# What is the fairest solution/formula for rewarding points in a hierarchical network?

Introduction

The nature of this hierarchical network is based on the concept of Multi-Level Marketing strategy.

Example 1 - Unfair Situation

Ancestor receives 1 point for every descendant present in its network.

• Parent (3 points)
• Child A (2 points)
• Grandchild A
• Grandchild B

Parent will always receive more points than each of its descendant >:(

So what I have in mind is decreasing the point each descendant gives as the level increases.

Example 2.1 - Fairer Situation

Formula: 1/0.5^(level-1)

Parent is level 0, child is level 1, grandchild is level 2 and so on...

Each child gives 1 point and each grandchild gives 0.5 point.

• Parent (2 points)
• Child A (2 points)
• Grandchild A
• Grandchild B

Example 2.2

When Child A invites 1 more person.

• Parent (2.5 points)
• Child A (3 points)
• Grandchild A
• Grandchild B
• Grandchild C

This is fairer to Child A who had invited 3 person than Parent who had only invited 1 person, while still giving some cuts to the Parent.

Update 1 - 12/11/2012 11:16AM

In the event a ranking element is involved

From Example 1, the Parent only needs to invite just one person and its descendants will be doing all the "dirty work" for it. The Parent will always rank first no matter how hard/many its descendant invites. This is unfair to the descendants.

From Example 2, even though the Parent is always receiving cuts from its descendants, it is possible for the descendants to score and rank higher than the Parent. This is fairer to the descendants.

Requirements

• Parent receives 1 point for every child that he/she has invited.

• Ancestors(Parent, Grandparent, Great Grandparent...) receive reduced points for every descendants(Child, Grandchild, Great Grandchild...) present in their network.

• Challenge: Any formula suggested must be able to prove that it indeed will guarantee fairness throughout the network.

Question: I'm looking for theory/study/statistical record to justify the reason why I use a certain formula instead of other formula (e.g 1/level). Is there any existing solution/formula that fulfills these requirements?

P.S. It does not have to be the formula given in Example 2, it was just an example.

Any help would be appreciated, thank you! :)

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Define fairness. – Hagen von Eitzen Nov 11 '12 at 18:30
@HagenvonEitzen I have added more explanation to my question. – Zany Nov 12 '12 at 3:20
What if you just count the immediate children and ignore further descendants? – Rahul Nov 12 '12 at 6:10
Yes, since feeding on the dirty work of the descendants is considered unfair, only the count of immediate children is fair. At least as long as we lack a quantifyable definition of fairness. Or why would we want to reward descendants in the first place? – Hagen von Eitzen Nov 12 '12 at 7:19
@RahulNarain & Hagen von Eitzen: My bad, I forgot to mention that the nature of this hierarchical network is based on the concept of Multi-Level Marketing strategy. Ancestors must receive a cut for every new people in its network. – Zany Nov 12 '12 at 9:01

The formula you used in examples 2.1 and 2.2 seems to do what you want. Whether you use $.5$ or some other number will depend upon what the creator of this multi-level marketing scheme thinks is fair. Supposing this is an actual multi-level marketing scheme the creator of the scheme might want $$\mathbf{RetailPriceOfGoods} - \mathbf{CostOfGoods} \geq \frac{1}{1 -c}P$$ where $P$ is the payment the actual seller gets. The factor $\frac{1}{1 - c}$ occurs because $${\sum}^{\infty}_{i = 0}c^{i} = \frac{1}{1 - c}.$$ Since you do not have a maximum number of generations the sum goes to infinity. The nice thing about this formula is that the payments depend only relative position in the hierarchy.
Rather than discuss "fairness" the remainder of this answer will focus on likely behavior of the agents. I will suppose that the agents wish to maximize the number of points their have. If this is not the case then the formulas do not matter. If the value of $c$ is small then the parent will get negligible points from grand children and later generations. But then the parent can be expected to expend the most effort recruiting new children. If the value of $c$ is large then it may make sense for the parent to help children and later generations to become better at recruiting their own children. This leads to cooperation between generations.