Axioms for Constant Function Market Makers Jan Christoph Schlegel1 Mateusz Kwa snicki2 and Akaki Mamageishvili3 1Department of Economics City University of London London UK

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Axioms for Constant Function Market Makers

Jan Christoph Schlegel1, Mateusz Kwa´snicki2, and Akaki Mamageishvili3

1Department of Economics, City, University of London, London, UK∗

2Department of Pure Mathematics, Wroclaw University of Science and

Technology, Wroclaw, Poland

3Oﬀchain Labs, Zurich, Switzerland

Abstract

We study axiomatic foundations for diﬀerent classes of constant-function auto-

mated market makers (CFMMs). We focus particularly on separability and on dif-

ferent invariance properties under scaling. Our main results are an axiomatic charac-

terization of a natural generalization of constant product market makers (CPMMs),

popular in decentralized ﬁnance, on the one hand, and a characterization of the Log-

arithmic Scoring Rule Market Makers (LMSR), popular in prediction markets, on

the other hand. The ﬁrst class is characterized by the combination of independence

and scale invariance, whereas the second is characterized by the combination of inde-

pendence and translation invariance. The two classes are therefore distinguished by

a diﬀerent invariance property that is motivated by diﬀerent interpretations of the

num´eraire in the two applications. However, both are pinned down by the same sepa-

rability property. Moreover, we characterize the CPMM as an extremal point within

the class of scale invariant, independent, symmetric AMMs with non-concentrated

liquidity provision. Our results add to a formal analysis of mechanisms that are cur-

rently used for decentralized exchanges and connect the most popular class of DeFi

AMMs to the most popular class of prediction market AMMs.

1 Introduction

One of the ﬁrst and so far most successful applications of Decentralized Finance (DeFi),

ﬁnancial applications run on permissionless blockchains, are so-called Automated Market

Makers (AMMs). They are used to trade cryptocurrencies algorithmically without relying

on a custodian or trusted third party. The average daily volume traded on AMMs used

in DeFi reached $3.5B in 2022, while the total value locked in DeFi protocols at the time

∗Corresponding author: jansc@alumni.ethz.ch

arXiv:2210.00048v4 [cs.GT] 14 Feb 2023

of writing this paper1is estimated to be $45B. The considerable volume of assets traded

on AMMs contrasts with their simple economic design which is notably diﬀerent from the

design of traditional ﬁnancial exchanges. The state of a typical AMM used in DeFi consists

of the current inventories of the traded tokens. Trades are made such that some invariant of

these inventory sizes is kept constant. Traders who want to exchange tokens of type Afor

tokens of another type B, add Atokens to the inventory and in return obtain an amount

of Btokens from the inventory so that the invariant is maintained. The simplicity of these

so-called constant function market makers (CFMMs) stems from the limited storage and

computation capabilities of smart contract blockchains. In comparison to more conven-

tional market designs such as limit order books, updating a simple function and inventories

is relatively inexpensive to implement and scalable on smart contract blockchains such as

Ethereum. Moreover, AMMs allow market participants to provide liquidity passively with-

out having to trade themselves. This makes liquidity provision in AMMs a popular way of

getting (negative) exposure to volatility while being compensated by a steady stream of fee

income.

While CFMMs proved to be very popular and reliable, the construction of invariants to

deﬁne them seems in many ways ad-hoc and not based in much theory. In this paper, we ﬁll

this gap and propose an axiomatic approach to constructing CFMMs. The approach is, as

in any axiomatic theory, to formalize simple principles that are implicitly or explicitly used

when constructing trading functions in practice and to check which classes of functions

satisfy these principles, beyond those functions already used in practice. The constant

product market maker (CPMM), which makes the market in such a way that the product of

inventories of the diﬀerent traded assets is kept constant, has been particularly focal in DeFi

applications. The CPMM was originally introduced by Uniswap, the ﬁrst and so far most

popular DeFi AMM by trading volume.2Our ﬁrst main result (Corollary 1) characterizes a

natural superclass of the (multi-dimensional) CPMM by three natural axioms. The CPMM

rule is characterized by being trader optimal within this class (Theorem 2).

In contrast to the case of AMMs for DeFi, there is a much more developed theory for

AMMs for prediction markets (see the literature review, Section 1.1), for which AMMs

have originally been introduced (Hanson, 2003). There are notable diﬀerences in the way

that AMMs for prediction markets and those for DeFi function in practice, which has to do

1January 20th, 2023, data obtained from deﬁlama.com.

2A natural generalization of the CPMM, the multi-dimensional weighted geometric mean is for example

used by Balancer, (Martinelli and Mushegian, 2019), the third largest DeFi AMM by trading volume.

The CPMM was used by Uniswap before its update to Version 3 in 2021 and is still in use in many

other AMMs. The CPMM was replaced in V3 by a new design (Adams et al., 2021) that abandons

fungibility of liquidity to allow for customized liquidity positions that can be chosen by individual liquidity

providers. Fungibility of liquidity positions is captured by the scale invariance axioms that we discuss

subsequently. The second largest DeFi AMM by trading volume, the Curve protocol (Egorov, 2019), is

speciﬁcally designed to trade highly correlated assets (such as stable coins or diﬀerent representations of

the same cryptocurrency). Our second main axiom, independence, is violated by the curve AMM. As we

will argue, our main characterization implies separability of the market-making function, which is mainly

appealing for situations where the price correlation between the traded tokens is imperfect. However, for

the two-dimensional case, Theorem 4 characterizes a broad class (including the two-dimensional version of

the curve formula) of generally non-separable CFMMs.

with the diﬀerent nature of the assets traded in them: In a prediction market, the assets

are artiﬁcially created Arrow-Debreu securities that allow traders to bet on an outcome.

Moreover, traders exchange assets against a num´eraire rather than swapping between assets.

However, there are also notable similarities that allow for a common analysis of the two

applications, as we discuss below. Similarly as for the DeFi context, for prediction markets,

one rule has become focal in theory and practice. This is Hanson’s logarithmic scoring rule

market maker (LMSR) Hanson (2007) which maintains an invariant that is the logarithm

of a sum of exponentials of the inventories.3Our second main result characterizes the

LMSR by three natural axioms (Corollary 3). To the best of our knowledge, this is the ﬁrst

axiomatic characterization of this rule. Our results give a possible normative justiﬁcation

for the use of the CPMM and of the LMSR rule. Moreover, they highlight a hidden

commonality between the two rules; the main axiom driving both characterization results

is the same independence axiom.

Independence requires that the terms of trade for trading a subset of assets should

not depend on the inventory levels of not-traded assets. In the case of smooth liquidity

curves, this is equivalent to requiring that the exchange rates for traded asset pairs do not

depend on the inventory levels of assets not involved in the trade. If AMMs are deﬁned

through cost functions, as is customary in the prediction market context, independence

(equivalently) means the following: suppose a trader trades in a subset of assets. Then

whether two trades involving these assets cost the same should be independent of previous

trades made in other assets. Our characterization results combine independence with one

of two invariance properties that derive their normative appeal through the diﬀerent role

the num´eraire plays in DeFi and in prediction markets.

In the DeFi application, the num´eraire is an ”LP token” which is a derivative product

that is a claim to a fraction of the pooled liquidity and the accrued trading fees of the

AMM. A desirable property of these liquidity positions is that they are fungible. Practically,

the fungibility of liquidity positions allows tokenizing them in order to use them in other

applications, for example as collateral or to combine them with other assets to new ﬁnancial

products. This is an instance of what is usually called the ”composability” property of DeFi

protocols. Fungibility of liquidity positions requires scale invariance, or, in a stronger form,

homogeneity of the trading function. Geometrically, scale invariance means that liquidity

curves through diﬀerent liquidity levels can be obtained from each other, by projection along

rays through the origin, analogous to how homothetic preferences in consumer theory can

be constructed. Scale invariance of the inventory has been assumed or discussed in previous

work, usually in the stronger form of (positive) homogeneity (Capponi and Jia, 2021) or

(positive) linear homogeneity (Othman, 2021; Angeris et al., 2022a) and most popular

AMMs used in practice, including the CPMM, satisfy it.

In the prediction market application, the num´eraire is external to the AMM (usually it is

3In its original formulation, the LMSR is deﬁned by a logarithmic scoring rule. AMMs deﬁned by

scoring rules can be equivalently described by a cost function (Chen and Pennock, 2007) which can be

shown to take the form of a sum of exponentials for the logarithmic scoring rule. See Footnote 8 for the

equivalence between the two deﬁnitions. Describing AMMs in terms of cost of trading and in terms of the

value of the inventory is equivalent up to change in sign as we discuss below.

a regular currency such as US dollars). Traders exchange assets against the num´eraire rather

than swapping between assets. The counterparty at settlement is the prediction market

organizer who sells the securities to traders and therefore is the sole liquidity provider.

Assets in a prediction market are Arrow-Debreu securities and the market is suﬃciently

complete so that traders can hold a risk-less portfolio consisting of one unit of each security

so that the portfolio always pays out one unit of num´eraire in every state of the world. As

a consequence of this, AMMs for a prediction market usually satisfy a diﬀerent invariance

property: translation invariance, requiring that making a risk-less trade consisting of the

same amount of each of the assets always costs the same independently of the state of the

AMM. In the case of a smooth trading function, this equivalently means that marginal

prices of the traded assets measured in the num´eraire always add up to one (see e.g. Chen

and Pennock (2007)). Translation invariance of AMMs is usually assumed in the literature

on AMMs for prediction markets, see Section 1.1, and AMMs used in practice, including

the LMSR, satisfy it.

The combination of scale invariance or translation invariance and independence imply

that the terms of trade are fully determined by the inventory ratio resp. the inventory

diﬀerence of the pair traded. Under scale invariance, at the margin, percentage changes

in exchange rates are proportional to percentage changes in inventory ratio. Under trans-

lation invariance, at the margin, percentage changes in exchange rates are proportional

to changes in the diﬀerence of the two inventories. Therefore, combining independence

and scale invariance, we obtain the class of constant inventory elasticity4market makers

(CEMMs) (Theorem 1). This general class contains as special cases constant products,

weighted geometric means as well as weighted means. On the other hand, by combining

independence and translation invariance, we obtain LMSR rules or constant sum market

makers (Theorem 3). Both classes also contain AMMs with non-convex liquidity curves5

which are usually not appealing in practice. We can, however, combine the two axioms in

either characterization with other axioms to obtain characterization results for convex liq-

uidity curves: If, in addition to scale invariance and independence, we impose that liquidity

curves are convex, then the elasticity in the above characterization is positive; alternatively,

if we require un-concentrated liquidity (liquidity curves should not intersect with the axes)

then the elasticity in the above characterization is positive but smaller or equal to 1 (Corol-

lary 1). Similarly, if, in addition to translation invariance and independence, we impose

convexity of liquidity curves,6then the liquidity parameter bin the LMSR formula is non-

negative (where the constant sum market maker corresponds to the limit of the LMSR as

4The inventory elasticity of an AMM is a measure of how marginal prices change with the ratio of the

inventories of the traded assets. If the inventory elasticity is when the AMM holds xunits of asset Aand

yunits of asset B, then changing the inventory ratio y/x by 1% corresponds to changing the exchange rate

of Aand Bby 1/%. For example, for the constant product rule, the inventory elasticity is 1 since the

exchange rate is simply given by the inventory ratio y/x.

5For CEMMs, the non-convex case corresponds to negative inventory elasticity, where increases in

inventory ratio correspond to decreases in exchange rates. For LMSRs the non-convex case corresponds to

a negative liquidity parameter b.

6Convexity of liquidity curves follows from concavity of the trading function which is equivalent to the

convexity of the cost function which is a standard assumption in the prediction market literature.

bapproaches inﬁnity, Corollary 3).

Finally, if we further require symmetry in market making, the AMMs in the class of

scale invariant, independent AMMs with un-concentrated liquidity can be ranked by the

curvature of their liquidity curves which determines how favorably the terms of trade are

from the point of view of traders; the CPMM is characterized by being trader optimal

within this class (Theorem 2).

The above characterizations are obtained for the case of more than two assets traded in

the AMM. For the case of exactly two assets, the independence axiom is trivially satisﬁed

and we generally obtain a much larger class of trading functions satisfying the above axioms

(Theorem 4). The class can no longer be completely ranked by the convexity of the induced

liquidity curves. However, if we focus on separable CFMMs we obtain the same kind of

characterizations (Theorems 5 and 7 and Corollaries 5 and 7) as in the multi-dimensional

case, as well as the same kind of optimality result for the CPMM (Theorem 6). In the

two-dimensional case, separability of the trading function is a consequence of an additivity

property for liquidity provision that we call Liquidity Additivity.

The axiomatic approach leads us to considerations and classes of functions familiar

from other ﬁelds in economics, consumer theory, and production theory in particular where

constant elasticity functions play an important role.A main technical contribution of the

present paper is to derive the constant elasticity functional form on the one hand and the

LogSumExp form on the other hand from two natural properties, without relying on any

diﬀerentiability or on convexity assumptions.

The paper is organized in the following way. In the next subsection, we brieﬂy review

the literature. In Section 2 we introduce notation and various axioms. In Section 3, we

provide characterization results for the case of more than two assets. In Section 4, we

provide results for the case of two assets. In Appendix A, we show that the axioms used in

the various characterizations are logically independent. In Appendix B we show how our

result fails to hold for discontinuous liquidity curves.

1.1 Related Work

Automated market makers have ﬁrst been analyzed scientiﬁcally in the context of predic-

tion markets (Hanson, 2003, 2007; Othman and Sandholm, 2010; Chen and Pennock, 2007;

Chen et al., 2008; Abernethy et al., 2011; Ostrovsky, 2012; Othman et al., 2013). The

original formulation of AMMs uses scoring rules (Hanson, 2003). Traders who interact

with the AMM obtain a net pay-oﬀ after settlement that is determined by a scoring rule

that measures the accuracy of the trader’s prediction in comparison to the realized state.

In the case of the LMSR the score is logarithmic. Chen and Pennock (2007) observe that

AMMs can alternatively be deﬁned in terms of cost functions. In this paper, we work with

trading functions that are equivalent to cost functions (see below). Translation invariance

is generally necessary for the equivalence between scoring rule AMMs and cost function

AMMs (Chen and Pennock, 2007). However, in subsequent work to ours, Frongillo et al.

(2023) show that the (not translation invariant) CPMM in two dimensions can be described

by a family of scoring rules where each scoring rule describes trading along a ﬁxed liquidity

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AxiomsforConstantFunctionMarketMakersJanChristophSchlegel1,MateuszKwasnicki2,andAkakiMamageishvili31DepartmentofEconomics,City,UniversityofLondon,London,UK*2DepartmentofPureMathematics,WroclawUniversityofScienceandTechnology,Wroclaw,Poland3OchainLabs,Zurich,SwitzerlandAbstractWestudyaxiomaticfound...

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Axioms for Constant Function Market Makers Jan Christoph Schlegel1 Mateusz Kwa snicki2 and Akaki Mamageishvili3 1Department of Economics City University of London London UK

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