The Efficiency of Generalized Nash and Variational Equilibria

Ankur A. Kulkarni

REVIEW 3 major objections 3 minor 32 references

The Efficiency of Generalized Nash and Variational Equilibria

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that in shared-constraint resource allocation games, generalized Nash equilibria and variational equilibria both have worst-case efficiency zero over a broad class of concave increasing utilities, and that generalized…

desk verdict The efficiency results are mostly right, but the theorems as stated quantify over a class where the efficiency ratio is sometimes undefined; the fix is easy and the conclusions survive. read the letter →

arxiv 1908.00702 v1 pith:ZIILJD7W submitted 2019-08-02 cs.GT cs.SYeess.SYmath.OC

Ankur A. Kulkarni This is my paper · ORCID

classification cs.GTcs.SYeess.SYmath.OC MSC 91A1091A8091B3290C33

keywords generalizedNashequilibriumvariationalshared-constraintgamesefficiencylossresourceallocationpriceofanarchyreservesocialwelfare

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

The paper asks how inefficient decentralized competition over a shared resource can be when players simultaneously choose consumption levels under a common capacity constraint, with no mechanism or auctioneer. It shows that over the class F of concave, continuously differentiable utilities that are increasing in each player's own allocation and have a concave aggregate with nonnegative gradient, the worst-case efficiency of both generalized Nash equilibria and variational equilibria is zero, and the bound is tight: for every ε∈(0,1] there is a game in F with an equilibrium of efficiency ε. It also characterizes the subclass F′ where every variational equilibrium is efficient, and shows that even there the generalized Nash equilibrium's worst-case efficiency remains zero. These results matter because they say the efficiency of decentralized resource allocation depends entirely on which equilibrium notion, and hence which shadow-price regime, governs play, rather than on the broad shape assumptions usually imposed on utilities.

What carries the argument

The machinery centers on the efficiency ratio Θ(x)/max_{z∈C} Θ(z), where Θ=Σφ_i is aggregate utility and C is the capacity simplex, and on the class F of admissible utility tuples. The linearization Lemma 4.1 bounds the efficiency of any feasible allocation by the efficiency of the same allocation in a linearized game, so the worst case over F collapses to the worst case over linear objectives L (Theorem 4.3). For the VE-versus-GNE comparison, the paper uses the variational-inequality characterizations: a VE solves VI(C,F), and when F is integrable with F=−∇Θ, the VE solves (SYS); Theorem 5.1 gives the exact functional form φ_i=(Σ_j η_j(x_{−j}))/(N−1)−η_i(x_{−i}) for this integrable case. The zero-efficiency examples are linear games whose GNE set is the entire simplex {x≥0, 1^T x=C}, and the remedies are the bounded-gradient class F[α,β], yielding worst-case efficiency α/β, and a reserve price π, yielding GNE efficiency at least π/max_i c_i.

What would settle it

A two-player game with C=1, φ1(x)=x1−x2 and φ2(x)=x2−x1 satisfies Assumption 2.1 but has Θ≡0; every allocation with x1+x2=1 is a VE and an efficient point, so the efficiency ratio is 0/0, directly contradicting the paper's claim that Assumption 2.1 ensures the solution of (SYS) is positive and finite and that all efficiencies lie in [0,1].

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Extended reading notes

Core claim

The central claim is a pair of zero worst-case efficiency theorems. Theorem 4.4 states that the worst-case efficiency of the variational equilibrium, and hence of the generalized Nash equilibrium, over F is zero, with tightness in that any ε∈(0,1] is realized by some game. Theorem 5.4 states that over F′, the subclass in which the identity −∇Θ=F holds, every variational equilibrium is efficient yet the worst-case GNE efficiency is still zero. The paper's constructive examples are linear games in which equilibria place all of the resource with the lowest-marginal-utility player, driving the efficiency ratio to zero, while the efficient allocation gives everything to the highest-marginal-utility player. The best-case efficiency of both concepts is one, attained in the perfectly competitive separable-utility setting where the variational equilibrium coincides with the competitive equilibrium.

Load-bearing premise

The load-bearing premise is the Section II assertion that Assumption 2.1 makes the solution of (SYS) 'positive and finite', so the efficiency ratio Θ(x)/max_z Θ(z) is well defined; this is asserted without proof, and it fails for utilities satisfying the assumption with Θ≡0.

Editorial extensions

If this is right

If the theorems are right, decentralized resource allocation without a mechanism offers no worst-case efficiency guarantee for either solution concept over the full class F.
When utilities depend only on one's own allocation, the VE is efficient and coincides with the competitive equilibrium, so efficiency is restored precisely when shadow prices are uniform.
Even in games where every VE is efficient, GNE can realize efficiency ε for any ε∈(0,1], so relying on GNE rather than VE is the key source of potential inefficiency.
Restricting the aggregate gradient to [α,β] gives a worst-case efficiency bound of α/β, showing that bounded marginal-utility spread restores a positive guarantee.
A reserve price that screens out players with marginal utility below π guarantees GNE efficiency at least π/max_i c_i, approaching unity as the reserve price approaches the top marginal utility.

Reading between the lines

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

I read the zero result as an equilibrium-selection statement: the failure mode is not competition itself but nonuniform shadow prices across players, since the VE with uniform prices is efficient on F′.
The α/β bound suggests a practical diagnostic for deployed systems: measure the minimal and maximal marginal utilities of the aggregate objective over the feasible set; their ratio is a directly measurable lower bound on equilibrium efficiency.
A testable extension would be to sample games from F with strictly positive aggregate gradients and simulate the distribution of GNE efficiencies under different tie-breaking rules; the worst case being zero does not reveal whether inefficient equilibria are typical or rare.
The reserve-price remedy parallels auction design: screening by minimum marginal valuation eliminates low-interest players but risks excluding everyone; the paper assumes at least one player remains, and a natural extension would quantify the trade-off between the reserve level and the probability of total exclusion.

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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 substantive results here are better than the formal statement. What is new: worst-case efficiency of both GNE and VE is zero over the full class F, and — more striking — GNE worst-case efficiency is still zero over the potential-game class F′ where every VE is efficient. The reduction to linear utilities is clean, the examples are simple and convincing, and the characterization of −∇Θ = F via the functions η_i is elegant. The reserve-price discussion in Section VI is a nice practical angle. I believe the central claims are correct for the games the author actually wants to talk about.

The soft spots are real but not fatal. The main one: Assumption 2.1 allows Θ ≡ 0, so the ratio Θ(x)/max_z Θ(z) in Definitions 2.4–2.5 is undefined for some Φ ∈ F. The paper tries to rule this out by saying "Θ(0) ≥ 0 and ∇Θ ≥ 0 ensures ... the solution of (SYS) is positive and finite," but that is false (e.g., ϕ1 = x1 − x2, ϕ2 = x2 − x1). The theorem statements should restrict F to a subclass with max_z Θ(z) > 0. The counterexamples all satisfy this, so the zero worst-case results survive intact. This is a fixable oversight, not a load-bearing flaw.

Second, the abstract says "the subclass" of games where all VE are efficient, but Theorem 5.1 only characterizes the sufficient condition −∇Θ = F; Example V.4 shows non-necessity. Change to "a subclass."

Third, Section VI.A claims the worst-case efficiency over F[α,β] is exactly α/β, but the text only proves the lower bound. The upper bound follows from a scaled version of Example V.5, but it needs to be stated explicitly. Also there is a small sign typo: "F(x) > 0" should be "∇iϕi(x) > 0" or "−F(x) > 0."

In short, this is a paper worth engaging with. The results are interesting and, after a straightforward tightening of the assumptions and a few exposition fixes, the claims will hold. I would send it to peer review rather than desk reject. It is the kind of negative benchmark that people working on generalized Nash games and price of anarchy should know about.

Referee Report

3 major / 3 minor

Summary. The paper studies the efficiency of generalized Nash equilibria (GNE) and variational equilibria (VE) in a class of shared-constraint resource allocation games. For utilities satisfying Assumption 2.1, it defines efficiency as the ratio of aggregate utility at an equilibrium to the aggregate utility at the social optimum, and then investigates best- and worst-case efficiency over the class F. The main results are that the worst-case efficiency of both GNE and VE over F is zero (Theorems 4.4 and 5.4), that the best-case efficiency is unity, that a certain class F′ of games has every VE efficient while the GNE worst-case efficiency over F′ is still zero, and that bounded gradients, alternative social objectives, or reserve prices can restore positive efficiency guarantees. The proofs use linearization arguments and explicit examples with linear utility functions.

Significance. If the claims are taken with the intended nondegenerate interpretation, the paper gives a sharp and somewhat surprising negative result: under mild concavity assumptions, decentralized shared-constraint competition provides no worst-case efficiency guarantee for either equilibrium concept. This contrasts with mechanism-based resource allocation (e.g., the 3/4 bound of proportional allocation) and reinforces the view that VE is a meaningful refinement of GNE. The paper's main strengths are its constructive examples, the elementary and self-contained nature of the derivations, and the absence of fitted parameters or target-dependent assumptions; the efficiency bounds follow from first-order conditions and the definitions of the solution concepts. The main caveat is that the formal class F, as stated, contains degenerate games for which the efficiency ratio is undefined, so the theorems need a domain restriction to be fully well-posed.

major comments (3)

[Section V, Theorem 5.1 and abstract] Assumption 2.1 permits the aggregate utility to be identically zero. For example, take N=2, phi_1(x)=x_1-x_2 and phi_2(x)=x_2-x_1; each phi_i is linear, concave, C^1, strictly increasing in x_i, phi_i(0)=0, and Theta=0 with nabla Theta=0. This Phi lies in F, yet max_{z in C} Theta(z)=0 and the efficiency ratio Theta(x)/Theta(x*) is 0/0 for every feasible x. Since Definitions 2.4-2.5 and Theorems 4.3-4.4 and 5.3-5.4 quantify over F, the worst-case and best-case efficiencies are not well-defined over the stated class. The statement in Section II that Assumption 2.1 'ensures that Theta is nonnegative on C and thereby the solution of (SYS) is positive and finite' is therefore false as written. This is load-bearing because the central zero-worst-case theorem quantifies over F. I recommend restricting F to games with max_{z in C} Theta(z) > 0, for example by adding the condition that Theta(x*) > 0 or that each component of Theta is strictly increasing, and then restating the theorems for this restricted class; the examples already satisfy such a positivity condition, so the zero-efficiency conclusion is preserved.
[Appendix, Lemma 4.1] The abstract and the introduction state that the paper characterizes 'the subclass of games where all VE are efficient.' Theorem 5.1, however, characterizes the stronger identity -nabla Theta = F, and the paper itself notes in the discussion after Theorem 5.1 and in Example V.4 that this identity is not necessary for every VE to be efficient. Example V.4 gives a family with every VE efficient even though equation (6) fails. The wording of the abstract and of Theorem 5.2 should be corrected to say that Theorem 5.1 provides a sufficient condition, or the paper should explicitly characterize the actual subclass of games in which every VE is efficient. As it stands, the abstract overstates the result.
[Appendix, Lemma 4.1] The proof of Lemma 4.1 divides by nabla Theta(x)^T x^ell and assumes that both nabla Theta(x)^T x and nabla Theta(x)^T x^ell are positive. Under Assumption 2.1, nabla Theta(x) can vanish at a point x that is a global maximizer of Theta over C, even when max Theta > 0. In that case x^ell may be arbitrary and the denominator nabla Theta(x)^T x^ell can be zero, so the displayed inequality is not meaningful. The lemma needs either an explicit positivity assumption on nabla Theta(x) or a separate argument for the zero-gradient case. Since Lemma 4.1 is the key input to Theorems 4.3 and 5.3, this gap should be closed before the worst-case reductions are fully rigorous.

minor comments (3)

[Examples IV.2 and V.5] The notation c is inconsistent with the definition of F in Section II. The paper defines F(x) = -(nabla_1 phi_1(x), ..., nabla_N phi_N(x))^T, but in Examples IV.2 and V.5 it writes c = F(x) after setting c_i = d_i^i. In fact c equals -F(x) under that definition. The equilibrium conditions later use -c + lambda, which is consistent with F = -c, so the intended mathematics is clear, but the notation should be corrected to avoid confusion.
[Section VI-A] The sentence 'Assumption 2.1 ensures that F(x) > 0' has the wrong sign: under Assumption 2.1 each F_i(x) = -partial phi_i/partial x_i is strictly negative because phi_i is strictly increasing in x_i. The intended statement is that -F(x) > 0 (i.e., the marginal utilities are positive), which is what is needed to conclude that every GNE satisfies 1^T x = C.
[Throughout] There are numerous typographical errors that should be cleaned up, including 'in zero' in the abstract, 'the worst case efficiency of the VE over is in fact zero' in Section IV, 'compatible wth' in Section II, 'Newtor' in the references, and 'an model' in Section II. None of these affect the mathematics, but they detract from readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the efficiency bounds follow from KKT conditions, concavity, and explicit constructions, with no fitted parameters or self-citation used as a load-bearing premise.

full rationale

The paper's efficiency analysis is self-contained. Definitions 2.1 and 2.2 state GNE and VE through the KKT conditions of the players' optimization problems, and Definition 2.3 defines efficiency through the aggregate-utility maximization (SYS). Lemma 4.1 lower-bounds the efficiency ratio using concavity of Theta and the linearized system (SYS_l(x)); the proof in the appendix uses only the first-order inequality Theta(x*) <= Theta(x) + grad Theta(x)^T(x* - x), nonnegativity of Theta(0) and grad Theta, and the optimality of x_l. No parameter is fitted to any target efficiency value, and no theorem assumes the conclusion it proves. The worst-case results are established by explicit linear examples (Example IV.2 and Example V.5) in which a specific equilibrium is computed and its efficiency is shown to approach zero; these are constructions, not re-statements of the definition. The best-case result is obtained by observing that the VE's KKT conditions coincide with those of (SYS) in the perfectly competitive case, again an identification of two independently defined optimization problems. The characterization in Theorem 5.1 derives the functional form (6) from the identity -grad Theta = F using elementary algebra, and Theorem 5.4 follows from the linear example, not from the characterization. The author's own prior work [10], [11] is cited for the interpretation of the VE as a refinement of the GNE and for the origin of the solution concepts, but this interpretation is not used as a premise in any bound; the theorems would stand unchanged without those citations. The citation to Slade [28] merely notes prior appearance of a stationarity equivalence and is not load-bearing. There is one notable non-circular formal gap: the text claims 'By Assumption 2.1, Theta(0) >= 0 and grad Theta >= 0. This ensures that Theta is nonnegative on C and thereby the solution of (SYS) is positive and finite,' but Assumption 2.1 permits games with Theta identically zero, making the efficiency ratio 0/0 and the infima in Definitions 2.4-2.5 ill-defined over F. That is a well-definedness/correctness issue in the quantification over F, not a circularity: the counterexamples used for the zero bound have positive aggregate gradients and positive optimum value, so the intended quantitative conclusion is not the product of an input-output identity. Overall, no step in the derivation reduces to its own input or to a self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central results rely on the concavity and monotonicity class F, the KKT/VI equivalences, and Rosen existence. The key under-specified input is the implicit positivity of the social optimum, which the paper asserts but does not prove. The reserve-price remedy explicitly assumes at least one player survives the price. No invented entities and no fitted parameters appear; the ε in the examples is an adversarial construction parameter, not a fitted value.

assumptions (5)

standard math KKT conditions characterize GNE, VE, and efficient allocations for convex feasible sets.
Used in Definitions 2.1, 2.2 and for the characterization of (SYS). Valid because the shared constraint is linear and Slater's condition holds.
standard math Rosen's existence theorem guarantees nonempty VE and GNE for concave games satisfying Assumption 2.1.
Cited as [25] in Section II to ensure the sets over which worst and best case efficiencies are taken are nonempty.
domain assumption Assumption 2.1: each ϕ_i is concave, C1, strictly increasing in x_i, ϕ_i(0)≥0, Θ is concave, and ∇Θ is componentwise nonnegative.
Defines the class F over which the worst and best case efficiencies are computed and is used in Lemma 4.1 and all main theorems.
domain assumption The social optimum Θ(x*) is positive and finite, so the efficiency ratio is well defined.
Asserted in Section II after Definition 2.3 but not implied by Assumption 2.1. This is the main formal gap; counterexamples put all resource on one player with positive gradient.
domain assumption For reserve price, at least one player remains with marginal utility above the reserve price.
Section VI.C explicitly excludes the case where the reserve price eliminates all players, which would yield the zero allocation and zero efficiency.

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Pith. "Pith review of The Efficiency of Generalized Nash and Variational Equilibria." pith.science (2026). https://pith.science/paper/ZIILJD7W

@misc{pith2026190800702,
  author       = {Pith},
  title        = {Pith review of: The Efficiency of Generalized Nash and Variational Equilibria},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZIILJD7W}},
  note         = {Machine review of arXiv:1908.00702}
}

abstract

Shared-constraint games are noncooperative $N$-player games where players are coupled through a common coupling constraint. It is known that such games admit two kinds of equilibria -- generalized Nash equilibria (GNE) and variational equilibria (VE) -- with two different economic interpretations. We consider such games in the context of resource allocation, where players move simultaneously to decide portions of the resource they can consume under a coupling constraint that the sum of the portions they demand be no more than the capacity of the resource. We clarify the worst case and best case efficiency of these kinds of equilibria over all games in a class. We find that the worst case efficiency of both solution concepts in zero and the best case efficiency is unity. Moreover, we characterize the subclass of games where all VE are efficient and show that even in this subclass but the worst case efficiency of GNE is zero. We finally discuss means by which zero worst case efficiency can be remedied.

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Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

Alpcan and T

T. Alpcan and T. Bas ¸ar. A game-theoretic framework for c ongestion control in general topology networks. In Proceedings of the 41st IEEE Conference on Decision and Control, 2002 , volume 2, pages 1218–1224, 2002

work page 2002
[2]

P . Dubey. Inefﬁciency of Nash equilibria. Mathematics of Operations Research , 11(1):1–8, February 1986

work page 1986
[3]

Engelbrecht-Wiggans

R. Engelbrecht-Wiggans. On optimal reservation prices in auctions. Management Science , 33(6):763–770, June 1987

work page 1987
[4]

Facchinei and C

F. Facchinei and C. Kanzow. Generalized Nash equilibriu m problems. 4OR: A Quarterly Journal of Operations Research , 5(3):173–210, 2007

work page 2007
[5]

Facchinei and J.-S

F. Facchinei and J.-S. Pang. Nash equilibria: The Variat ional Approach. In Convex Optimization in Signal Processing and Communication, chapter 12, pages 443–495. Cambridge University Press, Ca mbridge, 2009

work page 2009
[6]

R. Johari. Efﬁciency loss in market mechanisms for resource allocatio n. PhD thesis, Massachusetts Institute of Technology, 2004

work page 2004
[7]

Johari and J

R. Johari and J. N. Tsitsiklis. Efﬁciency loss in a newtor k resource allocation game. Mathematics of Operations Research , 29(3):407–435, 2004

work page 2004
[8]

F. P . Kelly, A. K. Maulloo, and D. K. H. Tan. Rate control fo r communication networks: Shadow prices, proportional fai rness and stability. The Journal of the Operational Research Society , 49(3):237–252, March 1998

work page 1998

Show all 32 references

[9]

F. Kelly. Charging and rate control for elastic trafﬁc. European Transactions on Telecommunications , 8(1):33–37, 1997

1997
[10]

A. A. Kulkarni and U. V . Shanbhag. On the variational equ ilibrium as a reﬁnement of the generalized Nash equilibrium . Automatica, 48(1):45–55, 2012

2012
[11]

A. A. Kulkarni and U. V . Shanbhag. Revisiting generaliz ed Nash games and variational inequalities. Journal of Optimization Theory and Applications , 154(1):1–12, 2012

2012
[12]

Kunniyur and R

S. Kunniyur and R. Srikant. End-to-end congestion cont rol schemes: Utility functions, random losses and ECN marks . Networking, IEEE/ACM Transactions on , 11(5):689–702, oct. 2003

2003
[13]

La and V

R. La and V . Anantharam. Charge-sensitive TCP and rate c ontrol in the internet. In Proceedings of IEEE INFOCOM. , volume 3, pages 1166–1175, mar 2000

2000
[14]

R. T. Maheswaran and T. Bas ¸ar. Efﬁcient signal proport ional allocation (ESPA) mechanisms: decentralized social welfare maximization for divisible resources. IEEE Journal on Selected Areas in Communications , 24(5):1000–1009, 2006

2006
[15]

R. T. Maheswaran and T. Bas ¸ar. Nash equilibrium and dec entralized negotiation in auctioning divisible resources . Group Decision and Negotiation, 12(5):361–395, 2003

2003
[16]

Mas-Colell

A. Mas-Colell. Noncooperative approaches to the theor y of perfect competition: Presentation. Journal of Economic Theory , 22(2):121–135, April 1980

1980
[17]

Mas-Colell, M

A. Mas-Colell, M. D. Whinston, and J. R. Green. Microeconomic Theory. Oxford University Press, USA, June 1995

1995
[18]

E. Maskin. Nash equilibrium and welfare optimality. The Review of Economic Studies , 66(1):23–38, January 1999

1999
[19]

Mo and J

J. Mo and J. Walrand. Fair end-to-end window-based cong estion control. IEEE/ACM Transactions on Networking , 8(5):556–567, 2000

2000
[20]

R. B. Myerson. Optimal auction design. Mathematics of Operations Research , 6(1):58–73, February 1981

1981
[21]

Narahari, D

Y . Narahari, D. Garg, R. Narayanam, and H. Prakash. Game Theoretic Problems in Network Economics and Mechanism Design Solutions. Springer, London, 1 edition, 2009

2009
[22]

Nisan, T

N. Nisan, T. Roughgarden, E. Tardos, and V . V . V azirani. Algorithmic Game Theory . Cambridge University Press, Cambridge, September 2007

2007
[23]

J. M. Ortega and W. C. Rheinboldt. Iterative Solution of Nonlinear Equations in Several V aria bles. Academic Press, New Y ork, January 1987

1987
[24]

Papadimitriou

C. Papadimitriou. Algorithms, games, and the internet . In Proceedings of the Thirty-third Annual ACM Symposium on The ory of Computing , pages 749–753, Hersonissos, Greece, 2001. ACM

2001
[25]

J. B. Rosen. Existence and uniqueness of equilibrium po ints for concave N -person games. Econometrica, 33(3):520–534, July 1965

1965
[26]

Roughgarden

T. Roughgarden. Selﬁsh Routing . PhD thesis, Cornell University, 2002

2002
[27]

Roughgarden and ´E

T. Roughgarden and ´E. Tardos. Bounding the inefﬁciency of equilibria in nonato mic congestion games. Games and Economic Behavior, 47(2):389–403, 2004

2004
[28]

M. E. Slade. What does an oligopoly maximize? The Journal of Industrial Economics , 42(1):45–61, 1994

1994
[29]

A. Smith. An Inquiry into the Nature and Causes of the W ealth of Nations . BiblioBazaar. Originally published in 1776, August 2008

2008
[30]

W. Vickrey. Counterspeculation, auctions, and compet itive sealed tenders. The Journal of Finance , 16(1):8–37, March 1961

1961
[31]

von Neumann and O

J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior . Princeton University Press, Princeton, 1944

1944
[32]

L. Walras. ´El´ ements d’´Economie Politique Pure: Ou, Th´ eorie de la richesse social e. originally published in 1874, 1952. DRAFT

1952

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alpcan and T

T. Alpcan and T. Bas ¸ar. A game-theoretic framework for c ongestion control in general topology networks. In Proceedings of the 41st IEEE Conference on Decision and Control, 2002 , volume 2, pages 1218–1224, 2002

work page 2002

[2] [2]

P . Dubey. Inefﬁciency of Nash equilibria. Mathematics of Operations Research , 11(1):1–8, February 1986

work page 1986

[3] [3]

Engelbrecht-Wiggans

R. Engelbrecht-Wiggans. On optimal reservation prices in auctions. Management Science , 33(6):763–770, June 1987

work page 1987

[4] [4]

Facchinei and C

F. Facchinei and C. Kanzow. Generalized Nash equilibriu m problems. 4OR: A Quarterly Journal of Operations Research , 5(3):173–210, 2007

work page 2007

[5] [5]

Facchinei and J.-S

F. Facchinei and J.-S. Pang. Nash equilibria: The Variat ional Approach. In Convex Optimization in Signal Processing and Communication, chapter 12, pages 443–495. Cambridge University Press, Ca mbridge, 2009

work page 2009

[6] [6]

R. Johari. Efﬁciency loss in market mechanisms for resource allocatio n. PhD thesis, Massachusetts Institute of Technology, 2004

work page 2004

[7] [7]

Johari and J

R. Johari and J. N. Tsitsiklis. Efﬁciency loss in a newtor k resource allocation game. Mathematics of Operations Research , 29(3):407–435, 2004

work page 2004

[8] [8]

F. P . Kelly, A. K. Maulloo, and D. K. H. Tan. Rate control fo r communication networks: Shadow prices, proportional fai rness and stability. The Journal of the Operational Research Society , 49(3):237–252, March 1998

work page 1998