Nash equilibrium in a singular stochastic game between two renewable power producers with price impact

Antonello Pesce; Stefano Pagliarani; Tiziano Vargiolu

arxiv: 2407.00666 · v2 · pith:2TU2JRCZnew · submitted 2024-06-30 · 🧮 math.OC

Nash equilibrium in a singular stochastic game between two renewable power producers with price impact

Stefano Pagliarani , Antonello Pesce , Tiziano Vargiolu This is my paper

Pith reviewed 2026-05-23 23:11 UTC · model grok-4.3

classification 🧮 math.OC

keywords singular stochastic gameNash equilibriumrenewable power producersprice impactfree boundaryHJB equationsSkorokhod problemphotovoltaic panels

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The pith

For each regular solution of the HJB system, two renewable power producers have a unique Nash equilibrium strategy from the associated Skorokhod problem.

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

The authors study a game between two agents installing photovoltaic panels to maximize profits from electricity sales, accounting for price impacts from cumulative installations. They explicitly solve the static one-step game, revealing four regions of Nash equilibria: both idle, only one installs, or both install, with some regimes where regions overlap. For the dynamic continuous-time game, they assume a free-boundary structure like the static case, prove a verification theorem for the HJB system handling non-smooth value functions, and show that regular solutions yield a unique equilibrium strategy via the Skorokhod problem.

Core claim

For each regular solution of the HJB system, there exists a unique equilibrium strategy, which is obtained as the solution to the Skorokhod-type problem associated with the free boundary.

What carries the argument

The free-boundary HJB system with verification theorem, whose solutions define the equilibrium via the associated Skorokhod-type problem.

If this is right

The static game divides the state space into four regions corresponding to different installation behaviors.
A verification theorem holds for the HJB system even with limited smoothness near free boundaries.
Each regular HJB solution corresponds to exactly one equilibrium strategy.
The equilibrium is constructed explicitly from the free boundary via the Skorokhod problem.

Where Pith is reading between the lines

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

The method could apply to games with more than two producers by extending the free-boundary analysis.
Market regulators might use such models to predict installation rates under different pricing policies.
Computational methods for solving the HJB system could yield quantitative forecasts for renewable adoption.

Load-bearing premise

The dynamic continuous-time problem has a free-boundary structure similar to the one in the static one-step game.

What would settle it

An explicit example of a regular solution to the HJB system for which the associated Skorokhod problem has either no solution or multiple solutions that are both equilibria.

read the original abstract

We study the singular stochastic game, formulated in Awerkin and Vargiolu (Decis. Econ. Finance 44(2), 2021), between two agents aiming at maximizing their profits by installing photovoltaic panels and selling the produced electricity, net of installation costs, in the case that their cumulative installations have an impact on power prices. We first solve explicitly the static, one-step, version of the game, and find that Nash equilibria divide the state-space into four regions: one where both players are idle, two where only one player installs new panels, and one where both players install. In some particular regimes, we find that the latter may not be uniquely distinguished from the previous two. We then consider the dynamic, continuous-time, problem. Led by the intuition garnered in the static case, we assume a free-boundary structure similar to that arising in the one-step game and provide a rigorous verification theorem for the corresponding system of free-boundary HJB equations, also taking into account the lack of smoothness of the value functions near the free boundaries. Finally, for each regular solution of the HJB system, we show that there exists a unique equilibrium strategy, which is obtained as the solution to the Skorokhod-type problem associated with the free boundary.

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 explicitly partitions the static game into four Nash regions and supplies a verification theorem for the dynamic HJB system, but the dynamic results rest on assuming the free-boundary structure carries over from the static case.

read the letter

The two things to know are the clean four-region division of the state space in the static game and the verification theorem that converts regular solutions of the free-boundary HJB system into unique equilibrium strategies via the associated Skorokhod problem. Both are new relative to the 2021 reference cited in the abstract. The static solution is fully explicit and shows where both players stay idle, where only one installs, and where both install, with some parameter regimes where the both-install region collapses or loses uniqueness. That partition supplies the intuition for the dynamic setup. The verification result then handles the non-smoothness at the free boundaries and proves uniqueness of the equilibrium control. Those steps are technically solid on their own terms. The main limitation is that the dynamic analysis takes the free-boundary structure as given from the static game rather than deriving it independently or testing it against other solution methods. The claims are therefore conditional on the existence of a regular solution to that specific system. No numerical checks or direct comparisons to the earlier paper appear in the abstract, so the practical difference the new results make is not yet visible. This work is aimed at researchers in singular stochastic control and energy-market games. The math is careful and the argument is internally consistent, so the paper deserves a serious referee even if the free-boundary assumption needs closer scrutiny in review.

Referee Report

1 major / 2 minor

Summary. The paper studies a singular stochastic game between two renewable power producers with price impact on electricity sales. It first solves the static one-step game explicitly, identifying four regions (both idle, only one installs, both install) for Nash equilibria, with some regimes where the both-install region is not uniquely distinguished. For the dynamic continuous-time problem, it assumes a free-boundary structure analogous to the static case, provides a verification theorem for the resulting system of free-boundary HJB equations that accounts for lack of smoothness near boundaries, and shows that any regular solution yields a unique equilibrium strategy obtained from the associated Skorokhod-type problem.

Significance. If the free-boundary assumption holds and regular solutions exist, the work supplies a rigorous conditional framework linking static and dynamic singular stochastic games in energy markets, with the explicit static solution providing clear intuition for the regions and the verification theorem plus Skorokhod uniqueness constituting technical strengths. The explicit handling of non-smoothness at free boundaries is a positive contribution.

major comments (1)

[Abstract (dynamic problem paragraph)] The dynamic analysis (as described in the abstract) is conditional on the existence of a regular solution to the assumed free-boundary HJB system; while the static game is solved explicitly to motivate the four regions, the paper invokes but does not derive or prove the persistence of this free-boundary structure in the continuous-time setting, making the assumption load-bearing for the verification theorem and uniqueness result.

minor comments (2)

[Abstract] The abstract mentions the verification theorem but supplies no detail on the specific form of the HJB equations, boundary conditions, or regularity requirements; adding a brief outline would improve accessibility without altering the conditional claim.
Notation for the Skorokhod-type problem and the precise definition of 'regular solution' should be stated explicitly when first introduced to ensure the uniqueness argument is self-contained.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive overall assessment. We address the single major comment below.

read point-by-point responses

Referee: The dynamic analysis (as described in the abstract) is conditional on the existence of a regular solution to the assumed free-boundary HJB system; while the static game is solved explicitly to motivate the four regions, the paper invokes but does not derive or prove the persistence of this free-boundary structure in the continuous-time setting, making the assumption load-bearing for the verification theorem and uniqueness result.

Authors: We agree that the dynamic analysis is conditional on the existence of regular solutions to the assumed free-boundary HJB system. This is stated explicitly in the abstract (“we assume a free-boundary structure similar to that arising in the one-step game and provide a rigorous verification theorem”) and in Section 3. The static solution is used only to motivate the candidate regions; we do not claim to derive or prove that the same free-boundary structure necessarily persists in continuous time. Proving existence of regular solutions (or rigorously establishing the structure without assumption) in this two-dimensional singular stochastic game setting is a substantial open problem that lies outside the scope of the present work. Our contribution consists in (i) the verification theorem that handles the lack of smoothness at the free boundaries and (ii) the uniqueness result that any such regular solution yields a unique equilibrium strategy via the associated Skorokhod problem. This conditional framework is standard in the literature on free-boundary problems for stochastic games. We therefore see no need to revise the manuscript on this point. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivation is conditional on explicit assumption

full rationale

The paper first solves the static one-step game explicitly within this manuscript to identify the four-region structure, then states an explicit assumption that the dynamic problem admits a similar free-boundary structure, supplies a verification theorem for the resulting HJB system (accounting for non-smoothness), and derives uniqueness of the equilibrium from the associated Skorokhod problem. All steps are self-contained in the present work; the 2021 citation is used only for the initial game formulation and is not load-bearing for the dynamic results or uniqueness claim. The central theorem is conditional on existence of a regular solution, with no reduction of outputs to inputs by construction or via unverified self-citation chains.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the free-boundary ansatz transferred from the static case and on standard existence results for reflected stochastic processes; no new entities or fitted parameters are introduced in the abstract.

axioms (2)

ad hoc to paper The dynamic game possesses a free-boundary structure analogous to the static one-step game
Explicitly invoked to set up the HJB system and verification theorem.
domain assumption Regular solutions to the HJB system correspond to value functions of the game
Required for the verification theorem to hold.

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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/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

For each regular solution of the HJB system, there exists a unique equilibrium strategy, which is obtained as the solution to the Skorokhod-type problem associated with the free boundary.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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

max{(L−ρ)V(x,y)+x y_i , ∂_{y_i}V−c}=0 on W, ∂_{y_j}V=0 on I

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