Time bounds for iterative auctions: A unified approach by discrete convex analysis

Murota, Kazuo; Shioura, Akiyoshi; Yang, Zaifu

doi:10.1016/j.disopt.2016.01.001

Cited by 16 publications

(27 citation statements)

References 21 publications

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“…It is shown in Murota et al (2016) that the behavior of the algorithm ASCEND-MINIMAL is exactly the same as the algorithm VICKREYENGLISH in Section 2.2. In particular, the set S computed in Step 2 of each iteration is equal to the the maximal set in excess demand at payoff q.…”

Section: Algorithm Ascendminimalmentioning

confidence: 72%

“…It is known (Ausubel (2006), Murota et al (2016)) that the set of minimizers of the Lyapunov function L coincides with the set of equilibrium price vectors in the assignment game. Moreover, the integrality of values w(i, j) implies the existence of an integral minimizer of the Lyapunov function L; in particular, the unique minimal and maximal minimizers of the Lyapunov function L are integral vectors.…”

Section: A Appendix: Review Of Price Adjustment Processes For Assignmentioning

confidence: 99%

“…We finally present the so-called two-phase algorithms for finding an equilibrium (see Murota et al (2016); see also Murota (2016)). A two-phase algorithm consists of the ascending phase and the descending phase, and the price vector is increasing in the ascending phase and then decreasing in the descending phase.…”

Section: Partnership Formation Problemmentioning

confidence: 99%

“…The set H 0 ⊆ R N given by (3.5) can be rewritten as (2006); Murota et al (2016)). For the assignment game (A, B, F, w), the algorithm DESCENDMINIMAL (resp., DESCENDMAXI-MAL) outputs the unique minimal (resp., maximal) price vector in the set H 0 given by (2.19).…”

Section: Partnership Formation Problemmentioning

confidence: 99%

“…For example, the combination of ASCENDMINIMAL and DESCENDMAXIMAL yields the following algorithm (Murota (2016); Murota et al (2016)). …”

Section: Partnership Formation Problemmentioning

confidence: 99%

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On the partnership formation problem

Shioura¹

2017

jMID

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In this paper we re-examine the partnership formation problem, which is a generalization of the classical assignment game. We show that the former problem can be transformed into the latter one in some sense; more precisely, we demonstrate that using an equilibrium in an associated assignment game, we can find an equilibrium in the partnership formation problem if it exists. Based on this, we devise an algorithm to compute an equilibrium of the partnership formation problem, and show that the proposed algorithm can be seen as a variant of the one by Andersson et al. (2014b).

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Section: Algorithm Ascendminimalmentioning

confidence: 72%

Section: A Appendix: Review Of Price Adjustment Processes For Assignmentioning

confidence: 99%

Section: Partnership Formation Problemmentioning

confidence: 99%

Section: Partnership Formation Problemmentioning

confidence: 99%

“…For example, the combination of ASCENDMINIMAL and DESCENDMAXIMAL yields the following algorithm (Murota (2016); Murota et al (2016)). …”

Section: Partnership Formation Problemmentioning

confidence: 99%

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On the partnership formation problem

Shioura¹

2017

jMID

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Walrasian equilibria from an optimization perspective: A guide to the literature

Bichler

Fichtl

Schwarz

2020

Naval Research Logistics

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An ideal market mechanism allocates resources efficiently such that welfare is maximized and sets prices in a way so that the outcome is in a competitive equilibrium and no participant wants to deviate. An important part of the literature discusses Walrasian equilibria and conditions for their existence. We use duality theory to investigate existence of Walrasian equilibria and optimization algorithms to describe auction designs for different market environments in a consistent mathematical framework that allows us to classify the key contributions in the literature and open problems. We focus on auctions with indivisible goods and prove that the relaxed dual winner determination problem is equivalent to the minimization of the Lyapunov function. This allows us to describe central auction designs from the literature in the framework of primal-dual algorithms. We cover important properties for existence of Walrasian equilibria derived from discrete convex analysis, and provide open research questions.

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A flow‐based ascending auction to compute buyer‐optimal Walrasian prices

Eickhoff,

McCormick,

Peis

et al. 2024

Networks

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We consider a market where a set of objects is sold to a set of buyers, each equipped with a valuation function for the objects. The goal of the auctioneer is to determine reasonable prices together with a stable allocation. One definition of “reasonable” and “stable” is a Walrasian equilibrium, which is a tuple consisting of a price vector together with an allocation satisfying the following desirable properties: (i) the allocation is market‐clearing in the sense that as much as possible is sold, and (ii) the allocation is stable in the sense that every buyer ends up with an optimal set with respect to the given prices. Moreover, “buyer‐optimal” means that the prices are smallest possible among all Walrasian prices. In this paper, we present a combinatorial network flow algorithm to compute buyer‐optimal Walrasian prices in a multi‐unit matching market with truncated additive valuation functions. The algorithm can be seen as a generalization of the classical housing market auction and mimics the very natural procedure of an ascending auction. We use our structural insights to prove monotonicity of the buyer‐optimal Walrasian prices with respect to changes in supply or demand.

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Time bounds for iterative auctions: A unified approach by discrete convex analysis

Cited by 16 publications

References 21 publications

On the partnership formation problem

On the partnership formation problem

Walrasian equilibria from an optimization perspective: A guide to the literature

A flow‐based ascending auction to compute buyer‐optimal Walrasian prices

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