Barrel Principle

Bond's "Barrel Principle" is inspired by the age-old adage that a barrel's capacity is determined by its shortest stave. In the context of the Bond social network, this principle governs the distribution of the Team Reward. Specifically, within each team formed by user invitation hierarchies, the reward a team receives is determined not by its highest or average contributor, but by its lowest. In essence, the contribution of the least contributing member sets the benchmark for the whole team's reward. This unique mechanism emphasizes the importance of collective contribution and encourages teams to ensure that every member is actively participating. The underlying idea is that for the team to maximize its reward, it needs to ensure that its weakest link is as strong as possible.

To put it in formulaic terms:

Let's define:

• $C_{min}(i)$​ as the contribution of the lowest contributing node in team $i$​.

• $R$​ as the total team reward.

For a specific team $i$​:

• Calculation of Contribution:

$Cmin(i)=Staking Time×Staking Amount​$

Where the Staking Time and Staking Amount belong to the node with the lowest contribution in team $( i )$.

• Calculation of Team's Reward:

The reward for team $i$​, denoted as $R_i$​, is:

$Ri=(Cmin(i)∑jCmin(j))×R​$

Where the sum in the denominator goes over all teams, representing the sum of contributions of the lowest contributing nodes in each team.

So, the Brarrel principle ensures that the reward a team gets is directly proportional to the contribution of its weakest (or least contributing) node. This incentivises each team to ensure that all of its members are contributing effectively, as the team's overall reward is determined by its least contributing member.

Example

1. We have three teams: Team A, Team B, and Team C.

2. Each team comprises three members.

3. For simplicity's sake, we're only looking at the contributions at a static point in time, not considering the staking duration.

Data

Suppose at a particular moment, the contributions of members in each team are as follows:

Team	Member 1	Member 2	Member 3	Lowest Contribution
A	10	5	7	5
B	8	9	6	6
C	11	10	12	10

Total reward pool: 300

Using the Barrel Principle to calculate the rewards:

$Reward_{Team A} = \frac{5}{5+6+10} \times 300$

$Reward_{Team B} = \frac{6}{5+6+10} \times 300$

$Reward_{Team C} = \frac{10}{5+6+10} \times 300$

As evident from the above, Team C, having the highest minimum contribution, receives the most significant reward, while Team A, with the lowest minimum contribution, gets the least.

This method of allocation incentivises each team to elevate its minimum contribution, thereby maximising team rewards.

Advanced Barrel Principle

Building on this, we can make some modifications to the Barrel Principle to ensure not only the minimum contribution within a team (representing the 'weakest link') is considered, but also the overall performance and disparity of wealth within the team.

Let's first standardise each metric, bringing them onto a similar scale. Subsequently, we can assign weights to these metrics for each team to achieve an aggregate score. The specific steps are as follows:

• Calculate Metrics: For every team, compute the following three metrics:

  • Minimum Contribution, denoted as $Min​$

  • Average Contribution, denoted as $Avg​$

  • Variance of Contribution, denoted as $Var​$

• Standardise Metrics: To make metrics comparable, we'll standardise each metric using the following formula: $\text{Standardized Value} = \frac{\text{Value} - \text{Minimum of all teams' values}}{\text{Maximum of all teams' values} - \text{Minimum of all teams' values}}$

• Assign Weights: Allocate weights to each metric, such as:

  • $w_{\text{Min}}$: weight for the minimum contribution

  • $w_{\text{Avg}}$: weight for the average contribution

  • $w_{\text{Var}}$: weight for the variance (Note: A smaller variance is preferable, so we'll subtract it from the final score)

• Compute Aggregate Score: For every team, the aggregate score is: $\text{Score} = w_{\text{Min}} \times \text{Standardized Min} + w_{\text{Avg}} \times \text{Standardized Avg} - w_{\text{Var}} \times \text{Standardized Var}$

• Allocate Rewards: The reward for each team is proportional to its aggregate score.

Sample Weight Allocation

To make sure that teams with less pronounced weaknesses, smaller variance, and high overall contributions are rewarded, we can set the weights as:

• $w_{Min}$=0.5​: Emphasising the team's weakest link.

• $w_{Avg}$=0.4​: Highlighting the team's overall commitment.

• $w_{Var}$=0.1​: Paying lesser attention to the wealth disparity within the team, though it's still factored in.

This approach ensures a focus on both the team's weakest link as well as its overall performance, with some consideration for the internal wealth disparity. Projects can adjust these weights based on actual conditions to reflect your priorities.

Supposing we have the following three teams: Team A, Team B, and Team C. For the sake of simplification, let's set the number of members in each team to five.

Now, let's set the initial contribution data for the three teams as:

Team	Member 1	Member 2	Member 3	Member 4	Member 5	Min	Avg	Var
A	x	7	6	8	7
B	8	9	10	7	8	7	8.4	1.0
C	4	3	5	3	4	3	3.8	0.8
D	8	9	10	7	8	7	8.4	1.0
E	8	9	10	7	8	7	8.4	1.0
F	8	9	10	7	8	7	8.4	1.0

Using the refined 'barrel principle' from the previous discussion, we'll compute the composite score:

1. Standardise the $Min$, $Avg$, and $Var$.

2. Use the weights: $w_{Min}=0.5​$, $w_{Avg}=0.4$​, and $w_{Var}=0.1​$.

We can have a fixed reward pool, say a total of 300. Each team's reward will be in proportion to its composite score.

Now, we will simulate the scenario, that is, increasing the minimum contribution for Team A, and observe the change in rewards, while the data for other teams remains unchanged.

So we have:

In Team A, as the contribution of 'Member 1' progressively increases within the range of 0 to 6, we observe a distinct slope change at a contribution level of 3. This shift isn't coincidental but closely relates to the relationship between Team A's minimum contribution and the overall minimum contribution amongst all teams.

Initially, Team A's minimum contribution is determined by 'Member 1', while the minimum contributions of the other teams remain unchanged during this period. However, as the contribution of 'Member 1' rises, when it reaches 3, both Team A and Team C have a minimum contribution of 3. Subsequently, as Team A's minimum contribution continues to increase, it surpasses that of Team C, meaning it's no longer the lowest contribution overall.

Thus, when Team A's "weakest link" is no longer the "weakest link" across all teams, we witness a significant shift in the standardised minimum value, which directly impacts Team A's overall score and reward. This "inflection point" marks a pivotal moment, emphasising that when a team's weakest link improves, surpassing other teams, the rate of reward growth for that team accelerates.

This mechanism, in essence, offers a potent incentive to users. As their team rises from being the overall weakest link, their efforts get substantially more rewarding. This ingenious design of the advanced barrel principle algorithm aims to motivate users to participate more actively and elevate their contributions.

Note: Standardisation is a relative process. When one team's metric surpasses those of other teams, its impact on the overall assessment shifts, subsequently affecting the scores and final rewards of all teams.

Comparison of Original and Advanced Barrel Principle

For comparison, we draw the reward curve calculated by the original barrel principle in the same scenario in the same coordinate system.

Let's dissect both methods one by one and then compare their merits and shortcomings.

1. Original Barrel Principle (Min-Based):

   • Reward Mechanism: This system solely zeroes in on the lowest contribution within the team. As this minimum contribution escalates, so does the team's reward.

   • Pros

     • Straightforward and easily grasped.

     • It underscores the team's collective spirit, suggesting a team's collective performance should be dictated solely by its weakest link.

   • Cons

     • It could overlook the contributions of other team members.

     • It doesn't encourage elevating the team's overall performance, just focusing on enhancing the weakest link.

2. Advanced Barrel Principle (Min, Avg, Var-based):

   • Reward Mechanism: It takes into account the team's minimum and average contributions, along with the variance. These three indicators together determine the team's reward.

   • Pros

     • More intricate, capturing various facets of the team's performance.

     • Motivates the team to improve in multiple areas, including elevating overall performance and minimising disparities among members.

   • Cons

     • Relatively complex, necessitating more data and computations.

Note:

1. Projects can opt for which barrel principle algorithm to employ based on their operational realities.

2. The reason the rewards from the Advanced Barrel Principle are lower than the Original algorithm is due to the weightage assigned to Min, Avg, and Var. To boost the rewards procured from the Advanced Barrel Principle, one could consider tweaking the weightage of the three parameters. For instance, when the weightage for Min, Avg, and Var is set at 0.4, 0.5, and 0.1 respectively, we have: