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: 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.