# On the Redistribution of Maximal Extractable Value:
A Dynamic Mechanism

Pedro Braga Georgios Chionas Piotr Krysta King’s College London University of
Liverpool Augusta University mailto:pedro.braga<EMAIL_ADDRESS>mailto:g.chionas<EMAIL_ADDRESS>mailto:pkrysta<EMAIL_ADDRESS>Stefanos Leonardos Georgios
Piliouras Carmine Ventre King’s College London Singapore University of
Technology King’s College London
mailto:stefanos.leonardos<EMAIL_ADDRESS>mailto:georgios<EMAIL_ADDRESS>mailto:carmine.ventre<EMAIL_ADDRESS>P. Krysta is also
affiliated with University of Liverpool.

###### Abstract

Maximal Extractable Value (MEV) has emerged as a new frontier in the design of
blockchain systems. The marriage between decentralization and finance gives
the power to block producers (a.k.a., miners) not only to select and add
transactions to the blockchain but, crucially, also to order them so as to
extract as much financial gain as possible for themselves. Whilst this price
may be unavoidable for the service provided by block producers, users of the
chain may in the long run prefer to use less predatory systems.

In this paper, we propose to make the _MEV extraction rate_ part of the
protocol design space. Our aim is to leverage this parameter to maintain a
healthy balance between miners (who need to be compensated) and users (who
need to feel encouraged to transact). Inspired by the principles introduced by
EIP-1559 for transaction fees, we design a dynamic mechanism which updates the
MEV extraction rate with the goal of stabilizing it at a target value. We
analyse the evolution of this dynamic mechanism under various market
conditions and provide formal guarantees about its long-term performance. Our
results show that even when the system behavior is provably chaotic, the
dynamics guarantee long-term liveness (survival) and robustness of the system.
The main takeaway from our work is that the proposed system exhibits desirable
behavior (near-optimal performance) even when it operates in out of
equilibrium conditions that are often met in practice. Our work establishes,
the first to our knowledge, dynamic framework for the integral problem of MEV
sharing between extractors and users.

##  1 Introduction

Smart contract blockchains, such as Ethereum [6], Cardano [24] and Tezos [18],
unveiled the ability to construct flexible financial products on top of
blockchains and led to emergence of _Decentralized Finance_ (DeFi). DeFi
supports most applications available in traditional finance such as asset
exchanges, trading of tokens through Decentralized Exchanges (DEXes), loans,
decentralized governance, and stablecoins (currency boards) [42].

However, block producers, e.g., miners in Proof of Work (PoW), or validators
in Proof of Stake (PoS) networks, have unilateral power to decide which
transactions to execute and in which order. This unfolds a rich strategy space
to extract additional profit, usually at the expense of users. _Maximal
Extractable Value_ (MEV) is any type of excess profit a block producer can
obtain by adding, ordering, or censoring transactions [10]. There are multiple
forms of MEV such as frontrunning, arbitrage and sandwich attacks in DEXes
[16], liquidations [20, 36], cross-chain MEV [33] and NFT-mint front running
[16]. While many of these practices have been prevented or regulated in
traditional finance (e.g., frontrunning is known to breach certain MiFID II
standards, such as, best execution — fca.org), the lack of centralized
authorities in the decentralized systems renders such mechanisms of little or
no practical relevance to the DeFi setting.

In current smart-contract blockchains, MEV constitutes an integral source of
concern. The reason is that even though MEV is produced by the economic
activity of users (MEV generators), in most cases, the extracted amount goes
to other agents of the system (MEV extractors). Resolving this tension is far
from straightforward. Extreme solutions, in which all MEV goes either back to
users or to block producers, are shown to adversely affect the behaviour of
the opposite group and end up compromising participation and system security
[37, 10, 7].

Hence, the question of how to share the welfare between these two groups,
i.e., users and block producers, constitutes a central problem in current
blockchain research and practice [13, 15]. The current debate around MEV
_sharing_ or mitigation involves two main schools of thought. The first is the
_anti-MEV_ approach, which states that MEV is harmful since it increases
network congestion, higher costs for users, information asymmetries and thus
its exploitation must be prevented or mitigated by adding additional ordering
constraints to the consensus layer [25, 23, 22]. The second is the _pro-MEV_
approach which argues that, despite its negative externalities, MEV is endemic
in permissionless blockchains, and it can present some benefits for market
participant such as balanced liquidity pools and in some cases improvement of
the quality of executing transactions [32, 21, 13, 7]. However, despite the
intense interest of the blockchain community to resolve this problem, to the
best of our knowledge, a rigorous proposal with provable guarantees remains
elusive.

##### Our Approach and Results

In this work, we take a neutral approach towards MEV and propose a mechanism
to balance the two extreme positions by enshrining MEV sharing in the protocol
design. The key element to achieve this is a variable MEV _extraction rate_
that determines the fraction of MEV that is retained by miners,111Due to its
common usage, we will also use the term _miners_ for simplicity as a shorthand
of all parties that are involved in the process of _block creation_ and MEV
extraction, such as validators, block proposers, builders, searchers, etc.
[31]. with the rest going back to users. For example, an MEV extraction rate
of $1$ ($0$, resp.) means that all MEV goes to miners (users, resp.).

The aim of the MEV extraction rate is to balance the participation of users
and miners, expressed in monetary terms, to a predetermined target ratio and
is updated after every block following a dynamic update rule. The intensity of
the updates is regulated by an _adjustment parameter_ that is part of the
design space of the mechanism. In simple terms, the rule increases (decreases,
respectively) the MEV extraction rate if the ratio of miner/user participation
is lower (higher, respectively) than desired. To measure deviations from the
target, we consider the winning bid of the auction held for the creation of
the last block as a proxy of the extracted MEV. The estimation of MEV is not
_the_ focus of our model. In any case, current research suggests that winning
bids in MEV-boost auctions [12, 14] or experimental methods [4, 37, 3, 19]
provide accurate estimates or tight lower bounds to the actual MEV. This
dynamic update rule is inspired by the design principles of EIP-1559 that have
proven successful in regulating transaction fee markets under similar
conditions [38, 39, 26, 27], and the trade-off between the goals of users
(bidders) and miners (auctioneers) as studied in the context of conventional
auctions [29, 11].222It is worth highlighting that to measure deviations from
its target and to adjust its updates accordingly, EIP-1559 uses as (on-chain)
proxy of current demand the size of the last block [2].

Our objective in this paper is to study the evolution and performance of the
above mechanism. In particular, to contribute to the ongoing debate about MEV,
our motivating questions are the following: (1) can a protocol-determined,
dynamically-updated MEV extraction rate balance the conflicting participation
incentives of users and miners, and (2) can the design principles of EIP-1559
be useful in accomplishing this task?

Our findings provide significant evidence that we can answer both questions
affirmatively. Specifically, our formal analysis of the long-term evolution of
the dynamic mechanism shows that the system exhibits one of the following
behaviors depending on the intensity of the updates, as these are determined
by the value of the adjustment parameter: (i) _low_ intensities: convergence
to an optimal MEV extraction rate, which is strictly between 0 and 1 that
achieves the target participation ratio, (ii) _intermediate_ intensities:
periodic behavior or provable chaos (Theorem 4) or (iii) _large_ intensities:
collapse to extraction rates of either $0$ or $1$ in which case one of the two
groups of this two-sided economy abandons the system. In Theorem 1 and Theorem
2, we establish formal threshold bounds on the intensity of the updates for
which the system either avoids the degenerate states of $0$ and $1$ and
remains _live_ , i.e., avoids regime (iii), or equilibrates (stabilizes) at
the target MEV extraction rate, i.e., reaches regime (i), respectively.

Turning to system performance, we mathematically quantify deviations from
target participation levels and show that they remain bounded for a broad
range of realistic market conditions and parameterizations (Theorem 5). This
means that the system maintains a healthy proportion between miners and users.
The main takeaway is that this desirable behavior is formally exhibited not
only when the system equilibrates, i.e., when it stabilizes at the target
extraction rate, a condition that may rarely be met in practice, but, also,
when it operates under _out-of-equilibrium (chaotic) conditions_.

Technically, our paper is most closely related to the literature on complex
dynamics that have been previously studied in various theoretical settings
[17, 34, 35] and, recently, also in market [8] and blockchain settings [26,
27]. Our current work extends these frameworks to include rigorous guarantees
on market-liveness and long-term system performance. In stark contrast to
conventional equilibrium analysis, which offers little or no insight on non-
equilibrium regimes, such guarantees are critical to ensure the long-term
survival and robustness of systems with strong practical orientation [28].

##  2 Model

We consider a two-sided market consisting of users (or MEV generators) who
submit transactions on the one side and miners (or MEV extractors) who process
these transactions on the other side. Here, the term _miners_ is used as a
general term to describe all MEV extractors including proposers, builders,
searchers. The two sides are described in terms of their total participating
stake, denoted by $U$ for users and by $M$ for miners. In other words, $U$ and
$M$ denote the maximum potential participation of users and miners,
respectively, measured in monetary terms (transaction volume for users and
locked stake for miners) during the time-period under consideration.333It is
safe to assume that the amounts $U$ and $M$ remain relatively stable in the
short/medium-term, but can undergo more significant changes in the long-term.
The impact of such changes in the dynamics of the current paper is explained
in the appendix.

The participation of users and miners in the market depends on the current MEV
extraction rate, denoted by $\lambda\in[0,1]$. Higher (lower) values of
$\lambda$ indicate more (less) aggressive extraction by miners and, hence, a
more (less) hostile environment for users. Equivalently, one may think of
$1-\lambda$ as the fraction of MEV that is returned to users through rebates,
direct payments or other on or off-chain vehicles. Each user and miner has a
certain tolerance on $\lambda$. Thus, if a user (miner) sees an MEV extraction
rate higher (lower) than what they can tolerate, they do not participate in
the market. The _tolerance distributions_ of users and miners are denoted by
$F(\lambda)$ and $G(\lambda)$, respectively. Based on the above, the users
that participate in the market for a given $\lambda$ are
$U\cdot\bar{F}(\lambda)$, where $\bar{F}(\lambda):=1-F(\lambda)$ and the
miners are $M\cdot G(\lambda)$. We assume that the support of $F,G$ is
included in $[0,1]$ and that both $F,G$ are strictly increasing and
differentiable with probability density functions $f=F^{\prime},g=G^{\prime}$,
respectively.444The strict monotonicity and differentiability assumptions are
common in the economics literature and only serve to avoid edge cases that
complicate the analysis without adding much intuition. Our results readily
extend to more complex distributions.

To incorporate the MEV extraction rate as part of the protocol design space,
we make the following design choices:

  * •

The MEV extraction rate is dynamically updated after every block. This
generates a sequence of MEV extraction rates, $(\lambda_{t})_{t\geq 0}$, where
$t$ indicates the block height (time).

  * •

The intensity of the updates between consecutive blocks is regulated by a
parameter $d$ that can be chosen by the designer.

  * •

The designer can set a desired (optimal) balance between users and miners in
the market denoted by $T$ (target). This means that the designer is seeking to
find a $\lambda^{*}$ such that $U\cdot\bar{F}(\lambda^{*})=T\cdot M\cdot
G(\lambda^{*})$.

  * •

The boundary MEV extraction rates of $\lambda=0$ and $\lambda=1$ correspond to
steady states in which the two-sided market collapses to having either only
miners ($\lambda=1$) or only users ($\lambda=0$).

Based on the above desiderata, and inspired by EIP-1559 transaction fee
design, we consider the following dynamic update rule to govern the evolution
of the MEV extraction rates, $\lambda_{t}$:

$\lambda_{t+1}:=\lambda_{t}+d\lambda_{t}(1-\lambda_{t})\cdot(U\cdot\bar{F}(\lambda_{t})-T\cdot
M\cdot G(\lambda_{t})).$

In the update function of the right hand side, the terms
$\lambda_{t},(1-\lambda_{t})$ ensure that $\lambda=0$ and $\lambda=1$ are
fixed points of the system and the term $U\cdot\bar{F}(\lambda_{t})-T\cdot
M\cdot G(\lambda_{t})$ serves as an indicator of the current performance of
the system, i.e., how far the system is from its objective. As mentioned
above, there are two critical parameters that a designer can utilize: $T$
which is the desired ratio between users and miners in the system, and
parameter $d$ which calibrates the intensity of the updates. We can simplify
the above formula as:

$\lambda_{t+1}=\lambda_{t}+\eta\lambda_{t}(1-\lambda_{t})\cdot\left(\bar{F}(\lambda_{t})-wG(\lambda_{t})\right)$

where $\eta=U\cdot d$ and $w=T\cdot\frac{M}{U}$. Finally, we can reformulate
the above formula as

$\lambda_{t+1}=\lambda_{t}+\eta\lambda_{t}(1-\lambda_{t})\cdot\Delta_{w}(\lambda_{t})$
($\lambda$-MEV)

where $\Delta_{w}(\lambda_{t}):=\bar{F}(\lambda_{t})-wG(\lambda_{t})$. The
important design parameters in ($\lambda$-MEV) are the adjustment quotient,
$\eta$, and the modified target participation ratio, $w$, which is equal to
the initial target ratio $T$ multiplied by ratio of the maximum users
participation over the maximum miners participation. For our purposes, it will
also be helpful to define the update function $h:[0,1]\to[0,1]$ as

$h(\lambda)=\lambda+\eta\lambda(1-\lambda)\Delta_{w}(\lambda).$ (1)

Whenever obvious, we will omit the dependence of $\Delta_{w}(\lambda)$ on $w$,
and we will simply write $\Delta(\lambda)$ or $\Delta(\lambda_{t})$ when time
is relevant.

##  3 Evolution of the Dynamic MEV Extraction Rates

The analysis of the above dynamical system, defined by ($\lambda$-MEV), will
be our main focus. Our first observation is that under our working assumptions
on $F$ and $G$, the target function $\Delta(\lambda)$ satisfies some useful
properties.

######  Property 1 (Properties of Target Function $\Delta(\lambda)$).

Let $w>0$, and assume that $F,G$ are differentiable and strictly increasing
with support in $[0,1]$. Then, the function
$\Delta(\lambda):=\bar{F}(\lambda)-wG(\lambda)$ satisfies the following
properties:

  1. (i)

$\Delta(\lambda)$ is strictly decreasing for $\lambda\in[0,1]$ with
$\Delta(0)=1$, and $\Delta(1)=-w$.

  2. (ii)

There exists a unique $\lambda^{*}\in(0,1)$ such that $\Delta(\lambda^{*})=0$.

  3. (iii)

There exists a function $K:[0,1]\to\mathbb{R}$ so that
$\Delta(\lambda)=(\lambda-\lambda^{*})\cdot K(\lambda)$, and
$\lim\limits_{x\to\lambda^{*}}K(x)<+\infty$.

###### Proof.

Properties $(i),(ii)$ and the existence part of $(iii)$ are straightforward
and their proof is omitted. To see why the second part of $(iii)$ holds,
observe that since $F,G$ are differentiable in $(0,1)$, de l’Hôpital’s rule
implies that

$\displaystyle\lim_{\lambda\to\lambda^{*}}K(\lambda)$
$\displaystyle=\lim_{\lambda\to\lambda^{*}}\frac{\Delta(\lambda)}{\lambda-\lambda^{*}}=\lim_{\lambda\to\lambda^{*}}\left(-f(\lambda)-wg(\lambda)\right)$
$\displaystyle=-f(\lambda^{*})-wg(\lambda^{*})<+\infty$

which holds since the probability distribution functions exist and are
continuous by assumption. ∎

Based on the above, our first observation is that, by construction,
($\lambda$-MEV) has 3 fixed points. These are $\lambda=0$, $\lambda=1$ and a
unique (under our assumptions) interior point $\lambda^{*}\in(0,1)$ such that
$\Delta_{w}(\lambda^{*})=0$. The first two fixed points are considered
_degenerate states_ of the system: the former corresponds to no MEV extraction
and, thus, miners are not willing to participate. The latter corresponds to a
state in which MEV extraction rate is so high that no user is willing to
interact with the blockchain.

We are primarily interested in analyzing the evolution of the dynamics with
respect to the third fixed point, i.e., the unique interior fixed point which
corresponds to the optimal state of the system, i.e., target levels of
user/miner participation. Before we proceed with our results about the
behavior of ($\lambda$-MEV), we introduce a necessary definition.

######  Definition 1 (Directionally Stable Fixed Point).

Let $\\{\lambda_{t}\\}_{t\geq 0}$ be a one-dimensional dynamical system
determined by a function $h:\mathbb{R}\to\mathbb{R}$, so that
$\lambda_{t+1}:=h(\lambda_{t})$, and let $\lambda^{*}$ be a fixed point of
$h$, i.e., $h(\lambda^{*})=\lambda^{*}$. Then $\lambda^{*}$ is called
_directionally stable_ for $\\{h_{t}\\}_{t\geq 0}$, if for every $t\geq 0$
such that $\lambda_{t}\neq\lambda^{*}$ it holds that
$\left(h(\lambda_{t})-\lambda_{t}\right)/(\lambda_{t}-\lambda^{*})<0$ where
$h(\lambda_{t})=\lambda_{t+1}$ for every $t\geq 0$.

In other words, if a fixed point is _directionally stable_ then the dynamical
system moves towards the direction of this fixed point at every iteration. We
will now proceed to show that in ($\lambda$-MEV), the fixed point
$0<\lambda^{*}<1$ is _directionally stable_.

######  Lemma 1.

The unique interior fixed point, $\lambda^{*}$ of the ($\lambda$-MEV) is
directionally stable, i.e., it holds that $h(\lambda)>\lambda$ whenever
$0<\lambda<\lambda^{*}$ and $h(\lambda)<\lambda$ whenever
$\lambda^{*}<\lambda<1$.

###### Proof.

First, for $\lambda>\lambda^{*}$ it holds that $\Delta_{w}(\lambda)<0$ since
$\Delta_{w}(\lambda^{*})=0$ and $\Delta_{w}$ is a decreasing function and
hence $h(\lambda)-\lambda=\lambda(1-\lambda)\Delta_{w}(\lambda)<0$. Thus,
$(h(\lambda)-\lambda)/(\lambda-\lambda^{*})<0$. Similarly, for
$\lambda<\lambda^{*}$ it holds that $\Delta_{w}(\lambda)>0$ and subsequently
$(h(\lambda)-\lambda)/(\lambda-\lambda^{*})<0$. This concludes the proof that
$\lambda^{*}$ is _directionally stable_. ∎

Directional stability secures that whenever a trajectory reaches a point close
to a degenerate state, i.e., $\lambda=0$ or $\lambda=1$, the next step will
move towards the fixed point $\lambda^{*}$. Due to the factors $\lambda$
(resp. $(1-\lambda)$), the updates close to the boundaries will be small. This
trend will continue until the trajectory reaches or overshoots the fixed point
$\lambda^{*}$.

An important implication of the previous observation is that the system can
reach the degenerate state $\lambda=0$ at time $t+1$ only when
$\lambda_{t}>\lambda^{*}$, and similarly, it can reach the degenerate state
$\lambda=1$ only when $\lambda_{t}<\lambda^{*}$. Further elaborating on this,
we can derive a condition on the intensity of system updates, to ensure that
these two boundary fixed points will _never_ be reached by the dynamics, for
any interior starting point, $\lambda_{0}\in(0,1)$.

######  Theorem 1.

(Market-Liveness) For any pair of tolerance distributions, $F,G$, target $w$,
and starting point $\lambda_{0}\in(0,1)$, it holds that $\lambda_{t}\in(0,1)$
for any $t>0$ as long as

$\eta<\min{\left\\{\frac{1}{w(1-\lambda^{*})},\frac{1}{\lambda^{*}}\right\\}}.$

where $\lambda^{*}$ is the unique solution of the equation $\Delta(\lambda)=0$
in $(0,1)$.

###### Proof.

We first consider the case of avoiding the fixed point at $\lambda=0$. If
$0<\lambda_{t}<\lambda^{*}$ for some $t\geq 0$, then by Lemma 1, it holds that
$\lambda_{t+1}=h(\lambda_{t})>\lambda_{t}$ which implies, in particular, that
$h(\lambda_{t})\neq 0$. Thus, for the dynamics to reach $0$, it must be the
case that $\lambda_{t}>\lambda^{*}$ and $h(\lambda_{t})\leq 0$ (recall that if
$h(\lambda_{t})<0$, then the $\lambda_{t+1}$ is projected back to $0$). To
exclude this case, we need to ensure that $h(\lambda_{t})>0$ for all
$\lambda>\lambda^{*}$ which can be written as

$\Delta(\lambda)>\frac{-1}{\eta(1-\lambda)},\quad\text{for all
}1>\lambda>\lambda^{*}.$ (2)

Since $\Delta(\lambda)=\bar{F}(\lambda)-wG(\lambda)>-w$, and
$\frac{-1}{\eta(1-\lambda^{*})}>\frac{-1}{\eta(1-\lambda)}$ for all
$1>\lambda>\lambda^{*}$, a sufficient condition for the inequality in (2) to
hold is that $-w>\frac{-1}{\eta(1-\lambda^{*})}$ or equivalently that
$\eta<\frac{1}{w(1-\lambda^{*})}$. The case of the fixed point at $\lambda=1$
is similar. In particular, if $\lambda^{*}<\lambda_{t}<1$ for some $t\geq 0$,
then, by Lemma 1, the dynamics will decrease in the next step. Thus, the only
case for ($\lambda$-MEV) to reach $1$ is that $\lambda_{t}<\lambda^{*}$ and
$h(\lambda_{t})\geq 1$. As before, to exclude this case, we need to ensure
that $h(\lambda_{t})<1$ for all $0<\lambda<\lambda^{*}$ which can be written
as

$\Delta(\lambda)<\frac{1}{\eta\lambda},\quad\text{for all
}0<\lambda<\lambda^{*}.$ (3)

Since $\Delta(\lambda)=\bar{F}(\lambda)-wG(\lambda)<1$, and
$\frac{1}{\eta\lambda}>\frac{1}{\eta\lambda^{*}}$ for all
$0<\lambda<\lambda^{*}$, a sufficient condition for the inequality in (3) to
hold is that $1<\frac{1}{\eta\lambda^{*}}$ or equivalently that
$\eta<\frac{1}{\lambda^{*}}$. Putting these two conditions together, we obtain
the claim, namely that
$\eta<\min{\left\\{\frac{1}{\lambda^{*}},\frac{1}{w(1-\lambda^{*})}\right\\}}$.
∎

Theorem 1 provides a sufficient condition for the values of the control
parameter $\eta$ for which ($\lambda$-MEV) never reaches the degenerate states
$\lambda\in\\{0,1\\}$. We next turn our attention to the evolution of the
system for lower values of $\eta$, i.e., for values of $\eta$ below the upper
bound of Theorem 1. As we show in Theorem 2, at the other extreme, i.e., for
sufficiently low values of $\eta$, the ($\lambda$-MEV) converge to the fixed
point $\lambda^{*}$.

######  Theorem 2 (Lazy Updates - Convergence to $\lambda^{*}$).

For any pair of tolerance distributions, $F,G$, target $w$ and starting point
$\lambda_{0}\in(0,1)$, the sequence $\\{\lambda_{t}\\}_{t>0}$ generated by
($\lambda$-MEV) converges to $\lambda^{*}$, for any control parameter $\eta>0$
such that

$\eta\leq\inf_{\lambda\neq\lambda^{*}}\frac{\left(\lambda^{*}+\lambda\right)}{\lambda^{2}(1-\lambda)K(\lambda)}.$

###### Proof.

We will prove that the function
$\Phi(\lambda):=(\ln{\lambda}-\ln{\lambda^{\ast}})^{2}$ decreases along any
trajectory of the dynamics, i.e., that $\Phi(h(\lambda))<\Phi(\lambda)$ for
any $\lambda\in(0,1)$ with $\lambda\neq\lambda^{*}$. This is equivalent to
showing that

$(\ln
h(\lambda)-\ln\lambda^{\ast})^{2}<(\ln\lambda-\ln\lambda^{\ast})^{2},\text{
for any }\lambda\neq\lambda^{\ast}.$ (4)

In other words, we will show that $\Phi(\lambda)$ acts as a potential function
for the ($\lambda$-MEV) which directly implies convergence. To proceed, we
rewrite the above inequality as $(\ln
h(\lambda)-\ln\lambda^{\ast})^{2}-(\ln\lambda-\ln\lambda^{\ast})^{2}=(\ln
h(\lambda)-\ln\lambda)\cdot(\ln
h(\lambda)+\ln\lambda-2\ln\lambda^{\ast})=\ln\left(\frac{h(\lambda)}{\lambda}\right)\cdot\ln\left(\frac{\lambda
h(\lambda)}{(\lambda^{\ast})^{2}}\right).$ We will consider two cases,
depending on whether $\lambda<\lambda^{\ast}$ or $\lambda>\lambda^{\ast}$:

  * •

$\lambda<\lambda^{*}$: In this case, it holds that $h(\lambda)>\lambda$, see
Definition 1 and Lemma 1, which implies that $h(\lambda)/\lambda>1$. Hence,
$\ln\left(\frac{h(\lambda)}{\lambda}\right)>0$. Thus, to obtain the desired
inequality, we need to select $\eta>0$ so that the argument of the second
$\ln$ on the right hand side of the above equation is less than $1$, i.e.,
$\lambda h(\lambda)/(\lambda^{*})^{2}<1$. Solving this inequality for $\eta$
yields the inequality
$\eta<\frac{(\lambda^{*}-\lambda)(\lambda^{*}+\lambda)}{\lambda^{2}(1-\lambda)\Delta(\lambda)}$,
where we used that $\Delta(\lambda)>0$ for all $\lambda<\lambda^{*}$. Since
this inequality must hold for any $\lambda<\lambda^{*}$, we obtain the
threshold
$\eta\leq\inf\limits_{\lambda<\lambda^{*}}\frac{(\lambda^{*}-\lambda)(\lambda^{*}+\lambda)}{\lambda^{2}(1-\lambda)\Delta(\lambda)}$.
It remains to show that the threshold on the right hand side remains strictly
positive for any distributions. Using Property 1, we can rewrite the threshold
as
$\inf\limits_{\lambda<\lambda^{*}}\frac{\lambda^{*}+\lambda}{\lambda^{2}(1-\lambda)K(\lambda)}$.
Since $\Delta(0)=1$, it follows that $K(0)=1/\lambda^{*}$ which implies that
$\lim\limits_{\lambda\to
0^{+}}\frac{\lambda^{*}+\lambda}{\lambda^{2}(1-\lambda)K(\lambda)}=+\infty$,
and
$\lim\limits_{\lambda\to(\lambda^{*})^{-}}\frac{\lambda^{*}+\lambda}{\lambda^{2}(1-\lambda)K(\lambda)}=\frac{2}{\lambda^{*}(1-\lambda^{*})\cdot\lim\limits_{\lambda\to(\lambda^{*})^{-}}K(\lambda)}$.
Thus, since $\lim\limits_{\lambda\to(\lambda^{*})^{-}}K(\lambda)$ exists and
is less than $+\infty$ by Property 1, the claim follows.

  * •

$\lambda>\lambda^{*}$: in this case, it holds that $h(\lambda)<\lambda$, see
Definition 1 and Lemma 1, which implies that $h(\lambda)/\lambda<1$. Hence,
$\ln\left(\frac{h(\lambda)}{\lambda}\right)<0$. Thus, to obtain the desired
inequality in (4), we need to select $\eta>0$ so that the argument of the
second $\ln$ on the right hand side of the above equation is larger than $1$,
i.e., $\lambda h(\lambda)/(\lambda^{*})^{2}>1$. Solving this inequality for
$\eta$ yields the same inequality as above (note that this time,
$\Delta(\lambda)<0$).

Putting the two cases together, we have that the ($\lambda$-MEV) converge to
$\lambda^{*}$ whenever
$0<\eta\leq\inf\limits_{\lambda\neq\lambda^{*}}\frac{(\lambda^{*}+\lambda)}{\lambda^{2}(1-\lambda)K(\lambda)}$
as claimed. ∎

Together with Theorem 1, Theorem 2 paints the picture for extreme values of
update intensities, i.e., either sufficiently high (degenerate market), in
which the system collapses by reaching either of the two boundary fixed
points, or sufficiently low (lazy updates), in which case the system
stabilizes to the unique optimal state. Intuitively, the sufficiency condition
of Theorem 2 suggests that convergence to $\lambda^{*}$ occurs when the
intensity of the updates is small enough _in comparison_ to current market
conditions as these are captured by function $\Delta(\lambda)$, i.e., the
distributions of users’ and miners’ tolerances. However, low values of the
control parameter $\eta$ correspond to a system with slow responses and come
at the trade-off of slow responses to otherwise fast-changing market
conditions. Thus, such values may be irrelevant for practical applications
[40].

This leaves open the case in which this condition is not satisfied.
Interestingly, as we show next, in this case, the system may exhibit
arbitrarily complex behavior. Specifically, in Theorem 4, we prove that for
any value of $\eta$, there exist distributions $F,G$, satisfying our
assumptions, for which the system does not converge to $\lambda^{*}$. In
particular, based on the aggressiveness of the updates, the system can become
provably chaotic in the sense of _Li-Yorke chaos_ [30]. Counterintuitively, as
we show in Section 4, this chaotic behavior is not necessarily detrimental to
system performance. Although chaotic updates in the MEV extraction rates make
their evolution unpredictable, they keep the system non-degenerate, in the
sense that they remain bounded away from $0$ and $1$, thus, ensuring
participation of both users and miners. Before we proceed to prove that the
($\lambda$-MEV) become Li-Yorke chaotic, we introduce new necessary notations
and definitions. Our technical tools in this part extend the framework of
[26].

######  Definition 2 (Periodic Orbit and Points).

Consider a dynamical system, $(\lambda_{t})_{t\geq 0}$, defined by a function
$h:\mathbb{R}\to\mathbb{R}$ so that $\lambda_{t+1}=h(\lambda_{t})$ for any
$t\geq 0$. A sequence $\lambda_{1},\lambda_{2},\dots,\lambda_{k}$ is called a
_periodic orbit of length $k$_ of the dynamical system, if
$\lambda_{t+1}=h(\lambda_{t})$ for $1\leq t\leq k-1$ and
$h(\lambda_{k})=\lambda_{1}$. Each point
$\lambda_{1},\lambda_{2},\dots,\lambda_{k}$ is called periodic point of least
period $k$.

######  Definition 3 (Li-York chaos [30]).

Let $X=[L,U]$ be a compact interval in $\mathbb{R}$ and let $h:X\to X$ define
a discrete-time dynamical system $(x_{t})_{t\geq 0}$ on $X$, so that
$x_{t}\vcentcolon=h^{t}(x_{0})$ for any $x_{0}\in X$. The dynamical system
$(x_{t})_{t\in N}$ is called _Li-Yorke chaotic_ if it holds that

  1. 1.

For every $k=1,2,\dots$ there is a periodic point in $X$ with period $k$.

  2. 2.

There is an uncountable set $S\subseteq X$ (containing no periodic points),
which satisfies the following conditions:

     * •

For every $x\neq y\in S$, $\lim\limits_{t\to\infty}\sup\lvert
h^{t}(x)-h^{t}(y)\rvert{}>0$ and $\lim\limits_{t\to\infty}\inf\lvert
h^{t}(x)-h^{t}(y)\rvert{}=0$.

     * •

For every points $x\in S$ and $y\in X$, $\lim\limits_{t\to\infty}\sup\lvert
h^{t}(x)-h^{t}(y)\rvert{}>0$.

Famously, a sufficient condition for a system to be _Li-Yorke chaotic_ is to
have a periodic point of period of $3$ [30]. This implication relies on the
Sharkovsky Ordering and and it is actually a special case of Sharkovsky’s
theorem which is presented next [41].

######  Definition 4 (Sharkovsky Ordering [5]).

The following ordering of the set $\mathbb{N}$ of positive integers, is known
as the Sharkovsky ordering: $3>5>7>9>\cdots$, $3\cdot 2>5\cdot 2>7\cdot
2>9\cdot 2>\cdots$, $3\cdot 2^{n}>5\cdot 2^{n}>7\cdot 2^{n}>9\cdot
2^{n}>\cdots$, and $\cdots>2^{n}>\cdots>2^{2}>2>1$. This is a total ordering,
i.e., every positive integer appears exactly once in the list.

######  Theorem 3 (Sharkovsky [41]).

Let $X\subset\mathbb{R}$ be a compact interval and let $h:X\to X$ be a
continuous function. If $h$ has a periodic point of least period $m$, then $h$
has a periodic point of least period $l$ for every $l$ such that $m>l$ in the
Sharkovsky ordering.

Using these results, we will now formally prove that for every intensity,
$\eta$, in the MEV extraction rate updates, there exist tolerance
distributions $F,G$ such that ($\lambda$-MEV) become _Li-York chaotic_.

######  Theorem 4 (Chaotic Updates).

For every intensity level, $\eta>0$, there exist continuous distributions
$F,G$ such that the ($\lambda$-MEV) become Li-Yorke chaotic.

Figure 1: Bifurcation diagram for the ($\lambda$-MEV) with respect to the
adjustment intensity $\eta$ (left panel). The tolerance distributions of
miners and users are shown in the panel on the right. For low values of
$\eta$, the dynamics converge to $\lambda^{*}$ (Theorem 2), whereas for large
values of $\lambda$, the dynamics reach the boundary and get trapped at the
corresponding fixed point (in this case $0$) in which case, the two-sided
market collapses (Theorem 1). For intermediate values of $\eta$, the dynamics
are either provably chaotic (Theorem 4) or periodic with periods of difference
density (Section 3.1.1).

###### Proof.

The proof is constructive and proceeds by creating an instance for which the
dynamics have a periodic point of least period $3$. The instance depends on
the $\eta$ and thus, it can apply to any $\eta>0$. Specifically, we consider a
case in which both users and miners have the same tolerance distributions on
$[a,b]$ with $0<a<b<1$. In this case, the update function (1) can be written
as

$h(\lambda)=\begin{cases}\lambda(-\eta\lambda+\eta+1),&0\leq\lambda<a,\\\
\lambda(-\eta\lambda\Delta(\lambda)+\eta\Delta(\lambda)+1),&a\leq\lambda<b,\\\
\lambda(\eta w\lambda-\eta w+1)&b\leq\lambda\leq 1,\end{cases},$

with $\Delta(\lambda):=\frac{b+wa-\lambda(1+w)}{b-a}$. We will prove that for
any fixed $\eta>0$, there exist $a,b$ such that the dynamical system has a
periodic orbits of period 3. Since the update function, $h$ is continuous, we
will prove that by finding points
$\lambda_{0},\lambda_{1},\lambda_{2},\lambda_{3}$ such that

$\lambda_{1}=h(\lambda_{0}),\,\,\lambda_{2}=h(\lambda_{1})=h^{2}(\lambda_{0}),\,\,\lambda_{3}=h(\lambda_{2})=h^{3}(\lambda_{0})$

which satisfy $\lambda_{3}\leq\lambda_{0}<\lambda_{1}<\lambda_{2}$. Firstly,
for $0\leq\lambda<a$, the update function $h$ is increasing and
$\lambda<h(\lambda)$. Thus, there exists $0<\lambda_{0}<a$ such that
$h(\lambda_{0})=a$. Let $\lambda_{1}=a,\lambda_{2}=h(a)$ and
$\lambda_{3}=h^{2}(a)$. Then, $h(a)=a+\eta a(1-a)$ and, hence, $a<h(a)$.
Moreover $\Delta(h(a))=1-\eta c$, where $c=\frac{a(1-a)(1+w)}{b-a}$ and, as
$b\rightarrow a^{+}$, $c\rightarrow\infty$. Lastly, we compute

$h(h(a))=h(a)+\eta h(a)(1-h(a))\Delta(h(a)).$

After some trivial algebra, we get that

$h(h(a))=-(h(a)-(h(a))^{2})\eta^{2}c+(h(a)-(h(a)^{2}))\eta+h(a).$

We want to prove that $\lambda_{3}=h(h(a))\leq\lambda_{0}$ which is equivalent
to

$-(h(a)-(h(a))^{2})\eta^{2}c+(h(a)-(h(a)^{2}))\eta+h(a)\leq\lambda_{0}.$

This is true as $b\rightarrow a^{+}$, since $h(a)-(h(a))^{2}>0$. Thus we get
$\lambda_{3}\leq\lambda_{0}<\lambda_{1}<\lambda_{2}$, which concludes the
proof. ∎

Intuitively, the construction in the proof of Theorem 4 suggests that as the
tolerance of users and/or miners becomes more concentrated around some value,
then the dynamics are more likely to become chaotic since even small updates
in the MEV extraction rate have significant impact in the participation rates
of the system.

###  3.1 Visualizations

In order to gain a better intuition on the results of the previous section, in
this part, we provide visualizations of the ($\lambda$-MEV) that showcase the
various cases discussed above.

To instantiate the dynamics, we select specific instances of Beta
distributions for both users and miners because these distributions have the
desirable properties that (1) their support range is $[0,1]$, matching the
values of possible MEV extraction tolerance for miners and users, and (2) they
have the expressive capacity to capture many different forms of valuation
distributions. The results for a specific instance are provided in the
_bifurcation diagram_ in Figure 1.

In the bifurcation diagram, the horizontal axis corresponds to values of the
update instensity, $\eta$. It is straightforward to see, that for any fixed
value of $\eta$, the ($\lambda$-MEV) either converge (low values of $\eta)$ or
are restricted in a bounded area around $\lambda^{*}$. This bounded area
expands as $\eta$ increases till it reaches one of the boundaries (in this
case $\lambda=0$) which implies that the dynamics get trapped there (with dire
implications for the two-sided market).

The instance depicted in the left panel of Figure 1 captures all cases
described in our previous formal analysis. From Figure 1, we can also see some
regimes, e.g., around $\eta\approx 0.4$, in which the dynamics only have
_periodic orbits_. In fact, using Theorem 3, we can further argue about the
behavior of the system in these cases.

####  3.1.1 Regions of Periodic Orbits

To better understand the orbits of Equation $\lambda$-MEV, we visualize the
function $h^{(k)}(\lambda)$ and observe its fixed points,
$\lambda=h^{(k)}(\lambda)$ for various values of $k\in\mathbb{N}$. In
particular, Theorem 3 implies that if we can find $k\geq 2\in\mathbb{N}$ such
that the function $h(\lambda)$ has periodic points of least period $k$, but
_not_ of least period $k+1$, then the dynamics generated by $h$ have periodic
orbits with all possible periods that precede $k$ in the Sharkovsky ordering.
As mentioned above, such regimes are visible around $\eta=0.4$ in Figure 1.

In Figure 2, we visualize one such instance that corresponds to areas of
periodic behavior for $\eta=0.5$. Note that for $\eta$ slightly larger than
$0.6$, the bifurcation diagram shows a range of parameter values for which
apparently the only cycle has period $3$. In fact, according to Theorem 3,
there must be cycles of all periods there, but they are not stable and
therefore not visible on the computer-generated simulation.

Figure 2: Regions of periodic behavior for the instance of Figure 1 and
$\eta=0.6$. The panels indicate the existence of periodic points of least
period $k=5$ and $k=7$, but not of $k=3$ (the period $k$ is denoted in the
legends). According to Theorem 3, this means that for these parameters, the
($\lambda$-MEV) have periodic points of uncountably many periods but are not
provably chaotic. Similarly, we can create instances with periods of $k=4,2$
and $1$ but not of $k=8$ etc.

Moreover, in Figure 3, we visualize the long term behavior of ($\lambda$-MEV)
with respect to

Figure 3: Bifurcation diagram for the ($\lambda$-MEV) with respect to the
range of tolerance distributions. We observe here the route from chaos to
order as the range of the tolerance increases.

the range of tolerance distributions. In this case, we keep the adjustment
quotient $\eta$ fixed. Now, the horizontal axis corresponds to the varying
parameter $r$, which is the range of the tolerance distributions. For this
simulation, we use uniform distributions for the tolerances $F$ and $G$ on the
interval $[L_{u},U_{u}]=[0.5-r,0.5+r]$ and $[L_{m},U_{m}]=[0.4-r,0.4+r]$,
respectively. Intuitively, as the area between the probability of density
distributions $\bar{F}^{\prime}$ and $G^{\prime}$ increases, the dynamics
($\lambda$-MEV) tend to stabilize. Interestingly, Figure 3, serves also as a
visualization of Theorem 4, which states that for each adjustment quotient
$\eta$, there are tolerance distributions $F,G$ such that the ($\lambda$-MEV)
become chaotic. We note that, we use uniform distributions for simplicity,
nevertheless simulations with different distributions produce substantially
the same outcome.

##  4 Performance

In the previous section, we analyzed the long-term performance of
($\lambda$-MEV). We studied three properties regarding the evolution of the
system; conditions for convergence, conditions for not reaching the boundaries
and conditions for chaotic behavior. Having established these, we now turn to
the performance of the system in these regimes. As mentioned above, low values
of the control parameter, $\eta$, allow the dynamics to stabilize at the fixed
point $\lambda^{*}$. However, such values may lead to slow responses of the
system against fast-changing market conditions and may be rarely applicable in
practice. Moreover, the intensity of the updates is only relative to
prevailing market conditions which may be subject to continuous changes.

Thus, the important question that we need to study is whether the system can
exhibit desirable performance even when it is pushed beyond the stable regime,
into the densely periodic or chaotic regimes. This is critical to ensure near-
optimality, i.e., bounded deviations from target, long-term survival, i.e.,
market-liveness, and ultimately, persistence of the system under possibly
adverse conditions. Interestingly, our analysis will not only establish such
desirable behavior in out of equilibrium conditions, but also provide formal
guarantees (rigorous proofs) that this will be the case.

###  4.1 Bounded Deviations and Average-Case Performance

Remember that, based on Sharkovsky’s Theorem (Theorem 3) the further left a
period point of $m$ is in the Sharkovsky’s ordering (Definition 4), the denser
the periodic orbits are. Thus, to measure the performance of this system, we
study if the target ratio $w$ is achieved on average, or equivalently the
time-average behavior of the values of the target function, i.e.,
$\lim\limits_{T\to\infty}\frac{1}{T}\sum_{t=1}^{T}\Delta(\lambda)$.

In Figure 4, we visualize the values of $\Delta(\lambda_{t})$ for the instance
of Figure 1. As the control parameter $\eta$ increases, the average deviations
increase. This trend (deviation from average optimal performance) continues
until the dynamics hit the boundary leading to non-liveness. The simulations
suggest that optimal values the control parameter $\eta$, as far as achieving
the target $w$ on average is concerned, correspond to regimes with chaotic or
complex - periodic with dense periods - evolution of the MEV extraction rate.
From a practical perspective, such regimes ensure _market-liveness_ along with
bounded deviations from the target (see Lemma 2).

Figure 4: Performance of the ($\lambda$-MEV) in the instance of Figure 1, cf.
left panels in both Figures. The simulations use a burn-in period of $200$
iterations followed by $T=200$ iterations that are plotted in the bifurcation
diagrams. The right panel shows the values of the target,
$\Delta(\lambda_{t})$, for each $t=201,\dots,400$ in blue dots and the
averages over this period in light-colored dots (see also the legend). As
$\eta$ grows, the deviations from the target also grow till the dynamics hit
the boundary and get absorbed there.

In the remainder of this section, we quantify these empirical results
regarding system performance. Our first observation is that the sequence of
MEV extraction rates, $(\lambda_{t})_{t\geq 0}$, remains within a bounded
neighborhood of $\lambda^{*}$, which depending on the size of $\eta$, can be
non-trivial (i.e., strictly between $0$ and $1$) even in the chaotic or
densely-periodic cases. This is formally stated in Lemma 2.

######  Lemma 2 (Attracting range of $\lambda$.).

Consider an instance of the ($\lambda$-MEV) with $\eta,w>0$ and tolerance
distributions $F,G$ with $\lambda^{*}\in(0,1)$ such that
$\Delta(\lambda^{*})=0$ as usual. Then, for the sequence of MEV extraction
rates $(\lambda_{t})_{t\geq 0}$ generated by the ($\lambda$-MEV) it holds that

$\max{\left\\{0,\lambda^{*}-\frac{w\eta}{4}\right\\}}\leq\lambda_{t}\leq\min{\left\\{\lambda^{*}+\frac{\eta}{4},1\right\\}}~{}\text{for
all~{}}t\geq 0.$

###### Proof.

By Lemma 1, we know that $h(\lambda)>\lambda$ only if $\lambda<\lambda^{*}$.
Thus,
$h(\lambda)=\lambda+\eta\lambda(1-\lambda)\Delta(\lambda)<\lambda^{*}+\eta\cdot\frac{1}{4}\cdot
1=\lambda^{*}+\frac{\eta}{4}$, where we used that $\Delta(\lambda)\leq 1$ for
all $\lambda\in[0,1]$. Similarly, to determine the lower bound, we need to
consider $\lambda>\lambda^{*}$. In this case, we have that

$\displaystyle h(\lambda)$
$\displaystyle=\lambda+\eta\lambda(1-\lambda)\Delta(\lambda)>\lambda^{*}+\eta\lambda(1-\lambda)\cdot(-w)>\lambda^{*}-\frac{\eta
w}{4},$

where we used that $\Delta(\lambda)\geq-w$ for all $\lambda\in[0,1]$. ∎

Note that the conditions $\lambda^{*}-\frac{w\eta}{4}>0$ whenever
$\eta<\frac{4\lambda^{*}}{w}$ and $\lambda^{*}+\frac{\eta}{4}<1$ whenever
$\eta<4(1-\lambda^{*})$. By Lemma 2, these inequalities offer an alternative
set of market-liveness conditions to those of Theorem 1.

The result of Lemma 2 is practically relevant when $\eta$ is small enough, so
that both comparisons are resolved with the non-trivial bounds, rather than
with $0$ and $1$. For such values of $\eta$, the sequence $\lambda_{t}$
remains bounded between $\lambda^{*}-\frac{w\eta}{4}$ and
$\lambda^{*}+\frac{\eta}{4}$. This allows us to bound the possible values of
$\Delta(\lambda_{t})$, i.e., the possible deviations from the target
$\Delta(\lambda^{*})=0$. To do so, we proceed with a concrete expression for
function $\Delta(\lambda)$ that uses the class of beta distributions for the
tolerances of both miners and users.555As mentioned above, beta distributions
have the desirable properties that (1) their support range is $[0,1]$,
matching the values of possible MEV extraction tolerance for miners and users,
and (2) they have the expressive capacity to capture many different forms of
valuation distributions. The result regarding performance is provided in
Theorem 5.

######  Theorem 5 (Bounded Deviations).

Consider an instance of the ($\lambda$-MEV) with $F\sim\text{Beta}(a,b)$,
$G\sim\text{Beta}(c,d)$ and $\eta,w$ such that
$0<\lambda^{*}-\frac{w\eta}{4}<\lambda^{*}+\frac{\eta}{4}<1$. Then, we have

$|\Delta(\lambda_{t})|\leq\frac{\eta(1+w)}{4}\left(\max_{\lambda\in(p,q)}{\\{f(\lambda)\\}}+w\max_{\lambda\in(p,q)}{\\{g(\lambda)\\}}\right),$

where $f,g$ denote the probability density functions of $F,G$ and
$p=\lambda^{*}-\frac{w\eta}{4},q=\lambda^{*}+\frac{\eta}{4}$.

###### Proof.

Since $\Delta(\lambda_{t})$ is decreasing, we have by Lemma 2 that

$\Delta\left(\lambda^{*}-\frac{w\eta}{4}\right)\geq\Delta(\lambda_{t})\geq\Delta\left(\lambda^{*}+\frac{\eta}{4}\right),$

for all $\lambda_{t}$’s that lie in the attracting region of the dynamics,
i.e., in $[\lambda^{*}-\frac{w\eta}{4},\lambda^{*}+\frac{\eta}{4}]$ provided
that $\eta$ is such that
$0<\lambda^{*}-\frac{w\eta}{4},\lambda^{*}+\frac{\eta}{4}<1$. This implies
that

$\displaystyle|\Delta(\lambda_{t})|$
$\displaystyle\leq\max\left\\{\left|\Delta\left(\lambda^{*}-\frac{w\eta}{4}\right)\right|,\left|\Delta\left(\lambda^{*}+\frac{\eta}{4}\right)\right|\right\\}$
$\displaystyle\leq\left|\Delta\left(\lambda^{*}-\frac{w\eta}{4}\right)\right|+\left|\Delta\left(\lambda^{*}+\frac{\eta}{4}\right)\right|\,.$

Using the Mean Value Theorem, and the fact that $\Delta(\lambda^{*})=0$, we
have that

$\Delta\left(\lambda^{*}-\frac{w\eta}{4}\right)=\Delta\left(\lambda^{*}-\frac{w\eta}{4}\right)-\Delta(\lambda^{*})=-\Delta^{\prime}(x_{1})\cdot\frac{w\eta}{4},$

for some $x_{1}\in\left(\lambda^{*}-\frac{w\eta}{4},\lambda^{*}\right)$, and
similarly that
$\Delta\left(\lambda^{*}+\frac{\eta}{4}\right)=\Delta\left(\lambda^{*}+\frac{\eta}{4}\right)-\Delta(\lambda^{*})=\Delta^{\prime}(x_{2})\cdot\frac{\eta}{4}$,
for some $x_{2}\in\left(\lambda^{*},\lambda^{*}+\frac{\eta}{4}\right)$. Thus,
to obtain meaningful bounds for $\Delta(\lambda_{t})$, it suffices to bound
the derivative of the function $\Delta(\lambda_{t})$ for $\lambda_{t}$ in the
specified regions around $\lambda^{*}$. It holds that

$\displaystyle|\Delta^{\prime}(x)|$
$\displaystyle=|\bar{F}^{\prime}(x)-wG^{\prime}(x)|=|-f(x)-wg(x)|$
$\displaystyle\leq\frac{1}{B(a,b)}x^{a-1}(1-x)^{b-1}+\frac{w}{B(c,d)}x^{c-1}(1-x)^{d-1}=H(x).$

Note that all coefficients in this expression are positive. Thus, we can
complete the proof since

$\displaystyle|\Delta(\lambda_{t})|$
$\displaystyle\leq\left|\Delta\left(\lambda^{*}-\frac{w\eta}{4}\right)\right|+\left|\Delta\left(\lambda^{*}+\frac{\eta}{4}\right)\right|$
$\displaystyle\leq\frac{\eta
w}{4}\cdot\max_{x\in(p,\lambda^{*})}\\{H(x)\\}+\frac{\eta}{4}\cdot\max_{x\in(\lambda^{*},q)}\\{H(x)\\}$
$\displaystyle\leq\frac{\eta(1+w)}{4}\cdot\max_{x\in(p,q)}\\{H(x)\\}$

which is less or equal than

$\displaystyle\frac{\eta(1+w)}{4}\cdot$
$\displaystyle\left(\frac{1}{B(a,b)}\cdot\max_{x\in(p,q)}\\{x^{a-1}(1-x)^{b-1}\\}\right.\left.+\frac{w}{B(c,d)}\cdot\max_{x\in(p,q)}\\{x^{c-1}(1-x)^{d-1}\\}\right)$

as claimed. ∎

The next Lemma offers an exhaustive case analysis for the maximization
problems that are involved in Theorem 5. Our intention is to use this lemma
for values of $p=\lambda^{*}-\frac{w\eta}{4}$ and
$q=\lambda^{*}+\frac{\eta}{4}$, to find analytically the maxima of the
probability density functions $f$ and $g$ of the beta distributions as used in
Theorem 5.

######  Lemma 3 (Maxima of Beta distributions).

Let $p,q\in(0,1)$ and $p<q$. Suppose that we are given a function
$f:[0,1]\rightarrow{\mathcal{R}}$, such that $f(x)=x^{a-1}\cdot(1-x)^{b-1}$,
for some fixed constant values $a,b\in\mathcal{R}$, and for any $x\in[0,1]$.
Then we have the following: Suppose first that $a+b\not=2$.

  * •

If $2>a+b$ and

    * –

$\frac{1-a}{2-a-b}<p$, then $f$ has the maximum value in $[p,q]$ that is
$f(q)$; i.e. $\forall x\in[p,q]:f(x)\in(0,f(q)]$.

    * –

$p\leq\frac{1-a}{2-a-b}\leq q$, then $f$ has the maximum value in $[p,q]$ that
is $\tau=\max\\{f(p),f(q)\\}$; i.e. $\forall x\in[p,q]:f(x)\in(0,\tau]$.

    * –

$\frac{1-a}{2-a-b}>q$, then $f$ has the maximum value in $[p,q]$ that is
$f(p)$; i.e. $\forall x\in[p,q]:f(x)\in(0,f(p)]$.

  * •

If $2<a+b$ and

    * –

$\frac{1-a}{2-a-b}<p$, then $f$ has the maximum value in $[p,q]$ that is
$f(p)$; i.e. $\forall x\in[p,q]:f(x)\in(0,f(p)]$.

    * –

$p\leq\frac{1-a}{2-a-b}\leq q$, then $f$ has the maximum value in $[p,q]$ that
is $\gamma=f\left(\frac{1-a}{2-a-b}\right)$; i.e. $\forall
x\in[p,q]:f(x)\in(0,\gamma]$.

    * –

$\frac{1-a}{2-a-b}>q$, then $f$ has the maximum value in $[p,q]$ that is
$f(q)$; i.e. $\forall x\in[p,q]:f(x)\in(0,f(q)]$.

If $a+b=2$, then under $a>1$, function $f$ reaches it maximum at $x=q$, i.e.,
$\forall x\in[p,q]:f(x)\in(0,f(q)]$.

If $a+b=2$ and $a<1$, then function $f$ reaches it maximum at $x=p$, i.e.,
$\forall x\in[p,q]:f(x)\in(0,f(p)]$.

Finally, if $a+b=2$ and $a=1$, then $f(x)=1$ for all $x\in[p,q]$.

###### Proof.

Let us first observe that $f(x)>0$ for all $x\in[p,q]$. We have that

$\displaystyle f^{\prime}(x)$
$\displaystyle=(a-1)x^{a-2}(1-x)^{b-1}-x^{a-1}(b-1)(1-x)^{b-2}$
$\displaystyle=x^{a-2}(1-x)^{b-2}((2-a-b)x+a-1).$

We will now inspect the behavior of function $f$ for $x\in[p,q]\subset(0,1)$.
Then, because $x^{a-2}(1-x)^{b-2}>0$, unless $x\in\\{0,1\\}$, the monotonicity
of $f$ is determined by the sign of the linear function $(2-a-b)x+a-1$ of $x$.

Suppose first that $a+b\not=2$, then $f^{\prime}(\zeta)=0$ for
$\zeta=\frac{1-a}{2-a-b}$. If $a+b<2$, then the function $(2-a-b)x+a-1$ of $x$
is strictly increasing and its zero is $\zeta$. This implies that
$(2-a-b)x+a-1<0$ for $x<\zeta$ and $(2-a-b)x+a-1>0$ for $x>\zeta$. So function
$f$ is strictly decreasing when $x<\zeta$ and strictly increasing when
$x>\zeta$, achieving its minimum at $x=\zeta$.

We now consider the following cases:

  * •

$\zeta<p$: in this case function $f$ in increasing when $x\in[p,q]$, meaning
that it reaches its maximum at $x=q$

  * •

$p\leq\zeta\leq q$: in this case function $f$ in decreasing when
$x\in[p,\zeta]$, and increasing when $x\in[\zeta,q]$, meaning that it reaches
its maximum at $x=p$ or $x=q$, depending which of the two values $f(p)$,
$f(q)$ is larger.

  * •

$\zeta>q$: in this case function $f$ in decreasing when $x\in[p,q]$, meaning
that it reaches its maximum at $x=p$.

If $a+b>2$, then the function $(2-a-b)x+a-1$ of $x$ is strictly decreasing,
that is, $(2-a-b)x+a-1>0$ for $x<\zeta$ and $(2-a-b)x+a-1<0$ for $x>\zeta$.
Thus, function $f$ is strictly increasing when $x<\zeta$ and strictly
decreasing when $x<\zeta$, achieving its maximum at $x=\zeta$.

We now consider the following cases:

  * •

$\zeta<p$: in this case function $f$ in decreasing when $x\in[p,q]$, meaning
that it reaches its maximum at $x=p$.

  * •

$p\leq\zeta\leq q$: in this case function $f$ in increasing when
$x\in[p,\zeta]$, and decreasing when $x\in[\zeta,q]$, meaning that it reaches
its maximum at $x=\zeta$.

  * •

$\zeta>q$: in this case function $f$ in increasing when $x\in[p,q]$, meaning
that it reaches its maximum at $x=q$.

Let us finally consider the case when $a+b=2$. Then
$f^{\prime}(x)=x^{a-2}(1-x)^{b-2}(a-1)>0,$ when $a>1$ and $f^{\prime}(x)<0$
when $a<1$. In the former case, function $f$ is strictly increasing for
$x\in[p,q]$ and therefore reaches it maximum at $x=q$. In the latter case,
function $f$ is strictly decreasing for $x\in[p,q]$ and therefore reaches it
maximum at $x=p$. Finally, if $a+b=2$ and $a=1$, then $b=1$, therefore
function $f$ is a constant function $f(x)=1$ for all $x$. ∎

###  4.2 Expressiveness of the Dynamics

In this section, we demonstrate that the behavior of the dynamics
($\lambda$-MEV) is expressive enough, to capture the evolution of a family of
update rules. Firstly, we recall that ($\lambda$-MEV) have three fixed points:
$\lambda=0$ and $\lambda=1$, which correspond to the degenerate states of the
market, and the interior fixed point $\lambda^{*}$. However, when the dynamics
($\lambda$-MEV) tends to the boundaries, i.e., $\lambda\to 0$, or $\lambda\to
1$, the updates are slow because of the factors $\lambda$ (or $(1-\lambda)$,
respectively) in the update rule (1). However, the designer would want the
updates in one or both of the boundaries to be more aggressive in order to
quickly restore the desired balance between users and miners. Next, we show,
that the following update rules exhibit similar behavior compared to the
dynamic evolution of (1).

(a) $h(\lambda)=\lambda+\eta\lambda(1-\lambda)\Delta_{w}(\lambda)$

(b) $h_{1}(\lambda)=\lambda+\eta\lambda\Delta_{w}(\lambda)$

(c) $h_{2}(\lambda)=\lambda+\eta(1-\lambda)\Delta_{w}(\lambda)$

(d) $h_{3}(\lambda)=\lambda+\eta\Delta_{w}(\lambda)$

Figure 5: In all the above simulations, we use Normal distributions for both
the tolerance of users and miners on the extracted MEV rate. Specifically
$F\sim\mathcal{N}(\mu=0.4,\sigma^{2}=0.01)$ and
$G\sim\mathcal{N}(\mu=0.5,\sigma^{2}=0.01)$. In our toy example, the targeted
ratio between users and miners participation is $w=1.6$.

The purpose of the above simulations is to figure out if by removing the fixed
points $\lambda=0$ and $\lambda=1$ and allowing $\lambda$ to take arbitrary
value, the dynamics maintain a similar behavior. In order to do so, for all
the update rules $h,h_{1},h_{2},h_{3}$, the interior fixed point
$\lambda^{*}\in(0,1)$ must remain directionally stable. To achieve that, for
the update rule $h_{1}(\lambda)=\lambda+\eta\lambda\Delta_{w}(\lambda)$, for
each $\lambda$, it should hold $\lambda\geq 0$; otherwise, if $\lambda<0$, we
project this value to $0$. In order to check that this is mandatory, suppose
that we allow negative value and for some time $t$, $\lambda_{t}<0$. This
would imply that $h_{1}(\lambda_{t+1})<\lambda_{t}<0$. For that extraction
rate, it holds $\lambda_{t}<\lambda^{*}$, and thus the dynamics become
unstable since $h_{1}(\lambda_{t})-\lambda_{t}/\lambda_{t}-\lambda^{*}>0$.
Similarly, for the update rule $h_{2}$, in order to maintain the directional
stability of the interior fixed point $\lambda^{*}$ the updates need to be
projected to $(-\infty,1]$. For the update rule $h_{3}(\lambda)$, in order to
maintain the directional stability of $\lambda^{*}$, we do not have to
restrict the values of $\lambda$, because it actually is directionally stable,
see Figure 5(d). Moreover, by comparing the dynamic evolution of
($\lambda$-MEV) in Figure 5(a), with the dynamic evolution of the update rules
$h_{1},h_{2},h_{3}$ in Figure 5, we observe that, in the latter case, the
dynamics collapse faster to the boundaries $\lambda=0$ or $\lambda=1$ since
the updates are more aggressive. That is because, for $\lambda\in(0,1)$, it
holds that $\lambda(1-\lambda)<\lambda$ and $\lambda(1-\lambda)<1-\lambda$.
Moreover, for each of those update rules, we can prove similar bounds for the
respective Market Liveness, see Theorem 1 about the convergence to
$\lambda^{*}$, and Theorem 2 about the chaotic updates, and Theorem 4. That
is, we can also find the values of the parameters for which analogous theorems
hold for the systems $h_{1},h_{2}$ and $h_{3}$. We also remind that the former
theorems do not rely on the tolerance distributions $F,G$ and the results hold
regardless of the exact distribution.

##  5 Robustness of Findings

In this section, we provide further simulations to test the robustness of our
model. We remind that in the theoretical part, for our analysis, we considered
stationary tolerance distributions for users and miners. However, a shift in
the prevailing market conditions, dynamics, or dominant trends is a well
studied phenomenon in finance and it is called regime changes [1]. In
financial markets, a regime change refers to a transition from one market
regime to another, characterized by distinct behavior and patterns. Regime
changes can have a profound impact on various aspects of financial markets,
including asset prices, volatility, investor sentiment, and trading
strategies.

Figure 6: In this scenario, we consider a two-regime market with a threshold
$\theta=0.408$ on the extraction rate governing the transition. We handle two
control parameters based on the current extraction rate. The left panel shows
the evolution of ($\lambda$-MEV) for 5,000 iterations, which is divided in
groups of 100 epochs. The right panel shows the deviations from the target
$\Delta(\lambda)$ during those 100 epochs.

In Figure 6, we examine a realistic scenario with two market regimes. We
envision an environment where there are two pairs of tolerance distributions
$F_{1},G_{1}$ and $F_{2},G_{2}$. The transition between these two regimes is
governed by a threshold of the MEV extraction rate (set at $\theta=0.408$ in
the experiments). When the MEV extraction rate is below (above) $\theta$ users
and miners are in regime 1 (2). Building on this idea, we also adopt two
different intensity values, $\eta_{1},\eta_{2}$, for the updates based on the
MEV extraction rate, and two target values $w_{1}$ and $w_{2}$. It is known
that MEV extracted differs from one application to another. For instance, the
MEV generated in decentralized exchanges is regular, whereas liquidations and
NFT generate spikes of MEV. Thus, when a spike in MEV is observed, the updates
must be more aggressive in order to restore the equilibrium or return to the
bounded regions faster. In this scenario, we observe the evolution of
($\lambda$-MEV) for 5,000 iterations. Initially the control parameter is set
at $\eta_{1}=0.9$ and the target at $w_{1}=1.1$. If at a given time $t$ the
extraction rate becomes larger than the threshold, i.e., $\lambda_{t}>\theta$,
this triggers a change in the control parameter to $\eta_{2}=1.0$ to deal with
the new regime. Vice versa, when the ($\lambda$-MEV) undershoots the
threshold, the control parameter sets back to $\eta_{1}$.

We observe here that the extraction rate stabilizes and in fact it stabilizes
within one of the two regimes – see Figure 6 (left). We also see quite good
stabilisation of the target values in the right part of the figure. This
experiment suggests that our model shows reasonable stability even when the
market conditions change between two regimes.

Figure 7: In this scenario, we consider an adversarial setting where the
intensity of the updates and the target is randomly sampled in each epoch. The
upper panel shows the evolution of ($\lambda$-MEV) in 5000 iterations, that is
divided in 100 epochs. The bottom left panel shows the control parameters
randomly drawn over the 100 epochs. The middle panel shows the target values
drawn over the 100 epochs. The right panel shows the deviations
$\Delta(\lambda)$ from the target over the 5000 iterations

Now, in the second scenario, Figure 7, we consider a very adversarial setting.
This setting is akin to stress tests used by regulators and financial
institutions in classical financial markets. In this setting, we split the
evolution of ($\lambda$-MEV) in 100 epochs, where each epoch consists of 50
consecutive blocks. In each epoch, we sample randomly a control parameter
$\eta$ from the interval $[0.4,1]$ and target $w$ from the interval $[0.5,2]$.
This simulates very drastic changes in the parameter $\eta$ and $w$ – see the
yellow and green graphs in Figure 7. Moreover, every 10 blocks, we also
perturb the tolerance (beta) distributions $F,G$ by changing their respective
parameters $a_{u},b_{u}$ and $a_{m},b_{m}$. This perturbation emulates the
(high) volatility in demand that may characterise financial markets in certain
periods.

Even in this extreme stress test, the results support our theoretical
findings. The market remains in a safe zone, in the sense that the system
keeps fluctuating and is bounded around the respective target – see the top
blue chart in Figure 7. As it is shown in Lemma 2, the bounded area expands as
the control parameter $\eta$ increases. This is evident also in this
simulation if we compare the evolution of ($\lambda$-MEV) with the trajectory
of the random values drawn of the control parameter $\eta$. The evolution of
the target value (bottom right pink diagram in Figure 7) exhibits a more
challenging behavior, which on one hand shows how demanding our stress test is
whilst on the other supports the need to investigate more dynamic
environments.

##  6 Conclusion

Maximal Extractable Value constitutes a critical issue in smart-contract
blockchains. With the evolution of the MEV ecosystem, the whole process of
extracting excess value from the economic activity of users has been
systematized and benefits different kind of blockchain participants including
searchers, builders, validators and block proposers. This creates a
fundamental trade-off between users’ sovereignty over their own transactions
and revenue maximization for the entities who secure and process these
transactions. Accordingly, a lot of research is concentrated on designing
mechanisms that will return to users part of the excess value created by their
economic activity.

Our work establishes the first to our knowledge dynamic framework for the
integral problem of MEV sharing. Without arguing on whether MEV is desirable
or not, this framework treats MEV as an unavoidable phenomenon of current
blockchains and enables the market to decide dynamically on its exact level
through a protocol-based implementation. Using the design principles of the
EIP-1559 upgrade, our rigorous mathematical results provide the first formal
evidence that similar mechanisms with properly engineered target functions can
be successful in tackling blockchain optimization problems that extend beyond
the current use of EIP-1559 in regulating transaction fee markets. From a
practical perspective, our results suggest that dynamic MEV extraction can
achieve targets set by the protocol and can, thus, inform the ongoing
discussion about actively enshrining MEV extraction in protocol designs. Last
but not least, recent work [9] showed that separate pricing of resources in
blockchain fee markets can lead to better resource utilization and also
decrease the oscillatory behaviour of the respective dynamics. We expect that
similar results will hold in our analysis, if we decompose the adjustment
quotient in multiple ones. We leave such analysis as future work.

## Acknowledgments

This research was supported in part by the National Research Foundation,
Singapore and DSO National Laboratories under its AI Singapore Program (AISG
Award No: AISG2-RP-2020-016), grant PIESGP-AI-2020-01, AME Programmatic Fund
(Grant No.A20H6b0151) from A*STAR.

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