On randomized matching mechanisms

Ma, Jinpeng

doi:10.1007/bf01211824

Cited by 42 publications

(36 citation statements)

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“…By means of an example he showed that the random order mechanism does not always reach all stable matchings. Although Ma's (1996) result is true, we show that the probability distribution he presented -and therefore the proof of his Claim 2 -is not correct. The mistake in the calculations by Ma (1996) is due to the fact that even though the example looks very symmetric, some of the calculations are not as "symmetric."…”

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confidence: 72%

Corrigendum to “On randomized matching mechanisms” [Economic Theory 8(1996)377–381]

Klaus

Klijn²

2006

Economic Theory

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Summary: Ma (1996) studied the random order mechanism, a matching mechanism suggested by Roth and Vande Vate (1990) for marriage markets. By means of an example he showed that the random order mechanism does not always reach all stable matchings. Although Ma's (1996) result is true, we show that the probability distribution he presented -and therefore the proof of his Claim 2 -is not correct. The mistake in the calculations by Ma (1996) is due to the fact that even though the example looks very symmetric, some of the calculations are not as "symmetric." JEL classification: C78 For a description of the marriage model we refer to Roth and Vande Vate (1990). A marriage market is denoted by (M, W, P ) where M = {m 1 , . . . , m a } is a set of "men," W = {w 1 , . . . , w a } is a set of "women," and P is a preference profile. The set of stable matchings for (M, W, P ) is denoted by S(P ). We now recall the random order mechanism. Random Order (RO) MechanismInput: A marriage market (M, W, P ). Set R 0 := ∅, µ 0 such that for all i ∈ N , µ 0 (i) = i, and t := 1.Step t: Choose an agent i t from N \R t−1 at random. Set R t := R t−1 ∪ {i t }. Suppose i t = w ∈ W . (Otherwise replace w by m in Step t.) Stable Room ProcedureCase (i) There exists no blocking pair (m, w) for µ t−1 with m ∈ R t : Stop if t = n and define RO(P ) := µ t−1 . Otherwise set µ t = µ t−1 and go to Step t := t + 1. Case (ii)There exists a blocking pair (m, w) for µ t−1 with m ∈ R t : Choose the blocking pair (m * , w) for µ t−1 with m * ∈ R t that w prefers most. If µ t−1 (m * ) = m * , then define µ t such that µ t (w) := m * , µ t (m * ) := w, and for all i ∈ N \{w, m * }, µ t (i) := µ t−1 (i). Stop if t = n and define RO(P ) := µ t . Otherwise go to Step t := t + 1. If µ t−1 (m * ) = w ∈ W , then redefine µ t−1 (w) := m * , µ t−1 (m * ) := w, µ t−1 (w ) := w , and for all i ∈ N \{w, m * , w }, µ t−1 (i) := µ t−1 (i). Set w := w , and repeat the Stable Room Procedure. * We thank two anonymous referees for their helpful comments. B.

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confidence: 72%

Corrigendum to “On randomized matching mechanisms” [Economic Theory 8(1996)377–381]

Klaus

Klijn²

2006

Economic Theory

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“…Random serial dictatorship is a classic mechanism for one-sided domains that obtains a Pareto-optimal allocation by iterative arrival of agents in a uniform random order [1,6]. In two-sided instances, Ma [25] proposed the random-order mechanism, where agents arrive in uniform random order and blocking pairs are resolved in a best-response manner. This mechanism is known to arrive at stable matchings that are neither man-nor woman-optimal and has interesting structural and computational properties [13,14,25].…”

Section: Related Workmentioning

confidence: 99%

“…In two-sided instances, Ma [25] proposed the random-order mechanism, where agents arrive in uniform random order and blocking pairs are resolved in a best-response manner. This mechanism is known to arrive at stable matchings that are neither man-nor woman-optimal and has interesting structural and computational properties [13,14,25]. In terms of fully dynamic populations, Blum et al [13] study resolution chains of blocking pairs when in each round an arbitrary agent arrives or leaves.…”

Section: Related Workmentioning

confidence: 99%

Maintaining Near-Popular Matchings

Bhattacharya

Hoefer

Huang

et al. 2015

Lecture Notes in Computer Science

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We study dynamic matching problems in graphs among agents with preferences. Agents and/or edges of the graph arrive and depart iteratively over time. The goal is to maintain matchings that are favorable to the agent population and stable over time. More formally, we strive to keep nearpopular matchings with a small unpopularity factor by making only a small amortized number of changes to the matching per round. Our main result is an algorithm to maintain matchings with unpopularity factor (∆ + k) by making an amortized number of O(∆ + ∆ 2 /k) changes per round, for any k > 0. Here ∆ denotes the maximum degree of any agent in any round. We complement this result by a variety of lower bounds indicating that matchings with smaller factor do not exist or cannot be maintained using our algorithm.As a byproduct, we obtain several additional results of independent interest. First, our algorithm implies existence of matchings with small unpopularity factors in graphs with bounded degree. Second, given any matching M and any value α ≥ 1, we provide an efficient algorithm to compute a matching M with unpopularity factor α over M if it exists. Finally, our results show the absence of voting paths in two-sided instances, even if we restrict to sequences of matchings with larger unpopularity factors (below ∆).

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“…By assuming that each order is equally likely, we may calculate the probability of each stable matching being obtained. Ma [25] carried out this calculation for an instance, originally suggested by Knuth [21], and observed that not all stable matchings can be reached by this mechanism and there is a higher probability of reaching some stable matchings over others (although his calculation was not entirely correct as [19] pointed out). 4 In this paper we will also study this instance (Example 2 in Section 3) with respect to a different stochastic process.…”

Section: Introductionmentioning

confidence: 99%

Analysis of stochastic matching markets

Bíró

Norman

2012

Int J Game Theory

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Abstract. Suppose that the agents of a matching market contact each other randomly and form new pairs if is in their interest. Does such a process always converge to a stable matching if one exists? If so, how quickly? Are some stable matchings more likely to be obtained by this process than others? In this paper we are going to provide answers to these and similar questions, posed by economists and computer scientists. In the first part of the paper we give an alternative proof for the theorems by Diamantoudi et al. and Inarra et al. which imply that the corresponding stochastic processes are absorbing Markov chains. Our proof is not only shorter, but also provides upper bounds for the number of steps needed to stabilise the system. The second part of the paper proposes new techniques to analyse the behaviour of matching markets. We introduce the Stable Marriage and Stable Roommates Automaton and show how the probabilistic model checking tool PRISM may be used to predict the outcomes of stochastic interactions between myopic agents.In particular, we demonstrate how one can calculate the probabilities of reaching different matchings in a decentralised market and determine the expected convergence time of the stochastic process concerned. We illustrate the usage of this technique by studying some well-known marriage and roommates instances and randomly generated instances.

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On randomized matching mechanisms

Cited by 42 publications

References 3 publications

Corrigendum to “On randomized matching mechanisms” [Economic Theory 8(1996)377–381]

Corrigendum to “On randomized matching mechanisms” [Economic Theory 8(1996)377–381]

Maintaining Near-Popular Matchings

Analysis of stochastic matching markets

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