Discussion of "Allocative Efficiency and Mispricing in Automated Market Maker (AMM) and Continuous Double Auction (CDA) Markets" 

Paper by Roland Mestel, Stefan Palana, Erik Theissen, & Isabel Walchera
Discussion by: Andreas Park, University of Toronto

WBS Gillmore DeFi & Digital Currencies Conference
September 21–22, 2026 · London

A clean laboratory comparison

Human quotes in a CDA versus a machine-run constant-product AMM.

Private values pin down the efficient allocation.
Findings:

• Similar allocative efficiency
• Lower measured mispricing in AMM markets.
• With fees: fee shifts surplus to LPs and lower holistic efficiency.

Summary thoughts:

• convincing experimental results
• my interpretation: not AMM vs CDA but rather
  • free-flowing human interaction v forced-intermediated liquidity.
• suggestions on theoretical development and limits to allocative efficiency, the meaning of prices, the possible learnings from timing/dynamic behavior

Comment 1:
What is an AMM and what are we  comparing?

or: how one might want to frame the paper

Background: Some AMM Mechanics and Theory

An AMM is a pool of reserves + a pricing rule \(x\cdot y=k\). Three things to know:

Background: the effect of a "return" trade

Buy q units
Asset reserve falls to x − q.

Cash enters the pool.
The next marginal price rises.

Reverse the trade
The pool returns to its starting point.

Background: Mathematically, an AMM implements a specific limit-order book

A → B: buy q units

\(p(q)=\dfrac{xy}{(x-q)^2}\)

\(C(q)=\int_0^q p(z)\,dz=\dfrac{yq}{x-q}\)

Comment 1: What is actually being compared?

Continuous Double Auction

• Humans supply liquidity
• Humans choose quotes
• Post, revise, cancel or hit orders
• Coordination and learning matter

AMM

• Algorithm supplies liquidity
• Reserves determine quotes
• Subjects demand liquidity only
• No quote-setting problem

Comment 2:
Develop consistent theoretical prediction

or: I get the big picture, but some nuances matter and looking after them would strengthen the interpretation of the findings

Comment 2: What is the theretical welfare/surplus?

A simplified setting

• \(N\) intramarginal buyer units with values \(v_1 > \dots > v_N > P^*\) and
• \(N\) intramarginal seller units with costs \(c_1 < \dots < c_N < P^*\).
• The starting ("equilibrium") price is \(P^*\) and the total available rent is

\[\overline{W} = \sum_{i=1}^{N} (v_i - c_i) = \sum_{i=1}^{N} (v_i - P^*) + \sum_{i=1}^{N} (P^* - c_i).\]
 

Continuous Double Auction Setting

• Buyer \(i\) and seller \(j\) trade at price \(p\).
• Their profits are \(v_i - p\) and \(p - c_j\), and \((v_i - p) + (p - c_j) = v_i - c_j .\)
• If all \(N\) pairs trade, for any prices \(p_1, \dots, p_N\) realized surplus is \[\sum_{i=1}^{N} (v_i - p_i) + \sum_{i=1}^{N} (p_i - c_i) = \overline{W} ,\]

Automated Market Maker

• The pool holds \(a\) units of the asset and \(c = aP^*\) in cash, with invariant \(ac = k\).
• After \(n\) net buys (buys minus sells) the pool holds \(a - n\) units and \(c\,a/(a-n)\) in cash.
• The marginal price is \[p(n) = \frac{k}{(a-n)^2} .\]
• A buy that moves the pool from \(n\) to \(n+1\) costs \[\overline{\pi}_n = \int_{n}^{n+1} \frac{k}{(a-s)^2}\, ds = \frac{k}{(a-n)(a-n-1)} ,\] and a sell that moves the pool from \(n+1\) to \(n\) pays the same \(\overline{\pi}_n=\underline{\pi}_{n+1}\).
• For any sequence of trades ending at net position \(n\), total cash paid by buyers minus cash received by sellers equals the change in the pool's cash, \[\sum_{\text{buys}} \pi - \sum_{\text{sells}} \pi = \frac{c\,a}{a-n} - c = \frac{c\,n}{a-n} .\]

Comment 2: What is the theretical welfare/surplus?

Automated Market Maker

• Trader surplus is therefore \[\sum_{\text{buys}} (v - \pi) + \sum_{\text{sells}} (\pi - c) = \sum_{\text{buys}} (v - P^\ast) + \sum_{\text{sells}} (P^\ast - c) - \frac{c\, n^2}{a(a-n)} .\]

• If all \(N\) intramarginal units trade on each side and nothing else, then \(n = 0\), the pool's cash is back at \(c\), and \[\sum_{\text{buys}} (v - \pi) + \sum_{\text{sells}} (\pi - c) = \sum_{i=1}^{N} v_i - \sum_{i=1}^{N} c_i = \overline{W} .\]
• \(\to\) Any shortfall from \(W\) comes from an intramarginal unit that did not trade, or an extramarginal unit that did.
• For any sequence of trades ending at net position \(n\) the net cash paid to the pool is \(cn/(a-n)\), independent of the order of trades, and it decomposes as\[ \frac{c\,n}{a-n} = nP^* + \frac{c\,n^2}{a(a-n)} .\]

Comment 2: What is the theretical welfare/surplus?

Comment 2: What is the theory of allocative efficiency?

Big Picture

• In the simplified case we can pin down allocative efficiency by who doesn't trade
• But do we know whether theoretically everyone will/should trade?

A bit of theory-by-example on allocation

• Example 1: One buyer \(v\), one seller \(c\), with \(v - P^* > P^* - c\) and \(v>P^*>c\)

• CDA
  • Any \(p \in [c, v]\) is acceptable to both; nonempty since \(v > c\).
  • Trade is feasible regardless of any starting or reference price.
• AMM
  • Buyer can open iff \(v \geq \overline{\pi}_0\); seller can open iff \(c \leq \underline{\pi}_0\).
  • Both inside: no trade, ever.
• One feasible: after she moves, the other always trades.
  • Buyer opens at \(\overline{\pi}_0\); at imbalance 1 the bid is \(\underline{\pi}_1 = \overline{\pi}_0 > P^* > c\),
  • seller sells at \(\overline{\pi}_0\). Seller opens at \(\underline{\pi}_0\); at imbalance \(-1\) the ask is \(\overline{\pi}_{-1} = \underline{\pi}_0 < P^* < v\), buyer buys at \(\underline{\pi}_0\).
  • The opener pays the spread, the follower receives it: e.g., buyer first gives \((v - \overline{\pi}_0,\ \overline{\pi}_0 - c)\), seller first gives \((v - \underline{\pi}_0,\ \underline{\pi}_0 - c)\).
  • Each prefers to follow!
• Ex ante efficiency is strictly lower than CDA because \(\Pr(\underline{\pi}_0 < c,\ v < \overline{\pi}_0) > 0\)

Thoughts on allocation theory: Example 2

• Example 2: two buyers \(v_1 > v_2 > P^*\), two sellers \(c_1 < c_2 < P^*\).

• CDA: same as with 2 players: Every pairing has \(v_i > P^* > c_j\): all four units trade, theoretical afficiency =1 for any order and any prices.

• AMM
  • A type can still be locked out:
    • let \(v_1 \geq \overline{\pi}_0 > v_2\) and \(\underline{\pi}_0 < c_1 < c_2\).
    • Buyer 1 opens (imbalance 1), seller 1 sells at \(\underline{\pi}_1 = \overline{\pi}_0\) (imbalance 0).
    • Buyer 2 faces \(\overline{\pi}_0 > v_2\), seller 2 faces \(\underline{\pi}_0 < c_2\).
    • Two units never trade although \(v_2 > c_2\).
  • My hypothesis for a general condition (\(N\) units on each side) (may require explicit sequencing):
    • all units trade iff the number of inside buyers is at most the number of outside sellers, equivalently inside sellers \(\leq\) outside buyers.
    • An inside buyer can only buy at imbalance \(n \leq -1\), where \(\overline{\pi}_n \leq \underline{\pi}_0\), and every sell that lowers the imbalance from \(n \leq 0\) is at \(\underline{\pi}_n \leq \underline{\pi}_0\), hence by an outside seller.
    • Extreme types "unlock" marginal types.

Comment 3:
Are we using the right prices and are price measures comparable for CDA v AMM?

Thoughts on price benchmarks

• Prices

  • starting midprice: \(P^*\) in both, by construction. 
  • There is another candidate: the Fair price \(=\) equal split, \(\bar p = \frac{v+c}{2} \not= P^*\).
  • (the expected absolute deviation of the fair price from \(P^*\) is positive)
• CDA: a bargained price.
  • Smith measured deviations from \(P^*\) and explained them by excess rent
  • Deviation from \(P^*\) does not, by itself, establish an allocative loss. Maybe distinguish price convergence, surplus distribution, and allocation.
  • \(\bar p\) seems like a more natural benchmark; sum of fair prices in a trade is a fair price
• AMM
  • if both trade: ending marginal price is always \(P^*\).
  • Transaction prices are pre-set bids and asks
  • Average transaction price for the session can be determined by who trades first!
  • No price is by design "fair" \(\to\) prices are unrelated to \(\bar p\) but end-marginal price is \(P^*\)

Thoughts on allocation and prices: Example 2
 

Four trades, two buys and two sells but six price paths.  \[A = \overline{\pi}_{0}, ~A' = \overline{\pi}_{1}, ~B = \underline{\pi}_{0},~ B' = \underline{\pi}_{-1}\]

Ranking: \[ \tfrac{1}{2}(B+B') < B < P^\ast < \tfrac{1}{2}(A+B) < A < \tfrac{1}{2}(A+A'). \]

Thoughts on allocation and prices: Example 2

Four trades, two buys and two sells, but six price paths.

Illustration: default pool depth \(a=17.39\), \(P^\ast=100\), no fees. \[ A=\overline{\pi}_0=106.10,\quad A'=\overline{\pi}_1=119.89,\quad B=\underline{\pi}_0=94.56,\quad B'=\underline{\pi}_{-1}=84.81. \]

GD = geometric signed deviation; GAD = geometric absolute deviation from \(P^\ast\), using the paper’s definitions. Calculated from unrounded prices.

Ranking of arithmetic averages: \[ \tfrac12(B+B') < B < P^\ast < \tfrac12(A+B) < A < \tfrac12(A+A'). \]

Insights from the examples

• ex ante allocative efficiency of CDA is HIGHER
  • AMM "lock-outs" are determinable \(\to\) must check
• Mistakes in CDA lead to more efficiency loss than in AMM
  • if one player doesn't play ball in the CDA, a counterparty loses, too
  • in AMM single players can make mistakes
• Price benchmarks
  • without asymmetric info, info efficiency \(P^*\) is not very meaningful
  • "fair price" may be more natural for CDA
  • no concept of fair price in AMM (no split per se)
• AMM trades are complex timing problem and delay can be beneficial (related to knapsack problems)
  • \(\to\) tricky to write theory
  • \(\to\) test/document behavior as function of valuation

Comment 4:
Liquidity provision and fees

it's fine to discuss fees but a bit more background is needed

When outside prices move, arbitrage trades against the pool

buy & hold

AMM LP: concave relative to buy & hold

Asset price change

Portfolio value change

Some insights for AMMs fees?

Fees compensate LPs for mechanical liquidity provision and trading losses.

Impermanent loss measures underperformance versus holding after a price move.

Fees and depth interact through liquidity supply.

Here, pool size is exogenous.
A partial-equilibrium fee experiment.

Comment 4 (minor): A bit more discussion of liquidity provider risk

At what price should terminal inventory be valued?

• Standard impermanent loss uses an outside price move and arbitrage.
• Here, the private-value competitive benchmark stays fixed within a period.
• Maybe add more economic interpretation (LP needs to mark their LP portfolio to market at the end-price and can't get the fundamental value)

Comment 2: What is the theoretical basis for the allocative efficiency that we are looking for?

• General setup:
  • 4 buyers and 4vsellers with private values \(v_i>P^*\) and \(c_i<P^*\) trade 3 units each
  • Example 1: one buyer \(v\), one seller \(c\), with \(v - P^* > P^* - c\).

• Continuous Double Auction (CDA) outcome for allocation
  • Any price \(p \in [c, v]\) is acceptable to both; nonempty since \(v > c\).
  • Trade is feasible regardless of any starting or reference price.

Key Measures used

• Profit dispersion: how far individual profits are from what each trader would earn at \(P^*\),\[PD = \sqrt{\tfrac{1}{8}\sum_{\text{traders}} \big(\text{realized profit} - \text{profit at } P^*\big)^2 }.\]
• Geometric deviation: the average transaction price relative to \(P^*\), signed,\[GD = \exp\Big(\tfrac{1}{I}\sum_{i=1}^{I}\ln\tfrac{p_i}{P^*}\Big) - 1,\] over the \(I\) transactions in the period. Negative means prices were on average below \(P^*\).
• Geometric absolute deviation: the average distance of transaction prices from \(P^*\), unsigned,\[GAD = \exp\Big(\tfrac{1}{I}\sum_{i=1}^{I}\Big|\ln\tfrac{p_i}{P^*}\Big|\Big) - 1.\]
• In the AMM, \(\ln(p_i/P^*)\) is a function of the imbalance at the time of trade \(i\), so \(GD\) is average signed imbalance and \(GAD\) average absolute imbalance, in price units.

Key Measures used

• Profit dispersion: how far individual profits are from what each trader would earn at \(P^*\),\[PD = \sqrt{\tfrac{1}{8}\sum_{\text{traders}} \big(\text{realized profit} - \text{profit at } P^*\big)^2 }.\]
• Geometric deviation: the average transaction price relative to \(P^*\), signed,\[GD = \exp\Big(\tfrac{1}{I}\sum_{i=1}^{I}\ln\tfrac{p_i}{P^*}\Big) - 1,\] over the \(I\) transactions in the period. Negative means prices were on average below \(P^*\).
• Geometric absolute deviation: the average distance of transaction prices from \(P^*\), unsigned,\[GAD = \exp\Big(\tfrac{1}{I}\sum_{i=1}^{I}\Big|\ln\tfrac{p_i}{P^*}\Big|\Big) - 1.\]
• In the AMM, \(\ln(p_i/P^*)\) is a function of the imbalance at the time of trade \(i\), so \(GD\) is average signed imbalance and \(GAD\) average absolute imbalance, in price units.

Thoughts on allocation and prices: Example 2

• Example 2: buyers \(v_1 > v_2 > P^*\), sellers \(c_1 < c_2 < P^*\).

• CDA
  • Every pairing has \(v_i > P^* > c_j\): all four units trade, efficiency 1 for any order and any prices.
  • Pairwise fair prices average to the aggregate fair price for any pairing:
    
    \[\frac{1}{2}\sum_{\text{pairs}} \frac{v_i + c_j}{2} = \frac{\bar v + \bar c}{2}\]
  • The paper scores prices against \(P^*\), not against \(\frac{\bar v + \bar c}{2}\).

Thoughts on allocation and prices: Example 2

• Example 2: two buyers \(v_1 > v_2 > P^*\), two sellers \(c_1 < c_2 < P^*\).

• average AMM prices (assume all trade)
  • buy sell buy sell: all at \(\overline{\pi}_0\).
  • sell buy sell buy: all at \(\underline{\pi}_0\).
  • buy sell sell buy: \(\overline{\pi}_0, \overline{\pi}_0, \underline{\pi}_0, \underline{\pi}_0\), average \(\approx P^*\).
  • buy buy sell sell: \(\overline{\pi}_0, \overline{\pi}_1, \overline{\pi}_1, \overline{\pi}_0\)
  • Some facts:
    • Log deviation of a transaction from \(P^*\) is the imbalance at that moment, in price units;
    • GD is the mean imbalance,
    • GAD the mean absolute imbalance, floor \(\approx \frac{1}{a}\).
    • An ordering statistic: same allocation, same \(\bar p\), different "mispricing".

Where can allocative surplus disappear?

CDA: conditional on the traded units, payments cancel between buyers and sellers.

Fee-free AMM: exact back-and-forth trades restore the pool and net out its cash flows.

Sequence matters when it changes which units trade.
An unbalanced pool also leaves terminal inventory to value.

Lost gains come from inefficient units trading, or efficient units failing to trade.

A useful next step: account for missed gains unit by unit.

Efficiency and surplus sharing

G = total gains from trade available in competitive equilibrium

TE = trader profits net of fees / G

LE = LP profits (fees − impermanent loss) / G

HE = TE + LE

A fee transfer F lowers TE by F/G and raises LE by F/G.

HE removes pure transfers, given the LP valuation benchmark.
The paper’s lower HE with fees calls for an allocation decomposition.