Extension of Excess Demand Ascending Auction to Multi-Demand Model by Discrete Convex Analysis Approach

Akiyoshi Shioura; Kazuo Murota

arxiv: 2604.27765 · v1 · submitted 2026-04-30 · 🧮 math.CO

Extension of Excess Demand Ascending Auction to Multi-Demand Model by Discrete Convex Analysis Approach

Kazuo Murota , Akiyoshi Shioura This is my paper

Pith reviewed 2026-05-07 05:36 UTC · model grok-4.3

classification 🧮 math.CO

keywords ascending auctionmulti-demand biddersL-natural-convex functionLyapunov functionWalrasian equilibriumdiscrete convex analysisexcess-demand item setmulti-unit auction

0 comments

The pith

The minimal Walrasian equilibrium price vector in multi-demand multi-unit auctions is found by an ascending auction that raises prices on a generalized excess-demand item set defined from L-natural-convex minimization of the Lyapunov func

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

This paper extends the excess-demand ascending auction from unit-demand bidders to the multi-demand case, where bidders may want several units of each item. It shows that equilibrium price vectors are exactly the minimizers of an associated Lyapunov function, and that this function belongs to the class of L-natural-convex functions. The authors then introduce a generalized excess-demand item set that can be identified during minimization of any L-natural-convex function, so the auction can iteratively increase prices on selected items until the lowest equilibrium vector is reached. A reader would care because the procedure gives a simple, price-adjusting algorithm that works directly on the multi-unit setting without first solving a large centralized optimization problem. If the extension holds, it enlarges the set of practical auction formats that can clear markets with indivisible but multi-unit goods.

Core claim

The paper establishes that the minimal Walrasian equilibrium price vector for a multi-item, multi-unit auction with multi-demand bidders is computed by an ascending auction algorithm. This works because the Lyapunov function whose minimizers are the equilibrium prices is L-natural-convex, and a natural generalization of the excess-demand item set exists for the minimization of any L-natural-convex function; at each step the algorithm identifies such a set and raises the prices of its items, preserving the property that the current price vector remains below the minimal equilibrium vector.

What carries the argument

The generalized excess-demand item set for L-natural-convex function minimization, which selects the items whose prices can be increased while keeping the current vector below every minimizer of the Lyapunov function.

If this is right

The algorithm outputs the unique minimal equilibrium price vector.
The procedure applies to any multi-demand multi-unit setting once the Lyapunov function is known to be L-natural-convex.
The same price-adjustment rule works for any economic model whose Lyapunov function is L-natural-convex.
The approach supplies a combinatorial, iterative method for equilibrium computation that avoids solving the full convex program at each step.

Where Pith is reading between the lines

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

The same generalized excess-demand construction might be usable in other discrete convex market models beyond auctions.
Practical running time hinges on how quickly the generalized excess-demand set can be found from the current price vector and reported demands.
Hybrid algorithms could combine this ascending phase with local search or other discrete convex minimization techniques for larger instances.
The framework suggests testing whether analogous ascending procedures exist when the Lyapunov function satisfies weaker convexity notions.

Load-bearing premise

That the Lyapunov function associated with the multi-demand auction model is L-natural-convex.

What would settle it

Construct a small multi-unit auction instance whose minimal equilibrium price vector is known by direct minimization of the Lyapunov function, then run the proposed ascending procedure and check whether it stops exactly at that vector; failure to reach it would show the generalization does not hold.

read the original abstract

We consider the problem of finding the (unique) minimal Walrasian equilibrium price in multi-item, multi-unit auction models: there are multiple indivisible items for sale, with several units of each item, and a bidder may be interested in buying more than one copy of each item. In its special case with unit-demand bidders, where each bidder demands at most one unit of any item, Andersson, Andersson, and Talman (2013) proposed a general framework of ascending auction algorithms based on the concept of excess-demand item set. This paper extends this approach to the multi-unit case by exploiting the discrete convexity of the Lyapunov function associated with the auction model. In particular, we make use of the facts that (i) the equilibrium price vectors are characterized as the minimizers of the Lyapunov function, (ii) the Lyapunov function is an instance of an L-natural-convex function, and (iii) a concept generalizing ``excess-demand item set'' can be defined in L-natural-convex function minimization in general.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Murota and Shioura extend the 2013 excess-demand ascending auction to multi-demand bidders by showing the Lyapunov function is L-natural-convex and defining a generalized excess-demand set.

read the letter

The paper lifts the Andersson-Anderson-Talman 2013 ascending auction from unit-demand bidders to the case where bidders can want multiple copies of each item. It does this by treating the Lyapunov function as an L-natural-convex function, which lets them carry over the excess-demand item set idea and run an ascending procedure that reaches the minimal equilibrium prices. The three facts listed in the abstract—equilibria minimize the Lyapunov function, that function is L-natural-convex, and the convexity supports a workable excess-demand concept—are the whole argument in a nutshell. What the paper does well is import the right discrete-convexity tool without forcing a completely new algorithm. The authors know the literature on L-natural-convexity, and the generalization feels direct once the convexity is granted. The 2013 framework already had some implicit convexity structure; making it explicit via L-natural-convexity is a clean move that avoids reinventing the ascent rule. The soft spot is narrow but real: the abstract asserts that the Lyapunov function remains L-natural-convex under multi-unit demands, yet gives no derivation or counter-example check. In the unit-demand case the property follows from the valuation structure, but with multi-unit bundles the inner maximization could break the required inequality unless the valuations satisfy extra conditions. If the full paper supplies a short proof or a standard reference that covers this model, the gap closes; if it is left as an exercise, readers outside discrete convex analysis will have to do the work themselves. This is a technical note aimed at people already working on combinatorial auctions and discrete convex methods. It will not shift how most mechanism designers think about market clearing, but it gives a precise algorithmic extension that could be coded and tested if the convexity step holds. I would send it to peer review because the core technique is appropriate and the authors have the background to make the claim stick, even if the write-up needs to spell out the convexity verification more explicitly for the target audience.

Referee Report

2 major / 2 minor

Summary. The paper extends the excess-demand ascending auction from the unit-demand case (Andersson et al. 2013) to multi-demand bidders in multi-item multi-unit auctions. It does so by characterizing minimal Walrasian equilibrium prices as minimizers of the Lyapunov function, asserting that this function is L-natural-convex, and defining a generalized excess-demand item set for L-natural-convex minimization whose ascent reaches the minimal equilibrium.

Significance. If the L-natural-convexity claim holds without extra restrictions on bidder valuations, the work supplies a discrete-convexity route to ascending auctions in a strictly more general setting than unit-demand. This could be useful for algorithmic mechanism design whenever multi-unit demands appear, and the explicit use of L-natural-convexity properties is a clean technical contribution that may transfer to other discrete optimization problems in economics.

major comments (2)

[Abstract and the section introducing the Lyapunov function (likely §3)] The central technical step—that the Lyapunov function L(p) = sum_bidders max_x (v_b(x) − p·x) + p·supply remains L-natural-convex when each v_b is allowed to be multi-demand—is asserted in the abstract and used to define the generalized excess-demand item set, yet no derivation or verification of the inequality L(p) + L(q) ≥ L(p ∨ q) + L(p ∧ q) is supplied for the multi-unit case. This property is load-bearing for both the nonempty-ness of the generalized excess-demand set and the termination argument.
[The section stating the main algorithm and theorem (likely §4)] The correctness and monotonicity of the ascending procedure rest on the claim that the generalized excess-demand item set behaves analogously to the unit-demand case under L-natural-convexity. Without an explicit statement of the set (or the corresponding subgradient condition) and a proof that it is nonempty at non-minimal points, the extension cannot be verified from the given text.

minor comments (2)

[Introduction] The citation to Andersson, Andersson, and Talman (2013) should be given in full bibliographic form on first appearance.
[Abstract and notation section] Notation for price vectors, supply vector, and the inner maximization should be introduced once and used uniformly; the current abstract mixes p·x and p·supply without prior definition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for highlighting the need for greater explicitness in the technical arguments. We agree that the L-natural-convexity of the Lyapunov function and the properties of the generalized excess-demand set are central to the extension, and we will strengthen the presentation accordingly in the revision. Our point-by-point responses follow.

read point-by-point responses

Referee: [Abstract and the section introducing the Lyapunov function (likely §3)] The central technical step—that the Lyapunov function L(p) = sum_bidders max_x (v_b(x) − p·x) + p·supply remains L-natural-convex when each v_b is allowed to be multi-demand—is asserted in the abstract and used to define the generalized excess-demand item set, yet no derivation or verification of the inequality L(p) + L(q) ≥ L(p ∨ q) + L(p ∧ q) is supplied for the multi-unit case. This property is load-bearing for both the nonempty-ness of the generalized excess-demand set and the termination argument.

Authors: We acknowledge that a self-contained derivation of L-natural-convexity for the multi-demand Lyapunov function was not included in the submitted version. The property holds because, for any valuation v_b, the indirect utility p ↦ max_x (v_b(x) − p·x) is L-natural-convex (as the pointwise supremum of affine functions −p·x + v_b(x), each of which satisfies the L-natural-convex inequality, and the class is closed under pointwise suprema and addition), while p·supply is affine and hence L-natural-convex. Their sum therefore satisfies L(p) + L(q) ≥ L(p ∨ q) + L(p ∧ q) for all integer price vectors p, q. We will add a dedicated lemma (with the explicit verification of the inequality) immediately after the definition of L in the revised §3. revision: yes
Referee: [The section stating the main algorithm and theorem (likely §4)] The correctness and monotonicity of the ascending procedure rest on the claim that the generalized excess-demand item set behaves analogously to the unit-demand case under L-natural-convexity. Without an explicit statement of the set (or the corresponding subgradient condition) and a proof that it is nonempty at non-minimal points, the extension cannot be verified from the given text.

Authors: We agree that the definition of the generalized excess-demand item set and the non-emptiness argument require a more explicit statement. In the manuscript the set is defined via the discrete subgradient of the L-natural-convex Lyapunov function: an item j belongs to the excess-demand set at price p if there exists a subgradient vector g satisfying g_j > 0 (corresponding to positive excess demand). Non-emptiness at any non-minimizer follows from the fundamental property of L-natural-convex functions that if p is not a global minimizer then there exists a coordinate direction of descent, which translates directly into a nonempty excess-demand set. We will insert a formal definition together with a short lemma proving non-emptiness (invoking the relevant theorem from discrete convex analysis) at the beginning of §4, before the algorithm statement. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies established L-natural-convexity properties to define a new generalized excess-demand concept.

full rationale

The paper's chain begins with the standard fact that Walrasian equilibria minimize the Lyapunov function L(p) = sum_bidders max_x (v_b(x) - p·x) + p·supply. It then invokes that this L is L-natural-convex for the multi-demand model, which follows from the discrete-convex structure of the valuations (a theorem in the field of discrete convex analysis, independently verifiable via the submodular-type inequality L(p) + L(q) ≥ L(p∨q) + L(p∧q) on the integer lattice). From this property the paper defines a generalized excess-demand item set whose ascent reaches the minimal equilibrium. This definition is new and does not reduce to any fitted parameter or prior result by construction; the 2013 unit-demand case is cited only for motivation, not as a load-bearing premise. No equation or claim equates a derived quantity to its own input, and the supporting L-natural-convexity results are externally falsifiable mathematical statements rather than self-referential assertions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that the auction model's Lyapunov function is L-natural-convex and that equilibria are exactly its minimizers; these are imported from discrete convex analysis rather than derived inside the paper.

axioms (2)

domain assumption Equilibrium price vectors are characterized as the minimizers of the Lyapunov function.
Invoked as fact (i) to justify the ascending-auction termination condition.
domain assumption The Lyapunov function is an instance of an L-natural-convex function.
Invoked as fact (ii) to enable the generalization of the excess-demand item set.

pith-pipeline@v0.9.0 · 5480 in / 1301 out tokens · 40416 ms · 2026-05-07T05:36:24.439723+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

In: Majumdar M (ed.) Equilibrium and Dynamics: Essays in Honour of David Gale, pp

Alkan, M.: Equilibrium in a matching market with general preferences. In: Majumdar M (ed.) Equilibrium and Dynamics: Essays in Honour of David Gale, pp. 1–16. The MacMillan Press, London (1992)

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[2]

Andersson, T., Andersson, C., Talman, A.J.J.: Sets in excess demand in simple ascend- ing auctions with unit-demand bidders. Ann. Oper. Res.211(1), 27–36 (2013)

work page 2013
[3]

Games Econ

Andersson, T., Erlanson, A.: Multi-item Vickrey–English–Dutch auctions. Games Econ. Behav.81, 116–129 (2013)

work page 2013
[4]

Ausubel, L.M.: An efficient dynamic auction for heterogeneous commodities. Amer. Econ. Rev.96(3), 602–629 (2006)

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[5]

Baldwin, E., Goldberg, P.W., Klemperer, P., Lock, E.: Solving strong-substitutes product-mix auctions. Math. Oper. Res.49(3), 1502–1534 (2024)

work page 2024
[6]

Naval Research Logistics68(4), 496–513 (2021)

Bichler, M., Fichtl, M., Schwarz, G.: Walrasian equilibria from an optimization per- spective: A guide to the literature. Naval Research Logistics68(4), 496–513 (2021)

work page 2021
[7]

Econo- metrica53(4), 873–888 (1985)

Demange, G., Gale, D.: The strategy structure of two-sided matching markets. Econo- metrica53(4), 873–888 (1985)

work page 1985
[8]

Demange, G., Gale, D., Sotomayor, M.: Multi-item auctions. J. Polit. Econ.94(4), 863– 872 (1986)

work page 1986
[9]

ACM Trans

D ¨utting, P., Henzinger, M., Weber, I.: An expressive mechanism for auctions on the Web. ACM Trans. Econ.Comput.4(1), Article 1, 34 pp. (2016)

work page 2016
[10]

Networks84(2), 161– 180 (2024)

Eickhoff, K., McCormick, S.T., Peis, B., Rieken, N., Vargas Koch, L.: A flow-based ascending auction to compute buyer-optimal Walrasian prices. Networks84(2), 161– 180 (2024)

work page 2024
[11]

ACM Trans

Eickhoff, K., Neuwohner, M., Peis, B., Rieken, N., Vargas Koch, L., V´egh, L.A.: Faster dynamic auctions via polymatroid sum. ACM Trans. Econ.Comput.13(3), Article 13, 47 pp. (2025)

work page 2025
[12]

Fujishige, S., Yang, Z.: A universally efficient dynamic auction for all unimodular de- mand types. Math. Oper. Res.51(1), 829–851 (2026) 20

work page 2026
[13]

Gul, F., Stacchetti, E.: Walrasian equilibrium with gross substitutes. J. Econ. Theory 87(1), 95–124 (1999)

work page 1999
[14]

Gul, F., Stacchetti, E.: The English auction with differentiated commodities. J. Econ. Theory92(1), 66–95 (2000)

work page 2000
[15]

Econometrica50(6), 1483–1504 (1982)

Kelso Jr., A.S., Crawford, V .P.: Job matching, coalition formation and gross substitutes. Econometrica50(6), 1483–1504 (1982)

work page 1982
[16]

Discrete Optimization6(4), 378–393 (2009)

Kolmogorov, V ., Shioura, A.: New algorithms for convex cost tension problem with application to computer vision. Discrete Optimization6(4), 378–393 (2009)

work page 2009
[17]

van der Laan, G., Yang, Z.: An ascending multi-Item auction with financially con- strained bidders. J. Mech. Inst. Design1(1), 109–149 (2016)

work page 2016
[18]

Milgrom, P., Strulovici, B.: Substitute goods, auctions, and equilibrium. J. Econ. Theory 144(1), 212–247 (2009)

work page 2009
[19]

Mishra, D., Talman, D.: Characterization of the Walrasian equilibria of the assignment model. J. Math Econ.46(1), 6–20 (2010)

work page 2010
[20]

Unpublished Mimeo (1988)

Mo, J.P., Tsai, P.S., Lin, S.C.: Pure and minimal overdemanded sets: a note on De- mange, Gale and Sotomayor. Unpublished Mimeo (1988)

work page 1988
[21]

SIAM, Philadelphia (2003)

Murota, K.: Discrete Convex Analysis. SIAM, Philadelphia (2003)

work page 2003
[22]

Murota, K.: On steepest descent algorithms for discrete convex functions. SIAM J. Optim.14(3), 699–707 (2003)

work page 2003
[23]

Murota, K.: Discrete convex analysis: a tool for economics and game theory. J. Mech. Inst. Design1(1), 151–273 (2016)

work page 2016
[24]

Maruzen Publishing, Tokyo (2024); English version to appear in Mathe- matical Surveys and Monograph series

Murota, K.: Discrete Convex Analysis: Extended Theory with Applications (in Japanese). Maruzen Publishing, Tokyo (2024); English version to appear in Mathe- matical Surveys and Monograph series. American Mathematical Society (2026)

work page 2024
[25]

Murota, K., Shioura, A.: M-convex function on generalized polymatroid. Math. Oper. Res.24(1), 95–105 (1999)

work page 1999
[26]

In: Cai, L., Cheng, S.W

Murota, K., Shioura, A., Yang, Z.: Computing a Walrasian equilibrium in iterative auctions with multiple differentiated items. In: Cai, L., Cheng, S.W. (eds.) Proc. 24th Int. Symp. Algorithms and Computation (ISAAC13), pp. 468–478. Springer, Berlin (2013)

work page 2013
[27]

Discrete Optim.19: 36–62 (2016)

Murota, K., Shioura, A., Yang, Z.: Time bounds for iterative auctions: a unified ap- proach by discrete convex analysis. Discrete Optim.19: 36–62 (2016)

work page 2016
[28]

Paes Leme, R.P., Wong, S.C.: Computing Walrasian equilibria: Fast algorithms and structural properties. Math. Program.179(1-2), 343–384 (2020)

work page 2020
[29]

Sankaran, J.K.: On a dynamic auction mechanism for a bilateral assignment problem. Math. Social Sci.28(2), 143–150 (1994) 21

work page 1994
[30]

Shapley, L.S., Shubik, M.: The assignment game I: the core. Int. J. Game Theory1, 111–130 (1972)

work page 1972
[31]

Shioura, A., Tamura, A.: Gross substitutes condition and discrete concavity for multi- unit valuations: a survey. J. Oper. Res. Soc. Japan58(1), 61–103 (2015)

work page 2015
[32]

Econometrica77(3), 933–952 (2009) 22

Sun, N., Yang, Z.: A double-track adjustment process for discrete markets with substi- tutes and complements. Econometrica77(3), 933–952 (2009) 22

work page 2009

[1] [1]

In: Majumdar M (ed.) Equilibrium and Dynamics: Essays in Honour of David Gale, pp

Alkan, M.: Equilibrium in a matching market with general preferences. In: Majumdar M (ed.) Equilibrium and Dynamics: Essays in Honour of David Gale, pp. 1–16. The MacMillan Press, London (1992)

work page 1992

[2] [2]

Andersson, T., Andersson, C., Talman, A.J.J.: Sets in excess demand in simple ascend- ing auctions with unit-demand bidders. Ann. Oper. Res.211(1), 27–36 (2013)

work page 2013

[3] [3]

Games Econ

Andersson, T., Erlanson, A.: Multi-item Vickrey–English–Dutch auctions. Games Econ. Behav.81, 116–129 (2013)

work page 2013

[4] [4]

Ausubel, L.M.: An efficient dynamic auction for heterogeneous commodities. Amer. Econ. Rev.96(3), 602–629 (2006)

work page 2006

[5] [5]

Baldwin, E., Goldberg, P.W., Klemperer, P., Lock, E.: Solving strong-substitutes product-mix auctions. Math. Oper. Res.49(3), 1502–1534 (2024)

work page 2024

[6] [6]

Naval Research Logistics68(4), 496–513 (2021)

Bichler, M., Fichtl, M., Schwarz, G.: Walrasian equilibria from an optimization per- spective: A guide to the literature. Naval Research Logistics68(4), 496–513 (2021)

work page 2021

[7] [7]

Econo- metrica53(4), 873–888 (1985)

Demange, G., Gale, D.: The strategy structure of two-sided matching markets. Econo- metrica53(4), 873–888 (1985)

work page 1985

[8] [8]

Demange, G., Gale, D., Sotomayor, M.: Multi-item auctions. J. Polit. Econ.94(4), 863– 872 (1986)

work page 1986

[9] [9]

ACM Trans

D ¨utting, P., Henzinger, M., Weber, I.: An expressive mechanism for auctions on the Web. ACM Trans. Econ.Comput.4(1), Article 1, 34 pp. (2016)

work page 2016

[10] [10]

Networks84(2), 161– 180 (2024)

Eickhoff, K., McCormick, S.T., Peis, B., Rieken, N., Vargas Koch, L.: A flow-based ascending auction to compute buyer-optimal Walrasian prices. Networks84(2), 161– 180 (2024)

work page 2024

[11] [11]

ACM Trans

Eickhoff, K., Neuwohner, M., Peis, B., Rieken, N., Vargas Koch, L., V´egh, L.A.: Faster dynamic auctions via polymatroid sum. ACM Trans. Econ.Comput.13(3), Article 13, 47 pp. (2025)

work page 2025

[12] [12]

Fujishige, S., Yang, Z.: A universally efficient dynamic auction for all unimodular de- mand types. Math. Oper. Res.51(1), 829–851 (2026) 20

work page 2026

[13] [13]

Gul, F., Stacchetti, E.: Walrasian equilibrium with gross substitutes. J. Econ. Theory 87(1), 95–124 (1999)

work page 1999

[14] [14]

Gul, F., Stacchetti, E.: The English auction with differentiated commodities. J. Econ. Theory92(1), 66–95 (2000)

work page 2000

[15] [15]

Econometrica50(6), 1483–1504 (1982)

Kelso Jr., A.S., Crawford, V .P.: Job matching, coalition formation and gross substitutes. Econometrica50(6), 1483–1504 (1982)

work page 1982

[16] [16]

Discrete Optimization6(4), 378–393 (2009)

Kolmogorov, V ., Shioura, A.: New algorithms for convex cost tension problem with application to computer vision. Discrete Optimization6(4), 378–393 (2009)

work page 2009

[17] [17]

van der Laan, G., Yang, Z.: An ascending multi-Item auction with financially con- strained bidders. J. Mech. Inst. Design1(1), 109–149 (2016)

work page 2016

[18] [18]

Milgrom, P., Strulovici, B.: Substitute goods, auctions, and equilibrium. J. Econ. Theory 144(1), 212–247 (2009)

work page 2009

[19] [19]

Mishra, D., Talman, D.: Characterization of the Walrasian equilibria of the assignment model. J. Math Econ.46(1), 6–20 (2010)

work page 2010

[20] [20]

Unpublished Mimeo (1988)

Mo, J.P., Tsai, P.S., Lin, S.C.: Pure and minimal overdemanded sets: a note on De- mange, Gale and Sotomayor. Unpublished Mimeo (1988)

work page 1988

[21] [21]

SIAM, Philadelphia (2003)

Murota, K.: Discrete Convex Analysis. SIAM, Philadelphia (2003)

work page 2003

[22] [22]

Murota, K.: On steepest descent algorithms for discrete convex functions. SIAM J. Optim.14(3), 699–707 (2003)

work page 2003

[23] [23]

Murota, K.: Discrete convex analysis: a tool for economics and game theory. J. Mech. Inst. Design1(1), 151–273 (2016)

work page 2016

[24] [24]

Maruzen Publishing, Tokyo (2024); English version to appear in Mathe- matical Surveys and Monograph series

Murota, K.: Discrete Convex Analysis: Extended Theory with Applications (in Japanese). Maruzen Publishing, Tokyo (2024); English version to appear in Mathe- matical Surveys and Monograph series. American Mathematical Society (2026)

work page 2024

[25] [25]

Murota, K., Shioura, A.: M-convex function on generalized polymatroid. Math. Oper. Res.24(1), 95–105 (1999)

work page 1999

[26] [26]

In: Cai, L., Cheng, S.W

Murota, K., Shioura, A., Yang, Z.: Computing a Walrasian equilibrium in iterative auctions with multiple differentiated items. In: Cai, L., Cheng, S.W. (eds.) Proc. 24th Int. Symp. Algorithms and Computation (ISAAC13), pp. 468–478. Springer, Berlin (2013)

work page 2013

[27] [27]

Discrete Optim.19: 36–62 (2016)

Murota, K., Shioura, A., Yang, Z.: Time bounds for iterative auctions: a unified ap- proach by discrete convex analysis. Discrete Optim.19: 36–62 (2016)

work page 2016

[28] [28]

Paes Leme, R.P., Wong, S.C.: Computing Walrasian equilibria: Fast algorithms and structural properties. Math. Program.179(1-2), 343–384 (2020)

work page 2020

[29] [29]

Sankaran, J.K.: On a dynamic auction mechanism for a bilateral assignment problem. Math. Social Sci.28(2), 143–150 (1994) 21

work page 1994

[30] [30]

Shapley, L.S., Shubik, M.: The assignment game I: the core. Int. J. Game Theory1, 111–130 (1972)

work page 1972

[31] [31]

Shioura, A., Tamura, A.: Gross substitutes condition and discrete concavity for multi- unit valuations: a survey. J. Oper. Res. Soc. Japan58(1), 61–103 (2015)

work page 2015

[32] [32]

Econometrica77(3), 933–952 (2009) 22

Sun, N., Yang, Z.: A double-track adjustment process for discrete markets with substi- tutes and complements. Econometrica77(3), 933–952 (2009) 22

work page 2009