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STOCHASTIC PROPERTIES OF EIP-1559 BASEFEES
A PREPRINT
Ian C. Moore, PhD
ic3moore@gmail.com
Jagdeep Sidhu, MSc
Syscoin Core Developer
Blockchain Foundry Inc.
jsidhu@blockchainfoundry.co
May 11, 2021
ABS TRAC T
EIP-1559 is a new proposed pricing mechanism for the Ethereum protocol developed to bring stability
to fluctuating gas prices. To properly understand this as a stochastic process, it is necessary to develop
the mathematical foundations to understand under what conditions the base fee gas price outcomes
behave as a stationary process, and when it does not. Understanding these mathematical fundamentals
is critical to properly engineering a stable system.
Keywords EIP 1559 ·Base Fees ·Stochastic Processes ·Stationarity
1 Introduction
The current Ethereum pricing mechanism employs the price auction model where high-value use cases are prioritized
over lower ones. The problem with this approach is that as Ethereum has grown more popular, users have difficulty
estimating optimal gas fees. In 2018, EIP-1559 was proposed by Ethereum Founder, Vitalik Buterin, as a major change
to Ethereum’s transaction fee mechanism, and introduces two different types of fees, which is the base fee and inclusion
fee [
1
]. The ideas is the base fee is relatively stable while the inclusion fee serves as an additional tip offered to
compensate miners. This mechanism allows for the base fee to be adjusted after every block in accordance to demand.
Hence, when demand for gas is higher than the target gas price, the block size is adjusted upwards, and when demand
for gas is lower than the target gas, the block size is adjusted downwards. Blocks are allowed to grow as large as double
the target block size.
In this study, we investigate the stochastic properties of EIP-1559 base fees. As the main motivation behind EIP-1559
is to bring stability to gas fees, we look at the mathematical conditions that this mechanism must have to achieve
stationarity. We open up this discussion in Section 2 where we go over the EIP 1559 pricing mechanism. Using these
simulated gas demands, we analyze the stationary properties in Section 3. In Section 4 we simulate the gas demand.
Finally, we close off with our conclusions in Section 6.
2 EIP 1559 Pricing Mechanism
The basic premise of the new EIP 1559 proposal begins with setting the base fee which increases when the network
capacity exceeds the target per-block gas usage, and decreases when the capacity is below the target. The calculation of
the updated basefee, bkat the kth block, is as follows:
bk=bk−1fk(1)
fk= 1 + δk
c(2)
arXiv:2105.03521v1 [cs.GT] 7 May 2021
EIP-1559 Basefees A PREPRINT
where
δk=demandk−T argetGasF ee
and
c=T arget / BaseF eeM axC hange
; for the purpose of this study, we
assume
δk
to be a random stationary process. However before we do that, we run the abm1559 simulator [
2
] to show
that this is the case, under the assumption that new users are Poisson distributed.
Constant Value
Target Gas Fee 12,500,000
Base Fee Max Change 50
Initial Basefee 10,000,000,000
Table 1: EIP 1559 constants
3 Analyze Stationarity
Definition:
A time series
Xt
is called strictly stationary if the random vectors
(X1, ..., XN)T
and
(X1+τ, ..., XN+τ)T
have the same joint distribution for all sets of indices t1, ..., tnand for all integers τand n > 0. It is written as
(X1, ..., XN)TD
= (X1+τ, ..., XN+τ)T,(3)
where D
=means equal in distribution.
3.1 Augmented Dickey Fuller Test
Given the following higher-order autoregressive processes:
δyt=αyt−1+θ1δYt−1+... +θpδYt−p+t,(4)
the Augmented Dickey Fuller (ADF) test checks the existence of a unit root
α= 1
(ie, H0:
α= 1
) where the null
hypothesis is non-stationary. The unit root in (5) is characteristic of a time series that makes it non-stationary.
4 Gas Demand Simulations
To simulate gas demand
δk
in (2) we use the abm1559 simulator [
2
] where the following assumptions were made: (a)
users are Poisson distributed; and (b) every TX consumes 21000 Gas. For our analysis, we are interested in simulating a
year’s worth of gas costs, which is infeasible as running the full Ethereum chain simulator consumes a lot of overhead.
Hence, we used the simulator to generate a sample, which was analysed using Exploratory Data Analysis (EDA)
techniques, as shown in Figure 2. As we can see the gas demand is behaving as a normal random sample.
We also applied the ADF test to the simulated gas demands, shown in top image of Figure 1, and found it to be
statistically significant (ie, p-val = 2.33e-29). Hence, rejecting HO, which indicate simulated gas demands to be
stationary.
5 Gas Fee Simulations
In Section 4 we simulated gas demand
δk
in (2) and found them to be stationary and normally distributed with
µ
=
13641075, and variance
σ2
= 550980. To simulate gas prices using EIP 1559 we used these parameter estimates to
model a year’s worth of gas demands and fed it through the system described in (1) and (2). A series of these simulations
can be seen in Figure 3.
We tested gas demands for stationarity. In Table 3 we fail to reject to H0 for all ten tests indicating insufficient evidence
to conclude that the effect of stationarity exists in the gas fee simulations.
5.1 Problem Statement
For sake of comparison, let’s look at stationarity for an AR(1) process. Consider a standard first-order auto-regressive
process defined by the recursive equation:
xt=µ+α(xt−1−µ) + t−1t∼IID N (0, σ2)(5)
2
EIP-1559 Basefees A PREPRINT
Figure 1: Simulations of; (top) gas demands using random samples from a normal distribution; and (bottom) gas costs
using (1). As we can see, the gas cost simulations (bottom) are behaving like a random walk process
Figure 2: Testing normality of gas demands (top image of Fig. 1) using; (left) histogram with normal distribution fit of
demands; and (right) Normal Q-Q plot indicates sample and theoretical quantiles match
Figure 3: Gas cost simulations using EIP 1559; we can visual see the non-stationary behaviour in price over time, which
is indicative of a random walk.
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EIP-1559 Basefees A PREPRINT
Simulation ADF Statistic p-value Outcome
1 -1.506 0.531 not stationary
2 -1.260 0.647 not stationary
3 -2.170 0.217 not stationary
4 -2.432 0.133 not stationary
5 -1.844 0.359 not stationary
6 -1.323 0.618 not stationary
7 -1.645 0.459 not stationary
8 -1.662 0.451 not stationary
9 -1.124 0.705 not stationary
10 -1.930 0.318 not stationary
Table 2: ADF test on basefee simulations from Figure 3 to test null hypothesis (H0) that a unit root is present; when
unit root is present, then sample is considered to be non-stationary. As we can see, we fail to reject H0 for all ten tests
indicating insufficient evidence to conclude that the effect of stationarity exists.
The above process is stationary when
|α|
<1 so that this has a single root 1/
α
outside the unit circle. It can be shown that
if σ>0, then this process has the asymptotic stationary distribution:
x∞∼N(µ, σ2
1−α2)(6)
Considering the (state space like) simularities to the AR(1) process to the system of the system of (1) and (2); the
question is what are the constraints (if any) that time varying process
δt
must meet to ensure stationary outcomes for
the basefees.
5.2 Random Coefficient Autoregressive Models
Considering the problem statement in the previous section, we look to another relative to the AR(1) process, namely the
Random Coefficient Autoregressive of order 1, or RCA(1) process. The RCA(1) is similar to the AR(1) process in the
sense that the parameter
α
is allowed to vary with time. This is quite similar to the EIP 1559 system of (1) and (2). The
RCA(1) model is given by:
xk=α+βkxk−1+k,(7)
where
k
is an independent sequence of random variables with 0 mean and variance
σ2>0
;
βk
is an independent
sequence of random variables with mean
µβ
and variance
σ2
β
of the random coefficient
βk
, and variance
σ2
of the error
k
. Given (3) and (4) it is obvious that when
σ2
β= 0
, the RCA(1) in (5) becomes a AR series, and becomes a random
walk when µβ= 1 and σ2
β= 0, hence non-stationary.
EIP 1559 (1) fits into the framework of a Random Coefficient Autoregressive (RCA) model when
is negligibly small.
This is good news, as it allows us to apply the body of work on RCA processes to this problem. The next question is;
under what conditions does an RCA(1) process exhibit stationary behaviour. If we know this, then we can apply this
understanding to the EIP 1559 mechanism.
Theorem
Consider RCA(1) model (5) with
βk, k
, which is identically normally distributed. Then the sufficient
condition for the existence of a strictly stationary and ergodic solution is that:
ln(σ2
β)< ς + ln(2) −2Z1
0
1−exp[−λ(1 −w2)]
1−w2dw, (8)
where ς≈0.57721, denotes Euler’s constant and λ=µ2
β/2σ2
β
Proof: See Theorem 2 in [7]
6 Conclusion
The purpose of this study was to understand under what conditions the EIP 1559 pricing mechanism would behave as a
stationary process. We believe this as a critical first step to understanding the constraints to engineer stable system.
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EIP-1559 Basefees A PREPRINT
Figure 4: Region of strict stationarity and ergodicity for RCA(1) determined via (8). The annotated red dot represents
where our setup is in relation to the region where an RCA(1) process behaves as a strictly stationary process. Based on
this we can see that our setup was non-stationary.
Given the enormity of the Ethereum project, understanding these fundamentals are needed for setting up simulation
experiments to help capture all the edge cases that may arise so that the number of updates and patches are minimized
prior to release.
It was determined that under the parameter setting used in this study that the outcomes behaved as Brownian motion,
which is a non-stationary process. The RCA(1) was used as the mathematical framework to help understand this, as we
can go to the literature to help guide our understanding of this kind of process. This conclusion was based on a series of
simulations which we tested using the ADF, and Wang’s Theorem indicating the region of stationarity for an RCA(1)
process.
Ideally, we would like EIP-1559 to behave as a stationary process. Hence, more investigation is required to understand
the setup that is necessary to achieve these conditions before proper simulation studies are implemented.
References
[1]
V. Buterin, Blockchain Resource Pricing, Apr 2019. Accessed on: May 2021. [Online]. Available:
https://github.com/ethereum/research/blob/master/papers/pricing/ethpricing.pdf
[2]
B. Monnot, Agent-based Simulation Environment for EIP 1559, Accessed on: Apr 2021. [Online]. Available:
https://github.com/barnabemonnot/abm1559
[3]
T. Roughgarden, Transaction Fee Mechanism Design for the Ethereum Blockchain: An Economic Analysis of
EIP-1559, Dec 2020. Accessed on: Apr 2021. [Online]. Available: https://arxiv.org/abs/2012.00854
[4]
L. Trapani, Testing for Strict Stationarity in a Random Coefficient Autoregressive Model, Jan 2019. Accessed on:
Apr 2021. [Online]. Available: https://arxiv.org/abs/1901.01077
[5] C. Chatfield, The Analysis of Time Series: An Introduction. 4th Edition, Chapman and Hall, New York, 1989
[6] D. Nicholls and B. Quinn , Random Coefficient Autoregressive Models: An Introduction, Springer-Verlag, 1982
[7]
D. Wang, Frequentist and Bayesian Analysis of Random Coefficient Autoregressive Models, North Carolina State
University PhD Dissertation, 2003
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