SAR Academy

Please analyze the following mathematical system neutrally.

Assume there are 9,120 SAR units. For this mathematical example, assume each unit belongs to a different participant.

For comparison with Direct SAR, one direct participation costs $1.10 in total:
- $1.00 is the SAR principal unit that enters the rotation
- $0.10 is the Real SAR referral/fallback component and is outside the $1.00 rotation payout calculation

Every day, all 9,120 units are randomly reshuffled into positions in one binary tree.

After the shuffle, each $1.00 unit distributes value upward through its parent chain:

- ancestor level 1 receives 8%
- ancestor level 2 receives 8%
- ancestor level 3 receives 8%
- ancestor level 4 receives 8%
- ancestor level 5 receives 8%
- ancestor level 6 receives 8%
- ancestor level 7 receives 8%
- ancestor level 8 receives 8%
- ancestor level 9 receives 8%
- ancestor level 10 receives 8%
- ancestor level 11 receives 8%
- ancestor level 12 receives 12%

The maximum distribution is therefore 100% of the $1.00 unit.

If a unit has fewer than 12 ancestors above it, calculate only the distributions to ancestors that actually exist.

Any portion for which no ancestor exists is unallocated in this mathematical model.

Do not assume that this unallocated portion is returned to the participant, redistributed to other positions, or paid to anyone else.

Assume every participant has exactly one unit so that each participant receives one random tree position.

Please calculate:

A) What percentage of participants will receive any payout after one random shuffle?

B) What is the probability that one specific participant receives more than the full $1.10 Direct SAR participation cost in SAR payouts after one shuffle?

C) Show the different possible payout amounts according to tree position.

D) What is the expected average payout per participant after one shuffle?

E) Based only on the mathematics and the fact that every participant is randomly reshuffled each day, how fair would you rate this mechanism from 1 to 10?

Show your assumptions, formulas and calculations.

Do not treat this as an investment recommendation, income promise or guaranteed return. Analyze only the mathematical mechanism described above.

Now analyze exactly the same system with 100,000 SAR units.

For this mathematical example, assume each unit belongs to a different participant.

For comparison with Direct SAR, one direct participation costs $1.10 in total:
- $1.00 is the SAR principal unit that enters the rotation
- $0.10 is the Real SAR referral/fallback component and is outside the $1.00 rotation payout calculation.

All units are randomly reshuffled every day into one binary tree.

Each $1.00 unit distributes upward to its ancestors:

levels 1 through 11: 8% per level
level 12: 12%

Only existing ancestors receive a distribution.

Any portion for which no ancestor exists is unallocated in this mathematical model.

Do not assume that this unallocated portion is returned to the participant, redistributed to other positions, or paid to anyone else.

Please calculate:

A) The probability that one randomly selected participant receives any payout after one shuffle.

B) The probability that this participant receives more than the full $1.10 Direct SAR participation cost in SAR payouts.

C) The approximate percentage of participants who receive:
- $0
- between $0.01 and $1.10
- more than $1.10

D) The expected payout for one randomly selected participant.

E) Compare these results with a system containing 9,120 SAR units. Does increasing the number of units materially change an individual participant's probability, or mainly increase the size and depth of the tree?

F) Rate the mathematical fairness from 1 to 10 if all positions are reassigned randomly every day.

Explain the mathematics step by step and do not assume guaranteed earnings.

Analyze the mathematical fairness of two position systems.

Assume N participants, each with exactly one $1.00 SAR unit.

The payout tree and payout percentages are identical in both systems.

SYSTEM A — FIXED POSITIONS
Each participant receives one tree position and keeps that same position every day.

SYSTEM B — DAILY RANDOM RESHUFFLE
Every day all N units are uniformly and independently reshuffled into all N tree positions.

For each $1.00 source unit:
- ancestor levels 1–11 receive 8% each
- ancestor level 12 receives 12%
- only ancestors that actually exist receive a payout
- any undistributable portion is classified as unallocated
- do not assume that unallocated value is returned to a participant or redistributed

Please analyze:

A) Before each day begins, does every participant in System B have exactly the same probability of receiving every tree position?

B) Compare cumulative payout variance after 30, 100 and 365 days.

For System A, remember that a participant's daily payout is fixed by their permanent position.

For System B, assume each daily shuffle is statistically independent.

Show how the variance of cumulative outcomes grows in each system.

C) Does daily reshuffling eliminate permanent structural position advantage?

D) Does daily reshuffling guarantee equal earnings?
Clearly distinguish:
- equal opportunity
- equal expected value
- equal realized outcome

E) For N = 9,120, calculate the probability that one specific participant experiences at least once within:
- 30 days
- 100 days
- 365 days

each of the following:
1. the root position
2. any position receiving a payout
3. any position receiving more than the $1.10 total participation cost

F) Compare procedural fairness between:
- one permanently fixed position
- a fresh uniformly random position every day

Do not rate fairness based on whether payouts are equal.
Give separate scores for:
1. equality of opportunity
2. equality of realized outcomes

G) Explain whether repeated reshuffling reduces long-term positional inequality even though rare high-paying positions may still not be experienced by most participants during a finite period.

Use formulas and numerical results.
Do not make an investment recommendation or promise future earnings.