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Mean Field Stackelberg Game for Production and Carbon Emission Reduction with State Reflections

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A hard-cap carbon game has an asymptotically exact Stackelberg equilibrium, with a semi-explicit optimal price rule.

desk verdict Genuinely new hierarchical mean-field Stackelberg model with reflection, but the leader's verification rests on an unproved t-Lipschitz density regularity and several deferred proofs. read the letter →

arxiv 2608.12775 v1 pith:I74KRYON submitted 2026-08-13 math.OC

Lijun Bo , Zhen Liu , Jingfei Wang , Jiacheng Zhang This is my paper · ORCID
classification math.OC MSC 91A1691A2391A6593E2060H30
keywords carbonemissionstransboundarypollutionmeanfieldStackelberggamestatereflectionlocaltimeapproximateequilibriumreflectedgeometricBrownianmotionstochasticoptimalcontrol
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 tries to establish that a two-level carbon governance game remains solvable when cumulative emissions are forced below a hard cap. One central regulator sets a price-adjustment policy first; $n$ heterogeneous regions then choose production to maximize profits, and their emissions aggregate into a common reflected state that cannot exceed a deterministic ceiling. The authors prove that, for large $n$, these strategies form an approximate Stackelberg equilibrium: the error vanishes as $n$ grows, and the leader's optimal price adjustment has a semi-explicit formula. The formula is an expectation involving a reflected geometric Brownian motion and its local time, so it can be evaluated by Monte Carlo simulation. If the proof is right, hard emission caps can be embedded in hierarchical mean-field policy games rather than approximated only through soft penalties.

What carries the argument

The load-bearing object is the dynamic Skorokhod map: the reflected state is the unreflected process minus the running supremum of its excess over the cap, so the hard cap is enforced by a local-time process that increases only when emissions touch the ceiling. For the leader's problem, a Cole-Hopf transformation turns the nonlinear HJB equation into a linear Robin PDE whose probabilistic representation is built from a reflected geometric Brownian motion $Y^{t,x,z}$ together with its local time $R^{t,x,z}$. The regularity engine is Lemma 4.3: the running supremum of the auxiliary process $U^{t,z}$ has locally bounded densities, locally Lipschitz in time, under the original measure and under two Girsanov tilts. That regularity makes the gradient of the representation well defined and justifies the feedback formula for $u^{*,(n)}$; the final verification uses a generalized Itô formula for Sobolev-space solutions.

What would settle it

Choose a simple cap schedule (for instance a constant or linear $\Phi$), fix the parameters, and estimate the density of $\sup_{r\in[t,s]}U^{t,z}(r)$ near the boundary where the process touches the cap using Monte Carlo or an analytic calculation; if the density's derivative in $t$ or in the level is unbounded at any parameter point, Lemma 4.3 fails and the leader's optimality proof collapses.

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Extended reading notes

Core claim

The central claim is Theorem 4.5: the leader's strategy $u^{*,n}$ in (4.14) and the followers' strategies $q_i^{*,n}(u)$ in (3.8) constitute an $(\epsilon,0)$-Stackelberg equilibrium in the sense of Definition 2.2, with $\epsilon(n)\to 0$. The leader's optimal price adjustment has the probabilistic representation $$$u^{{*,(n)}}$(t,x,z)=-\frac{\mu x}{2c_P}\$nu^{{(n)}}$\Big(\frac1{2c}\Big)\, \frac{ \mathbb{E}\,[$e^{{-\eta(c_A\nabla_x R^{t,x,z}}$(T)+c_E\nabla_x $Y^{{t,x,z}}$(T))}]}{ \mathbb{E}\,[$e^{{-\eta(c_A R^{t,x,z}}$(T)+c_E $Y^{{t,x,z}}$(T))}]},$$ with the gradients expressed explicitly through the hitting time $\tau^t_{x,z}$ and the geometric factor $K^t(s)$. The proof follows the standard backward-induction route: solve the followers' mean-field game to obtain an $\epsilon$-Nash equilibrium, then verify the leader's control against that response, using a Malliavin-calculus regularity result for the running maximum of a transformed process.

Load-bearing premise

The proof of the leader's optimality rests on a regularity estimate: the probability density of the running maximum of a certain transformed emission process must be locally Lipschitz in time and in the starting value, under both the original measure and two change-of-measure tilts; if that estimate fails, the leader's optimal-control characterization and the equilibrium theorem lose their proof.

Editorial extensions

If this is right

  • For large region populations the equilibrium is asymptotically exact, so the leader can compute her price policy from one reflected-geometric-Brownian-motion expectation while regions follow the linear rule $q_i=(a_i+uX)/(2c_i)$.
  • The leader's layer is exactly optimal against the followers' approximate best responses: the only vanishing error is the followers' $\epsilon$-Nash error, so $\epsilon_2=0$ in the approximate Stackelberg equilibrium.
  • The mean-field consistency condition reduces to $m_q^*(t)=\nu(a/2c)+\nu(1/2c)u(t)X^*(t)$, which gives a closed feedback structure linking pricing, regional production, and the emission state.
  • Simulations show a policy trade-off: a looser long-run cap can trigger pre-emptive price cuts that contract output, while a faster-tightening cap can be paired with price support and higher abatement to keep emissions under the lower ceiling.

Reading between the lines

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

  • The same reflected-state Stackelberg machinery should transfer to other common-pool resource problems with hard ceilings and common noise, such as water quotas, fishery catch limits, or congestion caps, wherever the state is a positive multiplicative diffusion.
  • If a fully explicit density for the running supremum of the reflected geometric Brownian motion were found, the approximate equilibrium could be upgraded to an explicit one and the Malliavin regularity step would become unnecessary.
  • The 'looser cap contracts output' outcome is demonstrated for one cost structure; testing it across abatement cost functions and price-sensitivity parameters would reveal whether it is a general mechanism or a parameter-dependent artifact.
  • The leader's exact optimality assumes followers coordinate on the approximate-Nash strategies; in a decentralized rollout, followers' miscoordination would feed back into the leader's cost and reintroduce error at both levels.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper studies a finite-horizon Stackelberg differential game with one regulator (leader) and n heterogeneous regions (followers), where the average cumulative emission process is constrained by a deterministic cap Z through a reflected SDE with a local-time abatement process A. The leader chooses a price-adjustment process u; the followers choose production processes q_i to maximize profits. For fixed u, the followers' problem is solved as an MFG, yielding an epsilon(n)-Nash equilibrium. The leader's HJB equation is transformed via a Cole-Hopf change of variables into a linear Robin problem, solved probabilistically by a reflected geometric Brownian motion; the authors use Malliavin calculus for the density of the running supremum and a generalized Ito formula for verification, obtaining a semi-explicit optimal strategy u*. The paper concludes that the resulting strategies form an (epsilon,0)-Stackelberg equilibrium and presents Monte Carlo sensitivity analysis.

Significance. If correct, the paper would be a meaningful contribution: it would provide the first approximate Stackelberg equilibrium for a hard-cap, mean-field, common-noise emission game with a semi-explicit leader strategy and non-trivial policy trade-offs. The Malliavin-based treatment of the running supremum density for a reflected geometric Brownian motion is a genuine technical idea, and the numerical comparative statics are informative and falsifiable. However, the central leader-verification argument depends on a regularity property that is asserted but not proved, and the follower ANE proof outsources a key step to lecture notes. The contribution is therefore conditional on completing and verifying those components.

major comments (3)
  1. [Section 4.3, proof of Proposition 4.2 and Eq. (4.24)] The regularity p(s,u;t,z) locally Lipschitz in t is load-bearing but unproved. Lemma 4.3 establishes only local boundedness of the densities p, p^{Q1}, and p^{Q2}; it contains no t-Lipschitz estimate. In the proof of Proposition 4.2, the claim that P(sup_{r in [t,s]} V^{t,z}(r) <= m) is locally Lipschitz in t is justified only by 'local Lipschitz continuity of E_t', but no Lipschitz estimate for E_t in t is supplied, and E_t depends on t through W(t) and the solution Z^{t,z}. The subsequent Girsanov/time-change paragraph also states an identity between the laws of V^{t,z} and U^{t,z} under Q and P that is not established as written, since V^{t,z} is defined using W rather than the Q-Brownian motion. Consequently, the bound on the term L4/h in (4.24) does not close, Psi may fail to belong to W^{1,2,1;infinity}_{loc}([0,T] x A), and the verification in Corollary 4.4, and hence Theorem 4.5, is not proved as written.
  2. [Section 3.2, Lemma 3.3] Lemma 3.3 states the unique best response q^{*,u}(t) = (2c)^{-1}(a+u(t)X^{*,u}(t;m_q^u)) and the running-supremum representation of A^{*,u}, but its proof is omitted. This lemma is the foundation for the MFE in Lemma 3.5 and for the finite-n ANE in Theorem 3.6. The DPP/HJB verification for a reflected SDE whose drift depends on the control and whose regulator is the running supremum of a controlled diffusion is not a one-line argument. The authors need to provide the proof or a precise reference with conditions under which this verification is valid.
  3. [Section 3.3, proof of Theorem 3.6] The epsilon-Nash property is not actually verified in the proof. The argument establishes the state-process convergences (3.14) and (3.16) and then says that the conclusion follows 'by adopting a similar argument in the proof of Theorem 8.3 of Lacker (2018).' It does not demonstrate, for each i, the payoff comparison J_i^{(n)}(q^{*, (n)}(u)) >= sup_{q_i in U_i} J_i^{(n)}(q_i, q^{*, (n)}_{-i}(u)) - epsilon(n) with a quantified epsilon(n) -> 0, nor does it control the deviating player's payoff through the reflected dynamics (3.15), where A^{*,u,(n)}_{-i} itself depends on the deviating control q_i. Without this, the ANE half of the Stackelberg equilibrium is asserted rather than proved.
minor comments (4)
  1. [Section 3.2, Lemma 3.5] The symbol b in the displayed formula q^{*,u}(t) = (a - b m_q^{*,u}(t))/(2c) is undefined and is inconsistent with the subsequent formula (3.5). Please correct the expression or remove the undefined symbol.
  2. [Section 3.3, Eq. (3.12)] The auxiliary follower problem (3.12) contains an exponential discount factor e^{-rho t}, but the parameter rho is never defined and the discount factor is absent from the original follower objective (2.7). Please clarify the role of rho and explain why the discounted auxiliary payoff comparison is valid for the undiscounted finite-n game.
  3. [References and text] The in-text citation 'Nualart and Nualart 2018' does not match the reference list, which lists Nualart and Nualart (2006). Please harmonize the citation and the bibliographic entry.
  4. [Section 5, numerical analysis] The Monte Carlo figures would be more informative with the number of paths, time-discretization step, and confidence bands or standard errors reported; this would help assess whether the reported monotonicity patterns are numerically stable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the approximate Stackelberg equilibrium is derived from stated assumptions; the self-citations are background/technical references and are not load-bearing.

full rationale

The paper's central result, Theorem 4.5, is an existence theorem: for fixed model parameters, the leader's strategy (4.14) and the followers' strategies (3.8) are shown to form an (epsilon,0)-Stackelberg equilibrium in the sense of Definition 2.2. The derivation chain is self-contained in the sense required for non-circularity. First, the representative follower's best response is obtained by pointwise maximization of the quadratic payoff (Lemma 3.3), and the mean-field consistency condition E[q*|F_t^MF] = m is solved explicitly (Lemma 3.5). Second, the finite-n approximate Nash equilibrium (Theorem 3.6) follows from the L^2 convergence estimates proved in Appendix A and invokes Lacker's Theorem 8.3 for the final ANE comparison, an external reference. Third, the leader's HJB equation is solved via the Cole-Hopf transformation and a Feynman-Kac probabilistic representation (4.7); the regularity of the running-supremum density is attributed to Hayashi-Kohatsu, Nakatsu, Coutin-Pontier, and Nualart-Nualart, and the verification step in Corollary 4.4 is a standard Sobolev-space verification argument. No parameter is fitted to data, no 'prediction' is defined in terms of the target output, and no conclusion is forced by a self-citation chain. The self-citations (Bo et al. 2021, 2025, 2026) appear in model background, in the dynamic Skorokhod problem terminology, and as comparison points, but the load-bearing analytic arguments either are proved in the paper or rely on independent external references. There are proof-completeness concerns, notably the omitted proof of Lemma 3.3 and the asserted but not fully established local Lipschitz continuity of p(s,u;t,z) in t used in Proposition 4.2, but these are correctness risks rather than circular reductions of the conclusion to its inputs.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper's central theorem is a pure existence result for a stylized stochastic game; it introduces no new physical entities and fits no data. It carries domain assumptions about the economic structure, including the linear price rule, quadratic costs, and a common pollution stock with a fixed cap, and it relies on a set of standard technical results from stochastic analysis. The only hand-chosen numbers are in the illustrative Monte Carlo section. Hence the ledger is modest: no fitted constants, several modeling axioms, and several external theorems.

free parameters (1)
  • simulation calibration set = mu=1, sigma=0.1, delta=0.1, a=10, c=2, c_P=1, c_A=0.5, c_E=0.4, x0=3, z0=3.3, A_Z in {3.5,4.5,5.5}, B_Z in {0.9,1,1.1}
    Chosen by hand for homogeneous Monte Carlo experiments in Section 5; they do not enter the existence theorem, which only requires parameters to lie in a compact interval [m,M].
assumptions (6)
  • standard math Usual filtered probability space with Brownian motion W and filtration satisfying the usual conditions.
    Section 2 setup; all stochastic integrals and SDEs rely on this.
  • domain assumption Assumption (A1): empirical type measures converge to a limit law nu on O for all bounded continuous f.
    Section 3, used to define the mean-field limit and to get convergences (3.14) and (3.16).
  • domain assumption Type vectors and model parameters lie in a compact set [m,M]^2 and [m,M]; admissible controls are L2-bounded by constants C1 and C2.
    Section 2, guarantees uniform boundedness and integrability.
  • domain assumption Cap process Z solves dZ = Phi(Z)dt with Lipschitz Phi and Phi(0) >= 0, ensuring a nonnegative deterministic cap.
    Section 2.1; the cap trajectory is an input to the reflection mechanism.
  • standard math The dynamic Skorokhod problem has a unique solution with explicit representation in Lemma 3.2, cited from PiliPenko.
    Used to define the reflected state and local time process A.
  • standard math External technical results: density of the running supremum from Hayashi-Kohatsu, Nakatsu, and Coutin-Pontier; Malliavin calculus from Nualart; generalized Ito formula for Sobolev functions from Krylov; ANE approximation from Lacker.
    Invoked in Lemma 4.3, Proposition 4.2, and Theorem 3.6; they are standard but not re-proved in this paper.

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Pith. "Pith review of Mean Field Stackelberg Game for Production and Carbon Emission Reduction with State Reflections." pith.science (2026). https://pith.science/paper/I74KRYON

@misc{pith2026260812775,
  author       = {Pith},
  title        = {Pith review of: Mean Field Stackelberg Game for Production and Carbon Emission Reduction with State Reflections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I74KRYON}},
  note         = {Machine review of arXiv:2608.12775}
}
abstract

Global warming, driven by anthropogenic carbon emissions with transboundary pollution characteristics and irreversible damage, poses an existential threat to human society. This paper develops a novel two-level Stackelberg game with mean field interaction of controls and common noise which integrates hierarchical decision-making under a state-reflected emission dynamics. A central regulator (leader) adjusts product prices to guide $n$ heterogeneous competing regions (followers) while enforcing a hard emission cap via a reflection mechanism that models emergency reductions through a local time process. We establish the existence of an approximate Stackelberg equilibrium and perform sensitivity analysis via Monte Carlo simulations.

Figures

Figures reproduced from arXiv: 2608.12775 by the authors.

Figure 5.1
Figure 5.1. Simulations w.r.t. different values of the parameter [PITH_FULL_IMAGE:figures/full_fig_p019_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Simulations w.r.t. different values of the parameter [PITH_FULL_IMAGE:figures/full_fig_p020_5_2.png] view at source ↗

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Works this paper leans on

59 extracted references · 56 canonical work pages

  1. [1]

    (2000): Reflecting Brownian snake and a Neumann-Dirichlet problem

    Abraham, R. (2000): Reflecting Brownian snake and a Neumann-Dirichlet problem. Stoch. Process. Appl. 89(2), 239-260

    work page 2000

  2. [2]

    Aid, R. and S. Biagini (2022): Optimal dynamic regulation of carbon emissions market. Math. Finan. 33(1), 80-115

    work page 2022

  3. [3]

    Anand, K.S. and F.C. Giraud-Carrier (2020): Pollution regulation of competitive markets. Manag. Sci. 66(9), 4193-4206

    work page 2020

  4. [4]

    Andersson, J. and G. Atkinson (2020): The distributional effects of a carbon tax: The role of income inequality. Working paper 349, Grantham Research Institute on Climate Change and the Environment London, UK

    work page 2020

  5. [5]

    Ba s ar, T. and G.J. Olsder (1982): Dynamic Noncooperative Game Theory . SIAM, New York

    work page 1982

  6. [6]

    Budhiraja and A

    Bayraktar, E., A. Budhiraja and A. Cohen (2019): Rate control under heavy traffic with strategic servers. Ann. Appl. Probab. 29(1), 1-35

    work page 2019

  7. [7]

    Benchekroun, H. and G.M. Herran (2016): The impact of foresight in a transboundary pollution game. Euro. J. Oper. Res. 251(1), 300-309

    work page 2016

  8. [8]

    Camacho and B

    Bertinelli, L., C. Camacho and B. Zou (2014): Carbon capture and storage and transboundary pollution: A differential game approach. Euro. J. Oper. Res. 237(2), 721-728

    work page 2014

Show all 59 references
  1. [9]

    Huang and X

    Bo, L., Y. Huang and X. Yu (2024): A decomposition-homogenization method for Robin boundary problems on the nonnegative orthant. Electron. J. Probab. 200, 1-25

    2024

  2. [10]

    Liao and X

    Bo, L., H. Liao and X. Yu (2021): Optimal tracking portfolio with a ratcheting capital benchmark. SIAM J. Contr. Optim. 59(3), 2346-2380

    2021

  3. [11]

    Liu and S.H

    Bo, L., Z. Liu and S.H. Wang (2026): Mean field games for emission control of multiple regions. Automatica 188, 112975

    2026

  4. [12]

    Wang and X

    Bo, L., J. Wang and X. Yu (2025): Mean field game of controls with state reflections. Preprint, available at https://arxiv.org/abs/2503.03253

    2025 arXiv

  5. [13]

    Wang and X

    Bo, L., S. Wang and X. Yu (2024): A mean field game approach to equilibrium consumption under external habit formation. Stoch. Process. Appl. 178, 104461

    2024

  6. [14]

    Borkar, V. and A. Budhiraja (2005): Ergodic control for constrained diffusions: Characterization using HJB equations SIAM J. Contr. Optim. 43(4), 1467-1492

    2005

  7. [15]

    Zaccour and M

    Breton, M., G. Zaccour and M. Zahaf (2005): A differential game of joint implementation of environmental projects. Automatica 41(10), 1737-1749

    2005

  8. [16]

    Correani and F

    Cerqueti, R., L. Correani and F. Di Dio (2023): Environmental policies in a Stackelberg differential game. Naval Res. Logis. 70(8), 1128-1149

    2023

  9. [17]

    Wang and Z

    Chang, S., X. Wang and Z. Wang (2015): Modeling and computation of transboundary industrial pollution with emission permits trading by stochastic differential game. PloS one 10(9), e0138641

    2015

  10. [18]

    De and G.C

    Choudhury, M., S.K. De and G.C. Mahata (2023): A pollution-sensitive multistage production-inventory model for deteriorating items considering expiration date under Stackelberg game approach. Environ. Dev. Sustain. 25, 11849-11887

    2023

  11. [19]

    Coutin, L. and M. Pontier (2019): Existence and regularity of law density of a pair (diffusion, first component running maximum). Stats. Probab. Lett. 153, 130-138

    2019

  12. [20]

    L\'opez-P\'erez and G

    De Frutos, J., P.M. L\'opez-P\'erez and G. Mart\'in-Herr\'an (2021): Equilibrium strategies in a multiregional transboundary pollution differential game with spatially distributed controls. Automatica 125, 109411

    2021

  13. [21]

    Leocata and G

    Del Sarto, G., M. Leocata and G. Livieri (2024): A mean field game approach for pollution regulation of competitive firms. Preprint, available at https://arxiv.org/abs/2407.12754

    2024 arXiv

  14. [22]

    Dietz, S. and N. Stern (2015): Endogenous growth, convexity of damage and climate risk: how Nordhaus' framework supports deep cuts in carbon emissions. Econ. J. 125(583), 574-620

    2015

  15. [23]

    Dockner, E.J. and N. Van Long (1993): International pollution control: cooperative versus noncooperative strategies. J. Environ. Econ. Manag. 25(1), 13-29

    1993

  16. [24]

    J rgensen, S. and G. Zaccour (2001): Incentive equilibrium strategies and welfare allocation in a dynamic game of pollution control. Automatica 37(1), 29-36

    2001

  17. [25]

    (2016): Linear quadratic mean field type control and mean field games with common noise, with application to production of an exhaustible resource

    Graber, P.J. (2016): Linear quadratic mean field type control and mean field games with common noise, with application to production of an exhaustible resource. Appl. Math. Optim. 74, 459-486

    2016

  18. [26]

    Guo, X. and J.C. Zhang (2024): Ito's formula for flows of conditional measures on semimartingales. Preprint, available at https://arxiv.org/abs/2404.111671

    2024 arXiv

  19. [27]

    Hassler, P

    Golosov, M., J. Hassler, P. Krusell and A. Tsyvinski (2014): Optimal taxes on fossil fuel in general equilibrium. Econometrica 82(1), 41-88

    2014

  20. [28]

    Halkos, G. and G. Papageorgiou (2014): Controlling polluting firms: Nash and Stackelberg strategies. MPRA Paper No. 58947

    2014

  21. [29]

    E., and G.J

    Halkos, G. E., and G.J. Papageorgiou (2021): Some results on the control of polluting firms according to dynamic Nash and Stackelberg patterns. Economies 9(2), 77

    2021

  22. [30]

    Hayashi, M. and A. Kohatsu (2013): Smoothness of the distribution of the supremum of a multi-dimensional diffusion process. Potential Anal. 38, 57-77

    2013

  23. [31]

    Hitzemann, S. and M. Uhrig-Homburg (2018): Equilibrium price dynamics of emission permits. J. Finan. Quant. Anal. 53(4), 1653-1678

    2018

  24. [32]

    Hoel, M. and L. Karp (2001): Taxes and quotas for a stock pollutant with multiplicative uncertainty. J. Public Econ. 82(1), 91-114

    2001

  25. [33]

    Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change

    IPCC (2021): Climate Change 2021: The Physical Science Basis. Contribution of Working Group I to the Sixth Assessment Report of the Intergovernmental Panel on Climate Change . Cambridge University Press, Cambridge

    2021

  26. [34]

    (2009): Controlled Diffusion Processes

    Krylov, N.V. (2009): Controlled Diffusion Processes . Springer-Verlag, New York

    2009

  27. [35]

    Kunita (2019): Stochastic Flows and Jump-Diffusions

    H. Kunita (2019): Stochastic Flows and Jump-Diffusions. Springer-Verlag, New York

    2019

  28. [36]

    Kydland, F.E. and E.C. Prescott (1977): Rules rather than discretion: The inconsistency of optimal plans. J. Polit. Econ. 85(3), 473-491

    1977

  29. [37]

    Neuman and M

    Jaber, E.A., E. Neuman and M. Voss (2023): Equilibrium in functional stochastic games with mean-field interaction. Preprint, available at https://arxiv.org/abs/2306.05433

    2023 arXiv

  30. [38]

    Li and J

    Jiang, C., T. Li and J. Yu (2022): Approximating Nash equilibrium for production control with sticky price. Oper. Res. Lett. 50(3), 335-342

    2022

  31. [39]

    (2018): Mean field game and interacting particle systems

    Lacker, D. (2018): Mean field game and interacting particle systems. Lecture Notes available at https://www.columbia.edu/ dl3133/MFGSpring2018.pdf

    2018

  32. [40]

    Lashof, D.A. and D.R. Ahuja (1990): Relative contributions of greenhouse gas emissions to global warming. Nature 344, 529-531

    1990

  33. [41]

    (2019): Climate tipping points—too risky to bet against

    Lenton, T.M., et al. (2019): Climate tipping points—too risky to bet against. Nature 575, 592-595

    2019

  34. [42]

    (2014): A differential game of transboundary industrial pollution with emission permits trading

    Li, S. (2014): A differential game of transboundary industrial pollution with emission permits trading. J. Optim. Theor. Appl. 163, 642-659

    2014

  35. [43]

    (2016): Dynamic optimal control of pollution abatement investment under emission permits

    Li, S. (2016): Dynamic optimal control of pollution abatement investment under emission permits. Oper. Res. Lett. 44(3), 348-353

    2016

  36. [44]

    (1992): Pollution control: A differential game approach

    Long, N.V. (1992): Pollution control: A differential game approach. Ann. Oper. Res. 37(1), 283-296

    1992

  37. [45]

    Gillett, P.A

    Matthews, H.D., N.P. Gillett, P.A. Stott and K. Zickfeld (2009): The proportionality of global warming to cumulative carbon emissions. Nature 459, 829-832

    2009

  38. [46]

    Moon, J. and T. Başar (2018): Linear quadratic mean field Stackelberg differential games. Automatica 97, 200-213

    2018

  39. [47]

    (2013): Absolute continuity of the laws of a multi-dimensional SDEs with coefficients dependent on the maximum

    Nakatsu, T. (2013): Absolute continuity of the laws of a multi-dimensional SDEs with coefficients dependent on the maximum. Statist. Probab. Lett. 83(11), 2499–2506

    2013

  40. [48]

    Nualart, D. and E. Nualart (2006): Introduction to Malliavin Calculus . Cambridge University Press, Cambridge

    2006

  41. [49]

    (2022): A Stackelberg–Nash–Cournot equilibrium in a pollution reduction scheme

    Perera, R.S. (2022): A Stackelberg–Nash–Cournot equilibrium in a pollution reduction scheme. Inter. Game Theor. Rev. 24(2), 2150014

    2022

  42. [50]

    PiliPenko, A. (2014). An Introduction to Stochastic Differential Equations with Reflection. Lect. Pure Appl. Math. Potsdam University Press, Potsdam

    2014

  43. [51]

    Quick, J.C. and E. Marland (2019): Systematic error and uncertain carbon dioxide emissions from U.S. power plants. J. Air Waste Manag. Assoc. 69, 646-658

    2019

  44. [52]

    (2008): The economics of climate change

    Stern, N. (2008): The economics of climate change. Amer. Econ. Rev. 98(2), 1-37

    2008

  45. [53]

    (2013): The structure of economic modeling of the potential impacts of climate change: grafting gross underestimation of risk onto already narrow science models

    Stern, N. (2013): The structure of economic modeling of the potential impacts of climate change: grafting gross underestimation of risk onto already narrow science models. J. Econ. Liter. 51(3), 838-859

    2013

  46. [54]

    Van Der Ploeg, F. and A.J. De Zeeuw (1992): International aspects of pollution control. Envir. Resour. Econ. 2, 117-139

    1992

  47. [55]

    (2004): The conditional probability density function for a reflected Brownian motion

    Veestraeten, D. (2004): The conditional probability density function for a reflected Brownian motion. Comput. Econ. 24(2), 185-207

    2004

  48. [56]

    Wang, B. and M. Huang (2019): Mean field production output control with sticky price: Nash and social solutions. Automatica 100, 90-98

    2019

  49. [57]

    (2009): On modeling and interpreting the economics of catastrophic climate change

    Weitzman, M.L. (2009): On modeling and interpreting the economics of catastrophic climate change. Rev. Econ. Stats. 91(1), 1-19

    2009

  50. [58]

    (2007): Dynamically consistent cooperative solution in a differential game of transboundary industrial pollution

    Yeung, D.W.K. (2007): Dynamically consistent cooperative solution in a differential game of transboundary industrial pollution. J. Optim. Theor. Appl. 134(1), 143-160

    2007

  51. [59]

    Yeung, D.W.K. and L.A. Petrosyan (2008): A cooperative stochastic differential game of transboundary industrial pollution. Automatica 44(6), 1532-1544

    2008

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.