As discussed in Integrity in the Rewards Model, the economic model for DoubleZero rewards has two potential vulnerabilities, one of which is the inorganic traffic problem. Network contributors are potentially incentivized to send inorganic traffic over lines they operate, as the boost in rewards they receive outweighs the cost of sending traffic.
Conceptually, this is because there is a disconnect between fees and rewards. As discussed in Rewards to Network Contributors, the details of connectivity are abstracted away to the end user; but those connectivity details matter critically for rewards. Thus, a network contributor could send traffic over a long or high-throughput line they operate, which costs a fixed amount but earns them an outsized share of rewards.
This document explains how exactly to calculate the proportion of rewards that should be burned, to dissuade a network operator from sending inorganic traffic. This establishes the economic floor (that lower bounds the eventual burn rate); and in doing so, this restores economic integrity to the network.
Consider the following setting: a two-line three-city network, with Chicago, New York, and London. The Chicago - New York line offers an improvement of 8ms over the public internet; and the London - New York lines offers an improvement of 24ms over the public internet. There is 1 Gb of actual organic traffic between Chicago and New York, but there is no organic traffic between London and New York. For simplicity, suppose each gigabit of traffic pays exactly $1 in fees.
Under the proposal in Rewards to Network Contributors, the operator of the Chicago line gets 100% of the $1 fees, while the operator of the London line gets nothing (as he facilitates no traffic).
But the operator of the London line would be tempted to send inorganic traffic — say, 1 Gb — over his line. He pays $1 to do so, such that the collected fees for DoubleZero grow to $2. Critically, though, the rewards methodology would award the operator of the London line 75% of the value (as his line improved latency by three times as much as the operator of the Chicago line, for the same amount of traffic). The deviation is profitable: the London operator collects $1.50 in revenue against $1 in costs. Note that the Chicago operator has no such strategy available to him. His costs will exceed his increased revenue take, because he operates the shorter line.
This would lead to abuse of the DoubleZero system, unless a portion of the rewards were burned. Indeed, if 33% of the fees were removed here (i.e. $0.67 of the collected $2), then this particular deviation stops being profitable. The London operator would collect $1 of the remaining $1.33 in fees, which do not exceed his costs and so net him zero profit.
The optimal burn rate is the rate needed to dissuade any network contributor from profiting from sending inorganic traffic. More precisely, the rate is computed at the end of an epoch, when the total traffic over the network is observable and finalized — and so it is the rate needed to have made inorganic traffic unprofitable in retrospect. While the rate is calculated after the epoch completes, network contributors who understand the methodology ex-ante will not send inorganic traffic knowing it will be made uneconomic for them ex-post.
Formally, at the conclusion of each epoch, the DoubleZero system observes a set of traffic between locations. Any stream of observed traffic t can be split into o organic traffic and f inorganic (fake) traffic. For any split and from the perspective of any network contributor, a burn rate can be computed wherein it would be unprofitable for the network contributor to have sent inorganic traffic. The maximum rate across all splits and all contributors is the safest rate for the system, and preserves the equilibrium whereby network contributors do not send inorganic traffic during the epoch.
That condition can be written mathematically below for a single network contributor k. To introduce the notation in the condition, over n traffic types (e.g. origin and destination city, priority level, etc), there is organic and inorganic demand (indexed by i) that sum to the observed traffic. For simplicity, assume each transaction costs c to send, although this can be easily generalized to different fee tiers of traffic. The network contributor earns a share of all fees collected (both organic and inorganic), where the share function is S indexed by the contributor-specific k, i.e. the methodology in Rewards to Network Contributors. The collected fees are simply the sum of all traffic, multiplied by the fee c and net of the burn parameter $\tau$. Finally, the network contributor pays the fees for any inorganic traffic it sends. Thus, in the words, the condition is that the left-hand side — the share of total fees, net of the burn rate, and minus the costs of inorganic traffic — is weakly dominated by the right-hand side, i.e. the share of fees stemming from organic traffic only and net of the burn rate.
$$
S_k\left(o_1 + f_1, ..., o_n + f_n\right) \left(c\sum o_i + c \sum f_i\right) (1 - \tau) - c \sum f_i \le S_k\left(o_1, ..., o_n\right) \left(c \sum o_i\right) (1 - \tau)~~~\forall~o_i + f_i = t_i, k $$
To make this more readable and practical, I isolate the parameter $\tau$. I also denote o, f, and t as vectors of organic, inorganic, and total traffic respectively (and use i as a vector of 1’s when needed for summations). I finally let the share function S take in the vectors as arguments.
$$ \tau = \sup_{k, f \in [\vec{0}, t]} \left\{ 1 - \frac{f'i}{S_k(t)[t'i] - S_k(t-f)[(t-f)'i]} \right\} $$
This equation tells us precisely the burn parameter needed to restore economic integrity: it must be the maximum value across any breakdown of total traffic into organic and inorganic traffic and for any network contributor, such that the inorganic traffic weakly lowers profits to that contributor.
Finally, this procedure is more efficient by being performed at the completion of the epoch rather than throughput the epoch. The DoubleZero system has the benefit of full information in observing total traffic demand. Holding rewards through the epoch will be less efficient, as the system does not know the final value of total demand and so must remove rewards at more aggressive rates to safeguard against a wider range of scenarios.