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arxiv: 2606.12167 · v1 · pith:LBNNLC3Anew · submitted 2026-06-10 · 💻 cs.GT

Shared Infrastructure Investment and Pricing: Stackelberg Equilibria in Risk-Aware Take-or-Pay Contracts

Amal Sakr , Andrea Araldo , Tamer Ba\c{s}ar , Tijani Chahed This is my paper

Pith reviewed 2026-06-27 07:44 UTC · model grok-4.3

classification 💻 cs.GT
keywords Stackelberg equilibriumgeneralized Nash equilibriumCVaRtake-or-pay contractsinfrastructure pricingrisk aversionMobile Edge Computing
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The pith

A Stackelberg game yields an equilibrium where an infrastructure provider optimizes capacity and pricing against risk-averse firms committing to usage under revenue uncertainty.

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

The paper models shared infrastructure investment where one provider acts as leader to choose capacity and access prices that recover large upfront costs from firms facing uncertain revenues. Multiple firms act as followers that choose resource commitments influenced by pricing, costs, congestion, and exogenous factors, with their heterogeneous risk aversion captured by CVaR; their joint decisions form a generalized Nash equilibrium. The authors prove that a Stackelberg equilibrium exists in this setting, supply a polynomial-time algorithm that finds an approximate equilibrium with a bounded optimality gap, and derive a lower bound on the firms' probability of profit. Monte Carlo simulations of a mobile edge computing scenario show that greater follower risk aversion produces smaller deployed capacity, lower prices, lower leader profit, and higher follower profit probability.

Core claim

In the Stackelberg game with risk-aware take-or-pay contracting, the infrastructure provider jointly optimizes capacity dimensioning and access pricing while the firms optimize their resource usage commitments under CVaR, and the existence of an equilibrium is established in which the followers' decisions constitute a generalized Nash equilibrium; an approximate equilibrium can be computed in polynomial time with bounded gap, and a lower bound holds on followers' Probability of Profit.

What carries the argument

The Stackelberg game in which the InP leader jointly optimizes capacity dimensioning and access pricing while the CVaR-using firms as followers commit to future resource usage under uncertain revenues and congestion costs, with their responses forming a generalized Nash equilibrium.

If this is right

  • Existence of the equilibrium lets the provider plan capacity knowing that firms will settle on stable usage commitments.
  • The polynomial-time algorithm makes near-optimal pricing and capacity decisions feasible to compute for practical contract design.
  • The lower bound on Probability of Profit supplies firms with a guaranteed minimum chance of positive profit under the take-or-pay terms.
  • Higher follower risk aversion produces strictly smaller optimal capacity, lower prices, and lower leader profit.

Where Pith is reading between the lines

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

  • The same leader-follower structure with CVaR could be tested on shared cloud or energy infrastructure with similar revenue uncertainty.
  • If real revenue distributions deviate from the modeled uncertainty, the computed equilibrium and PoP bound would need recalibration.
  • Regulators could examine whether the derived lower bound on firm profit probability is high enough to encourage participation in shared infrastructure.

Load-bearing premise

The model assumes that firms' resource usage is jointly influenced by exogenous factors, infrastructure pricing, operational costs, and resource congestion, with heterogeneous risk aversion captured exactly by CVaR, allowing the followers' problem to form a generalized Nash equilibrium.

What would settle it

A specific numerical instance of the MEC setting in which no Stackelberg equilibrium exists or in which the polynomial-time algorithm produces a solution whose optimality gap exceeds the claimed bound.

Figures

Figures reproduced from arXiv: 2606.12167 by Amal Sakr, Andrea Araldo, Tamer Ba\c{s}ar, Tijani Chahed.

Figure 2
Figure 2. Figure 2: InP profit under different risk classes and CVs. RN MRA HRA ERA SP risk class 60 70 80 90 100 C * ( v C o r e s ) CV = 25% CV = 50% CV = 75% CV = 100% [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Optimal capacity C ∗ under different risk classes and CVs. RN MRA HRA ERA SP risk class 40 50 60 70 * ( $ / v C o r e - h o u r ) CV = 25% CV = 50% CV = 75% CV = 100% [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Lower bound on the PoP νˆi under different SP risk classes and CVs. realizations harder to absorb. For SP 2 and SP 3, the lower bound is sensitive to CV mainly under risk neutrality, whereas once they become risk-averse it remains almost unchanged and close to one across all CV levels. Overall, we show that CVaR-based risk aversion strengthens the PoP, while high revenue uncertainty primarily hurts SP 1. A… view at source ↗
Figure 6
Figure 6. Figure 6: InP profit when one SP changes risk class at CV = 100%. RN MRA HRA ERA Risk class assigned to changing SP SP 1 changes SP 2 MRA SP 3 MRA SP status 0.29 0.36 0.40 0.49 0.93 0.93 0.93 0.93 0.91 0.91 0.91 0.91 0.00 0.25 0.50 0.75 1.00 L o w e r b o u n d [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: Lower bound when SP 2 changes risk class at CV = 100%. RN MRA HRA ERA Risk class assigned to changing SP SP 1 MRA SP 2 MRA SP 3 changes SP status 0.36 0.36 0.38 0.39 0.93 0.93 0.93 0.93 0.36 0.91 0.91 0.99 0.00 0.25 0.50 0.75 1.00 L o w e r b o u n d [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: InP profit and capacity–pricing decisions for 3 and 10 SPs under varying [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Lower bound on the PoP for 3 and 10 SPs under heterogeneous risk classes [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
read the original abstract

We study a shared infrastructure deployed by an Infrastructure Provider (InP) and used by multiple firms that generate revenues through resource usage. We focus on a challenging setting where: (i) infrastructure deployment requires substantial upfront investment, which the InP must recover via payments by firms that depend on their uncertain future revenues; (ii) firms' resource usage is jointly influenced by exogenous factors, infrastructure pricing, operational costs, and resource congestion; and (iii) firms exhibit heterogeneous risk aversion. This setting is typical in emerging technologies, e.g., Mobile Edge Computing (MEC). We formalize this setting as a novel Stackelberg game with risk-aware take-or-pay contracting and firm-side operational and congestion costs, in which the InP acts as the leader and jointly optimizes capacity dimensioning and access pricing, while firms act as followers that share the infrastructure and commit upfront to future resource usage under uncertain revenues. Followers' heterogeneous risk aversion is modeled through Conditional Value-at-Risk (CVaR). We prove the existence of a Stackelberg equilibrium (SE), in which the followers' decisions constitute a generalized Nash equilibrium, and develop a polynomial-time algorithm that computes an approximate SE with a bounded optimality gap. We also derive a lower bound on the followers' Probability of Profit (PoP). Monte Carlo simulations for a MEC case study show that higher followers' risk aversion reduces infrastructure capacity, pricing, and leader profit, while increasing followers' PoP.

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.

Referee Report

0 major / 2 minor

Summary. The paper studies shared infrastructure investment and pricing in a Stackelberg game setting with risk-aware take-or-pay contracts. The infrastructure provider acts as the leader optimizing capacity dimensioning and access pricing, while multiple firms with heterogeneous risk aversion (modeled by CVaR) act as followers committing to resource usage under revenue uncertainty, with operational and congestion costs. The authors prove the existence of a Stackelberg equilibrium where followers' decisions form a generalized Nash equilibrium, develop a polynomial-time algorithm for an approximate SE with bounded optimality gap, derive a lower bound on the followers' Probability of Profit (PoP), and present Monte Carlo simulations for a Mobile Edge Computing case study showing the impact of risk aversion on capacity, pricing, leader profit, and PoP.

Significance. If the theoretical results hold, this paper makes a contribution to the literature on game-theoretic models for infrastructure sharing in emerging technologies such as MEC. The proof of SE existence under GNE with CVaR, the poly-time approximate algorithm with bounded gap, and the PoP lower bound are notable strengths. The simulations provide empirical support for the model's implications regarding risk aversion effects. These elements could inform both theoretical advancements and practical contract design in shared infrastructure settings.

minor comments (2)
  1. [Abstract] The abstract states that the algorithm computes an approximate SE 'with a bounded optimality gap' but provides no indication of the gap's functional form or dependence on problem parameters; adding this detail would strengthen the claim of a polynomial-time result.
  2. [Model section (inferred from abstract description)] The model formalization assumes that the followers' problem constitutes a generalized Nash equilibrium under the joint effects of exogenous factors, pricing, costs, and congestion; an explicit statement of the convexity or compactness conditions used to guarantee existence would aid verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a theoretical Stackelberg game model with CVaR risk aversion, proves existence of equilibrium (followers forming GNE), develops a polynomial-time approximation algorithm with bounded gap, and derives a PoP lower bound. These are standard mathematical derivations and proofs that do not reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. The model formalization and MEC simulations provide independent content supporting the claims, making the derivation self-contained against external game-theoretic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; all modeling elements (CVaR, congestion costs, take-or-pay) are described at a high level without numerical fits or new postulates.

pith-pipeline@v0.9.1-grok · 5813 in / 1263 out tokens · 23574 ms · 2026-06-27T07:44:31.685719+00:00 · methodology

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Reference graph

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