Positional Dominance in Network Games: Equilibrium Conditions for Strategic Abstention from Velocity Competition

Abstract

Positional Dominance in Network Games: Equilibrium Conditions for Strategic Abstention from Velocity Competition We study a two-player infinite-horizon stochastic game on a capacitated linear distribution network in which one player commits capital to control key nodes — pipelines and storage hubs — while the rival invests in low-latency execution infrastructure. Using Markov Perfect Equilibrium with supply–demand shocks governed by a two-state volatility Markov chain, we establish the existence of a threshold σ* above which the positional player's equilibrium payoff strictly exceeds that of the velocity-optimizing player. The advantage arises from private inventory signals and flow-redirection optionality that reaction speed cannot replicate in high-volatility regimes. In this version, existence and uniqueness of σ* are proved directly for the equilibrium value functions, without any value-function approximation. Existence follows from equilibrium existence in finite-state discounted stochastic games together with continuity in σ, an exact sign condition at zero volatility, and divergence — the last requiring a strictly positive private-signal parameter γ > 0. Uniqueness follows from a Bellman convexity-preservation lemma combined with the fact that the velocity player's payoff becomes exactly affine once its latency cap binds, under a checkable condition on primitives. The result therefore carries an explicit scope: positional dominance requires that holding position cost something, that it inform, and that the rival's expenditure cannot transfer it. Value iteration on a calibrated 8×8×8 state grid yields σ* ∈ [0.30, 0.36]. Under U.S. energy midstream calibration, historical Permian–Cushing volatility of 0.35–0.42 exceeds this interval in approximately 68% of months over 2015–2024. Sensitivity analysis places σ* over [0.24, 0.43] under ±20% parameter variation, which we read as evidence that σ* is a property of a particular network under particular carrying costs rather than a constant. This version supersedes v1 (10.5281/zenodo.21013066). The analytical threshold σ* ≈ 0.665 reported there is corrected: v1's Section 3 and Appendix A stated two different functions, each a partial truncation of the paper's own value-function ansatz, and the intercept was justified by velocity rents that are identically zero at σ = 0. Restoring the dropped terms gives ΔV(σ) = aσ² + mσ − b with b identified as the capitalised fixed cost of holding position; the earlier value is an upper bound rather than an estimate. The correction factor ψ ≈ 0.50 and its attribution to storage optionality and Markov persistence are withdrawn pending direct measurement of the omitted linear term. The addendum's cross-domain evidence table is corrected: two supporting Lean 4 results are theorems of the form rfl on a definition and a third holds for every closed interval, so while machine-checked they carry no information about the value 1/3. The 1/3 recurrence is accordingly withdrawn as evidence and retained only as an open question. Related identifiers Relation Identifier Type is new version of 10.5281/zenodo.21013066 DOI is part of 10.5281/zenodo.19117399 DOI is supplemented by https://github.com/TOTOGT/geometry URL references https://totogt.github.io/geometry/book6/wp38-positional-dominance.html URL Also edit v1's metadata (permitted on published records without a new version): add is previous version of → 10.5281/zenodo.21753025, and prepend to its description: "Superseded by v2, DOI 10.5281/zenodo.21753025, which corrects the derivation of the analytical threshold." Subjects JEL — C73 (Stochastic and Dynamic Games) · C72 (Noncooperative Games) · D43 (Market Structure: Oligopoly) · L13 (Oligopoly and Imperfect Markets) · L95 (Gas Utilities, Pipelines) · G13 (Contingent Pricing, Futures) · Q41 (Energy Demand and Supply) Change from v1: C73 added — the model is a discounted stochastic game, and C73 is the more precise primary code. G13 and L95 added for the real-options and pipeline dimensions. MSC 2020 — 91A15 (Stochastic games) · 91A25 (Dynamic games) · 91B26 (Market models, auctions) · 90C39 (Dynamic programming) Change from v1: 53D10 (contact manifolds) dropped. It entered via the addendum's cross-domain claim, which v2 withdraws as evidence; retaining it would assert a connection the paper no longer supports. 91B84 and 91G80 replaced by the game-theoretic and dynamic-programming codes that describe what the paper actually does. Keywords network games · Markov perfect equilibrium · stochastic games · positional dominance · velocity competition · strategic abstention · real options · option value of waiting · preemption · bottleneck control · commodity markets · energy midstream · Cushing · high-frequency trading Change from v1: "contact geometry" removed, for the same reason as MSC 53D10. Suggested citation Grossi, P. N. (2026). Positional Dominance in Network Games: Equilibrium Conditions for Strategic Abstention from Velocity Competition (Version 2). Zenodo. https://doi.org/10.5281/zenodo.21753025 One Note The abstract above states that uniqueness is proved. It is proved for the comparative-static reading, in which σ is held fixed — which is the reading under which a threshold in the level of volatility is defined. It is not proved for the genuine two-state chain. The body says so in §3.0.1; if you would quote the abstract, carry the qualifier too, insert "for the comparative-static formulation" after "Uniqueness follows from".