A Myersonian Framework for Optimal Liquidity Provision in Automated Market Makers

The purpose of an exchange is to enable the trade of two or more assets. At a stock exchange, for example, shares of a stock might be exchanged for US dollars via a mechanism known as a “limit order book” (LOB), in which buyers and sellers post “limit orders” (offers to buy or sell a specified quantity at a specified price) that are then matched greedily.

Every trade has two sides: for one user to buy a risky asset at a given price, there must be a corresponding seller willing to sell at that same price. Such trades can occur organically due to a coincidence of wants from two traders, but in practice many trades are enabled by professional market makers who continually match buy and sell orders whenever the price is right. This paper is about the profit-maximization problem faced by such a market maker. In particular, the market maker is continually updating their beliefs around the asset price, according to a Bayesian-like belief inference framework, from a prior belief in conjunction with observations by traders who report price estimates.

Digital assets secured by blockchains, such as cryptocurrencies, are also traded via exchanges. In centralized exchanges such as Coinbase or Binance, users hand control of their assets over to the exchange and, therefore, accept the credit risk of not getting them back. Such exchanges are typically based on the CLOB design. Decentralized exchanges (DEXs), on the other hand, operate purely programmatically (i.e., “on-chain”), are typically non-custodial (meaning that traders at all times have direct control of their assets in the sense that assets are not entrusted to a third party), and often depart from the CLOB model. There are several reasons for this departure: on-chain computation and storage can be expensive, and CLOBs need a lot of both; the markets for many digital assets (especially long-tail ones) are thin and have no dedicated market makers, in which case CLOBs may fail to provide an acceptable level of liquidity; and a general openness in the blockchain world to experimental designs. The DEX design space is large, and many different designs have been deployed (on Ethereum and other general-purpose smart contract platforms) over the past years.

For example, Uniswap (especially in its earliest iterations, v1 and v2) is a canonical example of an automated market maker (AMM). In an AMM (unlike in a CLOB), the roles of traders and market makers are clearly separated. A market maker, which in this context is called a liquidity provider (LP), deposits into the AMM some amounts of the assets being traded (and is, by default, passive thereafter). At all times, the AMM defines a marginal price as a function of its current reserves of those assets. (In the case of Uniswap v1 and v2, the spot price of an AMM is simply the ratio of the quantities of its two assets.) The AMM is willing to take either side of a trade (buy or sell) at its current marginal price, and in this sense is always providing liquidity to traders. The latest iteration of Uniswap (v3) blurs the line between AMMs and CLOBs by allowing a much wider range of LP demand curves than in v2 (to some extent approximating what can be expressed by limit orders in a CLOB), thereby encouraging an LP to more actively manage its demand curve as market conditions evolve.

The goal of this paper is to characterize the solution to the following fundamental question: What is the optimal liquidity demand curve for a profit-maximizing market maker? This question is relevant for a market maker in a CLOB, an LP of an expressive AMM such as Uniswap v3, or even the designer of a new AMM (in which case the market making liquidity demand curve is defined programmatically rather than submitted by a third-party LP). This paper offers a new model for reasoning about this question and a quite general answer, which resembles in many respects, and in fact generalizes, Myerson’s theory of optimal auctions [33]. To get a feel for our model and results, we next discuss a highly restricted but nonetheless illuminating special case, and give a very brief overview of our general theory.

The theory of market microstructure (see e.g., [35]) differentiates between “informed traders,” who have better information about the fundamental value of an asset than the market maker does, and “noise traders,” who trade for idiosyncratic reasons that do not reflect any private information about the fundamental value of the asset. Generally speaking, market makers lose money to informed traders due to adverse selection: by virtue of such a trader being willing to trade, it expects a profit, which, due to the zero-sum nature of common-value assets, comes at the expense of the market maker. Market makers can, however, generally profit from noise traders. In this section, we first consider (as an instructive warm-up) the special case in which all traders are noise traders (and hence adverse selection plays no role). Our general model, the results of which are briefly presented in the end of this section, and which is detailed in Section 3 accommodates arbitrary mixtures of informed and noise traders, and our general results must therefore reflect adverse selection effects in addition to the aspects identified in this section.

Fundamentally, a market maker must decide, as a function of the information available to it, how much of a risky asset they are willing to buy or sell at different prices. For this example, we will assume that the market maker knows the true value $p_{0}$ of the risky asset that it is trading.

An obvious strategy for such a market maker is to accept exactly the orders on which it makes money: all (and only) the buy orders with offered price more than $p_{0}$, and all (and only) the sell orders with offered price less than $p_{0}$. However, as we will see, this is not in general the profit-maximizing strategy for a market maker.

It will be convenient to encode a market maker’s strategy via the choice of a demand curve $g$, which maps prices to quantities (i.e., an amount of risky asset that will be held by the market maker). The interpretation is that the market maker would be interested in selling $g(p_{0})-g(p)$ units of the risky asset at the price $p$. (Note that this may be a negative number of units, in which case the market maker will be buying the risky asset from the trader.) A choice of $p_{0}$ and $g$ induces the allocation rule $x(p)=g(p_{0})-g(p)$, the amount of the risky asset allocated to a trader who reports a price of $p$.

Next, we assume that a trader reports a price ${\hat{p}}\sim\mathcal{D}$ that is drawn according to a distribution $\mathcal{D}$ known to the market maker. We assume that the trader is willing to buy or sell at most a bounded amount of the risky asset (normalized to be 1 only for notational convenience, since there is no difference whatsoever for any arbitrary finite bound on the results), and that the demand curve $g$ and price $p_{0}$ accordingly satisfy $g(p_{0})-g({\hat{p}})\in[-1,1]$ for all ${\hat{p}}$. When the noise trader reports their price ${\hat{p}}$, they receive an allocation $x({\hat{p}})=g(p_{0})-g({\hat{p}})$ of the risky asset and must remit a payment $y({\hat{p}}$) to the AMM for this provision. As we will see in Section  2, the natural notion of incentive-compatibility in this setting — that a utility-maximizing trader who values the risky asset at a price ${\hat{p}}$ will in fact report ${\hat{p}}$ to the AMM — uniquely pins down the payments (the $y({\hat{p}})$’s) as a function of the allocation rule (the $x({\hat{p}})$’s).¹¹1As the reader might guess, it is also essential that the allocation rule $x$ is monotone in ${\hat{p}}$, or equivalently that the demand curve $g$ is downward-sloping, with the market maker selling more and more of the risky asset as the going price gets higher and higher. See Section 2 for details.

With informed traders, a rational market maker will update its estimate $p_{0}$ of the risky asset’s true price following a trade (generally increasing it after a trader buys the risky asset from the market maker and decreasing it after a sell). Our general model in Section 3 accommodates this through a generic update rule $\pi(p_{0},{\hat{p}})$ that specifies the market maker’s new estimate as a function of its old estimate $p_{0}$ and the new information conveyed by a trader reporting the price ${\hat{p}}$. For example, $\pi$ might correspond to a Bayesian update with respect to some assumed information structure, and the update rule may come from and be revised according to a no-regret learning algorithm used by the market maker on successive rounds. In the simple special case of only noise traders being present, because the behavior of noise traders is (by assumption) independent of the true value of the risky asset, the market maker has no cause for updating its internal price (i.e., $\pi(p_{0},{\hat{p}})=p_{0}$ for all $p_{0}$ and ${\hat{p}}$).

What is the optimal strategy (i.e., choice of demand curve $g$) for the market maker? That is, which choice of $g$ maximizes

where the expectation is with respect to ${\hat{p}}\sim\mathcal{D}$, $x({\hat{p}})=g({p_{0}})-g({\hat{p}})$ is the allocation rule, and $y$ is the uniquely defined payment rule mentioned above?²²2Note that, obviously, the choice of the optimal mechanism does not need the knowledge of the reported price by the trader; only its probabilistic characteristics are known according to the distribution $\mathcal{D}$. (In the general case, the “$p_{0}"$ term in (1) is replaced by $\pi({p_{0}},{\hat{p}})$, with the market maker evaluating its portfolio according to its new belief about the value of the risky asset, conditioned on observing the trader’s report of ${\hat{p}}$.)

The results in this paper imply that the answer to this question is always defined by a posted-price-like mechanism with two prices (depending on $\mathcal{D}$), $p_{l}\leq p_{0}$ and $p_{h}\geq p_{0}$. These two prices split the price range into three intervals, and the optimal strategy is to always buy the maximum amount in the lowest price interval, always sell in the highest price interval, and refuse to trade in the middle price interval.

We will be interested in the length of this middle price interval, which we call the no-trade gap. Bigger no-trade gaps are generally worse from a welfare perspective, with welfare-increasing trades sacrificed in the name of higher profit to the market maker. The no-trade gap is reminiscent of the “bid-ask spread” in CLOBs, which occupies a central position in the study of traditional market-making in CLOBs.³³3In a CLOB, the “bid” and “ask” refer to the highest price of an outstanding buy order and the lowest price of an outstanding sell order, respectively. Because limit orders in a CLOB are always matched greedily, the bid-ask spread is always non-negative. Small spreads are generally viewed as a good thing, indicating a “highly liquid” market (with lower trading costs, at least for small quantities). Traditionally, theoretical justifications for non-trivial bid-ask spreads in CLOBs have relied on frictions (like transaction fees or inventory costs) or adverse selection (with a market maker needing to exploit noise traders to cover the losses to informed traders). In the current example, with neither frictions nor adverse selection, the no-trade gap arises for a different reason, namely the monopolist position of the market-maker. In our general model, with the possibility of informed traders, monopoly pricing and adverse selection both contribute to the no-trade gap. For example, no matter what the distribution $\mathcal{D}$ is, if all traders have perfect information (i.e., extreme adverse selection), then the no-trade gap encompasses the full price range.

The prices $p_{l}$ and $p_{h}$ that split the price range into three intervals, and hence the no-trade gap, are distribution-dependent. For example, intuitively, one might expect a smaller gap for distributions $\mathcal{D}$ that are tightly concentrated around $p_{0}$. In general, our analysis shows that these two prices can be characterized as roots of two functions that resemble the virtual valuation functions used in optimal auction theory [33].

We next give two example cases for pure noise-trading: if $\mathcal{D}$ is the uniform distribution on $[{p_{\text{min}}},{p_{\text{max}}}]$, then $p_{l}=({p_{\text{min}}}+{p_{0}})/2$ and $p_{h}=({p_{0}}+{p_{\text{max}}})/2$; the resulting optimal allocation rule is depicted in Figure 1a. If $\mathcal{D}$ is the exponential distribution with parameter $\lambda$ (on $[0,\infty)$), then $p_{h}=p_{0}+\tfrac{1}{\lambda}$. The lower price $p_{l}$ does not have a closed-form expression in this case, but can be easily approximated numerically (given ${p_{0}}$ and $\lambda$); Figure 2 enumerates some examples of how these lower prices vary, and an illustration of the optimal allocation rule is contained in Figure 1b.

The rest of this paper develops a similarly precise understanding of optimal market-maker strategies in a more general setting. We will now briefly present here the general results in an informal way. For the details, the reader is referred to Section 3.

The formal statements of the above results are Theorem 3.1 and Theorem 3.2 respectively. Finally, in Section 4, we examine how our general theory above applies in the interesting special case of a linear update rule. Through this special case, we naturally show that the structure of the no-trade gap captures precisely the complex effect that is the interpolation between the effects of adverse selection (observed when all traders have perfect information) and monopoly pricing (observed when all traders are noise traders). For further details, we refer the interested reader to Section 4.

We will now consider that traders interacting with AMMs have utility functions that emerge naturally from their true belief about the asset price versus what they bid to obtain a quantity of risky asset from the AMM. In particular, a trader who has a true belief of $p$ for the asset’s price, and submits a bid ${\hat{p}}$ to the AMM seeks to optimize (maximize) their utility defined by

because they consider that the asset’s true value to them is $p$ per unit, they obtain a quantity $x({\hat{p}})$ of it, and pay $y({\hat{p}})$ for this trade.

Is an AMM defined by an arbitrary demand curve $g(p)$ and an arbitrary payment rule $y({\hat{p}})$ incentive compatible? The answer is no, and the characterization of IC AMMs follows standard results of dominant-strategy incentive-compatible (DSIC) single-parameter auctions.

The next question to be answered is whether arbitrary payment rules $y({\hat{p}})$ are allowed to be requested by an IC AMM. Not surprisingly, as is the case for payments defined by an allocation rule of a DSIC mechanism, the payments need to be precisely defined by the demand curve for an IC AMM. The below formula is familiar from Myerson’s single-parameter DSIC auction design.

The above integrals are Riemann–Stieltjes integrals. In cases where $g(p)$ is differentiable, the differential takes the form $dg(p)=g^{\prime}(p)~{}dp$.

Note that the integral that defines the payments is non-negative if ${\hat{p}}\geq{p_{0}}$, and non-positive if ${\hat{p}}\leq{p_{0}}$. This is compatible with the direction of the trade, as defined by the allocation rule and Proposition 2.1, which is that $x({\hat{p}})\geq 0$ for ${\hat{p}}\geq{p_{0}}$, and $x({\hat{p}})\leq 0$ for ${\hat{p}}\leq{p_{0}}$.

First, we describe our model for how the market maker should optimize their allocation rule for the asset. From now on, the market maker will be assumed to be risk-neutral, i.e., be indifferent to “risk,” and only care to be profit-maximizing.

The Bayesian belief framework operates as follows: the market maker has a prior belief distribution on the asset price, and the (single) price it decides to use is ${p_{0}}$; for example, and most commonly, ${p_{0}}$ will be the mean of this prior distribution. A trader then comes to the AMM with an estimate ${\hat{p}}$ (that is the true belief of the trader, if the AMM is IC according to Definition 1). Traders are on the market to buy or sell up to 1 unit of the asset;⁵⁵5This is just for notational convenience. In fact, it generalizes easily for demand up to any arbitrary finite bound, and a subsequent normalization enables using exactly our results here with unit demand. that is, the allocation rule should satisfy $-1\leq x({\hat{p}})\leq 1$ for all ${\hat{p}}$. The trader’s estimate ${\hat{p}}$ is perceived by the market maker as one observation of an asset price estimate coming from a distribution $\mathcal{D}$ that is included in the market maker’s model. For this distribution $\mathcal{D}$, we assume that it has a continuous, positive density function $f({\hat{p}})$ on a compact support $[{p_{\text{min}}},{p_{\text{max}}}]$ and a cumulative density function (CDF) $F({\hat{p}})=\int_{p_{\text{min}}}^{\hat{p}}f(s)~{}ds.$⁶⁶6Our results can be extended to non-continuous density functions supported at possibly non-compact intervals, e.g., $(0,\infty)$, but for simplicity of exposition, we do not focus on handling these edge cases here, since they do not add to the general discussion points of the paper.

Applying Bayes’ theorem allows the market maker to specify the posterior distribution of the asset price, conditional on the observation from the trader. The market maker then sets this posterior distribution as their new prior for future observations (trades). We will assume that the way this posterior is used by the AMM is through valuing the quantity of asset given or obtained at a price of ${\pi}({p_{0}},{\hat{p}})$ which can be, for instance, the mean of the posterior distribution according to the Bayesian framework. (A very interesting special case as to how a linear dependence arises on the mean of the posterior with normally distributed prior and observational error is shown in Section 4.)

In general, we will call ${\pi}({p_{0}},{\hat{p}})$ the “update rule” and — for generality — we will not impose further constraints arising from the Bayesian framework on what it may be other than that:

Therefore, according to the above, the market maker’s next best estimate for the asset’s price is ${\pi}({p_{0}},{\hat{p}})$ conditioned on receiving the trader’s estimate ${\hat{p}}$. It is then natural to consider that, when selling (respectively, buying) some quantity $x({\hat{p}})$ of the asset from the trader, the AMM will not only consider the amount of money that they receive (respectively, give) but also their perceived best estimate of the value of the asset that they lost (respectively, gained). This, then, arises as an interesting contrast to traditional auction theory, where the revenue of the auction is only the payment that is made. Here, according to the above reasoning, it makes sense for the AMM to value their profit according to

Therefore, an AMM choosing an optimal strategy to provide liquidity will consider the strategy that maximizes their profit, among all feasible, incentive-compatible (for the traders) strategies.

By re-working Myerson’s optimal auction theory under the new objective that we want to optimize and the constraints that come with the definition of the allocation rule from an AMM, we get that:

We now turn our attention to optimizing the expected profit, i.e., finding the allocation rule (equivalently, the demand curve up to an arbitrary positive additive constant) which maximizes the expected profit.

Therefore, in any case, we observe that the optimal solution of Eqs. 7 and 8 has a well-specified no-trade gap around ${p_{0}}$ where the market maker is not willing to buy or sell any quantity of the asset to the trader. More specifically, the length of this no-trade gap is precisely $p_{1}-p_{2}$, where $p_{1}\geq{p_{0}}$, $p_{2}\leq{p_{0}}$ are as per the above analysis roots of the upper and lower virtual value functions respectively.

We can now specialize in the two interesting extreme cases that were examined in Section 1.2 to obtain the respective observations there: first, consider the pure noise trading case, where the traders are assumed to offer no information on the true asset price, and hence the market maker’s prior is static, i.e., in the case that the update rule is ${\pi}({p_{0}},{\hat{p}})={p_{0}},\ \forall{\hat{p}}\in[{p_{\text{min}}},{p_{\text{max}}}]$. In this case, we see that there is a non-trivial no-trade gap, since we have for the extremal values that $\phi_{u}({p_{0}})<0$, $\phi_{u}({p_{\text{max}}})>0$, $\phi_{l}({p_{\text{min}}})>0$, and $\phi_{l}({p_{0}})<0$ (see the beginning of Section 3.3 for the expressions), hence the respective roots are contained strictly in between their corresponding intervals (by continuity of the virtual value functions). This no-trade gap is akin to a bid-ask spread observed in traditional CLOB-based markets, and is attributable to the monopoly pricing power of the market maker. In addition, it is distribution-dependent and its length can be intuitively understood as being smaller under distributions $\mathcal{D}$, according to which the traders’ estimates ${\hat{p}}$ are distributed, that are more “concentrated” around ${p_{0}}$⁸⁸8With this statement, we intend to only give intuition around the length of the interval, because there can be pathological cases of pairs of distributions that do not follow this generic intuition, and hence it need not be true in full generality.. To mention an example that illustrates this intuition, the no-trade gap under pure noise trading and a uniform distribution supported on $[p_{\text{min}}^{(1)},p_{\text{max}}^{(1)}]$ will be smaller than the one under another uniform distribution supported on $[p_{\text{min}}^{(2)},p_{\text{max}}^{(2)}]$, where $p_{\text{min}}^{(1)}>p_{\text{min}}^{(2)}\text{ and }p_{\text{max}}^{(1)}<p_{\text{max}}^{(2)}$.

At the other extreme, consider the case where the traders always have perfect information about the true asset price, i.e., the market maker’s update rule is ${\pi}({p_{0}},{\hat{p}})={\hat{p}},\ \forall{\hat{p}}\in[{p_{\text{min}}},{p_{\text{max}}}]$. In this case, we observe that the two virtual value functions become

thus the optimal allocation rule is everywhere zero: $x^{\star}({\hat{p}})=0,\ \forall{\hat{p}}\in[{p_{\text{min}}},{p_{\text{max}}}]$. It is therefore the case that this corollary which may be phrased as a version of a no-trade theorem in the presence of perfect adverse selection is distribution-independent: that is, regardless of the distribution $\mathcal{D}$ of the traders’ asset price estimates ${\hat{p}}\sim\mathcal{D}$, if the market maker believes that the traders have perfect information, then their optimal allocation rule is to perform no-trade at all. More specifically: