Stochastic stability for roommate markets

Klaus, Bettina; Klijn, Flip; Walzl, Markus

doi:10.1016/j.jet.2010.07.006

Cited by 50 publications

(46 citation statements)

References 18 publications

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“…Constructing this instance in PRISM, we find there are 308 matchings and a single ergodic set which consists of the matchings {M 4 , M 5 , M 6 , M 7 }, where M 7 = {(1, 2), (3,8), (4,6), (5, 7)}. This corresponds to the results presented in [20].…”

Section: Analysing the Market Behavior With Automatasupporting

confidence: 67%

“…However, in this case the existence of a stable solution does not guarantee that there is a convergence to a stable solution when starting from any unstable state, as illustrated by Klaus et al [20]. So in this case absorbing states and ergodic sets may appear together in the Markov chain.…”

Section: Further Remarksmentioning

confidence: 99%

“…they form a new pair if they both would be better off). This is called the better response dynamics by Ackermann et al [1]; whilst, Klaus et al [20] refer to it as unperturbed blocking dynamics. What is the expected outcome of a matching market with myopic agents?…”

Section: Analysing the Market Behavior With Automatamentioning

confidence: 99%

“…This implies that any ergodic set (which is called absorbing set in [14] and [20]) of the corresponding Markov chain must contain some of the above matchings. Constructing this instance in PRISM, we find there are 308 matchings and a single ergodic set which consists of the matchings {M 4 , M 5 , M 6 , M 7 }, where M 7 = {(1, 2), (3,8), (4,6), (5, 7)}.…”

Section: Analysing the Market Behavior With Automatamentioning

confidence: 99%

“…[17]), and a matching M is in an ergodic set if and only if the limit probability of M , starting from the empty matching, is positive. Moreover, Klaus et al [20] prove that the absorbing sets consist of exactly those matchings that have positive probabilities in the limit distribution of a stochastic process where, starting from any matching, the agents make mistakes with small probabilities in their myopic blocking decisions. They called this process perturbed blocking dynamics.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Analysing the Market Behavior With Automatasupporting

confidence: 67%

Section: Further Remarksmentioning

confidence: 99%

Section: Analysing the Market Behavior With Automatamentioning

confidence: 99%

Section: Analysing the Market Behavior With Automatamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analysis of stochastic matching markets

Bíró

Norman

2012

Int J Game Theory

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Stochastic Coalitional Better-Response Dynamics for Finite Games with Application to Network Formation Games

Avrachenkov¹,

Singh

2019

Multilevel Strategic Interaction Game Models for Complex Networks

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We consider a coalition formation among players, in an n-player strategic game, over infinite horizon. At each time a randomly selected coalition makes a joint deviation, from a current action profile to a new action profile, which is strictly beneficial for all the players belonging to the coalition. Such deviations define a stochastic coalitional better-response (CBR) dynamics. The stochastic CBR dynamics either converges to a K-stable equilibrium or becomes stuck in a closed cycle. We also assume that at each time a selected coalition makes mistake in deviation with small probability. We prove that all K-stable equilibria and all action profiles from closed cycles, having minimum stochastic potential, are stochastically stable. Similar statement holds for strict K-stable equilibrium. We apply the stochastic CBR dynamics to the network formation games. We show that all strongly stable networks and closed cycles of networks are stochastically stable.

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Matching Dynamics with Constraints

Hoefer

Wagner

2014

Web and Internet Economics

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We study uncoordinated matching markets with additional local constraints that capture, e.g., restricted information, visibility, or externalities in markets. Each agent is a node in a fixed matching network and strives to be matched to another agent. Each agent has a complete preference list over all other agents it can be matched with. However, depending on the constraints and the current state of the game, not all possible partners are available for matching at all times. For correlated preferences, we propose and study a general class of hedonic coalition formation games that we call coalition formation games with constraints. This class includes and extends many recently studied variants of stable matching, such as locally stable matching, socially stable matching, or friendship matching. Perhaps surprisingly, we show that all these variants are encompassed in a class of "consistent" instances that always allow a polynomial improvement sequence to a stable state. In addition, we show that for consistent instances there always exists a polynomial sequence to every reachable state. Our characterization is tight in the sense that we provide exponential lower bounds when each of the requirements for consistency is violated. We also analyze matching with uncorrelated preferences, where we obtain a larger variety of results. While socially stable matching always allows a polynomial sequence to a stable state, for other classes different additional assumptions are sufficient to guarantee the same results. For the problem of reaching a given stable state, we show NP-hardness in almost all considered classes of matching games.

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Stochastic stability for roommate markets

Cited by 50 publications

References 18 publications

Analysis of stochastic matching markets

Analysis of stochastic matching markets

Stochastic Coalitional Better-Response Dynamics for Finite Games with Application to Network Formation Games

Matching Dynamics with Constraints

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