Market Clearing with Semi-fungible Assets

Guillermo Angeris; Tarun Chitra; Theo Diamandis

REVIEW 2 major objections 6 minor 6 references

Market Clearing with Semi-fungible Assets

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that market clearing for divisible semi-fungible assets reduces to one convex program whose dual variables are the clearing prices, that these prices respect the partial order, and that an externality payment rule makes…

desk verdict A genuinely useful convex-programming template for clearing divisible semi-fungible markets, with a sound but narrowly-scoped DSIC mechanism; the stress-test counterexample is wrong on inspection. read the letter →

arxiv 2505.19298 v1 pith:RFI4GL52 submitted 2025-05-25 cs.GT

Theo Diamandis , Tarun Chitra , Guillermo Angeris This is my paper · ORCID

classification cs.GT MSC 91B2690C2591A10

keywords marketclearingsemi-fungibleassetspartialordersconvexoptimizationdualitydominantstrategyincentivecompatibilityVCGpaymentsdecentralizedfinance

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper aims to show that markets for divisible, semi-fungible assets -- items that are similar but not identical, such as bonds with different ratings and yields or metals with different purity and delivery location -- can be cleared by a single convex optimization problem. Each buyer states one threshold property, and the market treats any item at least as good as that threshold as an acceptable substitute, which gives buyers access to more liquidity. The paper proves that the dual variables of the clearing program are market-clearing prices, that these prices respect the partial order so that better items never carry a lower marginal price, and that charging each buyer the externality she imposes on the others is dominant-strategy incentive compatible. If the paper is right, a combinatorial explosion of tradable portfolios becomes a tractable pricing and auction problem.

What carries the argument

The load-bearing object is the convex market-clearing problem (2) and its simplified dual (5), in which the per-item price $\lambda_i^\star$ is the pointwise maximum of the marginal values that willing buyers assign to item $i$. Buyer preferences are Boolean vectors $c_b$ supported on upward-closed sets of a directed acyclic graph of properties, and the identity $p_b \le p_{b'} \iff c_b \ge c_{b'}$ turns the partial order into elementwise inequalities that deliver price monotonicity. On the mechanism side, the max-of-sum welfare function $F(f) = \sup_x \sum_b f_b(x)$ and the externality payment rule support the dominant-strategy result; the payment is evaluated by solving the program once with all buyers and once without each buyer.

What would settle it

Find a buyer who accepts a worse item at the going price but rejects a strictly better item at that same price because of an unmodeled quality dimension, then solve (2) on that market and check whether the reported prices still satisfy $\lambda_i^\star \ge \lambda_{i'}^\star$ for $p_i \ge p_{i'}$ and whether the payment rule (8) still prevents profitable misreporting.

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Extended reading notes

Core claim

The central claim is that clearing a market for divisible semi-fungible assets is equivalent to solving the convex program maximize $U(x)$ subject to $x_b = c_b^T z_b$, $\sum_b z_b \le q$, $z_b \ge 0$, where $c_b$ encodes exactly the items whose properties are at least as good as buyer $b$'s stated threshold. Strong duality gives per-item prices $\lambda_i^\star = \max_b (\nu_b^\star c_b)_i$, and because buyers with weaker thresholds have superset preference vectors, these prices satisfy $\lambda_i^\star \ge \lambda_{i'}^\star$ whenever $p_i \ge p_{i'}$: better items have no lower marginal price. The payment rule $P_b(f) = F(f_{-b}) - \sum_{b' \neq b} f_{b'}(x^\star)$, which charges each buyer the welfare the other buyers lose from her participation, is proven dominant-strategy incentive compatible and reduces to the single-item second-price auction as a special case.

Load-bearing premise

Each buyer's preference is fully captured by one threshold property, and all buyers share the same partial order, so anyone who would accept a worse item would also accept any better item at the same price.

Editorial extensions

If this is right

One convex solve returns both the optimal allocation and a price for every distinct property in the market.
Because prices respect the partial order, a strictly better property is assigned a marginal price no lower than a worse one, a directly testable prediction in bond, commodity, or compute markets.
The B+1-solve externality payment rule is dominant-strategy incentive compatible under quasilinear utilities, so buyers cannot gain by shading their reported valuations.
Any item at least as good as a buyer's stated threshold is a valid delivery, so sellers see a larger pool of acceptable buyers and the market gains liquidity without bespoke matching.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A corollary the paper leaves open: the DSIC argument only needs the max-of-sum welfare structure, so the same payment rule should extend to the nonseparable restaking utility the paper conjectures about, provided the optimization remains convex.
The simplified dual suggests a tatonnement-style iteration -- update each item price to the maximum marginal value among willing buyers -- which could be implemented as a distributed or streaming pricing algorithm.
The model assumes one global partial order; applying it where buyers weigh quality dimensions differently would require weighted preference vectors $c \in [0,1]^n$, an extension the paper explicitly defers.
An empirical signature of the model is price monotonicity along the property order; markets that violate it would reveal either unmodeled dimensions or buyers whose preferences are not upward-closed.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things you should know before reading. The paper delivers a clean, genuinely reusable formulation: it turns market clearing for divisible assets with a partial order into a single convex program, gets prices from dual variables, and attaches a VCG-style DSIC payment rule. That is a real contribution, not incremental dressing. The second thing is that the stress-test note's counterexample does not hold up. It claims c=(1,2) models 'one unit of the better item equivalent to two units of the worse item,' but that equivalence requires c=(2,1) when p1≥p2. With the vector corrected, λ1≥λ2 follows immediately. More generally, the monotonicity proof for Boolean vectors extends to any nonnegative weight vector that respects the order (c_i ≥ c_i' whenever p_i ≥ p_i'): λ_i = max_b ν_b c_bi ≥ max_b ν_b c_bi' = λ_i'. The paper never states this extension, but the examples' weights are all monotone, so the advertised price ordering holds in every example they give.

What's new: the preference-vector embedding of upward-closed sets into a divisible-goods Eisenberg-Gale-style program is a neat bridge between the partial-order literature (mostly indivisible goods or discrete choice) and standard market clearing. The dual simplification to (5) and the price monotonicity are correct for the Boolean model. The DSIC proof is a valid VCG externality argument, brief but adequate, and it is honestly scoped: it covers quasilinear utilities and separable welfare, and the restaking example is explicitly labeled a conjecture.

Soft spots, in proportion. Equation (6) is asserted, not proven; it takes one line from the dual simplification, but the paper should show it. The formal model assumes Boolean c_b, while the examples in §5.1 use weighted c=(6,5,7); the paper flags the extension but doesn't connect it to the guarantees. That is a presentation gap, not a mathematical one. The examples are toy-sized and the 'real-world' sections are narrative; that is fine for a theory paper, but it means significance is moderate, not paradigm-shifting. Self-citations are contextual and not load-bearing.

Bottom line: this deserves a serious referee. I'd ask for a revision that proves the weighted monotonicity in one line, reconciles the examples with the formal model, and expands the proof of (6). For researchers in DeFi market design or batched auctions, this is a useful template and worth citing. I'd bring it to reading group and would cite it if I worked on semi-fungible markets.

Referee Report

2 major / 6 minor

Summary. The paper proposes a convex optimization framework for market clearing with divisible semi-fungible assets whose properties are partially ordered. Buyer b's preferences are encoded by a preference vector c_b (Boolean in the formal development: c_{b,i}=1 iff item i has properties at least as good as a threshold p_b), and her utility is evaluated on the scalar amount x_b = c_b^T z_b. The market-clearing problem (2) maximizes total utility subject to supply constraints. The authors derive the convex dual, interpret the dual variables as prices, and claim that the item prices respect the partial order, i.e., λ_i^* ≥ λ_i'^* whenever p_i ≥ p_i'. They then propose the VCG-style payment rule (8) and prove it is dominant-strategy incentive compatible for max-of-sum welfare functions. Examples cover yield-bearing assets, DeFi lending markets, and restaking networks.

Significance. If the claims hold, the paper offers a useful unification: market clearing and truthful mechanism design for a class of partially ordered divisible assets reduce to solving a convex program plus B auxiliary convex programs. The duality derivation is standard and appears correct; the DSIC proof is a valid VCG argument; the numerical examples are simple and checkable. The main limitation is that the price-monotonicity guarantee is established only for Boolean preference vectors, while the weighted extension and the main examples use non-Boolean vectors, so the advertised guarantee is broader than the proven result. The DSIC result is also proved only for separable objectives, with the nonseparable case explicitly left as a conjecture in §5.3.

major comments (2)

[§4.1, Eq. (6); §2.1; §5.1] The price-monotonicity claim λ_i^* ≥ λ_i'^* whenever p_i ≥ p_i' is proved only for Boolean preference vectors c_b ∈ {0,1}^n, because the proof relies on the support inclusion (1). Section 2.1 then proposes weighted vectors c ∈ [0,1]^n as an extension, and the examples in §5.1 use non-Boolean vectors such as c = (6,5,7) without stating any additional monotonicity assumption. For arbitrary weighted vectors, Eq. (6) does not imply the price ordering: with p_1 ≥ p_2, q = (1,1), a single buyer with u(x) = √x, and c = (0.5,1), solving (2) gives x = 1.5 and ν = 1/(2√1.5), so λ_1 = 0.5ν < λ_2 = ν, violating the advertised ordering. The natural fix is to impose the condition c_{b,i} ≥ c_{b,i'} whenever p_i ≥ p_i' for weighted vectors (which is satisfied by the purification example), and then the proof extends; alternatively, the paper should explicitly restrict the monotonicity theorem and the §5.1 examples to the Boolean model. As written, the central price-ordering guarantee does not cover the paper's own motivating settings.
[§4.2; §3; §5.3] The DSIC theorem is proved only for the max-of-sum welfare form F(f) = sup_x Σ_b f_b(x) in Eq. (7), which corresponds to separable utility U(x) = Σ_b u_b(x_b). The market-clearing problem (2), however, is stated for a general concave U, and the restaking example in §5.3 uses a nonseparable U for which the authors only conjecture that the same payment rule is DSIC. The abstract and the opening of §4.2 say that the mechanism gives DSIC payment rules for clearing 'these markets' without the separability caveat. The paper should state clearly that the DSIC guarantee applies to the separable case and that the nonseparable case remains open, so that the reader does not infer a broader result than the proof establishes.

minor comments (6)

[§1.2] There is a typo in 'existance' (should be 'existence').
[§3] The problem data are described as 'the items supplied to the market g ∈ R^n_+'; the symbol should be q to match the constraint and the rest of the paper.
[§4.1] In the 'Implementation and payments' paragraph, 'charge each user i their marginal utility ν_i^*' should refer to buyer b, not item index i, to avoid confusion with item prices.
[§2.1 and §5.1] The weighted extension is stated for c ∈ [0,1]^n, but the example vectors c = (6,5,7) have entries larger than 1; the paper should clarify whether these are intended as preference weights or as utility scalings, and adjust the notation accordingly.
[§5.3] The displayed formula for U(x) in the restaking example is ambiguous: the subscript b in x_{b,s} is not defined in the separable x notation used in §3, and the claimed concavity of U is not demonstrated (the term r_s x_{b,s}/Σ_b x_{b,s} is not obviously concave). Please clarify the notation and either prove concavity or state the assumption explicitly.
[§3, Discussion] The statement that concave utility 'allows for preferences with complements' appears backwards: concave utilities with U(0)=0 are subadditive and typically represent substitutes or diminishing returns, whereas complementarity requires superadditivity. Please clarify or remove this remark.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the price and DSIC results are derived from the model's own definitions via standard convex duality and VCG-style arguments; self-citations are contextual, not load-bearing.

full rationale

The paper contains no fitted parameters, no empirical predictions, and no load-bearing self-citation. Market-clearing prices in Section 4.1 follow directly from the Lagrangian of problem (2), and equation (6) is a consequence of the dual simplification, not an assumed relation; the price-monotonicity statement is then a short argument from the Boolean preference-vector definition and upward closure. The DSIC payment rule (8) is explicitly constructed and proved in Section 4.2 via the standard max-of-sum argument, with the proof reproduced in the text; the citation to Myerson is contextual, and self-citations to [DAE24] and [CP24] are used only for reinterpretation and an example, not as justification of the central claims. The weighted-preference examples in Section 5.1 use non-Boolean vectors that fall outside the formal model of Section 2.1, but that is a scope gap, not circularity, since the paper's advertised theorems are stated for Boolean vectors. Overall the derivation is self-contained against standard convex duality and VCG theory.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The model fits no parameters to data; q, c_b, and the utility functions are market inputs. The load-bearing premises are the common partial order with a universal interpretation, the threshold (upward-closed) representation of preferences, concavity and monotonicity of utility, and quasilinearity for the DSIC result. These are domain assumptions, not fitted constants.

assumptions (5)

domain assumption The set of properties P is partially ordered (transitive and antisymmetric), and if p <= p' then any buyer who accepts p also accepts p' at the same price.
This defines semi-fungibility and makes preference vectors and price monotonicity well-defined. Stated in Section 2, 'Ordering over properties'.
domain assumption Each buyer's preference is a single threshold property pb with preference vector cb in {0,1}^n, where (cb)_i = 1 iff p_i >= pb.
The model restricts buyers to upward-closed sets w.r.t. the global order; idiosyncratic tradeoffs cannot be mixed. Section 2.1, 'Embedding the ordering'.
domain assumption The aggregate utility U is concave, nondecreasing, U(0)=0, and strictly increasing at 0; buyer utilities depend only on the scalar xb = cb^T zb.
Needed for the convex program (2), for strong duality, and for the dual variables to be interpreted as marginal prices. Section 3, 'Utilities'.
domain assumption Buyers have quasilinear utility over money and the received allocation, so the VCG-style payment rule yields DSIC.
The DSIC proof in Section 4.2 assumes quasilinear utility; the paper claims it generalizes but does not formalize the generalization. Section 4.2, 'Payment rule' and 'Proof of DSIC'.
standard math Strong duality holds for problem (2) because constraints are linear and the feasible set is nonempty (Bertsekas, Prop. 5.3.1).
Invoked in Section 4.1 to identify dual variables with optimal prices.

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Cite this review

Pith. "Pith review of Market Clearing with Semi-fungible Assets." pith.science (2026). https://pith.science/paper/RFI4GL52

@misc{pith2026250519298,
  author       = {Pith},
  title        = {Pith review of: Market Clearing with Semi-fungible Assets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RFI4GL52}},
  note         = {Machine review of arXiv:2505.19298}
}

read the original abstract

As markets have digitized, the number of tradable products has skyrocketed. Algorithmically constructed portfolios of these assets now dominate public and private markets, resulting in a combinatorial explosion of tradable assets. In this paper, we provide a simple means to compute market clearing prices for semi-fungible assets which have a partial ordering between them. Such assets are increasingly found in traditional markets (bonds, commodities, ETFs), private markets (private credit, compute markets), and in decentralized finance. We formulate the market clearing problem as an optimization problem over a directed acyclic graph that represents participant preferences. Subsequently, we use convex duality to efficiently estimate market clearing prices, which correspond to particular dual variables. We then describe dominant strategy incentive compatible payment and allocation rules for clearing these markets. We conclude with examples of how this framework can construct prices for a variety of algorithmically constructed, semi-fungible portfolios of practical importance.

Figures

Figures reproduced from arXiv: 2505.19298 by the authors.

**Figure 1.** An example partial ordering on five yield-bearing assets with different ratings and yields, provided by different operators. We assume the operators are not comparable, whereas any buyer prefers a higher rating and higher yield, all else being equal. 2.1 The embedding Here, we discuss how to express buyer preferences for the items. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

**Figure 2.** Partial orders on the set of, say, node operators. Some users may be indifferent between the operators A and B, while others may be indifferent between A and C, and so on. Embedding the ordering. A buyer b can express her preferences for the asset by specifying a property pb ∈ P which will, in some sense, be the base item she is willing to purchase. In other words, given a list of n items i = 1, . . . , n with prop… view at source ↗

**Figure 3.** Three yield bearing assets with various ratings, and their associated partial ordering. different scenarios and look at the market clearing prices (dual variables) and the payment rules in each case. Homogenous market. First, consider the case where each buyer would accept any item. Buyers express utilities in units of yield. The resulting (identical) preference vectors are c1 = c2 = (6, 5, 7) where assets are index… view at source ↗

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Reference graph

Works this paper leans on

6 extracted references · 4 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Nonparametric option pricing under shape restrictions

[AD03] Yacine Aıt-Sahalia and Jefferson Duarte. “Nonparametric option pricing under shape restrictions”. In: Journal of Econometrics 116.1-2 (2003), pp. 9–47. [AR22] Eleni Aristodemou and Adam M Rosen. “A discrete choice model for partially ordered alternatives”. In: Quantitative Economics 13.3 (2022), pp. 863–906. [BCS15] Eric Budish, Peter Cramton, and ...

work page 2003

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How much should you pay for restaking secu- rity?

2011, pp. 152–157. [CP24] Tarun Chitra and Mallesh Pai. “How much should you pay for restaking secu- rity?” In: arXiv preprint arXiv:2408.00928 (2024). [CV07] Bruno Codenotti and Kasturi Varadarajan. “Computation of market equilibria by convex programming”. In: Algorithmic Game Theory (2007), pp. 135–158. [DAE24] Theo Diamandis, Guillermo Angeris, and Ala...

arXiv 2024

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Optimal auction design

[Mye81] Roger B Myerson. “Optimal auction design”. In: Mathematics of operations re- search 6.1 (1981), pp. 58–73. [New24] Bloomberg News. “It’s PhDs Against MBAs in Credit Investing as Quants Jump In”. In: Bloomberg (May 2024). Accessed: 2024-12-04. url: https : / / www . bloomberg.com/news/articles/2024-05-20/it-s-phds-against-mbas-in- credit-investing-...

arXiv 1981

[4] [2009]

Airline seat allocation with multiple nested fare classes

[BM93] Shelby L Brumelle and Jeffrey I McGill. “Airline seat allocation with multiple nested fare classes”. In: Operations research 41.1 (1993), pp. 127–137. [Bud+23] Eric Budish et al. Flow trading . Tech. rep. National Bureau of Economic Re- search,

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[KL17] Albert S Kyle and Jeongmin Lee. “Toward a fully continuous exchange”. In: Oxford Review of Economic Policy 33.4 (2017), pp. 650–675. [Lev24] Matt Levine. “You Want Some Stocks That Go Down”. In: Bloomberg Opin- ion (Oct. 2024). Accessed: 2024-12-04. url: https : / / www . bloomberg . com / opinion/articles/2024-10-28/you-want-some-stocks-that-go-do...

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url: https : / / defillama . com / protocol/eigenlayer. [Sta25c] DeFi Llama Staff. Morpho. Feb. 2025.url: https://defillama.com/protocol/ morpho. [Tea24a] EigenLayer Team. “Eigenlayer: The restaking collective”. In: White paper (2024). [Tea24b] Euler Team. “Euler Whitepaper”. In: (May 2024). [Vaz07] Vijay V Vazirani. “Combinatorial algorithms for market e...

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