Market Efficiency in Decentralized Exchanges

View Slides & Handout on Zenodo

This post is based on a seminar I gave at the Singapore Management University on September 30, 2026.

Does transparency imply efficiency?

Transparent reserves and a deterministic pricing rule make the pool observable, but do not by themselves eliminate price divergence or execution costs. On many public blockchains, pool reserves and submitted transactions are public. Yet, price still diverges from the external market price between blocks, and liquidity providers face costs and potential losses.

Why is a perfectly transparent market still not necessarily efficient?

We define market efficiency operationally: a market is efficient if a passive liquidity provider cannot be systematically disadvantaged relative to a benchmark that trades continuously at the external reference price.

In particular, we introduce the Grossman–Stiglitz paradox.

Definition (Grossman–Stiglitz Paradox). There cannot be an economic equilibrium in which all available information is fully reflected in prices at zero cost¹.

If prices reflected all available information at zero cost, no trader would be rewarded for trading the asset such that prices are brought back in line. We will discuss what such reward looks like in the context of DEXs.

Automated Market Makers

Liquidity Pools

A liquidity pool is a collection of funds locked in a smart contract to provide liquidity for trading on a DEX.

Constant Function Market Makers

Because the liquidity supply for a pair is transparent and known at all times, the price of the assets in the pool can be determined algorithmically based on the pool's reserves.

An automated market maker (AMM) is a smart contract that determines the price of assets in a liquidity pool based on the pool's reserves.

Most AMMs are constant function market makers (CFMMs), meaning that the price of assets in the pool is determined by a constant function of the pool's reserves, also called reserve invariant²³. We commonly classify the CFMMs based on the type of constant function they use to determine prices⁴:

1. Constant Product Market Maker (CPMM)
2. Constant Mean Market Maker (CMMM)
3. Constant Sum Market Maker (CSMM)
4. Hybrid Function Market Maker (HFMM), which combines CSMM and CMMM
5. Dynamic Automated Market Maker (DAMM) is an alternative to HFMM which is more suitable for volatile markets

Constant Product Market Maker

The CPMM is characterized by its constant $k$. In the fee-free CPMM model, the following holds at all times.

Figure "fig-cpmm-invariant" shows the invariant curves for different values of $k$. The negative slope at any point on the curve is the marginal price $P_{Y/X} = y/x = -dy/dx$.

CPMMs were notably used in early versions of the Uniswap protocol. In the context of two tokens, $\sqrt{k}=:L$ is called the liquidity of the pool. Given the price and invariant, we can derive the relationship between the price and the liquidity $L$ of the pool.

For a fee-free trade with $\Delta x/x$ small, we can approximate slippage as follows.

which is interpreted as, the average fill sits halfway between pre- and post-trade marginal price. The relative move in the marginal price itself, the price impact, is twice as large, $\approx 2\Delta x / x$.

Constant Mean Market Maker

A CMMM is another type of CFMM characterized by its constant $k$ and weights $(w_i)_{i\geq 1}$.

CMMMs generalize CPMMs and were used in protocols like Balancer⁵²⁴³. Unlike a CPMM, which implicitly fixes equal weights across two assets, a CMMM lets a pool encode target weights across its assets. Figure "fig-cmmm-invariant" compares invariant curves for different weight choices.

Maximum Extractable Value

Transaction Ordering

On a blockchain, transactions are bundled into blocks; who controls transaction inclusion and ordering depends on the chain's participants and its protocol.

Example (Sandwich Attack on CPMM Pool). CPMM pool $(x, y) = (100, 100)$, $k = 10,000$, fair price $P_{Y/X} = 1$. Trader $A$ buys $\Delta x = 10$, trader $B$ buys $\Delta x = 5$. Attacker $E$ buys before trader $A$, moving the pool price against $A$; $E$ then sells after $A$'s trade.

In the sandwich, attacker $E$ front-runs ($\Delta x = 5$) and back-runs $A$, extracting $1.27$ Y at $A$'s expense.

Key Insight. Public-chain data can reveal reserves and public-mempool trades, but ordering control depends on the chain's protocol.

Defining MEV

Maximum Extractable Value (MEV) is the maximum value that can be extracted through transaction selection or ordering within a block.

MEV is value captured by active participants, such as searchers or block producers. In the sandwich attack, the attacker extracts value from trader A, not directly from the LPs. Sandwich MEV is principally a trader-facing ordering cost; LVR instead measures LP underperformance from arbitrage against stale AMM prices.

Liquidity Provider Benchmarks

For liquidity providers, the challenge is how to compute the losses linked to providing liquidity passively. Arbitrage against stale prices can impose adverse-selection costs on LPs and create MEV opportunities for active traders.

Impermanent Loss

Impermanent Loss (IL) is the difference in the value of the assets in the liquidity pool compared to holding the assets outside the pool³. IL is increasingly referred to as Loss-Versus-Holding (LVH). We introduce the holding benchmark $H(t)$, which is the value of the pool's initial reserves as a function of time $t$. This is often called the HODL benchmark.

We construct the LVH as the difference between the HODL benchmark and the pool value.

Loss Versus Rebalancing

HODL is an incomplete benchmark for evaluating liquidity providers. A more credible alternative rebalances on external exchanges at fair market prices. Fair comparison: self-financing portfolio that continuously replicates the exact same risky asset inventory, but without execution slippage or adverse selection.

Loss Versus Rebalancing (LVR) is the difference in the value of the assets in the liquidity pool compared to optimally rebalancing the assets outside the pool⁶. We construct the frictionless self-financing rebalancing portfolio $V^*(t)$ using the same external price $P(t)$ that marks the pool, from another exchange without slippage or adverse selection.

The LVR is then defined as the difference between the frictionless rebalancing portfolio and the pool value.

For the CPMM when $P$ follows a Geometric Brownian Motion with volatility $\sigma$⁶:

Under these assumptions, cumulative LVR is non-decreasing. One interpretation is that it compensates arbitrageurs for enforcing price efficiency, consistent with the Grossman–Stiglitz paradox: the impossibility of a costlessly efficient market¹.

Comparing LP Benchmarks

Example (Numerical Comparison: LVH vs. LVR). CPMM pool $(x_0, y_0) = (100, 100)$, $k = 10,000$, $P_0 = 1$, $V(0) = 200$.

LVH recovers to zero when the price reverts, but LVR accumulates from both arbitrage events ($17.16 + 12.13 \approx 29.29$).

Key takeaways

• Transparent, on-chain price formation does not eliminate forms of arbitrage;
• CFMMs price assets algorithmically from pool reserves; CPMM, CMMM are the most common functions;
• Providing liquidity: inventory moves opposite to price, and losses accrue through arbitrage;
• HODL-based measures (IL/LVH) can recover to zero, but under the benchmark assumptions Loss Versus Rebalancing (LVR) captures the non-decreasing adverse-selection cost borne by passive LPs.

Key Insight. Transparent price formation does not mean that the market is efficient.

The lecture slides and handout are available on Zenodo.