are just pre-assumptions of the rational players in the OpenPad market and the code snippets may or may not be used in the production environment, and is given for transparency.

## Coding The Welfare of OpenPad Citizens

We'll be modelling the behaviour and distribution of OpenPad citizens as a function of how much they stake via digesting Pareto optimality and allocative efficiency with OpenPad dynamics such as openness, fairness and beyond. Here are basic Python numerical computing and visualization library imports.

Let's imagine that there is an IDO with a $XYZ token, with an IDO-preserved token supply of 1M. Then, for the sake of simplicity, let the deal size be$1M, setting the per token price as 1 $XYZ =$1. Then, to model the IDO participant's stakes, let's generate random Gaussian numbers with 3 welfare classes. Let's made up 1500 OpenPad citizens with relatively low welfare, a staked average of 50 $OPN tokens with 15 standard deviations. In the same way, let's generate 1500 citizens for the middle and high welfare class with (300, 2000) mean and (70, 1500) standard deviations. We form the data as a list of tuples where each element of the list contains 3 numbers: mean stake, std stake and size. ## Gaussian Modelling of OpenPad Citizens Let's create random gaussian vectors of 3 classes as follow. ## Gaussian Welfare Distributions of OpenPad Citizens Who doesn't love visualizations? We love. The figures are self-explanatory. ## Allocation Modelling Let's start an actual business. In the df_agg_stakes variable, we concat the 3 gaussian vectors. ## Linear Multiplier How most IDO launchpads calculate your allocation in certain deals is by taking the linear percentile of your stakes! Your unit of influence is directly proportional to your stakes. Simple, right? Most of the time, linear models are preferred as they are easy to explain and compute but not always the efficient ones. ## Quadratic (Decreasing) Multiplier We can be more liberal. Taking the square root of the stakes, then computing the per-citizen allocation gives superior multipliers to low-welfare people while quadratically decreasing the unit of influence as stakers purchase influence. ## Logaritmic Multiplier Here we are way closer to becoming technoliberalist. Taking the log of citizen stakes, then normalizing them to in the range [1, 100] gives crazy multipliers to low welfare stakers, whereas exponentially decreasing the unit of influence as stakers purchase influence. ## Sub-Quadratic Multiplier Okay, let's be more conservative. Sub-quadratic functions give a more conservative multiplier than direct quadratic functions, which can be scaled more to be more linear. However, as the number of citizens increases, to allow more distributed capital investment and more people to join, the systems can be sub-quadratic. ## Scaled Tangent Hyperbolic Sigmoidal and tangent hyperbolic functions are non-linear processes mostly used in neural network contexts but have certain hyperbolic properties that we can utilize, especially when they are scaled to the unit range. For the sake of representation, we utilized a scaled hyperbolic tangent function. ## Visualizing the Multipliers Let's visualize the gaussian-modeled and computed multipliers as follows. In the above figure, we can visualize the correlation between the number of$OPN tokens staked and the corresponding allocation in the $XYZ IDO. We can see that the linear model is just linear! HAHA! Let's leave the linear model aside. Being more serious, the scaled tangent hyperbolic function is most conservative among other functions, caring for every size of the welfare, empowering the low-welfare citizens via enabling higher multiplier whereas not much decreasing the high-welfare citizen's multiplier, which can be optimal IDO to IDO. You can compare the models we presented in the form of$XYZ token allocations numerically.
Num Linear IDO Token
Num Scaled Tanh IDO Token
81.37
1552.15
323.21
87.52
108.82
1795.01
404.22
117.05
132.29
1979.11
469.73
142.29
178.74
2300.46
592.08
192.24
225.48
2583.83
707.94
242.52
...
...
...
...
72258.96
46254.16
59911.93
62606.00
72330.73
46277.12
59957.70
62645.10
74154.12
46856.79
61117.03
63622.46
82528.56
49431.87
66360.19
67731.24
82880.36
49537.12
66577.68
67890.66

# A Future Work

## Machine Learning in IDO Allocations

Most members of our technical team have a machine learning background so it is natural to develop ML-backed algorithms for IDO allocation calculation. What can we do is infinitely many, but for the sake of representation, we provide a K-means unsupervised learning algorithm, which tries to find a set of clusters with the optimization criterion, which is to minimize the total squared error between the training samples and their representative prototypes. Let's fit the model.
We have 8 clusters with their centroids! The are several strategies we can deploy, but let's take a simpler example as follows.
What we do is calculate per-cluster allocations with linear percentile and the number of citizens in each cluster, and divide them to find per-citizen allocation.
Finally, we can compare the algorithm's multipliers as follows.

# References

Compound Hyperbolic
Compound Time Layer
Hyperbolic Asset Layer
The Economic Results: Effects & Analysis
The Hyperbolic Asset Layer Simulation
Fair Division of Resources
Coding The Welfare of OpenPad Citizens
IDO Variables