# Transaction Fees on a Honeymoon: Ethereum’s EIP-1559 One Month Later

Daniël Reijsbergen1, Shyam Sridhar2, Barnabé Monnot2, Stefanos Leonardos1,
Stratis Skoulakis1, Georgios Piliouras1 1Singapore University of Technology
and Design (SUTD), 2Ethereum Foundation
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Ethereum Improvement Proposal (EIP) 1559 was recently implemented to transform
Ethereum’s transaction fee market. EIP-1559 utilizes an algorithmic update
rule with a constant learning rate to estimate a _base fee_. The base fee
reflects prevailing network conditions and hence provides a more reliable
oracle for current gas prices.

Using on-chain data from the period after its launch, we evaluate the impact
of EIP-1559 on the user experience and market performance. Our empirical
findings suggest that although EIP-1559 achieves its goals _on average_ ,
short-term behavior is marked by intense, chaotic oscillations in block sizes
(as predicted by our recent theoretical dynamical system analysis [1]) and
slow adjustments during periods of demand bursts (e.g., NFT drops). Both
phenomena lead to unwanted inter-block variability in mining rewards. To
address this issue, we propose an alternative base fee adjustment rule in
which the learning rate varies according to an _additive increase,
multiplicative decrease (AIMD) update scheme_. Our simulations show that the
latter _robustly outperforms the EIP-1559 protocol_ under various demand
scenarios. These results provide evidence that _variable learning rate_
mechanisms may constitute a promising alternative to the default
EIP-1559-based format and contribute to the ongoing discussion on the design
of more efficient transaction fee markets.

##  I Introduction

Transaction fees have been an integral part of the emerging blockchain
economies and the topic of heated debates in the blockchain community [2, 3,
4]. Ethereum, the second largest blockchain platform in terms of market value
(as of September 6th, 2021), recently launched its long-awaited _London hard
fork_ which implements – among other protocol upgrades – Ethereum Improvement
Proposal (EIP) 1559, which aims to radically transform its transaction fee
market [5, 6].

In Ethereum’s original fee market, users selected a gas price that indicated
how much they were willing to pay for their transactions to be included on the
blockchain. This mechanism behaved like a (generalized) first-price auction in
which miners, the “auctioneers”, allocated the available transaction slots in
each block to the highest paying users or “bidders”. Unfortunately, it also
shared all of a first-price auction’s drawbacks [7, 8]: untruthful bidding,
excessive intra- and inter-block variation of transaction fees, and
(over-)bidding competition for faster inclusion.

The EIP-1559 transaction market reform was put forward to address these
shortcomings. After a period of consultation and implementation, EIP-1559 came
into effect on August 5, 12:33:42 PM UTC, at block 12,965,000 [9, 10]. Its
main goals are to allow for more flexibility during changing market conditions
by aiming for a long-term average target block size (half-full blocks) instead
of blocks that are consistently full, and to make transaction fees more
predictable (but not necessarily lower).

To achieve these goals, EIP-1559 sets a base fee that is algorithmically
updated after each block to reflect the prevailing market demand. Critical in
these updates is a hard-coded learning rate (currently set at $12.5\%$) which
calibrates the size of the adjustment according to the current block size.
Unlike the traditional formats in which all fees were transferred from users
to miners, the base fee is now _burnt_ [11], i.e., permanently removed from
the total supply of Ethereum tokens (ether or ETH). Alongside newly minted
tokens, miners are compensated for their work by an additional tip set by the
users, typically much lower than the base fee.

#### Motivation and Contributions

Due to Ethereum’s prominence in the blockchain ecosystem, EIP-1559 naturally
attracted widespread academic and community attention even before its launch.
In [12, 4], its economic properties were studied, proving that it is incentive
compatible in the sense that it is rational for users to follow a simple
bidding strategy whenever the current base fee is comparable to market demand,
i.e., does not under-price the limited supply of gas in a block. Meanwhile,
[13, 1] analyzed its dynamics under different demand scenarios and [14, 3]
initiated the discussion about alternative (EIP-1559-like) transaction fee
market mechanisms.

In this paper, we contribute to this growing literature on EIP-1559. We use
blockchain data from the first month after EIP-1559’s launch – the “
_honeymoon_ ” period during which legacy transactions are still accepted – to
evaluate EIP-1559’s impact on user experience and network performance. We use
this empirical data to simulate and argue about the performance of EIP-1559
under different learning rate configurations. Our contributions can be
summarized as follows.

  * •

We find that the current learning rate leads to slow adjustments during
periods of demand peaks (e.g., NFT [15] drops) and to intense oscillations in
block sizes during periods of stable demand.

  * •

Both slow adjustments and oscillations have a negative effect on the user
experience and, more importantly, on mining rewards. In particular, they can
potentially generate strategic incentives for miners and hence undermine the
stability of the mechanism.

  * •

We then use the empirical data to run simulations with different learning
rates. Our results confirm the intuition that higher (lower) learning rates
perform well (poorly) during demand peaks and poorly (well) during periods of
demand stability. On the positive side, all EIP-1559 variations achieve (in
fact, slightly overshoot) the target block size (50%) on average.

  * •

Based on the above, we propose an alternative mechanism that utilizes an
additive increase, multiplicative decrease (AIMD) update scheme to implement a
variable learning rate. Our empirical results suggest that the AIMD mechanism
combines the best of both worlds: it selects a low learning rate during
periods of stable demand and reacts quickly during demand peaks to reflect
increasing gas prices. The AIMD not only achieves the target block size on
average (half-full blocks) but also considerably dampens the oscillations
between almost full and almost empty blocks, which leads to more stable mining
rewards and a more uniform user experience.

While our empirical results do not prove the optimality of the proposed AIMD
mechanism under arbitrary conditions, they provide evidence that variable
learning rates may considerably improve upon the current EIP-1559 format and
may thus constitute a promising direction for future work in the ongoing
discussion about EIP-1559-based transaction fee mechanisms [14, 3].

#### Outline

The rest of the paper is structured as follows. In Section II, we present the
main elements of Ethereum’s blockchain protocol and EIP-1559. In Section III,
we describe our dataset and in Section IV our simulation environment. Section
V contains the evaluation of the simulation results. Section VI concludes the
paper with questions and directions for future work.

##  II Preliminaries

###  II-A Ethereum

Ethereum has a prominent position in the growing blockchain ecosystem, as the
first platform to support decentralized, self-executing pieces of software
known as smart contracts [16].

#### Block Creation

The creation and execution of smart contracts on the Ethereum blockchain is
achieved through transactions which are ordered and grouped into _blocks_.
Whenever a new smart contract is deployed or a function of an existing
contract is called, the operation is embedded in a transaction. The
transaction is then broadcast to consensus nodes – which in Ethereum’s current
_proof-of-work_ mechanism are called miners – through a peer-to-peer network
[17]. The miners vie for the right to create new blocks for which they earn
_block creation rewards_ [18]. Eventually, a new block that includes a set of
new, previously pending transactions is created and broadcast to the nodes of
the peer-to-peer network, who then update their view of the blockchain. Valid
blocks that are not propagated across the network quickly enough can be
superseded by a competing block – the losing blocks are called _uncles_.

#### Transaction Fees

The maximum size of the blocks that encapsulate these transactions is limited
as per the specifications of the protocol and is measured in terms of the
total amount of computational effort required for executing all transactions
in that block. It is chosen as a trade-off between 1) higher throughput and 2)
making it possible for more low-power nodes to participate. Thus, there is a
demand for block space among users that wish to get their transactions
included in the blockchain, which is supplied by the miners who create the
blocks. To find a balance between this supply and demand, users must pay a
_transaction fee_ to the miners for including their transaction in a block.
This provides a second form of protocol-supported payment for the miners,
alongside the block creation rewards.

#### Ethereum Gas

The unit that measures the amount of computational effort required to execute
transactions on the Ethereum blockchain is referred to as _gas_. For example,
creation of a new smart contract may require $500,000$ units of gas, but a
simple transfer of ETH tokens typically requires only $21,000$ units of gas.
The use of gas units (or simply gas) allowed for a uniform implementation of
transaction fees in Ethereum’s original fee market design: for each
transaction, the user would specify a _gas price_ that indicates the fee paid
by the user per unit of gas used to execute the transaction, and a _gas limit_
that indicates the maximum amount of gas that the transaction is able to
consume.

#### Ethereum’s Legacy Fee Market

Ethereum’s legacy fee market mechanism, thus, closely resembled that of a
_(generalized) first-price auction_. Users would publish a transaction with a
fee, miners would choose the highest-paying transactions, and each user would
pay what they bid. The higher the user was willing to pay in transaction fees,
the higher their chances of having their transactions included in a block.
However, such auctions are well-known to be highly inefficient, leading to
extreme fee volatility and frequent over-payments [8, 19]. This is especially
observed when the network’s transaction load is high, e.g., due to new project
launches or periods of token price volatility. To remedy these shortcomings,
EIP-1559 was proposed [7].

###  II-B EIP-1559 Transaction Fee Market

EIP-1559 is an upgrade on Ethereum’s design which aims to address the above
challenges by reforming the transaction fee market. With its introduction on
August 05, 2021 on Ethereum’s mainnet, the maximum block size was doubled from
roughly 15 million to $T=30$ million gas. However, the long-term average block
size that the system targets is half that limit, namely $T/2$.

#### Base Fee

The main element of EIP-1559 to achieve this target is a dynamically adjusted
_base fee_ , $b_{t}$, that is updated after every block, indexed by block
height $t>0$, according to the equation

$b_{t+1}=b_{t}\left(1+d\cdot\frac{G_{t}-T/2}{T/2}\right).$ (1)

Here, $d$ denotes the _learning rate_ (or _adjustment parameter_), currently
set at $d=0.125$, and $G_{t}$ the total gas used by transactions that were
included in block $t>0$. Intuitively, if $G_{t}>T/2$, i.e., if more
transactions than targeted are included in a block, then the base fee
increases. The increase is scaled by $d$ and is proportional to the amount by
which the current block load $G_{t}$ exceeds the target $T/2$. Similarly, if
$G_{t}<T/2$, then the base fee decreases, whereas if $G_{t}=T/2$, then the
base fee remains unchanged.

#### Bids, Tips, and Effective Gas Prices

According to EIP-1559, instead of a single gas price, users now submit two
parameters, $(f,p)$, as shown in Table I.

Parameter |  Description
---|---
$f$: max fee |  maximum amount per gas unit that the user is willing to pay for their transaction to be included
$p$: max priority fee |  maximum tip per gas unit that the user is willing to pay to the miner who includes their transaction
TABLE I: EIP-1559 user bid parameters.

If the max fee that a user is willing to pay for their transaction is less
than the algorithmically computed base fee, i.e., if $f<b_{t}$, then the
transaction cannot be included in a block. Conversely, if $f\geq b_{t}$, then
the miners can include the transaction and hence earn the so-called miner’s
tip, which is calculated as $\text{miner's tip}=\min{\\{f-b_{t},p\\}}$ In
practice, if the miner’s tip is very small (less than 2 Gwei per gas unit,
where 1 Gwei = $10^{-9}$ ETH), it may be beneficial for miners not to include
the transaction. Larger blocks are harder to transmit over the network, making
it easier for a competing block to reach other nodes faster, causing the miner
to lose out on the block creation rewards and fees (“uncle risk”).

Importantly (and in sharp contrast to the original fee market), the base fees
are not transferred to miners: instead they are _burnt_ and are permanently
removed from the total supply of ETH.111As of September 3rd, 2021, over
180,000 ETH were burned through EIP-1559 transactions (source:
watchtheburn.com). The sum of the base fee and tip fee taken per unit gas is
said to be the _effective gas price_ and is equivalent to the gas price that a
user would have paid to include their transaction before EIP-1559.

#### Legacy Transactions

EIP-1559 was introduced in a backward-compatible manner allowing both _legacy_
and native EIP-1559 transaction formats to be included. However, all
transactions included after block 12,965,000 are subject to the base fee,
regardless of their format. In particular, when a legacy transaction is
submitted to the network, the max priority fee, $p$, and the max fee, $f$, are
set by default equal to the gas price specified by the user. The miner then
earns $f-b_{t}$ if the transaction is included in their block, possibly
leading to overpayment at inclusion, compared to the same transaction
following the simple EIP-1559 bidding strategy. With an EIP-1559 transaction,
the user places a cap on the miner’s tip by setting a max priority fee,
whereas in a legacy transaction all the remaining fees, after subtracting the
base fee, go to the miner.

##  III Data

To analyze the impact of EIP-1559 during the honeymoon period, we use publicly
available data recorded on the Ethereum blockchain. Our dataset covers a
period starting from block $l=12935000$, which was mined on 3 August 2021,
18:43, until block $u=13079999$, which was mined on 23 August, 06:38. EIP-1559
came into effect in block 12965000, which was mined on 5 August 2021, 12:33
(the mentioned times are all in UTC). Our data set is described in detail in
Table II and visualized in the next paragraphs, which a focus on both “long-
term” trends (i.e., observations from the entire study period) and “short-
term” behavior (i.e., observations regarding sudden demand peaks or other
transient phenomena).

Parameter |  Description
---|---
$x\in\mathbb{N}$ |  transaction, identified by its hash
$t_{x}\in\mathbb{N}$ |  height of the block in which $x$ was included
$g_{t}\in[0,1]$ |  relative block size of the block at height $t$, i.e., the total used gas $G_{t}$ divided by the gas limit $T$
$b_{t}\in\mathbb{N}$ |  base fee in the block at height $t$
$\tau_{x}\in\\{0,1\\}$ |  type of $x$, which is “$0$” for legacy transactions and “$1$” for EIP-1559 transactions
$f_{x}\in\mathbb{N}$ |  gas price of $x$ for legacy transactions, and the max fee of $x$ for EIP-1559 transactions
$p_{x}\in\mathbb{N}$ |  priority fee of $x$ (for legacy transactions this is set equal to $f_{x}$)
$u_{x}\in\mathbb{N}$ |  gas limit of transaction $x$ (used as a proxy for the gas that was actually used by the transaction)
$X(t)\subset\mathbb{N}$ |  set of all transactions included in block at height $t$
TABLE II: Transaction information in the dataset.

###  III-A Long-term Trends

#### Average Gas Prices

First, we study the _average block gas prices_ during the sample period. To
compute this metric over time, we first compute the average gas price
$\hat{f}_{t}$ for each block at height $t$, where each transaction $x$ is
weighted by its gas limit $u_{x}$:

$\hat{f}_{t}=\left(\sum\nolimits_{x\in
X(t)}u_{x}f_{x}\right)/\left(\sum\nolimits_{x\in X(t)}u_{x}\right).$

We then divide the set $\\{l,l+1,\ldots,u\\}$ into $1000$ batches of size
$w=\frac{u-l+1}{1000}$, and compute the average $\hat{f}^{*}_{i}$ for each
batch $i\in\\{1,\ldots,1000\\}$ as

$\hat{f}^{*}_{i}=\frac{1}{w}\sum\nolimits_{t=l+(i-1)w}^{l+iw}\hat{f}_{t}.$ (2)

The result is displayed in Figure 1, which shows the evolution of the average
block gas, $\hat{f}^{*}_{i}$.

(a) Gas price

(b) Fraction of EIP-1559 transactions

Figure 1: Top: average (blue) and smoothed averaged (red) gas prices over
time. Bottom: average fraction (gray) and smoothed averaged fraction (red) of
EIP-1559 transactions over time.

We observe that the (average) gas prices exhibit sharp bursts. We have found
that these bursts typically correspond to so-called “NFT drops”, in which a
creator (or project) announces the release of a limited number of Non-Fungible
Tokens (NFTs) [15]. This leads to a demand surge where purchasers put in high
bids to be included first by miners. Other, lower-scale fluctuations are due
to diurnal variations in Ethereum’s usage over different geographic locations.
In Section III-B, we return to the base fee and gas price dynamics and examine
them in the locality of periods of both sharp demand increases and of
relatively stable demand.

#### Smoothed Average Gas Prices

To dampen the effect of the sharp bursts on the long-term trend of gas prices,
we apply a median smoothing filter to $\hat{f}^{*}_{i}$, which is unaffected
by large outliers unlike the moving average. To compute the filtered value
$m(\hat{f}^{*}_{i},\eta)$ of $\hat{f}^{*}_{i}$, where $\eta$ is the half-width
of the filter, we first determine the set

$V(\hat{f}^{*}_{i},\eta)=\\{\hat{f}^{*}_{j}\in\mathbb{N}:j\geq l,j\leq
u,|j-i|\leq\eta\\}.$

We then compute the median $m_{i}=m(\hat{f}^{*}_{i},\eta)$ as the value such
that at least $50\%$ of the elements in $V(\hat{f}^{*}_{i},\eta)$ are greater
or equal than $m_{i}$ and at least $50\%$ of the elements in
$V(\hat{f}^{*}_{i},\eta)$ are smaller or equal than $m_{i}$. The result with
half-width $\eta=10$ is displayed by the red line in Figure 1.

#### Fraction of EIP-1559 Transactions

We apply the same methodology as above to the weighted fraction,
$\hat{\tau}_{h}$, of EIP-1559 transactions (versus legacy transactions) which
for the block at height $t$ is given by

$\hat{\tau}_{t}=\left(\sum\nolimits_{x\in
X(t)}u_{x}\tau_{x}\right)/\left(\sum\nolimits_{x\in X(t)}u_{x}\right).$

The result is displayed in Figure 1(b). The gray line represents
$\hat{\tau}^{*}_{i}$ averaged, as in equation (2), over 1000 batches and the
red line represents the corresponding smoothed values. The $i$-th value in the
plot corresponds to the middle block of the $i$-th batch for
$i\in\\{1,2,\dots,1000\\}$.

We observe that EIP-1559 adoption increases steadily within our sampling
period to around $60\%$ at the end of August. The sharp increase around block
13051500 (8 August) corresponds to the (actual date of) adoption of EIP-1559
by the popular Ethereum wallet MetaMask. After that point, we observe strong
diurnal fluctuations as a result of differences in wallet usage over different
geographic locations.

###  III-B Short-term Behavior

We now take a more detailed view at the short-term behavior of the base fee
and gas price dynamics during periods of both sharp demand increases (e.g.,
NFT drops) and of relative demand stability.

#### Demand peaks

Figure 2(a) zooms in on one of the price bursts that are visible in Figure 1,
namely the one that occurs around block 13025775 that was mined on August 14,
2021, at 09:35. The displayed period covers 450 blocks between blocks 13025550
and 13026000.

(a) Burst period prices

(b) Stable period prices

(c) Stable period sizes

Figure 2: (a) Average gas prices (blue) and base fees (brown) per block in the
volatile period after block 13025550. (b) Average gas prices (blue) and base
fee per block (red) in Gwei, in the period of relatively stable demand that
directly followed the burst of (a). (c) Relative block sizes, $\hat{g}_{t}$,
in the same period of relatively stable demand as in (b).

Before the burst, average gas prices (depicted by the blue line without marks)
fluctuate between 30 and 60 Gwei, with several short bursts in which the
average fee jumps to around 120 Gwei before returning to around 45 Gwei, and
several empty blocks (in which the average fee is set to $0$). The average gas
price then shoots from 40.82 Gwei in block 13025736 to 207.33 and ultimately
to 1360.02 Gwei within the next two blocks.

To compare, the miner of block 13025736 earns a total of 2.019 ETH: 2 ETH for
the block creation reward, plus 0.137 in transaction fees minus 0.118 in burnt
fees. In this case, as is a typical for blocks outside price bursts, the
transaction fees are small compared to the block rewards [20]. However, the
miners of blocks 13025737 and 13025738 earn 3.472 ETH and 45.646 ETH,
respectively. In the latter case, the earnings from transaction fees are more
than 20 times greater than the block creation reward!

Concerning the base fee (depicted by the brown line with marks), we observe
that shortly after the price burst it increases rapidly until it reaches a
level slightly below the average gas price. After that point, it starts to
decline together with gas prices which gradually return to normal. In sharp
contrast to the above, block 13025768 is ultimately far from target, netting
0.844 ETH in transaction fees for 20% block utilisation (10% of the block gas
limit). Comparing this revenue with that of the next miner 13025769, who
received 4.868 ETH for 180% block utilisation (90% of the block gas limit),
13025768 misses out on some share of these fees. With a uniform process
generating the transaction arrival time, this share may be estimated to 2 ETH
(an equal split between 0.844 and 4.868 ETH), worth around $\$6,500$ USD at
the time. These points demonstrate an instability in inter-block mining
rewards under EIP-1559 that may require closer monitoring.

#### Demand Stability

Figure 2(b) displays the gas price and base fee dynamics for the 450-block
period of relatively stable demand that starts immediately after the burst of
Figure 2(a). During this period, both the gas price and the base fee remain
roughly stable: the former at around 40 Gwei and the latter slightly below 30
Gwei. However, despite the fact that the base fee remains within a relatively
narrow range of 10 Gwei, i.e., between 25 and 35 Gwei, block sizes fluctuate
wildly between the two extremes (empty to full and vice versa). This can be
seen in Figure 2(c), where the green line (without marks) displays the
relative block size as a number between $0$ (empty) and $1$ (full). This is
exactly the behavior predicted in [1, 13], and can be attributed to the
discrete nature of the updates and the size of $d$.

However, it is interesting to observe that block sizes do, in fact, seem to
reach (or, actually, only slightly overshoot) the target block size _on
average_. As shown in Figure 2(c), the median filter with half-width 30
applied to $\hat{g}_{t}$ (red line with marks) is indeed centered around $0.5$
(half-full blocks). Providing theoretical reasons for or empirically testing
this observation over longer periods of data is an interesting direction for
future work.

##  IV Simulations: Learning Rate in EIP-1559

As established by the short-term analysis in the previous section, there are
two opposing effects concerning the learning rate $d$. In particular, the
default $d$ (set at 0.125) is _too low_ during periods of short bursts (during
which the base fee lags in catching up to actual demand) but is also _too
high_ during periods of relative stability (where it induces intense
oscillations in block sizes despite keeping the base fee relatively stable).
While long-term trends suggest that block-sizes achieve the target of half-
full blocks _on average_ , these fluctuations have significant effects on
miners’ earnings and thus, potentially, on protocol stability.

As such, we investigate the impact of choosing a different $d$ (constant or
variable) on the network. To do so, we run a series of simulations in which we
observe the base fee after transaction bids are drawn randomly.

###  IV-A Constant Learning Rate

The main challenge in simulating (i.e., replaying using a stochastic model)
the evolution of the transaction fee market for different values of $d$ (or
alternative mechanisms) is to draw accurate estimates for the actual demand
and the users’ valuations from the available data. The difficulty stems from
the fact that observable bids – i.e., those included on the blockchain – do
not necessarily reflect users’ true valuations, but merely their
valuations/expectations about the current and future base fees and their
urgency to get their transaction included. Furthermore, they do not include
low-value bids that were not included.

#### Behavioral Model

As such, we make some assumptions – justified by our current data and
Ethereum’s previous simulations, see [6, 1, 13] – and leave a deeper
investigation of bids, e.g., with questionnaires to users, as an open
challenge for future research. Accordingly, we model the number of users in
each time slot using a Poisson demand with arrival rate $\lambda$ and a
distribution of valuations, $F$. As mentioned above, both $\lambda$ and $F$
are subject to change over time – e.g., due to the effects of weekends and
weekdays, day/night cycles in different regions, sudden bursts due to popular
contracts and NFT launches – and therefore challenging to infer from the
available data.

#### Execution Model

Given a Poisson distribution of transaction arrivals (i.e., demand) with time-
dependent arrival rate $\lambda_{t}$ and valuation distributions $F_{t}$, the
simulator executes the steps of Algorithm 1 in each time slot $t$.

Input: Demand parameters $\lambda_{t}$, $\hat{f}_{t}$, base fee $b_{t-1}$ at
block height $t-1$ and learning rate $d$.

Output: base fee at block height $t$.

1 $n_{t}\leftarrow\textsf{draw\\_num\\_transactions}(\lambda_{t})$

2 $m^{\prime}_{t}\leftarrow\textsf{draw\\_transactions}(n_{t},\hat{f}_{t})$

3 $m_{t}\leftarrow\textsf{m\\_update}(m_{t-1},m^{\prime}_{t})$

4 $B_{t}\leftarrow\textsf{B\\_update}(m_{t})$

5 $g_{t}\leftarrow|B_{t}|/T$

6 $b_{t}\leftarrow\textsf{b\\_update}(b_{t-1},d,|B_{t}|)$

Algorithm 1 Simulation of EIP-1559 Updates

Algorithm 1 first calls $\textsf{draw\\_num\\_transactions}(\lambda_{t})$,
which pseudo-randomly draws a value $n_{t}$ from the Poisson distribution with
mean $\lambda_{t}\equiv 3$ (adopted from [1]). The function
$\textsf{draw\\_transactions}(n_{t},\hat{f}_{t})$ then draws $n_{t}$ samples
from $F_{t}$ that represent the valuations of the transactions created at time
$t$, given the observed median-smoothed average gas price in the block at
height $t$, denoted by $\hat{f}_{t}$. These values are then stored in the set
$m^{\prime}_{t}$. We have observed that a mixture of uniformly distributed and
Pareto-distributed random samples, with means that depend on the smoothed
average fees, results in a good correspondence with the observed values’
distribution. We then draw the valuations of $2.75n_{t}$ transactions from a
uniform distribution near $0.75\hat{f}_{t}$, and draw the remaining
$0.25n_{t}$ valuations from a Pareto distribution with shape $1.35$ and scale
$\frac{1}{10}\hat{f}_{t}$. The reasoning behind this modelling choice is to
divide users into two broad classes. The first class consists of _regular_
users for whom the delay before the transaction appears on the blockchain is
relatively unimportant, and who simply bid a value that (they believe) ensures
that their transaction is eventually included. The second class consists of
_high-urgency_ users who make high bids to ensure that their transaction is
included rapidly, and whose valuations more closely resemble a heavy-tailed
distribution. The parameters were chosen to give a good correspondence with
the studied dataset, but we have found that minor variations did not change
the overall conclusions of our evaluation, cf. Table V.

After having drawn $m^{\prime}_{t}$, Algorithm 1 combines the transactions in
$m^{\prime}_{t}$ with the process $m_{t}$, which represents the _mempool_ ,
i.e., the set of transactions that are pending to be included in the
blockchain. The function $\textsf{m\\_update}(m,m^{\prime})$ returns a new
mempool given an existing mempool $m$ and a set $m^{\prime}$ of new
transactions. In our simulations, transactions remain in the mempool until
they are included. The function $\textsf{B\\_update}(m)$ is then executed by
the miner. $\textsf{B\\_update}(m)$ returns a vector $B$ with the max fees of
the transactions that have been included in the new block given the mempool
$m$. We denote the length of $B$ as $|B|$. Finally, the function
$\textsf{b\\_update}(b,d,|B|)$ returns the new base fee as a function of the
current base fee $b$, the learning rate $d$ and the block size $|B|$.

###  IV-B Variable Learning Rate: AIMD Updates

To address the issue that the current learning rate appears too high in
stable-demand periods and too low in periods with demand peaks, we also
propose and simulate a mechanism with a variable learning rate. In particular,
motivated by the effectiveness of such schemes to control internet congestion
[21], we study an _additive increase and multiplicative decrease_ (AIMD)
update scheme for the learning rate $d$.

Parameter |  Description
---|---
$g_{\textsc{AVG}}$ |  average relative block-size of previous blocks
$n\in\mathbb{N}$ |  nr. of previous blocks over which $g_{\textsc{AVG}}$ is computed.
$\alpha>0$ |  size of an additive increase
$\beta\in[0,1]$ |  factor of a multiplicative decrease
$\gamma\in[0,\frac{1}{2}]$ |  threshold for an additive increase
$d_{\min},d_{\max}$ |  minimum and maximum values for $d$.
TABLE III: Parameters of the AIMD algorithm.

According to the AIMD scheme, the learning rate is adjusted with respect to a
parameter, $g_{\textsc{AVG}}$, that keeps track of the average relative block
sizes ($g_{t}=G_{t}/T$, cf. Table II) during some pre-specified rolling
window. The intuition is the following: when $g_{\textsc{AVG}}$ is close to
the target block-size, $0.5$, then the base fee is close to the right value,
so the algorithm reduces the learning rate to reduce the size of the
oscillations. By contrast, if $g_{\textsc{AVG}}$ is too small or high, then
the base fee is apparently far away from its equilibrium value, and the
algorithm increases the learning rate.

#### Execution Model (AIMD)

The AIMD update scheme uses the parameters in Table III. It then replaces line
6 in Algorithm 1 with the following two lines

6 $d_{t}\leftarrow\textsf{d\\_update}(d_{t-1},g_{t-n+1},\ldots,g_{t})$

$b_{t}\leftarrow\textsf{b\\_update}(b_{t-1},d_{t},|B_{t}|)$

Algorithm 2 Modified Alg. 1 with AIMD updates

which update the learning rate parameter $d$ according to Algorithm 3
(following the intuition described above):

Input: Learning rate, $d_{t}$, window $n$, block-sizes
$g_{t-n+1},\ldots,g_{t}$, update parameters $\alpha,\beta,\gamma$.

Output: Learning rate, $d_{t+1}$.

1

2$g_{\textsc{AVG}}\leftarrow\frac{1}{n}\sum^{t}_{t-n+1}g_{i}$

3 if  _$g_{\textsc{AVG}} <\gamma\textnormal{{ or }}g_{\textsc{AVG}}>1-\gamma$_
then

4 $d_{\textsc{new}}\leftarrow\min\\{d_{\max},\alpha+d\\}$

5 else

6 $d_{\textsc{new}}\leftarrow\max\\{d_{\min},\beta d\\}$

return $d_{\textsc{new}}$

Algorithm 3 AIMD update routine d_update.

##  V Evaluation

We now proceed to compare the different update mechanisms described in the
previous section. We present simulations of the standard EIP-1559 update rule
with constant step sizes at $d=0.125$ (default), $d=0.25$ and $d=0.0625$ which
are representative of high and low rates, respectively, and of the AIMD
(variable step size) update rule with parameters $n=8$,
$\alpha=0.025,\beta=0.95,\gamma=0.25,d_{\min}=0.0125$ and $d_{\max}=1$. Note
that $n$ is selected to be low enough to reflect recent market conditions, but
high enough to reduce its sensitivity to temporal fluctuations in demand.

###  V-A Performance Metrics

To compare the performance of the different mechanisms, we use the two metrics
described in Table IV.

Parameter |  Description
---|---
$\hat{g}$ |  long-term average relative block size
$p_{g>0.95}$ |  long-term fraction of blocks whose relative size is greater than $0.95$.
TABLE IV: Performance metrics.

The first metric, $\hat{g}$, should be as close to $0.5$ as possible, as this
is the target set by the miners.

(a) $d=0.0625$

(b) $d=0.125$

(c) $d=0.25$

(d) AIMD

(e) $d=0.0625$

(f) $d=0.125$

(g) $d=0.25$

(h) AIMD

Figure 3: Simulation of average gas prices in Gwei (blue lines) and base fees
(red lines) in stable (top) and peak demand periods (bottom) of the EIP-1559
market under different learning rates. Unlike standard EIP-1559 mechanisms
with slow or fast learning rates ($d=0.125$ being the default), the AIMD
mechanism exhibits good performance in both scenarios.

When $\hat{g}$ is (considerably) larger than $0.5$, propagation and execution
time of new blocks increase, putting stress on the network of miners to
validate all new items. Conversely, when $\hat{g}$ is (considerably) lower
than $0.5$, network capacity is under-utilized. The situation where $\hat{g}$
is slightly below 0.5 more closely resembles the original target setting of
the Ethereum network before EIP-1559 with max block size of $T/2$ making such
states marginally preferable over ones where $\hat{g}$ is slightly above
$0.5$.

The second metric, $p_{g>0.95}$ should be as low as possible, even if
$\hat{g}$ approximates $0.5$ (e.g., due to full blocks counterbalancing empty
blocks). When block-sizes are consistently high, the transaction fee market
approaches the behavior of a first-price auction, losing some of the
advantages of EIP-1559 [12].222The reason that we do not consider $p_{g=1}$ is
purely technical: since not all transaction have the same size, in some cases
a block will only use, say, $99\%$ of its gas and yet it will still be
considered “full” in the sense that no other available transactions can be
added to it.

###  V-B Simulation Results

The effect of choosing different learning rates in the standard EIP-1559
mechanism is displayed in Figure 3. We visualize two scenarios: the stable
period from Figure 2(b) (top row), and the NFT drop from Figure 2(a) (bottom
row). The blue lines show the simulated demand which approximates the actual
demand, cf. Figures 2(b) and 2(a).

As intuitively expected, the oscillations of the base fee (brown line with
marks) in the stable scenario decrease as the learning rate decreases.
Similarly, in the NFT drop scenario, the fast learning rate mechanism,
$d=0.25$, swiftly adapts to the changing situation, whereas the other two
learning rate mechanisms generate long periods of discrepancies between the
base fee and market demand.

The important takeaway, however, concerns the AIMD mechanism (panels in the
last column), which turns out to share the advantages of both approaches. In
particular, when the system is relatively stable, it selects a learning rate
of $d=0.0125$ (lower than the current default of $d=0.125$) that increases
swiftly during demand peaks. This results in low oscillations of the base fee
during the stable period (top row) and a timely increase during the demand
peak period (bottom row). To emphasize this, we have also displayed in Figure
4 the block sizes and the evolution of the learning rate $d$ for the AIMD
scheme. We set $d=0.125$ as the initial value, and observe that in both
scenarios the learning rate decreases to its minimum values,
$d_{\min}=0.0125$. However, in the NFT drop scenario, the learning rate
rapidly increases to over 0.45, and then decreases again when the base fee
starts to reflect market conditions. In the stable scenario, we observe from
Figure 4(b) that although there are still oscillations, they are noticeably
less intense than in Figure 2(c).

To further quantify this performance, Table V contains the results for both
metrics, $\hat{g}$ and $p_{g>0.95}$, over three scenarios: the stable period
(450 blocks), the demand peak period (450 blocks), and our full dataset of
blocks after EIP-1559 (115,000 blocks).

| Stable periods | $\hat{g}$ |  $p_{g>0.95}$ |  | Demand peaks | $\hat{g}$ |  $p_{g>0.95}$ |  | Full dataset | $\hat{g}$ | $p_{g>0.95}$
---|---|---|---|---|---|---|---|---|---|---|---
| empirical | 0.523 | 0.261 |  | empirical | 0.528 | 0.281 |  | empirical | 0.523 | 0.244
simulation | $d=0.0625$ |  | 0.505
---
$\pm$0.000
| 0.150
---
$\pm$0.005
simulation | $d=0.0625$ |  | 0.511
---
$\pm$0.000
| 0.222
---
$\pm$0.004
simulation | $d=0.0625$ |  | 0.505
---
$\pm$0.000
| 0.186
---
$\pm$0.000
$d=0.125$ |  | 0.510
---
$\pm$0.000
| 0.214
---
$\pm$0.004
$d=0.125$ |  | 0.513
---
$\pm$0.000
| 0.232
---
$\pm$0.004
$d=0.125$ |  | 0.510
---
$\pm$0.000
| 0.232
---
$\pm$0.000
$d=0.25$ |  | 0.520
---
$\pm$0.000
| 0.255
---
$\pm$0.002
$d=0.25$ |  | 0.522
---
$\pm$0.000
| 0.265
---
$\pm$0.003
$d=0.25$ |  | 0.520
---
$\pm$0.000
| 0.264
---
$\pm$0.000
AIMD |  | 0.499
---
$\pm$0.001
| 0.058
---
$\pm$0.003
AIMD |  | 0.481
---
$\pm$0.002
| 0.126
---
$\pm$0.003
AIMD |  | 0.501
---
$\pm$0.000
| 0.047
---
$\pm$0.000
TABLE V: Empirical evaluation of the different EIP-1559 mechanisms with low,
default ($d=0.125$), and high learning rates in terms of the performance
metrics $\hat{g}$ (average block size) and $p_{g>0.95}$ (probability that a
block is more than $95\%$ full). The AIMD mechanism has the best performance
in terms of $p_{g>0.95}$ in both the stable and unstable scenarios, and across
the entire post-EIP-1559 dataset (bold values).

In Table V, below each point estimate of $\hat{g}$ and $p_{g>0.95}$ we also
display the half-width of a 95% confidence interval for the corresponding
value based on the normal distribution. For each entry in the table, we
conducted $20$ simulation experiments – despite the small sample size, we note
that the variance is fairly low since each sampled estimate of $\hat{g}$ and
$p_{g>0.95}$ is itself an average over at least 450 blocks. For each scenario,
we also display the empirical result, i.e., the values of $\hat{g}$ and
$p_{g>0.95}$ that were observed in the data set.

Importantly, the AIMD has a low number of full blocks in all three scenarios
and hence exhibits the best performance in terms of $p_{g>0.95}$ by a wide
margin when compared to the other methods. AIMD updates also achieve the
desired average relative block size ($\hat{g}\approx 0.5$). Consistent with
our empirical observations, all standard EIP-1559 updates also approximate,
yet, consistently overshoot by a small margin, the target block-size on
average, but perform poorly on the $p_{g>0.95}$ metric. This suggests the
presence of more than 20% almost full blocks counterbalanced by a similar
amount of almost empty blocks.

###  V-C Discussion

Given the desirable properties showcased by the AIMD over the standard
EIP-1559 update rules in the above simulations, it is natural to ask whether
the AIMD update rule could actually fit in the current scope of EIP-1559. In
particular, the default, more moderate EIP-1559 update that corresponds to a
fixed $d=0.125$ aims to achieve the following goals: (i) generate more
predictable fees against comparable market conditions, (ii) adapt fees to
prevailing demand, (iii) fight inflation by burning excessive fees instead of
transferring them to miners, and (iv) tame strategic behavior by protocol
participants, both miners and users. In the following, we discuss how an AIMD
update rule would perform along these axes.

#### Predictability and Adaptation

Concerning the first two goals, there is a certain amount of trade-off between
short-term fee predictability and swiftly responding to changing market
conditions. However, the high-level goals of next-block inclusion and
invariant block sizes tip the scale towards more responsive fee mechanisms.

#### Excessive Fees

Concerning the goal of controlling inflation, it is tempting to think that a
slow-adapting mechanism may burn more fees by allowing full blocks during
demand peaks. However, users’ valuations are typically skewed towards higher
values [2, 22] and higher fees at periods of increased demand can actually
reap higher amounts of burned coins.

#### Strategic Behavior

The fourth goal, that of mitigating strategic behavior by miners or users, is
more intricate and merits a separate discussion. Apart from general attacks on
the PoS consensus layer [23, 24], there is an open debate on the
vulnerabilities of the EIP-1559 fee market (in its original format) against
exacerbated versions of existing or novel attack vectors [18]. One such
example is the _zero-base fee attack_ , in which a coalition of miners who
control 51% of the network’s resources collectively drive the base fee down to
zero by continuously mining empty blocks. At that point the only determining
factor for transaction inclusion are miner’s tips, essentially moving the
system back to the previous first price auction model. While the practical
feasibility of such an attack is still unclear, it is a necessary to look at
the effects of the more responsive AIMD update rule on miner’s incentives via
a more comprehensive threat model. We leave this as a direction for future
work.

(a) AIMD: demand peak

(b) AIMD: stable demand

Figure 4: Block sizes and learning rate evolution under the variable learning
rate AIMD scheme during the demand peak (first and second panel) and stable
demand periods (third and fourth panels), respectively. AIMD adjusts the
learning rate during the demand peak and uses a lower than default learning
rate to dampen the oscillations (in particular, empty or almost empty blocks)
during the stable demand period (cf. EIP-1559 performance in Figures 2(a) and
2(b).

##  VI Conclusions & Open Questions

Using on-chain data from the first month after its launch, we studied the
early stages of Ethereum’s EIP-1559 fee market upgrade. We found evidence that
the current fixed learning rate – the parameter that is used to adjust the
base fee after each block based on network congestion – achieves the target
block-size on average (or rather, slightly overshoots it). However, it may
have a considerable destabilizing effect on mining earnings – via slow
adjustments during demand bursts and intense oscillations in block occupancies
during periods of stable demand – and on the user experience. Such phenomena
may lead to strategic behavior and ultimately pose a threat to the stability
of the fee market.

Our simulations suggest that a promising direction to improve upon the current
EIP-1559 format is the use of variable learning rates. In particular, we
studied an additive increase, multiplicative decrease (AIMD) update scheme for
the learning rate that resulted in superior performance over standard EIP-1559
formats across periods of both demand bursts and demand stability.

Given this range of results, a number of interesting open questions emerge.
Specifically, it will be instructive to monitor the impact of EIP-1559 over
longer time spans (i.e., beyond the “honeymoon” period), to study different
learning rate update schemes (e.g., multiplicative increase, additive
decrease, or exponential updates) and to derive accurate estimates of
Ethereum’s recurring demand conditions to systematically optimize the
parameters of the proposed AIMD scheme. Finally, as mentioned above, it will
be interesting to develop a strategic threat model to analyze how miners’ and
users’ incentives change between the default EIP-1559 mechanism and the
proposed AIMD update rule. Any progress in the above directions will
contribute to the ongoing discussion about EIP-1559 and to the design of more
efficient transaction fee mechanisms in general.

## Acknowledgment

This research/project is supported in part by the National Research
Foundation, Singapore under its AI Singapore Program (AISG Award No:
AISG2-RP-2020-016), NRF 2018 Fellowship NRF-NRFF2018-07, NRF2019-NRF-ANR095
ALIAS grant, grant PIE-SGP-AI-2018-01, AME Programmatic Fund (Grant No.
A20H6b0151) from the Agency for Science, Technology and Research (A*STAR) and
the Ethereum Foundation. It is also supported by the National Research
Foundation (NRF), Prime Minister’s Office, Singapore, under its National
Cybersecurity R&D Programme and administered by the National Satellite of
Excellence in Design Science and Technology for Secure Critical
Infrastructure, Award No. NSoE DeST-SCI2019-0009.

## References

  * [1] S. Leonardos, B. Monnot, D. Reijsbergen, S. Skoulakis, and G. Piliouras, “Dynamical Analysis of the EIP-1559 Ethereum Fee Market,” in _ACM AFT’21_ , 2021.
  * [2] D. Easley, M. O’Hara, and S. Basu, “From mining to markets: The evolution of Bitcoin transaction fees,” _Journal of Financial Economics_ , vol. 134, no. 1, pp. 91–109, 2019.
  * [3] M. X. Ferreira, D. J. Moroz, D. C. Parkes, and M. Stern, “Dynamic Posted-Price Mechanisms for the Blockchain Transaction-Fee Market,” in _ACM AFT’21_ , 2021.
  * [4] T. Roughgarden, “Transaction Fee Mechanism Design,” in _EC’21_ , New York, NY, USA, 2021, p. 792, [arXiv].
  * [5] V. Buterin, E. Conner, R. Dudley, M. Slipper, I. Norden, and A. Bakhta, “EIP1559: Fee market change for ETH 1.0 chain,” [Online], 2019.
  * [6] Robust Incentives Group, “abm1559 notebooks,” [Online], 2021.
  * [7] V. Buterin, “EIP-1559 FAQ,” [Online], 2021.
  * [8] E. Maskin, “Auctions and Efficiency,” in _Advances in Economics & Econometrics_. Cambridge University Press, 2001, pp. 1–24.
  * [9] T. Beiko, “London Mainnet Announcement,” [Online], 2021\.
  * [10] Ethereum Foundation, “EF-Supported Teams: Research & Development Roundup,” [Online], 2021\.
  * [11] K. Karantias, A. Kiayias, and D. Zindros, “Proof-of-burn,” in _Financial Crypto’20_. Springer, 2020, pp. 523–540.
  * [12] T. Roughgarden, “Transaction Fee Mechanism Design for the Ethereum Blockchain: An Economic Analysis of EIP-1559,” [arXiv], 2020.
  * [13] B. Monnot, Q. Hum, C. S. M. Koh, S. Leonardos, and G. Piliouras, “Ethereum’s Transaction Fee Market Reform of EIP 1559,” _WINE’20 Workshop on Blockchain_ , 2020.
  * [14] V. Buterin, “Make EIP 1559 more like an AMM curve,” Ethereum Research [Online], 2021\.
  * [15] L. Kugler, “Non-fungible tokens and the future of art,” _Communications of the ACM_ , vol. 64, no. 9, pp. 19–20, 2021.
  * [16] V. Buterin, “Ethereum white-paper: A next-generation smart contract and decentralized application platform,” [Online], 2014\.
  * [17] G. Wood, “Ethereum: A secure decentralised generalised transaction ledger,” [Online], 2014.
  * [18] V. Buterin, D. Reijsbergen, S. Leonardos, and G. Piliouras, “Incentives in Ethereum’s Hybrid Casper Protocol,” in _IEEE-ICBC’19_. USA: IEEE, May 2019, pp. 236–244.
  * [19] N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani, _Algorithmic Game Theory_. USA: Cambridge University Press, 2007.
  * [20] M. Carlsten, H. Kalodner, S. M. Weinberg, and A. Narayanan, “On the instability of Bitcoin without the block reward,” in _CCS’16_ , 2016, pp. 154–167.
  * [21] Y. R. Yang and S. S. Lam, “General aimd congestion control,” in _Proceedings 2000 International Conference on Network Protocols_. IEEE, 2000, pp. 187–198.
  * [22] G. Huberman, J. D. Leshno, and C. Moallemi, “Monopoly without a Monopolist: An Economic Analysis of the Bitcoin Payment System,” _The Review of Economic Studies_ , vol. online, pp. 1–30, 03 2021, rdab014.
  * [23] M. Neuder, D. J. Moroz, R. Rao, and D. C. Parkes, “Low-cost attacks on Ethereum 2.0 by sub-1/3 stakeholders,” 2021.
  * [24] C. Schwarz-Schilling, J. Neu, B. Monnot, A. Asgaonkar, E. N. Tas, and D. Tse, “Three Attacks on Proof-of-Stake Ethereum,” 2021.