arxiv: 2605.14485 · v1 · submitted 2026-05-14 · 💰 econ.TH

Recognition: 1 theorem link

· Lean Theorem

Efficient liability assignment under shock propagation

Jens Gudmundsson , Jens Leth Hougaard , Kohmei Makihara , Alexandros Rigos

Authors on Pith no claims yet

Pith reviewed 2026-05-15 01:38 UTC · model grok-4.3

classification 💰 econ.TH

keywords liability rulesshock propagationnetwork cascadesefficient path selectionShapley valueproportional allocationsequential games

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The pith

Proportional liability rules with network-only weights induce agents to select paths minimizing total cascade losses.

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

The model has agents embedded in a network choosing paths along which shocks propagate by successive edge cancellations, each incurring a loss until a terminal node is reached. The total systemic cost is the sum of these individual losses. The paper characterizes liability rules that assign each agent's payment as a share of the realized total losses, where the shares depend only on the fixed network structure. Under these rules the induced sequential game leads agents to choose the path that minimizes aggregate losses rather than their private costs.

Core claim

The paper's main axiomatic result identifies the family of liability rules that set each agent's liability proportional to total realized losses with weights depending solely on the network structure. A concrete member of this family is obtained by a path-based procedure that assigns equal weight to every non-sink agent along each possible path and aggregates these contributions; the resulting weights coincide with the Shapley value of an associated path-counting cooperative game and can be computed in polynomial time.

What carries the argument

Proportional liability rules whose agent weights are derived from a path-counting procedure equivalent to the Shapley value of a cooperative game on the network.

If this is right

Agents will select the globally efficient path in the sequential-move game induced by the rule.
The liability weights can be computed once from the network topology and applied to any realized shock.
The same weights implement efficiency for any loss structure that matches the modeled cascade process.
The family of rules is fully characterized by the axioms of proportionality and network dependence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Regulators could precompute the weights for real-world supply-chain or financial networks and embed them in contracts.
The approach separates the design of the allocation rule from the details of any particular shock distribution.
Empirical tests on observed cascade data could check whether the polynomial-time weights predict actual loss-sharing patterns.

Load-bearing premise

Agents choose paths to minimize their own expected liability payments given the cascade mechanism exactly as modeled.

What would settle it

In a controlled network with two paths of different total loss, observe whether agents select the lower-total-loss path once the proportional liability rule with the proposed weights is imposed.

Figures

Figures reproduced from arXiv: 2605.14485 by Alexandros Rigos, Jens Gudmundsson, Jens Leth Hougaard, Kohmei Makihara.

**Figure 1.** Figure 1: There are two paths from source s to sink t: one with many unit-loss edges and one (thick) single-edge path with loss 1 + ε. If the source’s liability increases only with the direct loss associated to their choice, then s prefers canceling edge s → 1 over s → t, leading to an inefficient path with total loss m > 1 + ε. which coalitional worth increases in path count.3 Further, the weights can be computed i… view at source ↗

**Figure 2.** Figure 2: Solid edges have positive losses, dashed edges have zero losses. [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗

**Figure 3.** Figure 3: A graph in which the number of paths grows exponentially with the number of [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 4.** Figure 4: Kernel density estimates of individual liabilities under the fixed-weight rule b [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗

**Figure 5.** Figure 5: Graph for Observation 1. Next, define the loss function ℓ ′ ∈ L that matches ℓ except for ℓ ′ (it) = 0. Under ℓ ′ , path P ′ is the unique efficient path. Yet, as losses on P ′′ are unchanged, by realized-loss dependence, ϕi (P ′′, ℓ′ ) = ϕi (P ′′, ℓ) = 0 ≤ ϕi (P ′ , ℓ′ ). That is to say, i does not strictly prefer the efficient P ′ to the inefficient P ′′, contradicting efficient implementation at ℓ ′ . 3… view at source ↗

read the original abstract

We study a model in which shocks propagate along a path chosen by agents embedded in a network. When a shock hits an agent, the affected agent cancels one of her outgoing edges. This cancellation cascades sequentially along a chosen path until reaching a terminal agent, resulting in a systemic cost equal to the sum of individual cancellation losses. A liability rule determines agent payments for realized losses, and we seek to implement efficient path selection in the induced sequential-move game. Our main axiomatic result characterizes a family of rules, which set each agent's liability to be proportional to the system's total realized losses with agent weights depending only on the network structure. We propose a way to set such weights based on a simple path-based procedure that assigns equal importance to all non-sink agents along each path and then aggregates these contributions across paths. These weights coincide with the Shapley value of an associated "path-counting" cooperative game and can be computed in polynomial time. A simulation study illustrates the mechanics of our approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean Shapley-value construction for network-dependent weights in proportional liability rules, with a polynomial-time procedure, but the subgame-perfect equilibrium claim needs tighter verification.

read the letter

The main thing to know is that the authors characterize proportional liability rules whose weights depend only on network structure and coincide with the Shapley value of a path-counting game. They give an explicit procedure that treats non-sink agents equally along each path and aggregates across paths, and they show this can be computed in polynomial time. A simulation illustrates the mechanics. That is the concrete new piece: a workable way to set the weights without fitting parameters or solving fixed-point equations. The axiomatic part is straightforward and the link to cooperative game theory is a natural fit for this setting. The simulation helps make the rule feel operational rather than purely theoretical. The soft spot sits in the implementation claim. The weights are derived ex ante from the full set of paths, but the game is sequential and agents observe cancellations as play proceeds. Nothing in the description confirms that these fixed weights remain marginal-contribution weights inside every proper subgame. If an agent can steer toward a higher-total-loss continuation that still lowers her own share, the equilibrium may not select the socially minimal path. The abstract states the result, so the full proof will have to close that gap. This paper is for mechanism-design researchers who work on networks, contagion, or supply-chain liability. A reader who wants an explicit, computable rule rather than an existence argument will find something usable here. It deserves peer review because the construction is new, the computation is attractive, and the equilibrium question is fixable with more detail rather than fatal to the idea.

Referee Report

2 major / 2 minor

Summary. The paper models shock propagation in a network where agents sequentially cancel outgoing edges along a chosen path, generating a cascade with total systemic loss equal to the sum of individual cancellation costs. It provides an axiomatic characterization of a family of proportional liability rules (each agent's payment is its weight times total realized loss) whose weights depend only on network structure, shows that these weights coincide with the Shapley value of an associated path-counting cooperative game, proves the weights are computable in polynomial time, and presents simulations illustrating the mechanics.

Significance. If the Shapley weights derived from the path-counting game indeed induce subgame-perfect equilibria that select the socially minimal-loss path in the sequential cancellation game, the result supplies a computationally tractable, axiomatically justified method for designing liability rules that internalize network externalities without requiring central intervention. The polynomial-time computation and the clean separation between the cooperative-game weights and the realized path are genuine strengths.

major comments (2)

[§4] §4 (Implementation): the claim that the ex-ante Shapley weights of the path-counting game remain marginal-contribution weights inside every proper subgame after partial cascades is not verified. Because liability is always (w_i / Σw) × realized total loss, an agent's continuation payoff depends on the entire downstream path; the manuscript does not show that the fixed weights continue to equal the conditional marginal contributions once some edges have already been cancelled.
[§3.2] §3.2 (Axiomatic characterization): the proof that proportional rules with network-dependent weights implement efficiency assumes that agents anticipate the full cascade when choosing which edge to cancel, yet the argument does not explicitly check that the path-counting Shapley value satisfies the required incentive-compatibility condition in every history of the sequential game.

minor comments (2)

[Simulation study] The simulation section reports qualitative behavior but omits network sizes, number of Monte-Carlo draws, and quantitative efficiency gaps; adding these details would strengthen the illustration.
[§2] Notation for the path-counting game (Definition 2) could explicitly distinguish the ex-ante characteristic function from the conditional characteristic function that would arise after a partial cascade.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments on the implementation in §4 and the axiomatic characterization in §3.2 are well-taken, and we will strengthen the manuscript by providing explicit verifications as detailed in our point-by-point responses below.

read point-by-point responses

Referee: [§4] §4 (Implementation): the claim that the ex-ante Shapley weights of the path-counting game remain marginal-contribution weights inside every proper subgame after partial cascades is not verified. Because liability is always (w_i / Σw) × realized total loss, an agent's continuation payoff depends on the entire downstream path; the manuscript does not show that the fixed weights continue to equal the conditional marginal contributions once some edges have already been cancelled.

Authors: We agree that the subgame property requires explicit verification. The path-counting cooperative game is defined such that the value of a coalition is the number of paths it covers, which has an additive structure over paths. This implies that after a partial cascade (i.e., some edges cancelled), the remaining game is equivalent to a subgame on the residual network, and the Shapley value restricted to remaining agents preserves the proportionality. However, to address the referee's concern directly, we will add a new proposition in §4 that proves the fixed weights equal the conditional marginal contributions in every subgame. This will be a partial revision as the core argument is already present but not stated explicitly for subgames. revision: partial
Referee: [§3.2] §3.2 (Axiomatic characterization): the proof that proportional rules with network-dependent weights implement efficiency assumes that agents anticipate the full cascade when choosing which edge to cancel, yet the argument does not explicitly check that the path-counting Shapley value satisfies the required incentive-compatibility condition in every history of the sequential game.

Authors: The proof in §3.2 proceeds by showing that the proportional rule with Shapley weights makes each agent's payoff equal to its marginal contribution to the total loss, which by the properties of the Shapley value ensures that the efficient path is selected in equilibrium. We acknowledge that the incentive-compatibility in every history is not checked step-by-step. In the revision, we will include an inductive argument showing that at every history, given the continuation equilibrium, no agent benefits from choosing a suboptimal edge, because deviating would change the realized path in a way that reduces its own weighted share by more than the gain. This verification will be added to the manuscript. revision: yes

Circularity Check

0 steps flagged

Axiomatic characterization uses independent path-counting game; no reduction to fitted inputs or self-referential definitions

full rationale

The paper's core result is an axiomatic characterization of proportional liability rules whose weights depend only on network structure. It then defines a separate path-counting cooperative game on the network (aggregating equal contributions along each path) whose Shapley value supplies those weights. This game is constructed directly from the graph and does not reference realized losses, liability payments, or equilibrium outcomes, so the weights are not obtained by fitting to the target result. The sequential-game implementation claim is presented as a separate verification step rather than a definitional identity. No self-citation chain, ansatz smuggling, or renaming of known results is required for the central derivation; the path-counting construction stands as an independent, polynomial-time computable object.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard cooperative game theory axioms for the Shapley value and domain assumptions about the network cascade process; no free parameters or invented entities are introduced beyond the model primitives.

axioms (2)

standard math Standard axioms of cooperative game theory (efficiency, symmetry, dummy, additivity) that characterize the Shapley value
Invoked when stating that the path-based weights coincide with the Shapley value of the path-counting game.
domain assumption The shock propagation follows a deterministic cascade along the chosen path with additive individual cancellation losses
Core modeling assumption stated in the abstract that defines the systemic cost and liability base.

pith-pipeline@v0.9.0 · 5469 in / 1526 out tokens · 56377 ms · 2026-05-15T01:38:37.345766+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

weights coincide with the Shapley value of an associated path-counting cooperative game

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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