UBI Allocation Curve

Universal Basic Income (UBI) and Paid Job Compensation for Computing Providers (CP) in Swan Chain

Table of Contents

• Introduction

• Compensation Model

• Algorithm Implementation

• Visualization of Income Over Time

  • Interpretation of the Plots

  • Data Points Illustration

  • Impact of the Design

  • Conclusion

  • Future Work

• Appendix

  • Sample Calculations

Introduction

Swan Chain is a decentralized network that connects computing providers with users requiring computational resources. To foster early network growth and incentivize CPs to join and contribute resources, a dual compensation mechanism has been designed:

1. Universal Basic Income (UBI): Provides CPs with a predictable token income when their resources are underutilized.

2. Paid Jobs: Offers market-priced compensation for computational tasks requested by users.

This mechanism ensures a fair and gradual distribution of tokens to providers, supporting the network's expansion until it reaches a critical mass of user-paid tasks. Importantly, the UBI distribution rate is influenced by the resource usage rate, and CPs earn market-based compensation when engaged in paid jobs.

---

Compensation Model

$$I(x) = A \cdot x^{B} \cdot e^{-C x} \cdot (1 - u(x)) + P_{\text{market}}(x) \cdot u(x)$$

The total daily income $I(x)$for a computing provider on day $( x )$ comprises two components:

1. UBI Income $y_{\text{UBI}}(x)$

2. Paid Job Income $y_{\text{Paid}}(x)$

The UBI income is modeled using a gamma-like function adjusted by the resource usage rate $u(x)$:

$$y_{\text{UBI}}(x) = A \cdot x^{B} \cdot e^{-C x} \cdot (1 - u(x))$$

Where:

• $A = 20,000$ (Scaling factor)

• $B = 0.3100$ (Growth rate exponent)

• $C = 0.0017$ (Decay rate constant)

• $x$ is the day number, starting from 1

• $u(x)$ is the resource usage rate on day$( x )$ (ranging from 0 to 1)

The paid job income depends on the market demand and the resource utilization:

$$y_{\text{Paid}}(x) = P_{\text{market}}(x) \cdot u(x)$$

Where:

• $P_{\text{market}}(x)$ is the total market value for computational resources on day $( x )$

• $u(x)$ represents the proportion of a CP's resources utilized by paid job

Defining $u(x)$

(1) Calculate the total duration of real GPU orders across the network

$$T_{day}= \sum\limits_{i}Task_{ECP,i}(GPU_k) \times f_k + \sum\limits_{j}Task_{FCP,j}(GPU_k) \times f_k \times W_{FCP}$$

Where:

• $Task_{\text{FCP},i}(GPU_k)$ represents the time that the $i$-th FCP task uses $GPU_k$.

• $Task_{\text{ECP},j}(GPU_k)$ represents the time that the $j$-th ECP task uses $GPU_k$.

• $f_k$ represents the earnings growth factor

• $W_{FCP}$ represents the FCP resource bonus ratio, currently set at a constant value of 1.2

NOTE: The value of $W_{FCP}$, 1.2, means that if the same configuration of servers is deployed for FCP, it will generate 20% more earnings than ECP.

（2）Calculate the total available usage time for all GPUs in the network

$$T_{total} = \sum\limits_k N_{ECP}(GPU_k) \times 24 \times f_k + \sum\limits_k N_{FCP}(GPU_k) \times 24 \times f_k * W_{FCP}$$

Where:

• $N_{\text{FCP}}(\text{GPU}_k)$represents the number of $\text{GPU}_k$ in FCP

• $N_{\text{ECP}}(\text{GPU}_k)$ represents the number of $\text{GPU}_k$ in ECP.

(3) Calculate $u(x)$

$$u(x) = \frac{T_{\text{day}}}{T_{\text{total}}}$$

Defining Total Market Value $P_{\text{market}}(x)$

$P_{\text{market}}(x)$ represents the cost in Swan Tokens when all GPUs in the CP are fully utilized:

$$P_{market}{(x)} = \sum\limits_k N_{ECP}(GPU_k) \times Price({GPU_k}) \times 24 + \sum\limits_k N_{FCP}(GPU_k) \times W_{FCP} \times Price({GPU_k}) \times 24$$

Where:

• $Price(GPU_k)$ is the price of $GPU_k$.

• $W_{FCP}$ represents the FCP resource bonus ratio, currently set at a constant value of 1.2

• $N_{\text{FCP}}(\text{GPU}_k)$represents the number of $\text{GPU}_k$ in FCP

• $N_{\text{ECP}}(\text{GPU}_k)$ represents the number of $\text{GPU}_k$ in ECP.

Algorithm Implementation

The compensation mechanism proceeds as follows:

1. Initialization: Set day $(x = 1)$.

2. Determine Resource Usage Rate: Calculate $u(x)$ based on the CP's resource utilization by paid jobs.

3. Compute UBI Income:

$$y_{\text{UBI}}(x) = 20,000 \times x^{0.3100} \times e^{-0.0017 x} \times (1 - u(x))$$

1. Compute Paid Job Income:

$$y_{\text{Paid}}(x) = P_{\text{market}}(x) \times u(x)$$

1. Calculate Total Income:

$$I(x) = y_{\text{UBI}}(x) + y_{\text{Paid}}(x)$$

1. Distribute Income: Allocate $I(x)$ to CPs based on their resource contributions and utilization.

2. Increment Day: Increase $x$ by 1.

3. Repeat: Continue the process for each subsequent day.

This algorithm ensures that CPs are incentivized to contribute resources to the network, receiving UBI when their resources are underutilized and earning market-based compensation when engaged in paid jobs.

---

Visualization of Income Over Time

Scenarios

We consider three scenarios to illustrate how CPs' income evolves over time:

1. No Paid Jobs $u(x) = 0$ : CPs receive income solely from UBI.

2. Low Paid Job Demand $u(x) = 0.1$: CPs primarily earn UBI income with a small contribution from paid jobs.

3. Increasing Paid Job Demand: Resource usage rate $u(x)$ increases over time, shifting CPs' income from UBI to paid jobs.

Interpretation of the Plots

1. Total Income Over 720 Days

• Scenario 1: $u(x) = 0$

  • CPs receive income solely from UBI.

  • The total income decreases gradually over time due to the decay in the UBI function.

• Scenario 2: $u(x) = 0.1$

  • CPs receive slightly less UBI income than in Scenario 1 due to the 10% resource usage.

  • Paid job income contributes minimally, resulting in a slightly lower total income.

• Scenario 3: $Increasing u(x)$

  • Initially, total income is similar to Scenario 1.

  • As $u(x)$ increases, paid job income increases while UBI income decreases.

  • Total income remains relatively stable or increases slightly, demonstrating that paid job income offsets the reduction in UBI.

2. Income Components for Scenario 3

• UBI Income:

  • Decreases over time as resource usage rate $u(x)$ increases.

  • Reflects the transition from reliance on UBI to paid jobs.

• Paid Job Income:

  • Increases over time with the increase in $u(x)$ .

  • Compensates for the decrease in UBI income.

• Total Income Stability:

  • The sum of UBI and paid job income maintains income stability for CPs.

3. Resource Usage Rate $u(x)$ for Scenario 3

• Shows a smooth increase from 0 to 0.8 over 720 days.

• Reflects the gradual adoption of paid tasks in the network.

Data Points Illustration

Below is a table of computed token allocations for selected days:

Day	Daily UBI (y)	Cumulative UBI
1	19,966.03	19,966.03
30	54,549.22	1,261,976.56
60	64,262.68	3,062,143.25
90	69,246.55	5,072,341.49
120	71,941.60	7,194,431.61
150	73,261.06	9,375,212.61
180	73,666.56	11,581,013.65
210	73,430.22	13,788,817.87
240	72,728.28	15,982,188.47
270	71,682.24	18,149,084.82
300	70,379.70	20,280,565.34
330	68,885.86	22,369,958.88
360	67,250.50	24,412,305.58
390	65,512.29	26,403,963.32
420	63,701.70	28,342,321.28
450	61,843.01	30,225,585.83
480	59,955.70	32,052,616.78
510	58,055.51	33,822,799.99
540	56,155.17	35,535,946.61
570	54,265.01	37,192,212.48
600	52,393.39	38,792,032.93
630	50,547.09	40,336,069.55
660	48,731.55	41,825,166.37
690	46,951.10	43,260,313.71
720	45,209.18	44,642,617.97

Note: The "Cumulative UBI" column represents the definite integral of y(x) from day 1 to the specified day.

This table shows simulated data for UBI distribution calculated under the condition of U(x) = 0. The actual UBI release will dynamically change based on the network CP resource utilization rate.

---

Impact of the Design

Incentivizing Optimal Resource Utilization

• Adaptive Compensation: CPs are motivated to engage in paid jobs as they become available, earning higher income through market rates.

• Resource Availability: UBI ensures that CPs keep their resources available to the network, even during periods of low demand.

Sustainable Long-Term Distribution

• Transition to Market-Based Economy: As the network matures and paid job demand increases, CPs naturally shift from UBI reliance to market compensation.

• Controlled Token Issuance: The decreasing UBI allocation over time prevents token oversupply, maintaining economic stability.

Economic Implications

• Income Stability: CPs benefit from a combination of UBI and paid job income, smoothing income fluctuations.

• Market Alignment: Compensation reflects real-time network demand, promoting efficient resource allocation.

---

Conclusion

The combined UBI and paid job compensation model for Swan Chain computing providers effectively balances incentives, supporting early network growth while promoting efficient resource utilization. By dynamically adjusting CPs' income based on resource usage rates and market demand, the model ensures sustainable network development and economic stability as the network transitions to a mature, user-driven ecosystem.

---

Future Work

• Dynamic Market Pricing: Implement real-time market pricing mechanisms for paid jobs to reflect supply and demand accurately.

• Adaptive UBI Parameters: Explore methods to adjust UBI parameters ( A ), ( B ), and ( C ) based on network growth metrics.

• Enhanced Monitoring Tools: Develop systems to track resource usage and job completion accurately, ensuring fair compensation.

---

Appendix

Sample Calculations

Scenario 1: No Paid Jobs

Day 1:

• UBI Income: $y_{\text{UBI}}(1) = 20000 \times 1^{0.31} \times e^{-0.0017 \times 1} \approx 19,965.90 \, \text{tokens}$

• Paid Job Income: $0 \, \text{tokens}$ (No paid job demand)

• Total Income: $I(1) = 19,965.90 + 0 = 19,965.90 \, \text{tokens}$

Scenario 2: Low Paid Job Demand

Day 1:

• UBI Income: $y_{\text{UBI}}(1) = 20000 \times 1^{0.31} \times e^{-0.0017 \times 1} \approx 19,965.90 \, \text{tokens}$

• Paid Job Income: $y_{\text{Paid}}(1) = 50000 \times 0.1 = 5,000 \, \text{tokens}$

• Total Income: $I(1) = 19,965.90 + 5,000 = 24,965.90 \, \text{tokens}$

Scenario 3: Increasing Paid Job Demand

Day 360:

• UBI Income: $y_{\text{UBI}}(360) = 20000 \times 360^{0.31} \times e^{-0.0017 \times 360} \approx 13,898.99 \, \text{tokens}$

• Paid Job Income: $y_{\text{Paid}}(360) = 50000 \times 0.4 = 20,000 \, \text{tokens}$

• Total Income: $I(360) = 13,898.99 + 20,000 = 33,898.99 \, \text{tokens}$

Day 720:

• UBI Income: $y_{\text{UBI}}(720) = 20000 \times 720^{0.31} \times e^{-0.0017 \times 720} \times (1 - 0.8) \approx 2,477.66 \, \text{tokens}$

• Paid Job Income: $y_{\text{Paid}}(720) = 50000 \times 0.8 = 40,000 \, \text{tokens}$

• Total Income: $I(720) = 2,477.66 + 40,000 = 42,477.66 \, \text{tokens}$

Observations

• Income Stability: Despite the decrease in UBI income over time, total income remains stable or increases due to higher compensation from paid jobs.

• Incentive Alignment: Community participants (CPs) are incentivized to participate in paid jobs without experiencing significant income loss during transitions from UBI reliance to paid employment.

---