Feedback Nash equilibria for scalar N-player linear quadratic dynamic games

Benita Nortmann; Mario Sassano; Thulasi Mylvaganam

REVIEW 2 major objections 5 minor 1 cited by

Feedback Nash equilibria for scalar N-player linear quadratic dynamic games

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Feedback Nash equilibria of scalar N-player discrete-time linear quadratic games are exactly the intersections of 2^N auxiliary curves with a horizontal line, and the paper derives parameter conditions for how many equilibria exist.

desk verdict A genuinely useful graphical reduction for scalar discrete-time LQ games, with a couple of theorem-statement gaps that need fixing before the counting results can be taken at face value. read the letter →

arxiv 2411.19377 v1 pith:4U5DYSG3 submitted 2024-11-28 math.OC cs.SYeess.SY

Benita Nortmann , Mario Sassano , Thulasi Mylvaganam This is my paper · ORCID

classification math.OCcs.SYeess.SY MSC 91A2591A1093C5549N10

keywords feedbackNashequilibriumlinearquadraticgamesdiscrete-timedynamicscalardynamicsgraphicalrepresentationmultiplicitycoupledalgebraicequationsinfinite-horizon

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 proves a geometric way to count feedback Nash equilibria in infinite-horizon, discrete-time, linear quadratic games where one scalar state is driven by N players with scalar inputs. For such games, the algebraic conditions for a linear feedback Nash equilibrium are cubic in the strategy gains, and multiple equilibria can coexist. The paper shows that under mild parameter ordering every equilibrium with nonzero closed-loop gain corresponds to an intersection of one of $2^N$ auxiliary curves with a horizontal line at height $a$, the open-loop dynamics parameter. From the shape of those curves it derives parameter conditions for the number of equilibria, for instance $2^N-1$ equilibria for large $|a|$, at least one equilibrium when $\sigma_1>0$, and a unique equilibrium when $\sigma_N\ge 0$ and $|a|<1$. A reader cares because these conditions come from system and cost parameters alone, replacing the need to solve coupled cubic equations case by case.

What carries the argument

The load-bearing object is the family of $2^N$ auxiliary functions $f_\ell(\xi)=\hat f(\xi)+N\xi+\sum_{i=1}^N \tau_{\ell,i}\sqrt{\xi^2-\sigma_i}$ for sign tuples $\tau_\ell\in\{-1,1\}^N$. The function $\hat f(\xi)$ encodes the stabilizing closed-loop gain $a_{cl}=\hat f(\xi)$, and the square-root terms encode each player's possible equilibrium gain in (9). Intersections of $f_\ell$ with the horizontal line at level $a$ are exactly the feedback Nash equilibria of the game, and the asymptotic slopes, monotonicity intervals and definition gaps in Lemma 3 determine how many such intersections can occur for a given $a$.

What would settle it

For a concrete game with $\sigma_N<-1$, solve equation (10) graphically, recover the candidate gains from (9), and test each candidate against condition (6c). A candidate that fails (6c) while its intersection root satisfies (10) would demonstrate that the graphical count overcounts equilibria outside Assumption 2, and Theorem 2's counts cannot be applied there without the extra check.

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Extended reading notes

Core claim

The central claim, stated in Lemma 2, is that for the scalar $N$-player game with $\sigma_i=b_i^2 q_i/r_i$ ordered decreasingly and $\sigma_N>-1$, the linear feedback strategies $u_i^\star(k)=k_i x(k)$ with gains $k_i=-\xi-t_i\sqrt{\xi^2-\sigma_i}/b_i$ form a feedback Nash equilibrium with $a_{cl}\neq 0$ exactly when there exist signs $t_i\in\{-1,1\}$ and a real $\xi\neq 0$ solving $a=\hat f(\xi)+N\xi+\sum_{i=1}^N t_i\sqrt{\xi^2-\sigma_i}$, where $\hat f(\xi)$ is the stabilizing branch of $-\xi+\sqrt{\xi^2+1}$ selected by the sign of $\xi$. Introducing $\xi=\frac12(1/a_{cl}-a_{cl})$ turns the coupled equations (6) into this single scalar equation, so each equilibrium is a graphical intersection. Theorem 2 then reads off the number and properties of equilibria from the auxiliary functions $f_\ell(\xi)$, including a closed-loop bound $|a_{cl}^\star|\le|\sqrt{\sigma_1}-\sqrt{\sigma_1+1}|$ when $\sigma_1>0$ and the positive $\sigma_i$ are not all equal.

Load-bearing premise

The counting results rely on Assumption 2 ($\sigma_N > -1$), which guarantees that every algebraic solution of the coupled equations automatically satisfies the well-posedness condition (6c); if that fails, intersections of the auxiliary curves may not all be genuine feedback Nash equilibria.

Editorial extensions

If this is right

For sufficiently large $|a|$, exactly $2^N-1$ feedback Nash equilibria exist; the missing intersection corresponds to the one sign combination whose asymptotic slope does not meet the level line.
If $\sigma_1>0$, every open-loop parameter $a$ admits at least one equilibrium; for $a=0$ the trivial zero-gain strategy is counted separately as in Remark 3.
If all $\sigma_i\ge 0$ and the open-loop system is stable ($|a|<1$), the equilibrium is unique.
When $\sigma_1>0$ and the positive $\sigma_i$ are not all equal, every equilibrium closed loop satisfies $|a_{cl}^\star|\le|\sqrt{\sigma_1}-\sqrt{\sigma_1+1}|$.
In discrete time, a rapidly stabilizing open loop does not by itself guarantee uniqueness when $\sigma_N<0$, unlike the continuous-time scalar case; uniqueness then requires the additional hypotheses of items v and vi.

Reading between the lines

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

The same intersection picture could be turned into a homotopy or continuation method that counts equilibria as $a$ varies, which the paper does not explicitly pursue.
Items v and vi of Theorem 2 involve the internal variable $\xi$ in their inequalities, so they are not directly checkable from system parameters; deriving parameter-only reformulations would make those conditions fully a priori.
Remark 4 shows that coincident intersections can correspond to a single equilibrium rather than several, so any automatic enumeration must track degeneracies when parameters are tied; counting multiplicities is a separate question the paper leaves implicit.
Although the paper treats scalar state and inputs, the reduction through $\xi = \frac12(1/a_{cl}-a_{cl})$ suggests the geometric viewpoint might extend to systems whose closed-loop dynamics are scalarizable, at least as a heuristic.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The core of the paper is the reduction in Lemma 2: in the scalar case, with sigma_N > -1, every nontrivial FNE corresponds to an intersection of one of 2^N auxiliary functions (13) with the horizontal line at level a. That is a clean, honest algebraic elimination, and it converts a coupled cubic system into a one-dimensional graphical problem. The paper does good work here: it extends the two-player construction to N players, treats a wider range of cost parameters, and derives genuinely new counting statements (2^N - 1 equilibria for |a| large; at least one equilibrium when sigma_1 > 0; uniqueness under sigma_N >= 0 with |a| < 1). The debt to Engwerda's continuous-time study and to the authors' own two-player paper is acknowledged openly, and Remark 5 correctly points out where the discrete-time behavior differs from continuous time. The numerical example in Section 5 does what it claims: it shows 7 intersections for |a| large and illustrates the extra equilibria that appear when sigma_N < 0. On the technical side, Lemma 1's use of Assumption 2 to guarantee condition (6c) is explicit and, as far as I can tell, sound; I do not consider that a hidden flaw. The real soft spot is Theorem 2 items v and vi. Both contain inequalities involving the internal variable xi, which is defined as xi = (1/acl - acl)/2 and is not known before solving the game. As printed, those conditions are not checkable a priori, and the theorem's promise of parameter-based conditions is not fulfilled there. The stress-test note is right that the quantifier is missing: if the inequality is meant to hold for all xi in the relevant domain, the paper needs to say so and prove it; if it is meant only at the equilibrium xi, then items v and vi are restatements of the graphical construction, not conditions in terms of system parameters. The jump from Lemma 3(vi)'s non-equality condition to the strict inequality used in Theorem 2(v) also needs justification. Item 11 in Lemma 2's proof is garbled, though the intended identity is recoverable (the quadratic in b_i k_i with linear term 2 xi b_i k_i + sigma_i), so I would count that as a typo, not a substantive error. The asymptotic statements ('|a| >> 1', '|a| << 1') are informal; they should either be made quantitative or explicitly asymptotic. These issues do not sink the core method, and items i-iv and vii look solid to me, but as printed items v and vi do not deliver what they claim. Net: this is worth a serious referee and probably worth citing for the graphical reduction and the sigma_N >= 0 uniqueness result, but the author's revision should fix the quantifier problem in the two fragile items, correct the garbled equation, and sharpen the asymptotics before I would trust the full counting theorem.

Referee Report

2 major / 5 minor

Summary. The paper studies infinite-horizon, discrete-time, scalar-state, N-player linear quadratic dynamic games with quadratic costs. It recalls the standard characterization of linear feedback Nash equilibria through coupled algebraic equations, then introduces the change of variable xi = (1/acl - acl)/2 to eliminate the value functions. This leads to a single scalar equation, equation (10), and a graphical representation via 2^N auxiliary functions f_ell(xi). The paper uses this representation to state conditions, in Theorem 2, on the number of feedback Nash equilibria and on the closed-loop gain acl, and illustrates the results with a three-player numerical example.

Significance. The central reduction in Lemma 2 is appealing and, if correct, provides a clean algebraic picture of feedback Nash equilibria for a class of games where the coupled equations are otherwise difficult to analyse. The paper is self-contained from the stated algebraic equations, and the graphical reformulation is a genuine mathematical reduction rather than an assumption of the desired result. The comparison with the continuous-time scalar case in Remark 5 and the extension of the earlier two-player result to N players are useful contributions. The main value of the paper lies in the counting statements of Theorem 2; those statements are what must be made precise. The manuscript does not provide machine-checked proofs or code, but the algebraic derivations are sufficiently explicit to be verifiable by hand.

major comments (2)

[Theorem 2, items v and vi] Items v and vi are not checkable parameter conditions as stated because they contain the internal variable xi = (1/acl - acl)/2, which is itself defined through the unknown equilibrium closed-loop gain. For example, item v contains the condition (N-1)+|xi|/sqrt(xi^2+1) < sum_i |xi|/sqrt(xi^2-sigma_i) with no quantifier. If this is meant to hold at the equilibrium value of xi, then the condition cannot be verified before solving the game and the theorem reduces to a restatement of the graphical construction. If it is meant to hold for all xi in the relevant domain, the paper neither says so nor proves that this global condition follows from the parameter-only part of the statement. The same issue applies to the inequality condition in item vi.
[Theorem 2 proof, items v and vi; Lemma 3(vi)] The proof of items v and vi invokes Lemma 3(vi), but Lemma 3(vi) is a non-equality condition: it states that if (N-1)+|xi|/sqrt(xi^2+1) != sum_i |xi|/sqrt(xi^2-sigma_i) for all xi != 0, then f_L is strictly monotone for xi<0 and f_1 is strictly monotone for xi>0. Item v, however, uses the strict inequality '<', and the proof asserts strict decrease of these functions. The transition from 'monotone' to 'strictly decreasing', and the use of the strict inequality form, is not justified in the manuscript. As printed, the uniqueness claims in items v and vi are therefore not established.

minor comments (5)

[Lemma 2, equation (11)] Equation (11) is garbled in the typeset text: as printed, '0 = bi 2 k2 i + xi ki + sigma i 2bi' is dimensionally inconsistent and does not lead to the solutions in equation (9). The intended identity appears to be b_i^2 k_i^2 + 2 xi b_i k_i + sigma_i = 0, which should be corrected.
[Lemma 3(vi), proof] The derivative expression in the proof of Lemma 3(vi) is written as (N-1) +/- xi/sqrt(xi^2+1) + sum_i tau_{ell,i} xi/sqrt(xi^2-sigma_i). The sign of the xi/sqrt(xi^2+1) term depends on the branch of hat f, and the displayed formula should state explicitly which sign applies for f_1 and f_L respectively.
[Section 5, numerical example] The text refers to 'f1(ell), f2(ell), f7(ell) and f8(ell)' in the discussion of the second example; the notation should be f_ell(xi), or the functions should be identified consistently with the labels used in Figure 2.
[Remark 4] The sentence beginning 'or for any i != l, i != j, if the game is such that any sigma_l = sigma_j' is difficult to parse; the intended statement about repeated sigma values should be reformulated for clarity.
[Assumption 1] The phrase 'Assumption 1 can be introduced without loss of generality' is slightly imprecise: reordering the players is indeed without loss of generality, but the assumption itself is a convention, not a restriction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Lemma 2 is a reformulation of the standard coupled algebraic equilibrium conditions, and the graphical counting argument is self-contained; the xi-dependent inequalities in Theorem 2 items v and vi are a checkability and proof-gap issue, not a circular reduction.

full rationale

Lemma 2's iff statement is obtained by algebraically eliminating the costate variables p_i from Theorem 1 and parameterizing the feedback gains by t_i and xi; equation (10) is the same coupled equilibrium condition rewritten, not an independent assertion of the counting result. The auxiliary functions f_l in (13) are defined directly from this parameterization, and Lemma 3 derives their asymptotics, domains and monotonicity, so Theorem 2's counting statements follow by counting intersections of explicitly given functions. No parameter is fitted to any subset of equilibria and no 'prediction' is extracted from fitted data. The citations to Monti et al. and Nortmann et al. are background or the standard dynamic-programming characterization (Theorem 1); even though some coauthors recur, the cited theorem is published and independently checkable, and the present paper does not invoke a self-authored uniqueness theorem to forbid alternatives. The only notable defect is that items v and vi of Theorem 2 involve the internal variable xi inside the stated inequalities, and Lemma 3(vi) supplies a non-equality condition while Theorem 2 uses a strict inequality; this is a checkability and proof-gap concern, not a circular dependency, because the theorem's claims are conditional on those inequalities rather than being equivalent to the conclusion by construction. No circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper's constructive results rest on a standard external characterization of FNE (Theorem 1), a stated parameter assumption (sigma_N > -1) that guarantees the well-posedness condition (6c), and a relabeling assumption. No free parameters are fitted to data and no new entities are introduced.

assumptions (3)

domain assumption FNE are characterized by the coupled algebraic equations (6) with stability (5) and well-posedness (6c)
Theorem 1, cited from Monti et al. (2024), Theorem 3.2; the paper relies on this external characterization without reproving it.
domain assumption Assumption 2: sigma_N > -1 (equivalently b_i^2 q_i / r_i > -1 for all i)
Used in Lemma 1 to infer that (6c) holds automatically for all solutions of (5),(6a),(6b); Theorem 2 is stated under this assumption.
standard math Assumption 1: players are ordered so that sigma_1 >= ... >= sigma_N
Without loss of generality by relabeling players; used to order the auxiliary functions in Lemma 3.

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Cite this review

Pith. "Pith review of Feedback Nash equilibria for scalar N-player linear quadratic dynamic games." pith.science (2026). https://pith.science/paper/4U5DYSG3

@misc{pith2026241119377,
  author       = {Pith},
  title        = {Pith review of: Feedback Nash equilibria for scalar N-player linear quadratic dynamic games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4U5DYSG3}},
  note         = {Machine review of arXiv:2411.19377}
}

read the original abstract

Considering infinite-horizon, discrete-time, linear quadratic, N-player dynamic games with scalar dynamics, a graphical representation of feedback Nash equilibrium solutions is provided. This representation is utilised to derive conditions for the number and properties of different feedback Nash equilibria a game may admit. The results are illustrated via a numerical example.

Figures

Figures reproduced from arXiv: 2411.19377 by the authors.

**Figure 1.** Plot of the auxiliary functions for ˜q2 = 0.05 and q˜3 = 0, f1(ξ) (red), f2(ξ) (green), f3(ξ) (blue), fℓ(ξ), ℓ = 4, 5 (grey), f6(ξ) (cyan), f7(ξ) (dark green) and f8(ξ) (magenta), and their intersection points (yellow crosses) with the horizontal lines at a = ˜a (yellow) for ˜a = 0.3 and a˜ = −5. The black dashed lines indicate the linear asymptotes of fℓ(ξ), ℓ = 1, . . . , 8, and the grey dotted lines indicate ξ … view at source ↗

**Figure 2.** Plot of the auxiliary functions for ˜q2 = −0.8 and q˜3 = −0.9, f1(ξ) (red), f2(ξ) (green), f3(ξ) (blue), fℓ(ξ), ℓ = 4, 5 (grey), f6(ξ) (cyan), f7(ξ) (dark green) and f8(ξ) (magenta), and their intersection points (yellow crosses) with the horizontal lines at a = ˜a (yellow) for ˜a = 0.3 and a˜ = −5. The black dashed lines indicate the linear asymptotes of fℓ(ξ), ℓ = 1, . . . , 8, and the grey dotted lines indicate … view at source ↗

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Bridging Finite and Infinite-Horizon Nash Equilibria in Linear Quadratic Games
cs.MA 2025-08 conditional novelty 6.0 of 10

The finite-horizon Riccati recursion's fixed points equal infinite-horizon Nash equilibria, and its cycles, if any, are periodic Nash equilibria.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · cited by 1 Pith paper

[1]

& Olsder, G

Ba¸ sar, T. & Olsder, G. J. (1998), Dynamic Noncoop- erative Game Theory , Vol. 23 of Classics in Applied Mathematics, 2 edn, Society for Industrial and Ap- plied Mathematics

work page 1998
[2]

(1957), Dynamic Programming, Princeton University Press

Bellman, R. (1957), Dynamic Programming, Princeton University Press. 6

work page 1957
[3]

& Mylvaganam, T

Cappello, D. & Mylvaganam, T. (2022), ‘Distributed diﬀerential games for control of multi-agent systems’, IEEE Transactions on Control of Network Systems 9(2), 635–646

work page 2022
[4]

(2005), LQ dynamic optimization and dif- ferential games, John Wiley & Sons

Engwerda, J. (2005), LQ dynamic optimization and dif- ferential games, John Wiley & Sons

work page 2005
[5]

(2016), ‘Properties of feedback Nash equi- libria in scalar LQ diﬀerential games’, Automatica 69, 364–374

Engwerda, J. (2016), ‘Properties of feedback Nash equi- libria in scalar LQ diﬀerential games’, Automatica 69, 364–374

work page 2016
[6]

Etesami, S. R. & Ba¸ sar, T. (2019), ‘Dynamic games in cyber-physical security: An overview’, Dynamic Games and Applications 9, 884–913. Jørgensen, S., Quincampoix, M. & Vincent, T. (2007), Advances in Dynamic Game Theory: Numerical Meth- ods, Algorithms, and Applications to Ecology and Eco- nomics, Annals of the International Society of Dy- namic Games, ...

work page 2019
[7]

& Bur- det, E

Li, Y., Carboni, G., Gonzalez, F., Campolo, D. & Bur- det, E. (2019), ‘Diﬀerential game theory for versatile physical human–robot interaction’, Nature Machine Intelligence 1(1), 36–43

work page 2019
[8]

& Sas- sano, M

Monti, A., Nortmann, B., Mylvaganam, T. & Sas- sano, M. (2024), ‘Feedback and open-loop Nash equi- libria for LQ inﬁnite-horizon discrete-time dynamic games’, SIAM Journal on Control and Optimization 62(3), 1417–1436

work page 2024

Show all 16 references

[9]

& Astolﬁ, A

Mylvaganam, T., Sassano, M. & Astolﬁ, A. (2017), ‘A diﬀerential game approach to multi-agent collision avoidance’,IEEE Transactions on Automatic Control 62, 4229–4235

2017
[10]

& Mylvaganam, T

Nortmann, B., Monti, A., Sassano, M. & Mylvaganam, T. (2023), ‘Feedback Nash equilibria for scalar two- player linear-quadratic discrete-time dynamic games’, IF AC-PapersOnLine56(2), 1772–1777

2023
[11]

& Mylva- ganam, T

Nortmann, B., Monti, A., Sassano, M. & Mylva- ganam, T. (2024), ‘Nash equilibria for linear quadratic discrete-time dynamic games via iterative and data- driven algorithms’, IEEE Transactions on Automatic Control 69(10), 6561–6575

2024
[12]

& Pham, K

Pachter, M. & Pham, K. D. (2010), ‘Discrete-time linear- quadratic dynamic games’, Journal of Optimization Theory and Applications 146(1), 151–179

2010
[13]

P., Medanic, J

Papavassilopoulos, G. P., Medanic, J. V. & Cruz, J. B. (1979), ‘On the existence of Nash strategies and solu- tions to coupled Riccati equations in linear-quadratic games’,Journal of Optimization Theory and Applica- tions 28(1), 49–76

1979
[14]

& Sassano, M

Possieri, C. & Sassano, M. (2015), An algebraic geome- try approach for the computation of all linear feedback Nash equilibria in LQ diﬀerential games, in ‘IEEE Conference on Decision and Control’, IEEE

2015
[15]

P., Kishor, N

Singh, V. P., Kishor, N. & Samuel, P. (2017), ‘Dis- tributed multi-agent system-based load frequency control for multi-area power system in smart grid’, IEEE Transactions on Industrial Electronics 64(6), 5151–5160

2017
[16]

Starr, A. W. & Ho, Y. C. (1969), ‘Nonzero-sum diﬀeren- tial games’, Journal of Optimization Theory and Ap- plications 3, 184–206. 7

1969

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

& Olsder, G

Ba¸ sar, T. & Olsder, G. J. (1998), Dynamic Noncoop- erative Game Theory , Vol. 23 of Classics in Applied Mathematics, 2 edn, Society for Industrial and Ap- plied Mathematics

work page 1998

[2] [2]

(1957), Dynamic Programming, Princeton University Press

Bellman, R. (1957), Dynamic Programming, Princeton University Press. 6

work page 1957

[3] [3]

& Mylvaganam, T

Cappello, D. & Mylvaganam, T. (2022), ‘Distributed diﬀerential games for control of multi-agent systems’, IEEE Transactions on Control of Network Systems 9(2), 635–646

work page 2022

[4] [4]

(2005), LQ dynamic optimization and dif- ferential games, John Wiley & Sons

Engwerda, J. (2005), LQ dynamic optimization and dif- ferential games, John Wiley & Sons

work page 2005

[5] [5]

(2016), ‘Properties of feedback Nash equi- libria in scalar LQ diﬀerential games’, Automatica 69, 364–374

Engwerda, J. (2016), ‘Properties of feedback Nash equi- libria in scalar LQ diﬀerential games’, Automatica 69, 364–374

work page 2016

[6] [6]

Etesami, S. R. & Ba¸ sar, T. (2019), ‘Dynamic games in cyber-physical security: An overview’, Dynamic Games and Applications 9, 884–913. Jørgensen, S., Quincampoix, M. & Vincent, T. (2007), Advances in Dynamic Game Theory: Numerical Meth- ods, Algorithms, and Applications to Ecology and Eco- nomics, Annals of the International Society of Dy- namic Games, ...

work page 2019

[7] [7]

& Bur- det, E

Li, Y., Carboni, G., Gonzalez, F., Campolo, D. & Bur- det, E. (2019), ‘Diﬀerential game theory for versatile physical human–robot interaction’, Nature Machine Intelligence 1(1), 36–43

work page 2019

[8] [8]

& Sas- sano, M

Monti, A., Nortmann, B., Mylvaganam, T. & Sas- sano, M. (2024), ‘Feedback and open-loop Nash equi- libria for LQ inﬁnite-horizon discrete-time dynamic games’, SIAM Journal on Control and Optimization 62(3), 1417–1436

work page 2024