Alchemy of A Pyraimd Scheme: Transmutating Business Opportunity Into A Negative Sum Wealth Transfer
Alchemy of A Pyraimd Scheme: Transmutating Business Opportunity Into A Negative Sum Wealth Transfer
Alchemy of A Pyraimd Scheme: Transmutating Business Opportunity Into A Negative Sum Wealth Transfer
December 3, 2019
∗ The analyses and conclusions set forth are those of the authors and do not
necessarily reflect the views of the Commission, or any individual Commissioner.
3 Model
Consider a risk neutral firm and N risk neutral agents organized in a fixed
and exogenous social graph. Consistent with the stylized structure of an
MLM, we consider an inverted tree emanating from the firm, where each
agent is a node with direct links to M unique downstream agents (except
K
M k = N . For a tree
P
end nodes). There are K levels in the tree, such that
k=1
with M k nodes at each level and K levels, there are M K end-nodes. Each
agent in level k faces the same incentives, so we generically refer to all agents
in level k as agent k.
Contact to any node in the graph can only be activated through the
previous direct link in the tree, so that the firm can only link to the first
level agents, a first level agent can only link to her M second level nodes,
and so on. If a first level agent bridges a link to the second level, then the firm
can observe and extend the contract to the second level, and so on through
all K levels and all N agents. Attempted recruitment, i.e. bridging the direct
link, is costly at a constant amount per contact, γ > 0. Bridging indirect
links — including for the firm — is assumed to have an infinite cost.11
The firm offers contacted agents a fixed, uniform contract. We incorporate
a number of stylized observations about typical MLM contracts into this
10
See Lafontaine and Slade (2002) for an discussion of the franchising problem.
Economists often describe the principal-agent problem in a moral hazard framework,
where the principal does not observe the agent’s effort but the observable outcome de-
pends on the agent effort and some noise such as local demand fluctuation (see reviews
in Prendergast (March 1999) and Mookherjee (2006)). Franchising is often thought as a
two-sided moral hazard problem, adding the principal’s effort (in brand-wide marketing)
as the unobservable on the agent’s side.
11
This structure also assumes that the cost of recruiting more than M participants is
prohibitive.
10
11
3.1 Beliefs
In the context of a real world MLM, agents would have limited, and poten-
tially biased, information about how participation in the firm’s offer would
affect them. For simplicity, we limit the model to two sources of uncertainty:
demand uncertainty and recruiting uncertainty. The first of these is common
to most business opportunities where entrepreneurs do not necessarily know
ex ante the demand for the products they wish to sell. The second, as its
name suggests, is unique to recruiting-based business opportunities where
the existence of available recruits may not be known ex ante.
In terms of the recruitment uncertainty, at best, agents might know ex
ante the number of their potential contacts M , but they would not typi-
cally know their place in the network, k relative to K, or overall network
parameters N or K (either of which would be sufficient to determine the
other). Because of this, participants typically would not have enough infor-
mation to calculate the probability of being an end-node or not, which is
a crucial value for them to be able to evaluate their expected profit from
participation. Even given this information, typical participants may not be
able to correctly calculate the probability of success either due to innumer-
acy or incorrect information or deceptive practices. Recent empirical work
by Bosley et al. (2018), and the consistent presence of deception in schemes
that have been found to be pyramids, suggests a likelihood of inaccurate
beliefs by participants. We examine the effect of incorrect beliefs about the
availability of recruits on the firm’s offer, and agent outcomes. Given the
uncertainty, beliefs about success are plausibly k-independent and therefore
may be homogeneous across all participants, as we assume here to simplify.
Participants have exogenous ex ante beliefs µ̂ about the probability that
12
We use retail loosely here to encompass both discount purchases for self consumption
and selling to an outside consumer
12
N − MK
µ= .
N
As we lay out below, we will be simplifying the model by examining the
case where the population size goes to infinity. We do this by fixing M , and
letting K go to infinity. In that case, for a fixed M we define µ as the limit:
N − MK 1
µ = lim = . (1)
K→∞ N M
N
To see this, note that N − M K can be written as M
− 1.13 Substituting this
in to µ yields µ ≡ NN−M
M
, and simplifying yields:
1 1
µ= − . (2)
M N
As K → ∞, N goes to infinity and the final term goes to zero.
Agents also face demand uncertainty. Consumption value for the firm’s
product, w, (through own consumption or through retail sale) is unknown
by agents prior to purchase, and is drawn from {0, v} where v > 0 and the
probability of v is δ ∈ [0, 1].14 We assume that agents have homogenous,
exogenous prior beliefs about δ equal to δ̂. Given these beliefs, we assume
agents are risk neutral and have expected value for the product Ew = δ̂v.
We further characterize uninformed beliefs in two categories relative to
accurate beliefs. We call an agent’s belief about the recruiting opportunity
“optimistic” if they believe the probability that they are not an end node
13
PK−1 N
Add and subtract 1 from N − M K . The terms N − M − K + 1 = k=0 M k = M .
N
Returning the subtracted 1 yields M − 1.
14
Enforcement concerns have highlighted differences between consumption that is inter-
nal to the distribution network, and external to that network. The basic function of the
chained recruiting mechanism is not dependent on that distinction, therefore we abstract
away from that here.
13
1 M
πF (p, λ) = p(1 − λ) + (λp − γ). (4)
N N
This formulation highlights the fact that the firm’s profit per participant is
the product price, net of the reward needed to incentivize agents to recruit
an agent to pay that price, plus an adjustment term to account for the fact
that the firm does not need to compensate itself with λp for recruiting its M
15
The homogenous beliefs presented here allow us to examine the overall mechanism
and expected outcomes in a relatively simple model. The presence of heterogeneous beliefs
seems likely to allow some more informed participants to join the firm in profiting from
inaccurate beliefs. In order to maintain the simple model, we leave this to future work.
14
15
16
Per participant firm profit is the firm’s direct revenue from its distributors,
net of recruiting rewards for all successful recruiting. When the firm’s price
is greater than the expected value of the product, recruiting rewards must
include an upward adjustment to compensate for expected retail losses in
order to keep all recruiters indifferent between participating and not.
From Equations 11, when the expected retail margin is constrained to
be weakly positive — p ≤ δ̂v — the firm’s per participant profit is strictly
increasing with respect to direct price and the firm will charge the highest
price it can under that constraint, that is, p = δ̂v.
In the absence of such a constraint, the firm’s willingness to charge more
than δ̂v is a function of how much it will need to reward participants to
overcome their belief that they may be unsuccessful in recruiting, and to
therefore balance against their expected loss from unprofitable retail. For a
given belief µ̂, the marginal profit associated with increasing p when p > δ̂v
is:
17
4 Analysis
In this section, we draw a distinction between demand loss that is typically a
risk in business opportunities, and transfer loss, which is a risk specific to the
MLM mechanism. We examine more closely the firm’s offer to agents with
optimistic beliefs. We contextualize this by also showing the results when
agents are pessimistic.
18
Equation 13 illustrates the typical demand loss that entrepreneurs might face
in any business opportunity. In this context, the first term is participant
profit stemming from deviation of their information about the probability of
positive demand from true. Because participants may over- or under-estimate
the value of the product, they may be induced to over-pay, or induce the firm
to accept a lower wholesale price, relative to the true expected value.
The second term is participant profit associated with an incorrect belief
about the probability of recruiting (still with the perception of weakly prof-
itable retail sales). Participants may over or underestimate the probability
of finding a recruit, which in this case is synonymous with whether demand
can be increased by M . Both terms are related to inaccurate beliefs about
an ultimately exogenous issue: whether or not demand exists (from µ̂), and
whether or not the value is positive (from δ̂) if it does exist. Optimism
about either δ or µ leads to expected losses by the participant, as they then
would be, respectively, overcharged for the product and under-compensated
for recruiting new sellers. The effect is ambiguous when beliefs over the ex-
pected value of the good and the recruiting opportunity are mixed between
pessimistic and optimistic.
Based on 13, we use the following definition.
Actual ex ante recruiting profit for participants when the firm sets the
price above retail value — i.e., when the firm offers an unprofitable retail
19
20
21
Proposition 3. When agents are optimistic and the firm is profit maximiz-
ing, average and thus total agent profit from participation in the mechanism
is negative. The firm will profitably induce losses on participants as a group
in two ways:
22
23
This isolates the transfer loss that is captured by other participants and the
firm. It does not include demand loss from optimistic beliefs about demand,
or dissipated recruitment costs.
Each of the successful participants pay in I, and after recruiting their M
contacts, get µ̂1O I in return. Their average net positive transfer is:
O 1
Tk<K (µ̂ ) = − 1 µI > 0. (21)
µ̂O
If transfers the firm induced with accurate beliefs, the total payments to
the successful participants would match the total payments from the unsuc-
cessful ones — TK = Tk<K = (1 − µ)I. There would be no transfer loss,
as shown above, because the risk would exactly be compensated by the re-
ward. However, the optimism of participants allows the firm to profit from
inducing transfers by under-compensating, under the presumption of homo-
geneous ex ante beliefs, to the successful participants. The firm can then
We are changing notation slightly, to define I as I = I˜ − δ̂v, where I˜ is the arbitrary
19
24
5 Discussion
Our model highlights the importance of participant beliefs in an MLMs profit-
maximization problem. A key result of the analysis is that even when a
product could be marketed through a network of distributors efficiently (i.e.
delivering consumption value in excess of all production and distribution
costs), optimism on the part of participants may allow the MLM to set
prices and rewards such that the average return across participants is neg-
ative. In part, this result is the typical story of one party in a transaction
exploiting the systematic biases of the other for profit. However, because
the recruiting mechanism sets up a situation where the payment extracted
from a participant, p, is proportional to the rewards she could earn if she
successfully recruits, M λp, participants will always accept an increase in p
and corresponding increase in rewards if they agree to participate in the first
place. The operator of the MLM will find it profitable to increase p when
participants are optimistic about the odds of successfully recruiting because
the expected payments out for the operator are less than the payments in.
In other words, optimism on the part of participants allows the operator of
the MLM to induce a negative sum transfer mechanism where a portion of
the total transfer payments in are paid to recruiters in the form of rewards,
25
26
27
28
29
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