# Compounded Tokenized Incentives for Token Sales

The technical paper for Finceptor's allocation policy: Compound Hyperbolic

**Compound Hyperbolic **

**Compound Hyperbolic**

**Compound hyperbolic** is a state-of-the-art investment allocation calculation model of Finceptor, considering both time and token assets as input and producing the allocation amount of the users as output.

βThe 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., $FINC), creating a positive-sum incentive alignment between investors, projects owners and Finceptor.

## Compound Time Layer

## Hyperbolic Asset Layer

## The Economic Results: Effects & Analysis

#### The Derivation of Plain Language Allocation Policy

** "The more or longer the stake $FINC, the larger the allocation will be"**

#### Low Entry Barriers

#### 100% Guaranteed Allocation

#### Pre- & Post-IDO incentives of $FINC staking & unstaking

#### The incentives for extra capital contribution

#### 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, itβs 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 β 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

βHence, 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 Finceptor 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 Finceptor 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 Finceptor market and the code snippets may or may not be used in the production environment, and is given for transparency.

### Coding The Welfare of Finceptor Citizens

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

### IDO Variables

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 Finceptor citizens with relatively low welfare, a staked average of 50 $FINC 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 Finceptor Citizens

Let's create random gaussian vectors of 3 classes as follow.

### Gaussian Welfare Distributions of Finceptor 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 $FINC 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.

## 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

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