# Intersystem Bell-like States in Classical Polarization Optics

Partha<EMAIL_ADDRESS>
Tagore Centre for Natural Sciences and Philosophy, Kolkata, India

###### Abstract

A method is proposed to produce a classical optical state that is ‘intersystem
nonseparable’ and a close analog of the $\phi^{+}$ Bell state. A derivation of
the CHSH-Bell inequality is sketched within the framework of classical
polarization optics using noncontextuality for factorizable states as an axiom
rather than any hidden variable theory, and it is shown that the classical
state violates this inequality.

##  1 Introduction

There is some controversy in the literature as to whether the nonseparability
that has been observed in classical light should be called ‘entanglement’
because entanglement is a term coined by Schrödinger specifically to express a
purely quantum nonseparability that persists in spite of arbitrary spatial
separation of the sub-systems [1] (2019). It is true that most instances of
nonseparability observed with classical light so far are of a local nature,
involving different modes (DoFs) of the same optical beam [2, 3]. They have
been termed ‘intra-system noneparability’ or ‘classical entanglement’ as
opposed to ‘quantum entanglement’ which is believed (by the majority of
physicists) to imply nonlocality [4]. It is also generally believed that only
intrasystem nonseparability has a classical analogy, though there is no proof.

It is therefore important to investigate whether ‘intersystem nonseparability’
also has any classical analog, and if so, what it implies. In this paper I
will first propose a method of producing an intersystem nonseparable state in
classical polarization optics, and then explore to what extent it is quantum-
like.

To describe polarized light it is most convenient to use normalized Jones
vectors

$|J\rangle=\frac{1}{\sqrt{\langle J|J\rangle}}\left(\begin{array}[]{c}E_{x}\\\
E_{y}\end{array}\right)$ (1)

with $E_{x}=A_{0}\hat{e}_{x}{\rm exp(i\phi_{x})}$ and
$E_{y}=A_{0}\hat{e}_{y}{\rm exp(i\phi_{y})}$ the complex transverse electric
fields, $\hat{e}_{x}$ and $\hat{e}_{y}$ the unit polarization vectors, and
$\langle J|J\rangle=|E_{x}|^{2}+|E_{y}|^{2}=A_{0}^{2}$, the intensity $I_{0}$.
This is a column vector in a 2-dimensional Hilbert space of polarization
states. The Jones vector for horizontally ($x$ axis) polarized light is

$|h\rangle\equiv|1\rangle=\left(\begin{array}[]{c}1\\\ 0\end{array}\right),$
(2)

and that of vertically polarized ($y$ axis) light is

$|v\rangle\equiv|0\rangle=\left(\begin{array}[]{c}0\\\ 1\end{array}\right).$
(3)

Jones vectors for other linear polarization states can be written as linear
combinations of these basis vectors.

##  2 How to Produce a Classical Bell-like State

Let us now briefly recollect the standard procedure for producing the first
Bell state in quantum mechanics. One uses a two-qubit circuit consisting of a
Hadamard gate H and a CNOT gate. The Hadamard gate applied to a general qubit
gives $H|q\rangle=\frac{1}{\sqrt{2}}[|0\rangle+|1\rangle]$. This state
combined with a $|0\rangle$ state produces the state
$\frac{1}{\sqrt{2}}[|0\rangle\otimes|0\rangle+|1\rangle\otimes|0\rangle]$, and
the CNOT gate produces the Bell state
$\frac{1}{\sqrt{2}}[|0\rangle\otimes|0\rangle+|1\rangle\otimes|1\rangle]$.

Figure 1: Schematic diagram to produce a classical analog of the first Bell
state.

Let us now see how this circuit can be realized in classical optics. Consider
two quasi-monochromatic sources $S_{1}$ and $S_{2}$ of classical light of
equal intensities but different mean angular frequencies $\omega_{1}$ (denoted
by red in the figure) and $\omega_{2}$ (denoted by blue in the figure). Let
the beam emitted by $S_{1}$ be unpolarized and that emitted by $S_{2}$, also
unpolarized, be converted to a vertically polarized beam $|0\rangle_{2}$ by
passing through a vertical polarizer VP. These correspond to the qubit
$|q\rangle$ and $|0\rangle$. Let the unpolarized beam (red) be incident on a
polarizing beam splitter PBS as shown in the figure. It acts like a Hadamard
gate, producing the polarization state
$\frac{1}{\sqrt{2}}[i|0\rangle_{1}^{a}+|1\rangle_{1}^{b}]$, the reflected
light being vertically polarized ($|0\rangle$) and the transmitted light
horizontally polarized ($|1\rangle$). Let the second beam (vertically
polarized $|0\rangle_{2}$ and blue) be incident on an ordinary beam splitter
BS. The part of the beam transmitted by BS is combined with the horizontally
polarized beam transmitted by PBS by means of a dichroic mirror $DM_{2}$. The
part of the beam reflected by BS is first reflected by a lossless mirror
$M_{1}$, passes throgh a half-wave plate HWP, gets converted into a
horizontally polarized beam $|1\rangle_{2}$ which is then reflected by a
lossless mirror $M_{2}$ and combined with the vertically polarized red beam
reflected by PBS by means of a dichroic mirror $DM_{1}$.The beam splitter BS
and the half-wave plate HWP thus act like a CNOT gate, and the final
polarization state of the output beams $a$ and $b$ is

$|\phi^{+}\rangle=\frac{i}{\sqrt{2}}\left[|0\rangle_{1}^{a}\otimes|0\rangle_{2}^{b}+|1\rangle_{1}^{b}\otimes|1\rangle_{2}^{a}\right].$
(4)

This is a nonseparable/nonfactorizable state. It is precisely what we were
looking for, namely ‘intersystem nonseparability’ of classical light, a
nonseparability that persists in spite of arbitrary spatial separation of the
sub-systems. The three other Bell-like nonseparable states can be produced by
using variations of this method.

Let us see to what extent such states are quantum-like. Shimony [5] has listed
the following three criteria that any system S produced for testing its
quantum nature must satisfy:

(I) In any state of a physical system S there are some eventualities which
have indefinite truth values.

(II) If an operation is performed which forces an eventuality with indefinite
truth value to achieve definiteness … the outcome is a matter of chance.

(III) There are ‘entangled systems’ (in Schrödinger’s phrase) which have the
property that they constitute a composite system in a pure state, while
neither of them separately is in a pure state.

The state (4) clearly satifies criterion (III). As regards the other two
criteria, it would be useful to consider thermal light. In particular, let
$S_{1}$ and $S_{2}$ be two thermal sources which are wide-sense stationary. In
a beam of thermal light the value of the ‘optical field’ is unpredictable. The
observable optical field of such a light beam is the space-time dependent
electric field vector

$\vec{E}(r,t)=\hat{h}E_{h}(r,t)+\hat{v}E_{v}(r,t)$ (5)

where $\hat{h}$ and $\hat{v}$ are orthogonal and arbitrarily oriented
‘horizontal’ and ‘vertical’ polarization directions, and the amplitudes
$E_{h}(r,t)=\sum_{n}h_{n}(t)\phi_{n}(r)$ and
$E_{v}(r,t)=\sum_{l}v_{l}(t)\phi_{l}(r)$ are statistically completely
uncorrelated, being elements of a stochastic function space. Scalar products
of the vectors in this space correspond physically to observable correlation
functions such as $\langle E_{v}E_{h}\rangle$, the angular bracket denoting
expectation value. For unpolarized light $\langle E_{v}E_{v}\rangle=\langle
E_{h}E_{h}\rangle$ and $\langle E_{v}E_{h}\rangle=0$. The $\phi_{n}(r)$ are
orthonormal spatial mode functions with $\int
dr\phi^{*}_{n}(r)\phi_{l}(r)=\delta_{n,l}$, and $h_{n}(t)$ and $v_{l}(t)$
represent stochastic random coefficients whose origin one can assign to
distant and unknowable dipole radiators.

Since the value of $\vec{E}$ in thermal light is unpredictable at any time,
and a definite truth value is obtained only when the field is appropriately
detected, Shimony’s criteria (I) and (II) are also satisfied. Nevertheless,
this picture of the optical field does not imply that it is quantized, only a
non-deterministic viewpoint is employed to extract predictions from the
randomly unknown ensemble of field potentialities.

Since the red beam with frequency $\omega_{1}$ is unpolarized,

$\displaystyle\langle E_{1v}(r,t),E_{1h}(r,t)\rangle$ $\displaystyle=$
$\displaystyle 0,$ (6) $\displaystyle\langle
E^{*}_{1v}(r,t),E_{1v}(r,t)\rangle$ $\displaystyle=$ $\displaystyle\langle
E^{*}_{1h}(r,t),E_{1h}(r,t)\rangle$ (7)

for all $(r,t)$ [6]. Since this unpolarized red beam splits into vertical and
horizontal components at some point within the beam splitter PBS (as shown in
Fig. 1), the first equation (6) implies that the red beams in paths $a$ and
$b$ are completely uncorrelated. In other words, the red beam will exhibit
anticoincidence on the beam splitter, a quantum-like feature. The second
result (7) implies that the measured intensities of the vertical and
horizontal components are always equal everywhere in the beams.

Now, the classical electrical fields in the two paths $a$ and $b$ (Fig. 1) are
(see Appendix)

$\displaystyle\vec{E}^{a}(r_{a},t)$ $\displaystyle=$
$\displaystyle\hat{v}E^{a}_{1v}(r_{a},t)+\hat{h}E^{a}_{2h}(r_{a},t),$ (8)
$\displaystyle\vec{E}^{b}(r_{b},t)$ $\displaystyle=$
$\displaystyle\hat{h}E^{b}_{1h}(r_{b},t)+\hat{v}E^{b}_{2v}(r_{b},t).$ (9)

They fluctuate stationarily in the wide-sense. Since the sources $S_{1}$ and
$S_{2}$ are independent, the following first-order cross correlations vanish,
namely

$\displaystyle\langle E^{a*}_{1v}(r_{a},t),E^{a}_{2h}(r_{a},t+\tau)\rangle$
$\displaystyle=$ $\displaystyle\langle
E^{b*}_{2v}(r_{b},t),E^{b}_{1h}(r_{b},t+\tau)\rangle=0,$ (10)
$\displaystyle\langle E^{a*}_{1v}(r_{a},t),E^{b}_{2v}(r_{b},t)\rangle$
$\displaystyle=$ $\displaystyle\langle
E^{a*}_{2h}(r_{a},t),E^{b}_{1h}(r_{b},t)\rangle=0.$ (11)

Eqn (10) implies that red and blue lights (which, by experimental design,
always have orthogonal polarizations in each path) are completely uncorrelated
in each path. These results guarantee randomness of outcomes (red/blue) in
each path/end, another quantum-like feature. Eqn (11) shows that the
vertically polarized red beam in path $a$ and the vertically polarized blue
beam in path $b$ are uncorrelated, and so also are the horizontally polarized
red beam in path $b$ and the horizontally polarized blue beam in path $a$, in
spite of the light being in the nonseparable polarization state (4). This
guarantees that ‘superluminal signalling’ does not occur at the statistical
level, which is also a quantum-like feature.

These results demonstrate to what extent intersystem nonseparability in
classical optics is similar to that in quantum optics.

##  3 CHSH-Bell Inequality

The question that arises therefore is: does intersystem nonseparability of
classical light imply any Bell violation which is usually taken as a measure
of nonlocality? To see that, consider a classical separable state like

$\frac{1}{\sqrt{2}}(\left[|1\rangle\otimes|0\rangle+|0\rangle\otimes|0\rangle\right]=\left[|1\rangle+|0\rangle\right]\otimes|0\rangle$
(12)

and joint polarization measurements on the two sides $a$ and $b$ at angles
$(\theta_{a},\theta_{a^{\prime}})$ and $(\theta_{b},\theta_{b^{\prime}})$.
Instead of local realism, assume only noncontextuality which is the
requirement that the result of a measurement is predetermined and not affected
by how the value is measured, i.e. not affected by previous or simultaneous
measurement of any other compatible or co-measureable observable. Treating the
normalized classical Jones vectors mathematically as qubits, one can then
quite easily derive the CHSH inequality [9, 10]

$-2\leq
E(\theta_{a},\theta_{b})-E(\theta_{a},\theta_{b^{\prime}})+E(\theta_{a^{\prime}},\theta_{b})+E(\theta_{a^{\prime}},\theta_{b^{\prime}})\leq+2$
(13)

where $E(\theta_{a},\theta_{b})$ are expectation values. [The outcomes need
not be $\pm 1$ as is usually assumed. It is sufficient that they lie between
$\pm 1$ which is guaranteed by Malus’ Law in classical polarization optics
with normalized Jones vectors [11].] If the nonseparable state (4) is used,
one gets

$E(\theta_{a},\theta_{b})=\cos(\theta_{a}-\theta_{b}),$ (14)

and for a specific set of values
$\theta_{a}=\pi/2,\theta_{a^{\prime}}=0,\theta_{b}=\pi/4,\theta_{b^{\prime}}=-\pi/4$
one gets $S=2\sqrt{2}$ which violates the CHSH inequality.

The original derivations of the Bell and CHSH inequalities involved the use of
local hidden variables (local realism). Hence, in such a framework a CHSH
violation would rule out local hidden variable theories. If the hidden
variables are noncontextual (Einstein realism), nonlocality would follow. To
avoid nonlocality, one must assume a reality that is contextual (as Bohr did
in terms of his complementarity idea).

As we have just seen, Bell-CHSH inequalities can also be derived within the
framework of classical polarization optics using noncontextuality. Hence, a
CHSH violation with spatially separated subsystems in classical polarization
optics would imply a violation of noncontextuality rather than of locality
[12, 13], signalling a radical change in the very notion of classical realism,
namely a reality that is contextual.

##  4 Concluding Remarks

I have proposed a method to produce a classical optical state that is (a)
‘intersystem nonseparable’, (b) statistically close to a Bell state and (c)
violates the Bell-CHSH inequality without violating locality. Hence,
intersystem nonseparability per se does not imply nonlocality.

Having said that, it is important to point out a crucial difference between
classical and quantum nonseparable states, namely that the statistics can
never be sub-Poissonian with classical light.

##  5 Appendix

To see that the electric fields (8) and (9) indeed represent the polarization
state (4), consider the corresponding states

$\displaystyle|{\cal{E}}^{a}\rangle$ $\displaystyle=$ $\displaystyle
e^{a}_{1v}|0\rangle^{a}_{1}+e^{a}_{2h}|1\rangle^{a}_{2},$ (15)
$\displaystyle|{\cal{E}}^{b}\rangle$ $\displaystyle=$ $\displaystyle
e^{b}_{1h}|1\rangle^{b}_{1}+e^{b}_{2v}|0\rangle^{b}_{2},$ (16)

where $e^{a}_{1v}$ etc are the normalized electric field components. Then, the
normalized tensor product of the two independent light beams is

$\frac{1}{\sqrt{2}}\left[e^{a}_{1v}e^{b}_{2v}|0\rangle^{a}_{1}\otimes|0\rangle^{b}_{2}+e^{b}_{1h}e^{a}_{2h}|1\rangle^{b}_{1}\otimes|1\rangle^{a}_{2}\right].$
(17)

The normalized electric field coefficients are usually omitted in writing
Bell-like states.

To produce the nonseparable polarization state

$|\psi^{+}\rangle=\frac{i}{\sqrt{2}}\left[|0\rangle_{1}^{a}\otimes|1\rangle_{2}^{b}+|1\rangle_{1}^{b}\otimes|0\rangle_{2}^{a}\right],$
(18)

the experimental set up needs to be a variant of the one proposed here, namely
one using the scheme
$H|q\rangle\otimes|1\rangle=\frac{1}{\sqrt{2}}[|0\rangle+|1\rangle]\otimes|1\rangle=\frac{1}{\sqrt{2}}[|0\rangle\otimes|1\rangle+|1\rangle\otimes|1\rangle]$
followed by a CNOT gate. This means replacing the vertical polarizer VP in the
path of the $S_{2}$ beam (Fig. 1) by a horizontal polarizer VH. Then the
electrical fields in the two paths are

$\displaystyle\vec{E}^{a}(r_{a},t)$ $\displaystyle=$
$\displaystyle\hat{v}E^{a}_{1v}(r_{a},t)+\hat{v}E^{a}_{2v}(r_{a},t),$ (19)
$\displaystyle\vec{E}^{b}(r_{b},t)$ $\displaystyle=$
$\displaystyle\hat{h}E^{b}_{1h}(r_{b},t)+\hat{h}E^{b}_{2h}(r_{b},t),$ (20)

corresponding to the polarization state (18).

##  6 Acknowledgement

The author is grateful to C. S. Unnikrishnan, G. Raghavan and members of the
School of Quantum Technologies, DIAT, Pune for helpful discussions on the
difference between quantum and classical entanglement.

## References

  * [1] D. Paneru, E. Cohen, R. Fickler, R. W. Boyd and E. Karimi1, arXiv:1911.02201 [quant-ph] (2019) and references therein
  * [2] X-F Qian, B. Little, J. C. Howell and J. H. Eberly, ‘Violation of Bell’s Inequalities with Classical Shimony-Wolf States: Theory and Experiment’, arXiv:1406.3338 [quant-ph] (2014).
  * [3] X-F Qian, B. Little, J. C. Howell, AND J. H. Eberly,
bf 2 (7), 611-615 (2015).

  * [4] A. Aiello, F. Töppel1, C. Marquardt, E. Giacobino and G. Leuchs, New J. of Phys. 17, 043024 (2015).
  * [5] A. Shimony, Brit. J. Phil. Sci. 35, 25-45 (1984).
  * [6] X-F Qian, B. Little, J. C. Howell, and J. H. Eberly, Optica 2 (7), 611-615 (2015).
  * [7] C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach, Wiley, New York (1998).
  * [8] E. Wolf, Introduction to the Theory of Coherence and Polarization of Light, Cambridge Univ. Press (2007).
  * [9] J. F. Clauser, M. A. Horne, A. Shimony and R. A. Holt, Phys. Rev. Lett. 23 (15), 880–4, (1969).
  * [10] J.S. Bell, Physics 1 (3), 195-200 (1964), reproduced as Ch. 2 of J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press (2004).
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# An Analysis of Intent-Based Markets

Tarun Chitra
Gauntlet
<EMAIL_ADDRESS>Kshitij Kulkarni
UC Berkeley
<EMAIL_ADDRESS>Mallesh Pai
Rice University and SMG
<EMAIL_ADDRESS>Theo Diamandis
MIT
<EMAIL_ADDRESS>
###### Abstract

Mechanisms for decentralized finance on blockchains suffer from various
problems, including suboptimal price execution for users, latency, and a worse
user experience compared to their centralized counterparts. Recently, off-
chain marketplaces, colloquially called ‘intent markets,’ have been proposed
as a solution to these problems. In these markets, agents called _solvers_
compete to satisfy user orders, which may include complicated user-specified
conditions. We provide two formal models of solvers’ strategic behavior: one
probabilistic and another deterministic. In our first model, solvers initially
pay upfront costs to enter a Dutch auction to fill the user’s order and then
exert congestive, costly effort to search for prices for the user. Our results
show that the costs incurred by solvers result in restricted entry in the
market. Further, in the presence of costly effort and congestion, our results
counter-intuitively show that a planner who aims to maximize user welfare may
actually prefer to restrict entry, resulting in limited oligopoly. We then
introduce an alternative, optimization-based deterministic model which
corroborates these results. We conclude with extensions of our model to other
auctions within blockchains and non-cryptocurrency applications, such as the
US SEC’s Proposal 615.

##  1 Introduction

Some of the most popular applications used on blockchains are decentralized
exchanges, which allow users to trustlessly swap assets without a third-party
intermediary. Such exchanges have facilitated over $2 trillion of transactions
since 2018, and their usage continues to increase [Lla23]. However,
decentralized exchanges (DEXs) still have worse user experience and price
execution than centralized exchanges (CEXs), as they tend to be less capital
efficient. That is, it takes more passive capital to facilitate trades on a
DEX than on a CEX.

Decreased capital efficiency on DEXs results from their reliance on continuous
arbitrage to synchronize prices. The most popular type of DEX, the Constant
Function Market Maker (CFMM), was first shown to synchronize prices via
arbitrage with an external market in 2019 [Ang+19, AC20]. Recent work has
demonstrated that this continuous arbitrage causes liquidity providers to face
large losses and adverse selection costs, as their pricing and liquidity are
completely public at all times [Mil+22, AEC21, CDM23]. Moreover, the
permissionless nature of CFMMs has empirically led to excess liquidity
fragmentation [LPZ23] and maximal extractable value, or MEV, which does not
have a precise analogue in the centralized world [XFP23, KDC22, Dai+20].
Despite these shortcomings, CFMMs still command substantial trading volume
[Blo23], and new mechanisms for improved capital efficiency, such as
concentrated liquidity CFMMs [Ada+21], have narrowed the gap between
decentralized and centralized exchanges.

#### Intent-based markets.

Recently, there have been numerous proposals for settlement of on-chain
transactions collectively known as _intents_ that claim to close the gap
between centralized and decentralized exchanges. Intent-based mechanisms allow
users to specify precise conditions or covenants under which their
transactions can be executed. Transactions are then executed on a blockchain
by third parties called _solvers_. Cryptographic primitives ensure that
solvers cannot violate the user-specified conditions, and the validators of
the blockchain verify that these conditions were satisfied. CFMMs, conversely,
deterministically quote a price based on public liquidity, and therefore, in
contrast to intent-based markets, their execution price is known publicly.

Intents were first described by Anoma [Yin23] as a way of credibly executing
asynchronous transactions across multiple blockchains.These mechanisms
generalize ‘request-for-quote’ (RFQ) systems [Mor21, OZ21, O‘H10]. In
particular, intents allow users to express arbitrary covenants, whereas RFQ
systems do not. These covenants are guaranteed to hold throughout the
execution of the transaction on multiple blockchains; a transaction is only
settled after all blockchains have agreed on execution via a communication
protocol.

Solvers are assumed to be rational economic actors who seek to satisfy a
user’s order only if it benefits them. Importantly, they hold or can acquire
private information about the state of external markets that unsophisticated
users cannot acquire. The central question is, therefore, whether a solver is
incentivized to execute the user’s order at ‘optimal’ prices. In this sense,
intent markets resemble principal-agent problems in which the user is a
principal who employs an agent—the solver—to execute their order under the
intent covenants.

#### Example intents.

To provide concrete examples of user intents, we consider a user who holds $1$
unit of token $T_{1}$ on blockchain $B_{1}$ (for example, the ETH token on the
Ethereum blockchain). The user specifies an intent to convert the token into
the maximum amount of output token $T_{2}$ on the same blockchain (for
example, USDC on the Ethereum blockchain). We call this the _maximum output_
intent. As another example, consider the same user who now wants to move their
token on blockchain $B_{1}$ to another blockchain $B_{2}$ (for example, from
the Ethereum blockchain to the Solana blockchain) only if the realized
variance of the token on a set of predefined venues on chain $B_{2}$ is less
than a threshold. We call this the _variance limited transfer_ intent.

While these two intents are superficially similar, only one can be easily
verified after execution. In the first case, the validator is not able to
verify that the user did, in fact, receive the maximum available output of
token $T_{2}$ from all possible external venues. In the second case, however,
a validator on chain $B_{2}$ is able to verify that the realized variance of
the token did meet the conditions specified by the user. In this work, we do
not focus on the issue of verifiability, but rather consider the economic
incentives of solvers in fulfilling these intents.

#### Intents in practice: UniswapX.

Several intent-based marketplaces are already live in production in 2024. The
first example we consider is UniswapX [Ada23]. Since its launch in late 2023,
UniswapX has processed over $3 billion in notional volume [“@p24]. This
protocol incentivizes off-chain agents to provide users with better execution
prices than those provided by CFMMs and, therefore, implements the maximum
output transfer intent introduced earlier. In UniswapX, solvers participate in
a Dutch auction to provide price improvement over the price quoted by a CFMM.
Solvers in UniswapX are referred to as ‘fillers’. The solver who wins the
auction then commits to filling the order at an execution price equal to their
bid. The winning solver has to deliver the specified number of tokens within a
fixed-time interval. If the solver delivers the assets within the time-
interval, the transaction is settled: the user receives the delivered assets
less a payment to the solver for finding a better price. Should the solver be
delinquent and not deliver the assets, they are penalized and the user’s trade
is sent to a CFMM such as Uniswap. We will use this Dutch auction setup as a
starting point for our analysis of solvers’ incentives shortly.

#### Intents in practice: other protocols.

We now review several other intent protocols. CoW Swap [Pro23, Leu21, RGM23],
another intent-based protocol that implements the maximum output intent, has
processed over $30 billion since its inception in 2021 [CoW23]. In CoW Swap,
solvers participate in an auction to execute user orders at a uniform clearing
price, also known as a ‘batch auction’.

Another variant of an intent market is the so-called ‘oracle extractable
value’ market [Sen23, Lam24, Ben23]. These markets run auctions that sell the
right to execute a trade immediately after a price update is sent on-chain via
a price oracle [AC20, LSZ21]. Such markets allow for decentralized finance
protocols (who are the principals, akin to users in UniswapX) to outsource
trade execution to agents and realize a welfare gain.

While there has been an immense amount of activity in building more general
intent-based systems for purposes other than trading (e.g. SUAVE, Essential,
and Anoma [Cha23]), these systems do not currently have any formal guarantees.
Further, the economic incentives of the participants in these systems have not
been studied.

#### Intents without blockchains.

Another fulfillment model that is similar to intents but has been proposed for
traditional equities markets is the US Security and Exchange Committee’s
Proposal 615 [Sec22]. This rule aims to make Payment for Order Flow (PFOF)
relationships more competitive. These relationships, between brokerage firms,
who purchase shares on behalf of their users, and wholesalers who execute
transactions and settle trades for brokerages, have been controversial [Ang21,
BPS23, Wel22].

Currently, private agreements exist between brokerages and wholesalers that
diminish competition, as noted in [Sec22]: “Broker-dealers route more than 90%
of marketable orders of individual investors in NMS stocks to a small group of
six off-exchange dealers, often referred to as ‘wholesalers.’. The wholesaling
business is highly concentrated…” The SEC’s rule proposes the construction of
a Dutch auction wherein a wholesaler who receives orders from a brokerage must
auction right to execute the orders at improved prices before fulfilling it
themselves. This extra auction is similar to blockchain intents like UniswapX
or CoW Swap, but differs in the precise execution guarantees given to the
user. We discuss this auction and the applicability of our model to analyzing
it in §4.

#### Competition.

Despite the interest in intent-based systems, these systems are not
necessarily guaranteed to perform significantly better than existing atomic
mechanisms such as CFMMs. Understanding conditions under which these systems
provide improvements to existing mechanisms is therefore crucial to designing
welfare-maximizing intent systems. Moreover, reliance on a network of solvers
for fulfillment and execution in an intent market raises some questions:
first, will the market remain competitive or concentrate to a few specialized
actors? And second, how would such concentration affect user welfare?

Figure 1: Evolution of competition within UniswapX over time [“@p24]. Left:
the market share of different participants, in terms of the volume of orders
filled. Right: the same data evolving over time. The data set has over 2,000
unique addresses that have participated, despite fewer than 20 addresses
participating in January and Feburary 2024.

#### Empirical signatures of reduced competition in UniswapX.

In Figure 1, we see the evolution of competition within UniswapX [“@p24]. The
left panel shows the percentage of volume processed by a particular solver and
shows extreme concentration, with the top 3 solvers accounting for
$\approx$79% of daily volume transacted. The right panel shows that the number
of entrants into the auction has also decreased dramatically. While over 2,000
unique addresses have participated as UniswapX fillers, with most of them
participating in the earlier days of the protocol during Summer 2023, only 12
unique addresses participated in January 2024 (which can be viewed on the
dashboard [“@p24]). This concentration of fillers suggests that the UniswapX
intent market suffers from a lack of competition, which may translate to user
welfare losses. We note that the filler role is currently permissioned by
Uniswap Labs [Ada23], so there is not free entry within this market. However,
the heavy concentration to 12 out of 2,000 participating addresses suggests
that the market has barriers to competition even among the existing set of
entrants.

#### This paper.

In this paper, we formulate two simple models of solvers bidding in an auction
for a single user’s intent to receive the best possible price when converting
one token into another. We aim to understand the solvers’ economic incentives
to fulfill the user’s intent.

Our first model is probabilistic, relying on classical auction theoretic
tools, where the standard auction setting is complicated by two features.
First, the solvers initially face up-front entry costs, which correspond to
infrastructure set-up costs that are endemic to any entity participating in
intent markets. Second, they face congestion costs that are incurred upon
entry into the market—these costs potentially depend on the number of other
entrants, and are faced while competing with other solvers. For example, the
setting may require participants to exert costly effort to find liquidity to
satisfy the user’s order in external markets. Further, participants may have
to invest in finding this liquidity prior to winning this auction, or indeed
may need to expend this effort just to know how much to bid.

We study the effects of these costs in the context of a Dutch auction. We
start by assuming there is a universe of $n$ possible solvers who may enter
into the auction. For each solver, entry costs are drawn i.i.d. from a
distribution $F_{C}$. As a function of these costs, an equilibrium number
$k^{*}$ solvers enter the auction, and then search for prices to satisfy the
user’s intent, which are drawn i.i.d. from a distribution $F$. This search is
affected by the congestion costs and therefore, so is the solvers’ bid in the
Dutch auction. In this model, we have the following results:

First, in the presence of only entry costs but no congestion costs, we show
that the equilibrium number of entrants $k^{*}$ into the solver market is
$O(\sqrt{n})$ when the solvers’ prices are distributed exponentially, and
$O(n^{1/3})$ when they are distributed uniformly on $[0,1]$. This means that
the percentage of potential solvers who participate in the auction goes to
zero as $n\rightarrow\infty$ since $\frac{k^{\star}}{n}\rightarrow 0$. We
further show that for sufficiently fat-tailed price distributions, such as the
Pareto distribution, one can recover $k^{\star}=n$, although the user may not
be able to realize large welfare increases due to significant bid shading.

Second, when congestion costs are present, the distribution of prices is
modified to a distribution $F(p,e)$, where $e$ is costly, congestive effort
exerted by the solver. Under congestive costs (such as those that might occur
while searching for inventory in illiquid external markets), the effort that a
solver exerts in the search for better prices for the user degrades, and,
therefore, so does the user’s welfare. In practice, this result suggests that
intent markets may provide welfare gains over CFMMs in highly liquid markets
where congestion costs are small, but they likely do not do so in markets for
extremely illiquid assets. Our results therefore connect costs faced by
solvers to the welfare gains that intent-based markets can promise to users.

Our second model takes a deterministic approach, which is based on an
optimization framework. This model naturally suggests the Dutch auction as a
means to solve the social welfare problem. Although it introduces different
modeling assumptions compared to our first model, the deterministic model
offers two advantages: it can be easily parameterized, suggesting a fruitful
avenue for future empirical work; and it likely has an easier generalization
to the multi-asset case, which we also leave to future work.

Our analysis also implies several interesting features of intent markets.
Entry costs may cause restricted entry into the markets. Further, when solvers
are subject to congestion, limited oligopoly can be better for user welfare
than free entry; fewer entrants will have stronger incentives to invest in
higher effort due to less congestion. The welfare gains for the user from
additional investment from fewer entrants may outweigh the losses to the user
from increased bid-shading in an oligopolistic market. Understanding the
underlying costs faced by solvers is therefore important for an intent
platform to optimally design the market microstructure.

##  2 An auction-theoretic model

We begin with the simplest setting for an intent market: a single
decentralized trade between two tokens. Consider a user who desires to swap
$1$ unit of token $T_{1}$ into as many units of token $T_{2}$ as possible
(this is an example of the maximum output intent introduced earlier). The user
is faced with a choice to trade either with an infinitely liquid public market
(an idealization of a CFMM) that quotes a price $p^{*}$, or with one of $n$
possible solvers, where each solver $i$ has access to inventory of token
$T_{2}$, at a price $p_{i}$ in token $T_{1}$. The user only trades with a
solver if they offer a price improvement above the public market. We assume
that $p_{i}$ are distributed i.i.d. according to a distribution with CDF
$F(p)$ and density $f(p)$. This price could be either the price of their
inventory or the price of an external market in which they have infinite or
arbitrarily large liquidity relative to the user’s demand. The user, in an
attempt to elicit the best price for their tokens and improve upon the prices
available from the public market, runs an auction (such as the Dutch auction
found in UniswapX [Ada23]). We will examine several settings, in increasing
order of complexity.

#### Entry costs, costly effort, and congestion.

Solvers face a sequence of choices in attempting to fill a user’s order.
First, they must choose whether or not to enter the auction in the first
place. This choice is often driven by initial setup and infrastructure costs
that are required to bid on and execute a user’s orders. We call these _entry
costs_. These entry costs determine an equilibrium number $k^{\star}$ of the
universe of $n$ solvers that will enter the auction. Upon making the choice to
enter, the solvers then must decide how to bid. This choice, in our model, is
driven by the fact that solvers expend costly effort to search for a price to
win and satisfy a user’s order, which modifies the distribution of prices to
$F(p,e)$, where $e$ is the effort level chosen by the solver. In this case,
solvers may try to simultaneously access a set of external markets, and
therefore, effort may cause congestion. We seek to understand how the solvers’
bidding behavior changes as a function of these two kinds of costs they face.
In summary, we have the following settings and their associated results:

  1. 1.

_No cost of entry, no effort costs_ : No cost of entry implies that the
equilibrium number of entrants into the solver market satisfies $k^{\star}=n$.
Classical revenue equivalence [Kri09] applies and user welfare is maximized.

  2. 2.

_Costly entry_ : In equilibrium, the intent market has an asymptotically
negligible number of entrants $k^{*}$ (that is, $k^{*}/n\rightarrow 0$ as
$n\rightarrow\infty$) for reasonable distributions like the exponential and
uniform distributions. For the Pareto distribution, however, in anticipation
of large prices, we may have $k^{*}=n$ entrants.

  3. 3.

_Costly effort_ : We demonstrate cases welfare is reduced due to effort
causing congestion. This indicates that a planner may in fact want to restrict
entry into such a market.

These results demonstrate that entry and effort costs may lead to oligopoly,
defined as the combination of reduced competition ($k^{\star}=o(n)$) and
reduced welfare or auction revenue. Critically, in the case of effort costs, a
social planner may wish to limit entry, as a closed or permissioned solver
market may achieve better outcomes from the perspective of a welfare-
maximizing planner. Understanding when this occurs in practice, and how such
permissioned markets can be implemented is an important design consideration
that we leave for future work.

###  2.1 Dutch auctions for intents

The Dutch action has been implemented by by various protocols, including
UniswapX [Ada23], to execute a single user’s swap intent in a market with
multiple solvers. It is well-known that the Dutch auction is strategically
equivalent to a sealed-bid first price auction [Kri09]. Therefore, we analyze
the solvers’ strategies in the setting of a first price auction, where the
highest bidding solver wins at the highest price. Each solver submits a price
$\tilde{p}_{i}\leq p_{i}$, which is their bid. According to the rules of the
first price auction, the highest bid gets filled as long as it is better than
$p^{*}$. The net profit of a solver from being filled therefore becomes:

$p_{i}-\tilde{p}_{i}.$

If $k$ of the universe of $n$ possible solvers are present in the auction,
revenue equivalence yields that

$(p_{i}-\tilde{p}_{i})F^{k-1}(p_{i})=\int_{p^{*}}^{p_{i}}F^{k-1}(x)dx,$

or, equivalently,

$\tilde{p}_{i}=(p_{i})-\frac{\int_{p^{*}}^{p_{i}}F^{k-1}(x)dx}{F^{k-1}(p_{i})}.$

This formula has a very simple interpretation: in a first price auction, in
the absence of entry or congestion costs, each solver shades their bid
according to the competition they face in the auction.

#### Interim profits.

In this case, the interim expected profit of a solver with price $p_{i}$ in a
setting with $k$ other solvers can be written as:

$S(p_{i},k)=\int_{p^{*}}^{p_{i}}F^{k}(x)dx.$

We can therefore write the ex-ante expected profit of a solver as

$S(k)=\int_{0}^{\infty}S(p,k)f(p)dp.$

Note that:

$\displaystyle S(k)$ $\displaystyle=\int_{0}^{\infty}S(p,k)f(p)dp$
$\displaystyle=-\int_{0}^{\infty}S(p,k)\frac{d(1-F(p))}{dp}dp$
$\displaystyle=-\left(S(p,k)(1-F(p))\big{|}_{p=0}^{p=\infty}-\int_{0}^{\infty}S^{\prime}(p,k)(1-F(p))dp\right)$
$\displaystyle=\int_{p^{*}}^{\bar{p}}F^{k}(p)(1-F(p))dp.$

Therefore, by observation, $S$ is decreasing and satisfies increasing
differences in $k$. Note that $S:\mathbf{N}\rightarrow\mathbf{R}$ can be
extended to the positive reals using e.g., linear interpolation between
successive integers. The resulting function is convex, and we abuse notation
to consider this extended function in the sequel.

#### Exponential distribution.

First suppose that that $F$ is the exponential distribution with rate
$\lambda$ and $p^{*}=0$. Then,

$\displaystyle S(k)$ $\displaystyle=\int_{0}^{\infty}(1-\exp(-\lambda
p))^{k}\exp(-\lambda p)dp$
$\displaystyle=\frac{1}{\lambda}\int_{0}^{1}x^{k}dx$
$\displaystyle=\frac{1}{(k+1)\lambda},$

where the second equality is by substituting $1-\exp(-\lambda p)=x$.

#### Uniform distribution.

Now suppose that $F$ is the uniform distribution on [0,1] and $p^{*}=0$. Then,

$S(k)=\int_{0}^{1}p^{k}(1-p)dp,$

which in turn implies that

$S(k)=\frac{1}{(k+1)(k+2)}.$

#### Pareto distribution.

Finally, suppose that $F$ is the generalized Pareto distribution with CDF
$F(x)=1-\left[1+\left(\frac{x-\mu}{\sigma}\right)^{1/\gamma}\right]^{-\alpha}$,
where $x>\mu$, and $\sigma,\gamma,\alpha>0$, and that $p^{*}=0$. We consider
this distribution in the regime $\gamma/\alpha>1$, for which its mean is
infinity. In this case, we have that

$\displaystyle S(0)$ $\displaystyle=\int_{0}^{\infty}1-F(p)dp$
$\displaystyle=\mathop{\bf E{}}_{p\sim f}[p]$ $\displaystyle=\infty.$

We will use this fact shortly to show that in this case, the market admits all
$n$ potential solvers as entrants.

###  2.2 Costly entry

Now, we assume that prior to bidding in the auction, every solver must pay an
entry cost of $c_{i}$ if they choose to bid. These costs correspond to the
setup and infrastructure costs considered previously. For now, we consider the
case that the solver does not know how many other solvers have entered or
their oracle price; they choose to enter only based on their own price and
cost. Costs are nonnegative and distributed according to a given distribution
$F_{C}$. In the subsequent, we will always assume that $F_{C}$ has a density
$f_{C}$ for whom $f_{C}(0)$ is finite and nonzero. Under these entry costs, we
will show that expected number of entrants in the auction, $k^{\star}$, is
small relative to the universe of possible entrants, $n$.

#### Market size as a function of costs.

The above setup allows us to study entry into the market by providing a
threshold cost $\bar{c}$ such that it is not profitable in expectation for a
solver to enter the auction if their cost exceeds the threshold. The maximum
cost solver who will choose to enter this market will be one with cost
$\bar{c}$ where $\bar{c}$ solves:

$\displaystyle\sum_{k=0}^{n}\left({n\choose
k}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)S(k)=\bar{c}.$ (1)

Recall that solvers enter the auction if their cost is less than the threshold
cost $\bar{c}$. This happens with cumulative probability $F_{C}(\bar{c})$.
Therefore, once the threshold $\bar{c}$ has been found, the expected number of
entrants into the solver market can be computed as:

$\displaystyle k^{*}=nF_{C}(\bar{c}).$

We now show how $k^{*}$ depends on $n$ for various realistic distributions
$F$.

#### Exponential distribution.

For the case of exponentially distributed prices with rate $\lambda$, we have
that the left hand side of the above equation (1) can be written as:

$\displaystyle\sum_{k=0}^{n}\left({n\choose
k}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)S(k)=\frac{1}{(n+1)\lambda
F_{C}(\bar{c})}(1-(1-F_{C}(\bar{c}))^{n+1}).$

This equation is derived in Appendix A. So, for exponentially distributed
prices we must have that

$\displaystyle\frac{1}{(n+1)\lambda
F_{C}(\bar{c})}(1-(1-F_{C}(\bar{c}))^{n+1})=\bar{c},$

which implies

$\displaystyle\frac{1}{(n+1)\lambda}=\frac{\bar{c}F_{C}(\bar{c})}{(1-(1-F_{C}(\bar{c}))^{n+1})}.$

From this equation, we straightforwardly have that $\bar{c}$ must be
decreasing to 0 as $n\nearrow\infty$. This is because $\bar{c}$ asymptoting to
a constant would result in the left hand side going to 0 while the right hand
side stays constant. Now suppose that $\bar{c}=g(n)$ for some decreasing
function $g$ and take $n$ large so that we can do Taylor approximations. Then,
the above equation becomes

$\displaystyle\frac{1}{(n+1)\lambda}=\frac{(g(n))^{2}f_{C}(0)}{(1-(1-f_{C}(0)g(n))^{n+1})}.$

This implies that $g(n)=O(\frac{1}{\sqrt{n}})$ since for this rate of
convergence the denominator of the right hand side converges to $1$ because
$\left(1-f_{C}(0)g(n)\right)^{n+1}\rightarrow 0$ and the numerator matches the
left hand side. Recall that the expected number of entrants into the solver
market, $k^{*}$, equals $nF_{C}(\bar{c})$. Therefore, for large $n$, we have
$k^{*}=O(\sqrt{n})$ for exponentially distributed values.

#### Uniform distribution.

Now, for the case of prices uniformly distributed between $[0,1]$, we have
that the left hand side of equation (1) is:

$\displaystyle\sum_{k=0}^{n}\left({n\choose
k}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)S(k)$
$\displaystyle=\frac{1}{(n+1)(n+2)F_{C}^{2}(\bar{c})}(1-(1-F_{C}(\bar{c}))^{n+2}-(n+2)F_{c}(\bar{c})(1-F_{C}(\bar{c}))^{n+1})$

This equation is derived in Appendix B. Therefore, for uniformly distributed
prices, we must have that

$\displaystyle\frac{1}{(n+1)(n+2)F_{C}^{2}(\bar{c})}(1-(1-F_{C}(\bar{c}))^{n+2}-(n+2)F_{c}(\bar{c})(1-F_{C}(\bar{c}))^{n+1})=\bar{c},$

which implies

$\displaystyle\frac{1}{(n+1)(n+2)}=\frac{\bar{c}F_{C}^{2}(\bar{c})}{(1-(1-F_{C}(\bar{c}))^{n+2}-(n+2)F_{c}(\bar{c})(1-F_{C}(\bar{c}))^{n+1})}.$

Similar arguments as in the case of exponentially distributed prices now tell
us that $\bar{c}=O(n^{-\frac{2}{3}})$ and therefore the expected number of
entrants is $k^{*}=O(n^{1/3})$.

#### Pareto distribution.

Finally, when $F$ is the generalized Pareto distribution (in the regime where
$\gamma/\alpha>1$), recall that this distribution has infinite mean, which
means in particular that $S(0)$, the expected interim profits with no other
participating solvers, is infinity. This immediately allows us to conclude
that $\bar{c}$ in this case is also infinity, as we have:

$\displaystyle(1-F_{C}(\bar{c}))^{n}S(0)\leq\sum_{k=0}^{n}\left({n\choose
k}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)S(k)=\bar{c}.$

Because $S(0)=\infty$, the only $\bar{c}$ that makes the above inequality hold
is $\bar{c}=\infty$. Therefore, because $F_{C}(\bar{c})\rightarrow 1$ as
$\bar{c}\rightarrow\infty$, we have that the total number of entrants into the
auction is $k^{*}=nF_{C}(\bar{c})=n$.

#### Connections to extreme spacings.

The quantity $S(k)$, for a given distribution $F$, is intimately related to
the expected difference between the highest and second highest price in a
sample of $k+1$ prices drawn i.i.d. from $F$. This gap is also known as the
_extreme spacing_ [MCT09]. For a distribution with CDF $F$ and PDF $f$, define
for any $k\in{\mbox{\bf N}}$ the following quantity:

$ES(k)\equiv\mathop{\bf E{}}_{p_{i}\sim f}[p_{k:k}-p_{(k-1):k}],$

where $p_{k:k}$ and $p_{k-1:k}$ denote the largest and second-largest of $k$
i.i.d. draws according to $f$. We show in Appendix C that if $p^{*}=0$, then
for any distribution, we have that

$\displaystyle(k+1)S(k)=ES(k+1).$

For the generalized Pareto distribution, it was shown in [MCT09, Theorem 6.1]
that $ES(k)=O(k^{\gamma/\alpha})$, which allows us to conclude
$S(k)=O(k^{\gamma/\alpha-1})$. Such bounds may allow one to derive rates for
entry into the auction for a wider class of price distributions, which we
leave for future work.

Figure 2: Ratio of the expected revenue to the expected highest price,
$\mathop{\bf E{}}[p_{n-1:n}]/\mathop{\bf E{}}[p_{n:n}]$, for the Pareto
distribution, versus the number of solvers, $n$. The ratio of the medians of
$p_{n-1:n}$ and $p_{n:n}$ is plotted for reference.

#### Discussion and numerical results.

Our first set of results on costly entry demonstrate several interesting
features of intent markets. For both the exponential and uniform
distributions, entry costs can cause a much smaller fraction of potential
solvers to enter the auction than would otherwise enter in the absence of
costs. For reference, when $n=100$, one may have as few as $k^{*}=10$ entrants
in the case of exponentially distributed prices, and as few as $k^{*}=5$
entrants in the case of uniformly distributed prices. These results have a
natural economic interpretation: solvers decide to enter the auction depending
on whether they believe there is an opportunity to receive net expected profit
that exceeds their threshold costs. In the case of the exponential
distribution, the probability that a solver receives a price that is much
better than another solver decays exponentially. The situation is even worse
in the case of the uniform distribution, where there is an upper bound on the
best price available to the solvers.

This can be remedied in the case of the Pareto distribution, where even being
a single solver in the market yields infinite expected profits. Therefore, in
anticipation of these large profits, we can expect $n$ entrants into the
market. We note, however, that even in this case, this does not mean that the
user is able to realize a large price, as there may be significant bid shading
in the auction. We demonstrate this effect in Figure 2, where we plot the
ratio of the expected second highest price (equivalent to the expected revenue
of the auction, which in turn is the expected price the user receives) to the
expected highest price, $\mathop{\bf E{}}[p_{n-1:n}]/\mathop{\bf
E{}}[p_{n:n}]$, for $n$ i.i.d. draws of prices $p_{i}$ from the Pareto
distribution. This ratio tells us how much the price the user can expect to
receive from a solver auction with $n$ participants falls short of the
theoretical highest price that could be achieved by any solver. The ratio of
the medians of $p_{n-1:n}$ and $p_{n:n}$ are also plotted for reference.

Our simulation is done by setting $\sigma=100$, $\alpha=0.95$, $\gamma=1$,
$\mu=0$, which corresponds to a standard Pareto distribution with CDF
$F(x)=1-\left(\frac{x_{m}}{x}\right)^{\alpha}$, with scale parameter
$x_{m}=100$ and shape parameter $\alpha=0.95$. For each $n$, we compute the
expectations of $p_{n-1:n}$ and $p_{n:n}$ by averaging over $10000$ trials.
Code for this simulation can be found at

https://github.com/kkulk/intent.

We see that as $n$ gets large, the ratio $\mathop{\bf
E{}}[p_{n-1:n}]/\mathop{\bf E{}}[p_{n:n}]$ approaches zero. This means that
even with a large number of solvers present in the auction, the user is able
to realize a price that is only a small fraction of the theoretical highest
price available to any of the solvers. This suggests that intent markets in
assets with heavy tails may fail to realize user welfare that is much greater
than that afforded by public mechanisms such as CFMMs.

###  2.3 Costly effort

So far, we have considered markets where costs of entry were fixed for each
potential entrant, independent of the number of entrants, and similarly the
distribution of prices for each entrant is exogenously given. Now, instead,
suppose that entrants do not simply receive a price upon entry but have to
invest costly effort. This can be viewed as investment into infrastructure or
technology in order to be able to participate in an auction—an example: a
high-frequency trader who has to spend millions of dollars to achieve a
sufficiently low message latency to successfully participate in an auction.
Further, this can also correspond to the costs of sourcing liquidity for
assets that the solver does not initially hold. As we will see, if these
effort costs cause congestion (as they might in low-liquidity assets), user
welfare may be harmed.

#### Effort-dependent prices and costs.

Formally, suppose now that $k^{*}$ entrants have entered the auction (perhaps
by the process of costly entry introduced in the previous section). Each
entrant $i$ simultaneously chooses an investment level $e_{i}\in{\mbox{\bf
R}}_{+}$. This investment results in the entrant drawing a price $p_{i}\sim
F(p,e_{i})$ for a CDF $F$ dependent on the amount of investment. We make the
assumption that investments improve values in the sense of first-order
stochastic dominance: $F(p,e)\leq F(p,e^{\prime})$ for any $p$ and
$e>e^{\prime}$.

Effort is both costly and congestive. We model the cost of taking effort $e$
when the total number of entrants is $k^{\star}$ by $c(k^{\star},e)$ where $c$
is increasing and convex in both arguments. A special case of interest is when
$c(\cdot)$ is constant in the first argument, so the cost of effort is not
affected by other other entrants, i.e., there is no congestion cost. Note that
this setup is a symmetric game between the $k^{\star}$ solvers that enter the
auction and thus admits a symmetric equilibrium, which we search for next.

#### Equilibrium effort.

We now solve for the equilibrium investment $e^{\star}$ by each entrant in
this model. To do so, we pick a solver $i$ and assume that all $k^{\star}-1$
competing solvers play the equilibrium strategy of drawing $p\sim
F(x,e^{\star})$. Given this strategy, the expected revenue of solver $i$ with
with price $p$ is:

$S(p,k^{\star},e^{\star}):=\int_{0}^{p}F(x,e^{\star})^{k^{\star}-1}dx$

This expression corresponds to the probability that solver $i$’s value is
higher than the remaining $k^{\star}-1$ solvers. Therefore, given that all the
other entrants invest $e^{\star}$, entrant $i$ must pick their level of
investment $e_{i}$ to maximize their expected profit:

$e_{i}=\mathop{\rm argmax}_{e\in\mathbf{R}_{+}}\int
S(p,k^{\star},e^{\star})f(p,e)dp-c(k^{\star},e).$

Assuming that $e^{\star}>0$, we must also have $e_{i}$ satisfy a first order
condition at $e=e^{\star}$. This condition, assuming that the densities and
costs are differentiable, is:

$\left(\int S(p,k^{\star},e^{\star})\frac{\partial f(p,e)}{\partial
e}dp-\frac{c(k^{\star},e)}{\partial k}\right)\Bigg{\rvert}_{e=e^{\star}}=0.$
(2)

This equation implicitly describes $e^{\star}$ as a function of $k^{\star}$,
which allows us to measure congestion and effort’s impacts on net revenue.

#### Properties of the equilibrium.

We now consider the following specific example: $F(p,e)=p^{e}$ for
$p\in[0,1]$. Note that therefore $f(p,e)=ep^{e-1}$. Informally, an investment
of $e$ results in $e$ draws from a uniform distribution, with the entrant’s
value being the maximum of these. We assume from our results in §2.2 that
$k^{\star}=o(n)$ solvers have entered the auction.

Further, suppose $c(k^{\star},e)=\alpha(k^{\star})\frac{e^{2}}{2}$, where
$\alpha(\cdot)$ is an increasing function which parametrizes the
‘congestiveness’ of the underlying market as a function of the number of
entrants in the market. This functional form matches one’s intuition for
inventory cost: if all the entrants have to carry inventory in an intent
market, then the existence of other entrants drives up the costs of sourcing
and carrying the inventory. In this case we have

$S(p,k^{\star},e^{\star})=\frac{p^{(k^{\star}-1)e^{\star}+1}}{(k^{\star}-1)e^{\star}+1}$

Substituting this into our first order condition (2) we therefore have

$\int_{0}^{1}\frac{p^{(k^{\star}-1)e^{\star}+1}}{(k^{\star}-1)e^{\star}+1}(p^{e^{\star}-1}+e^{\star}p^{e^{\star}-1}\ln
p)dp-\alpha(k^{\star})e^{\star}=0$

Solving this integral yields the following equation:

$\displaystyle\frac{1}{(1+e^{\star}k^{\star})^{2}}-\alpha(k^{\star})e^{\star}=0\hskip
5.69046pt\implies$ $\displaystyle\hskip
5.69046pt\alpha(k^{\star})e^{\star}(1+e^{\star}k^{\star})^{2}=1.$

Since $\alpha(\cdot)$ is nondecreasing, we must have that the solution
$e^{\star}$ is decreasing in the number of entrants $k^{\star}$. Note that we
can write the left hand side as

$\displaystyle\frac{\alpha(k^{\star})}{k^{\star}}\left(e^{\star}k^{\star}(1+e^{\star}k^{\star})^{2}\right)=1$

Observe that the function $e^{\star}k^{\star}(1+e^{\star}k^{\star})^{2}$ is
increasing in $e^{\star}k^{\star}$. This implies that, viewing $e^{\star}$ as
an implicit function of $k^{\star}$, if $\alpha(k^{\star})$ is linear (i.e.
$\alpha(k^{\star})=\Theta(k^{\star})$), we have that $e^{\star}k^{\star}$ is
constant in $k^{\star}$ or $e^{\star}=c/k^{\star}$ for some constant c.
Conversely, if $\alpha(k^{\star})$ grows faster (slower) than linear, (i.e.,
$\alpha(k^{\star})/k^{\star}$ is increasing (decreasing) in $k^{\star}$), then
we have that $e^{\star}k^{\star}$ is decreasing (increasing) in $k^{\star}.$
Since $k^{\star}=\omega(1)$, this implies that if
$\alpha(k^{\star})=\Omega(k^{\star})$ then $e^{\star}=o(1)$ in $k^{\star}$
(and hence, $n$). In other words, a larger universe of searchers leads to
vanishing expected equilibrium effort as
$\frac{e^{\star}}{k^{\star}}\rightarrow 0$.

#### Welfare as a function of effort.

We now analyze the welfare of the user as a function of effort $e^{\star}$ and
equilibrium participation $k^{\star}$. Note that user welfare is maximized
when the expected revenue of the auction is maximized, since that is the
(expected) price that the user receives. By revenue equivalence, the expected
revenue equals the revenue of a second-price auction, so we can write the
revenue as:

$\displaystyle\mathsf{Rev}(e^{\star},k^{\star})=$
$\displaystyle\int_{0}^{1}pk^{\star}(k^{\star}-1)(1-F(p,e^{\star}))f(p,e^{\star})(F(p,e^{\star}))^{k^{\star}-2}dp$
$\displaystyle=$
$\displaystyle\int_{0}^{1}pk^{\star}(k^{\star}-1)(1-p^{e^{\star}})(e^{\star}p^{e^{\star}-1})(p^{e^{\star}})^{k^{\star}-2}dp$
$\displaystyle=$
$\displaystyle\frac{{e^{\star}}^{2}(k^{\star}-1)(k^{\star})}{(1+e^{\star}(k^{\star}-1))(1+e^{\star}k^{\star})}$
$\displaystyle=$
$\displaystyle\frac{{e^{\star}}(k^{\star}-1)}{(1+e^{\star}(k^{\star}-1))}\times\frac{{e^{\star}}k^{\star}}{(1+e^{\star}k^{\star})}$
(3)

We make two claims about this formula:

  1. 1.

If $\alpha(k^{\star})=o(k^{\star})$, then $\mathsf{Rev}(e^{\star},k^{\star})$
increases in $k^{\star}$

  2. 2.

If $\alpha(k^{\star})=\omega(k^{\star})$, then
$\mathsf{Rev}(e^{\star},k^{\star})$ decreases in $k^{\star}$

The first claim states that if the congestion cost is sublinear in
$k^{\star}$, then the planner has no reason to restrict entry into the
auction, since every new entrant increases their revenue. On the other hand,
the second claim states that if the congestion cost is superlinear in
$k^{\star}$, then the user would prefer to _restrict entry_ , i.e. a limited
oligopoly makes the user better off.

We show that the two claims are correct using elementary arguments. The first
claim is equivalent to the statement: if $\alpha(k^{\star})=o(k^{\star})$,
then $\frac{\partial\mathsf{Rev}(e^{\star},k^{\star})}{\partial k^{\star}}>0$.
To show this, note that if $\alpha(k^{\star})=o(k^{\star})$, then
$e^{\star}k^{\star}$ is increasing in $k^{\star}$ and so is
$e^{\star}(k^{\star}-1)$. Therefore both terms in (3) are increasing in
$k^{\star}$. Similarly, the second claim states that if
$\alpha(k^{\star})=\omega(k^{\star})$ then
$\frac{\partial\mathsf{Rev}(e^{\star},k^{\star})}{\partial k^{\star}}<0$.
Since $\alpha(k^{\star})=\omega(k^{\star})$ implies that $e^{\star}k^{\star}$
is decreasing in $k^{\star}$ and $e^{\star}(k^{\star}-1)$ is decreasing in
$k^{\star}$. This implies
$\frac{\partial\mathsf{Rev}(e^{\star},k^{\star})}{\partial k^{\star}}<0$.

##  3 An optimization approach

Here, we present an alternative, deterministic model of the intent market,
which follows from an optimization framework. Instead of considering prices
for each solver, drawn from some distribution, we equip each solver with a
utility and cost function. This approach has a number of advantages. First,
these functions can be easily parameterized, and, as a result, the approach
lends itself to empirical studies and model fitting with real-world data.
Second, modeling with utility functions may be more appropriate in some
scenarios, particularly when there is a lot of heterogeneity among the
solvers. Finally, this approach immediately suggests a Dutch auction-like
mechanism, analogous to that considered in the probabilistic model. We also
show how, under certain assumptions, the deterministic framework implies
similar results as those in the previous section, such as those involving the
impact of congestion.

In particular, we turn to convex optimization for our deterministic approach,
which is similar in spirit to the linear programs presented in [KP04] in the
context of auctions. We outline how social welfare maximizing allocations may
be reached via a primal-dual mechanism, which can be interpreted as a Dutch
auction. As in the previous section, we limit our focus to two-asset trades.
In the general multiple asset case, one must use a more complicated mechanism,
as asset prices may be negatively correlated. We leave this case to future
work.

###  3.1 Social welfare

#### Solvers.

Each solver $k=1,\dots,n$ is willing to provide, for given amount $x$ of token
$T_{1}$, an amount $\alpha_{k}x$ of $T_{2}$. To do this, the solver incurs a
cost $c_{k}(x)$ and gains utility $u_{k}(x)$, where the function $u_{k}$ may,
for example, reflect the solver’s relative preference for holding $T_{1}$
versus $T_{2}$. We restrict the cost function $c_{k}:{\mbox{\bf
R}}_{+}\to{\mbox{\bf R}}_{+}\cup\\{\infty\\}$ to be convex, the utility
function $u_{k}:{\mbox{\bf R}}_{+}\to{\mbox{\bf R}}_{+}$ to be concave, and
both to be defined over the nonnegatives—negative trades are not allowed.
Infinite values are used to encode constraints; a trade $x$ such that
$c_{k}(x)=\infty$ is unacceptable (e.g., it is outside solver $k$’s budget).
We assume that these functions are closed and proper. Thus, the net utility
for solver $k$ to tender $\alpha_{k}x_{k}$ units of $T_{2}$ to the user in
exchange for $x$ units of $T_{1}$ is given by

$U_{k}(x_{k})=-\alpha_{k}x_{k}-c_{k}(x_{k})+u_{k}(x_{k}),$

where this utility is measured in terms of token $T_{2}$, which we take as the
numeraire. This utility is concave by construction.

#### User.

We consider a user who wishes to trade $\delta$ units of token $T_{1}$ for a
maximum amount of token $T_{2}$, using a combination of these solvers and an
on-chain CFMMs. The user can find the optimal trade with the network of CFMMs
by solving the _optimal routing problem_ [Ang+22, Dia+23]. Because this
problem is a convex optimization problem with a concave objective, we can
partially maximize [BV04, §3.2.5] to define an aggregate forward exchange
function $G:{\mbox{\bf R}}_{+}\to{\mbox{\bf R}}_{+}$, which maps an input of
token $T_{1}$ to the maximum output of token $T_{2}$ using all of the on-chain
CFMMs. Note that this function is concave and nondecreasing. The user’s
utility is then given by

$U_{p}(x)=G(\delta-{\textstyle\sum_{k=1}^{K}x_{k}})+\sum_{k=1}^{K}\alpha_{k}x_{k},$

which is concave and nondecreasing.

#### Social welfare maximization.

Putting these utility functions together, we can define the problem of
maximizing social welfare:

maximize $\displaystyle G(\delta-y)+\sum_{k=1}^{n}u(x_{k})-c_{k}(x_{k}),$ (4)
subject to $\displaystyle y=\sum_{k=1}^{n}x_{k}.$

with variables $y\in{\mbox{\bf R}}$ and $x_{k}\in{\mbox{\bf R}}$ for
$k=1,\dots,n$. Note that the transfer from each solver $k$ to the user, given
by $\alpha_{k}x_{k}$, has canceled out in the objective.

###  3.2 Dutch auction

The Lagrangian of the social welfare maximization problem (4) is

$L(x,\tilde{x},\nu)=G(\delta-y)+\nu
y+\sum_{k=1}^{n}u_{k}(x_{k})-c_{k}(x_{k})-\nu x_{k}.$

The dual function is then

$\displaystyle h(\nu)$ $\displaystyle=\sup_{y\geq 0}\left\\{G(\delta-y)+\nu
y\right\\}+\sum_{k=1}^{n}\sup_{x_{k}\geq
0}\left\\{u_{k}(x_{k})-c_{k}(x_{k})-\nu x_{k}\right\\}.$

Referring back to the utilities, we can interpret these terms simply: they are
exactly the utility for the user and agent respectively when the universal
price of the asset is $\nu$. Since the function $G$ is nondecreasing, we can
conclude that the price $\nu\geq 0$. Otherwise if $\nu<0$, letting $y=-t$ we
have

$G(\delta+t)-\nu t\geq G(\delta)+\lvert{\nu}\rvert t,$

and the right hand side goes to infinity as $t\to\infty$ (equivalently, as
$y\to-\infty$).

#### Optimality conditions.

From the Lagrangian, at optimality we have primal feasibility
($y^{\star}=\sum_{k=1}^{n}x_{k}^{\star}$) and the dual conditions

$\displaystyle g(\delta-y^{\star})$ $\displaystyle=\nu^{\star}$ $\displaystyle
u_{k}^{\prime}(x_{k}^{\star})-c_{k}^{\prime}(x_{k}^{\star})$
$\displaystyle=\nu^{\star}\quad\text{for}~{}k=1,\dots,n$

(When $u_{k}$ or $c_{k}$ is not differentiable for some $k$, a similar
argument holds using subgradient calculus.) Putting these together, we have
that

$u_{k}^{\prime}(x_{k}^{\star})-c^{\prime}_{k}(x_{k}^{\star})=g(\delta-\textstyle\sum_{k=1}^{n}x_{k}).$

In other words, the marginal cost of getting asset 2 from the public CFMMs
should exactly equal each participating solver’s net marginal utility for
providing it. These optimality conditions allow us to also characterize when a
solver will not participate for a given price $\nu$. Any solution to the
solver subproblem, denoted by $\tilde{x}_{k}$, which lies on the interior of
the domain ($\tilde{x}_{k}>0$) must satisfy

$u^{\prime}(\tilde{x}_{k})-c^{\prime}(\tilde{x}_{k})=\nu.$

Thus, if $\nu$ is set too high, the solver will simply return the zero trade:
$\tilde{x}_{k}=0$. Furthermore, this condition implies that when the costs to
the solver are zero, it will report its true marginal utility (i.e., its true
price), corroborating the results in §2.2.

#### A dual approach.

We can solve this social welfare problem with a mechanism akin to a dutch
auction. As a consequence of strong duality, which holds under relatively weak
conditions [BV04, §5.2.3], we can solve the primal problem (4) by solving the
dual problem. The dual problem is

minimize $\displaystyle h(\nu).$

Since this problem is convex and single dimensional, we can solve it with an
iterative method that can be interpreted as a dutch-auction-like mechanism.
The first derivative of $h$ is

$h^{\prime}(\nu)=\tilde{y}-\sum_{k=1}^{n}\tilde{x}_{k},$

where $\tilde{y}$ is the optimizer of $G(\delta-y)+\nu y$ and
$\\{\tilde{x}_{k}\\}$ are the optimizers of the solvers’ subproblems, i.e.,
the amount they will quote given a price $\nu$. Since this function is
unimodal, we can optimize it with a simple strategy: we start with a large
$\nu$ and query each solver to get $\\{\tilde{x}_{k}\\}$, which is exactly the
solution to the solver utility-maximization subproblem above. Then, we compute
the gradient $h^{\prime}(\nu)$. If this gradient is greater than zero, we
decrease $\nu$ and repeat. When the gradient is sufficiently close to zero, we
are done. In practice this looks like a Dutch auction. Note that, at
termination, the $\tilde{y}$ and $\\{\tilde{x}_{k}\\}$ are exactly the
solutions to the original social welfare problem (4). The dual setup above can
also be extended to the case of $n$ assets with independent solver costs.
However, the proposed mechanism does not extend directly, as there is no
longer a total ordering on prices. A coordinate descent type mechanism may
work in this multiple asset case.

#### Interpretation.

Recall that $\alpha_{k}$ is the price a solver quotes to the user. It is clear
that for any rational solver

$u_{k}(\delta)\geq\alpha_{k}\delta\implies
u_{k}(\delta)/\delta\geq\alpha_{K}.$

In other words, $\alpha_{k}x_{k}$ is a linear underestimator of the solver’s
concave utility function $u_{k}$. The mechanism above, then, elicits the
solver’s true price for a given order size, and it terminates when the
solver’s marginal price is equal to that of the CFMM. However, instead of
proposing an order size for each of the $n$ solvers, the mechanism instead can
propose one price, which it broadcasts to all solvers. The solvers respond to
this price with a proposed order.

###  3.3 Extension: congestion

From this optimization framework, we also see that positively correlated costs
among solvers will reduce the user’s welfare. Now assume that the solver cost
functions are not independent. We model congestion by introducing a new cost
function $\tilde{c}_{k}:{\mbox{\bf R}}^{n}_{+}\to{\mbox{\bf
R}}_{+}\cup\\{\infty\\}$ which is increasing in all arguments and matches the
old cost function $c_{k}$ when that solver is the only one quoting an order.
In other words, we must have that

$\nabla\tilde{c}(x_{1},\dots,x_{n})>0$

and

$\tilde{c}_{k}(0,\dots,x_{k},\dots 0)=c_{k}(x_{k})$

for any order $x_{k}$. These conditions imply that marginal costs are higher
in this regime:

$\tilde{c}_{k}^{\prime}(x_{1},\dots,x_{n})>\tilde{c}_{k}(0,\dots,x_{k},\dots
0)=c_{k}^{\prime}(x_{k}),$

where there is at least one solver $k^{\prime}\neq k$ with $x_{k^{\prime}}>0$.
As a result, from the dual conditions, we have that

$\nu^{\star}=u_{k}^{\prime}(x_{k}^{\star})-\tilde{c}_{k}^{\prime}(x_{1}^{\star},\dots,x_{K}^{\star})\leq
u_{k}^{\prime}(x_{k}^{\star})-c_{k}^{\prime}(x_{k}).$

Thus, the price $\nu^{\star}$ when congestion costs are present is less that
the price with independent costs. In other words, the user will receive
reduced output from the solvers if their costs are positively correlated. This
result mirrors that of the probabilistic model: under congestion costs, user
welfare decreases.

##  4 Model Extensions

As discussed in the introduction, there are both non-blockchain focused intent
markets as well as more complex intent-like systems within the blockchain
world. We will briefly discuss the applicability of our model to those
scenarios and extensions that can be used for analyzing these systems.

#### Stock order-by-order auctions.

The US Securities and Exchange Commission (SEC) recently released a proposal
for an order flow execution auction within the US stock market [Sec22]. One
rationale for implementing such a system is to improve the fairness of ‘price
improvement’ that a retail stock trader receives [Sec22]. The majority of
retail, individual stock traders within the US use brokerages such as
Robinhood or Vanguard to execute their trades. These brokerages outsource the
task of finding an optimal execution price for a user to wholesalers, who
aggregate order flow from brokerages and institutions (such as endowments or
pension funds) to execute trades. Currently, there is no competitive market
for brokerages to engage wholesalers—they instead construct private agreements
which are known as payment for order flow [Wel22]. These agreements have come
under scrutiny by the US government, as the lack of a competitive market is
thought to have led to market failures during the Gamestop panic of 2021
[Ang21].

The SEC’s proposal is to have wholesalers run an auction whose initial price
is set at the price tendered by the wholesaler. The wholesaler would run the
auction for 100 milliseconds and any regulated entity, such as an accredited
investor, could participate in the auction. To win the auction, the wholesaler
would have to participate explicitly, as their initial price does not serve as
a reserve price. In contrast, the UniswapX Dutch auction guarantees the user
at least the amount of output quoted publicly by a CFMM, even if the auction
does not clear.

Due to latency constraints, it is assumed that the wholesaler would be running
many of these auctions simultaneously. It is an interesting extension of our
model to understand the degree to which auctions in multiple markets, where
price movements may be correlated, result in improved prices for users.

#### Proposer-builder separation.

One of the most common auctions settled on blockchains is the so-called
‘proposer-builder separation’ (PBS) auction [PR23, GPR23, BGR24]. The seller
in this auction is the proposer: an agent who has a temporary monopoly to
propose a valid block within a blockchain. The proposer runs an auction to
solicit blocks from agents known as builders who source transactions and
construct valid blocks. A number of works have studied the centralizing effect
within PBS auctions [BGR24, GPR23]. However, none of the prior works analyzed
the impact of congestion on auction performance.

PBS auctions fit into our framework as they are single item first price
auctions. Block builders, who source transactions to earn fees or rewards from
block construction (i.e. the aforementioned MEV [KDC22, BGR23, KDC23]),
compete for order flow from exchanges and wallets that generate transactions.
Similar to PFOF, these builders often offer incentives to users to send their
transactions to them, and may exert costly effort to acquire user order flow.
One notable difference is that our model assumes a uniform congestion cost
across agents. In the PBS auction, different agents will have varied costs and
congestion is more heterogeneous across agents. Modifying our model to include
hetereogeneity between the agents is future work and can be used to analyze
historical auction data to measure reductions in competition.

##  5 Conclusion

In this paper, we model the incentives that solvers face in intent-based
markets. Our first model is a probabilistic model of an auction under two
complicating features: (i) entry costs incurred by solvers while setting up
infrastructure and (ii) congestion costs faced by solvers while attempting to
search for liquidity at desirable prices. In our single intent model (single
item auction model), we demonstrate that entry costs reduce the number of
solvers that participate in the market in equilibrium. If, furthermore,
participants have to invest costly effort into improving their ability to find
prices, and face congestion costs while doing so, we also find the existence
of limited oligopoly. These results demonstrate the counterintuitive fact that
planners of intent markets may want to limit entry in order to maximize user
welfare.

Our second model is an optimization-based deterministic model. This model
enables the explicit analysis of the social welfare maximization problem faced
by a planner who considers both the user’s and the solvers’ utilities. This
approach has several benefits: (i) it allows for fitting utility functions to
data and (ii) suggests a natural Dutch auction-like primal-dual mechanism to
solve the welfare maximization problem.

We leave as future work a generalization of the single intent model to more
complicated intents which may include transacting in multiple markets.
Further, it is an interesting question whether numerical methods may be used
to estimate whether solvers face congestion costs in these markets, and the
degree to which users face welfare losses. Finally, in our modeling so far,
motivated by practice, we have restricted attention to solvers competing in an
auction to fulfill the user’s intent, where the profits from the auction
provide the incentives for solvers to enter and invest. However, certain kinds
of congestive investment costs may be difficult to observe and even more
challenging for users to directly contract [GH92]. Recent developments in the
literature provide insight into how a principal can robustly incentivize
effort by an agent (e.g., [Car15]). Another direction of future work,
therefore, is whether such contracts can be implemented in intent markets.

Our work implies that designers and planners of intent markets need to
thoroughly vet the various costs that solvers face. If the goal of an intent
network is to maximize welfare while also allowing free entry into the system,
designers should be careful to construct auctions such that bidders minimize
contention in sourcing inventory.

## Acknowledgments

We would like to thank Malcolm Granville for valuable feedback on an earlier
version of this paper. We would also like to thank Karthik Srinivasan and
Barry Plunkett for helpful discussions.

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##  Appendix A Exponential value distributions

We now derive the left hand side of equation (1) in the case where $F$ is the
exponential distribution with rate $\lambda$ (that is, $F(p)=1-\exp(-\lambda
p)$ for $p\geq 0$). We have:

$\displaystyle\sum_{k=0}^{n}\left({n\choose
k}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)S(k)$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda}\sum_{k=0}^{n}\left({n\choose
k}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)\frac{1}{k+1}$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda}\sum_{k=0}^{n}\left(\frac{n!}{k!(n-k)!}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)\frac{1}{k+1}$
$\displaystyle=$
$\displaystyle\frac{1}{\lambda}\sum_{k=0}^{n}\left(\frac{n!}{(k+1)!(n-k)!}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{(n+1)\lambda
F_{C}(\bar{c})}\sum_{k=0}^{n}\left(\frac{(n+1)!}{(k+1)!(n-k)!}F_{C}^{k+1}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{(n+1)\lambda
F_{C}(\bar{c})}\sum_{k=1}^{n+1}\left(\frac{(n+1)!}{(k)!(n+1-k)!}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n+1-k}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{(n+1)\lambda
F_{C}(\bar{c})}(1-(1-F_{C}(\bar{c}))^{n+1})$

##  Appendix B Uniform value distributions

We now derive the left hand side of equation (1) in the case where $F$ is the
uniform distribution on $[0,1]$ (that is, $F(p)=p$ for $p\in[0,1]$). We have:

$\displaystyle\sum_{k=0}^{n}\left({n\choose
k}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)S(k)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\left({n\choose
k}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)\frac{1}{(k+1)(k+2)}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\left(\frac{n!}{k!(n-k)!}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)\frac{1}{(k+1)(k+2)}$
$\displaystyle=$
$\displaystyle\sum_{k=0}^{n}\left(\frac{n!}{(k+2)!(n-k)!}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{(n+1)(n+2)F_{C}^{2}(\bar{c})}\sum_{k=0}^{n}\left(\frac{(n+2)!}{(k+2)!(n-k)!}F_{C}^{k+2}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{(n+1)(n+2)F_{C}^{2}(\bar{c})}\sum_{k=2}^{n+2}\left(\frac{(n+2)!}{(k)!(n-k)!}F_{C}^{k}(\bar{c})(1-F_{C}(\bar{c}))^{n-k}\right)$
$\displaystyle=$
$\displaystyle\frac{1}{(n+1)(n+2)F_{C}^{2}(\bar{c})}(1-(1-F_{C}(\bar{c}))^{n+2}-(n+2)F_{c}(\bar{c})(1-F_{C}(\bar{c}))^{n+1})$

##  Appendix C Extreme spacing

In this section, we derive the equality $(k+1)S(k)=ES(k+1)$ for any CDF $F$
and PDF $f$, where $p^{*}=0$. To see this, observe the following chain of
equalities:

$\displaystyle ES(k+1)$ $\displaystyle=\mathop{\bf E{}}_{p_{i}\sim
f}[p_{k+1:k+1}-p_{k:k+1}]$
$\displaystyle=\int_{0}^{\infty}\left((k+1)F(p)^{k}f(p)-(k+1)kF(p)^{k-1}(1-F(p))f(p)\right)pdp$
$\displaystyle=(k+1)\int_{0}^{\infty}F(p)^{k-1}f(p)\left(F(p)-k(1-F(p))\right)pdp$
$\displaystyle=(k+1)\int_{0}^{\infty}F(p)^{k-1}f(p)\left((k+1)F(p)-k\right)pdp$
$\displaystyle=(k+1)\int_{0}^{\infty}f(p)\left((k+1)F(p)^{k}-kF(p)^{k-1}\right)pdp$
$\displaystyle=(k+1)\int_{0}^{\infty}\frac{d\left(F(p)^{k+1}-F(p)^{k}\right)}{dp}pdp$
$\displaystyle=(k+1)\int_{0}^{\infty}(F(p)^{k}-F(p)^{k+1})dp$
$\displaystyle=(k+1)\int_{0}^{\infty}F(p)^{k}(1-F(p))dp,$

where the second to last equality follows from integration by parts.