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College Admissions and the Stability of Marriage
The American Mathematical Monthly
... We explore the problem of two-sided matching markets, where two distinct groups of agents must be matched with one another [20]. Such markets have diverse real-world applications, ranging from labor markets, like online crowd-sourcing platforms such as Amazon Mechanical Turk, to online dating services [8,9,19]. In these contexts, the challenge lies in designing a matching algorithm that respects the preferences of both sets of agents. ...
... In their seminal work, Gale and Shapley introduce the concept of stable matchings [8]. Here, agents are matched one-to-one according to ordinal preferences such that no two agents have the incentive to deviate from the proposed solution. ...
... The stable marriage problem, is a well studied problem in twosided markets in which agents from the two sides are matched such that no pair of agents prefers to deviate from the matching [8,17], making the solution a stable matching. In earlier work on stable matchings, preferences of agents in the market are assumed to be known. ...
We consider a learning problem for the stable marriage model under unknown preferences for the left side of the market. We focus on the centralized case, where at each time step, an online platform matches the agents, and obtains a noisy evaluation reflecting their preferences. Our aim is to quickly identify the stable matching that is left-side optimal, rendering this a pure exploration problem with bandit feedback. We specifically aim to find Probably Correct Optimal Stable Matchings and present several bandit algorithms to do so. Our findings provide a foundational understanding of how to efficiently gather and utilize preference information to identify the optimal stable matching in two-sided markets under uncertainty. An experimental analysis on synthetic data complements theoretical results on sample complexities for the proposed methods.
... Unlike the mechanism known in the literature for the problem without predictions, we crucially do not consider declarations in a fixed order. Instead, our mechanism draws inspiration from the well-known deferred acceptance algorithm algorithm by Gale and Shapley [1962]. ...
... Mechanism design without money has a rich history spanning over fifty years, being deeply rooted in economics and social choice theory. As will be evident in Section 4, the seminal works of Gale and Shapley [1962], Roth [1982] and Hatfield and Milgrom [2005] on stable matching are particularly relevant to our study. However, our work aligns more closely with the agenda of approximate mechanism design without money set forth by Procaccia and Tennenholtz [2013] and, in particular, the subsequent work by Dughmi and Ghosh [2010]. ...
... We introduce our mechanism, called Boost, for bipartite matching with predictions (BMP + ) in the private graph model. Our mechanism is inspired by the deferred acceptance algorithm by Gale and Shapley [1962]. Boost is parameterized by some ≥ 1, which we term the confidence parameter. ...
... I N their seminal 1962 paper, Gale & Shapley introduced the two-sided matching problem: We are given sets of men and women, each with preferences over members of the opposite gender; we seek a stable matching, i.e., an assignment of partners such that no one finds his or her partner unacceptable, and no two agents mutually prefer each other to their assigned partners. Gale & Shapley (1962) showed that stable matchings can be found via the following deferred acceptance algorithm, under which men propose marriage in sequence, and women defer accepting prospective partners until the full sequence of proposals has been completed. ...
... We start by introducing the marriage model of Gale & Shapley (1962): There are finite sets M and W of men and women; we denote by I ≡ M ∪ W the set of agents. We assume that each man m ∈ M has a complete, transitive, and strict preference ordering ≻ m over W ∪ { / 0}, where / 0 denotes an outside option that represents the possibility of remaining unmatched. ...
... The main result of Gale & Shapley (1962) shows that the deferred acceptance algorithm introduced in the Introduction always produces a stable outcome. ...
One of the oldest results in the theory of two-sided matching is the entry comparative static, which shows that under the Gale-Shapley deferred acceptance algorithm, adding a new agent to one side of the market makes all the agents on the other side weakly better off. Here, we give a new proof of the entry comparative static, by way of a well-known property of deferred acceptance called respect for improvements. Our argument extends to yield comparative static results in more general settings, such as matching with slot-specific preferences.
... Since problem (38) is obtained by approximating problem (10) at a feasible point set {x n in }. The approximation can be further improved by successively approximating problem (10) based on the optimal solution {x n in } obtained by solving problem (38) in the previous approximation. ...
... Since problem (38) is obtained by approximating problem (10) at a feasible point set {x n in }. The approximation can be further improved by successively approximating problem (10) based on the optimal solution {x n in } obtained by solving problem (38) in the previous approximation. Therefore, the proposed successive approximation approach can be described in the following. ...
... 2: Compute the objective value in (38), denoted as Φ[0]. Set t = 1. ...
This paper investigates the application of non-orthogonal multiple access (NOMA) in millimeter wave (mmWave) communications by exploiting beamforming, user scheduling and power allocation. Random beamforming is invoked for reducing the feedback overhead of considered systems. A nonconvex optimization problem for maximizing the sum rate is formulated, which is proved to be NP-hard. The branch and bound (BB) approach is invoked to obtain the optimal power allocation policy, which is proved to converge to a global optimal solution. To elaborate further, low complexity suboptimal approach is developed for striking a good computational complexity-optimality tradeoff, where matching theory and successive convex approximation (SCA) techniques are invoked for tackling the user scheduling and power allocation problems, respectively. Simulation results reveal that: i) the proposed low complexity solution achieves a near-optimal performance; and ii) the proposed mmWave NOMA systems is capable of outperforming conventional mmWave orthogonal multiple access (OMA) systems in terms of sum rate and the number of served users.
... Then, we formulate the proposed cache-enabled mobility management framework as a dynamic matching game, so as to provide a distributed solution for mobility management in HetNets, while taking the dynamics of the system into account. To solve the formulated dynamic matching problem, we first show that conventional algorithms such as the deferred acceptance algorithm adopted in [18] and [19], fail to guarantee a dynamically stable HO between MUEs and SBSs. Therefore, we propose a novel distributed algorithm that is guaranteed to converge to a dynamically stable HO policy in dense HetNets. ...
... Matching theory is a mathematical framework that provides polynomial time solutions for combinatorial assignment problems such as (28a)-(28f) [18]. In a static form, a matching game is defined as a two-sided assignment problem between two disjoint sets of players in which the players of each set are interested to be matched to the players of the other set, according to their preference profiles. ...
... algorithm [18], presented in Algorithm 1, is guaranteed to find a two-sided stable association µ * between MUEs and SBSs. ...
One of the most promising approaches to overcome the uncertainty and dynamic channel variations of millimeter wave (mmW) communications is to deploy dual-mode base stations that integrate both mmW and microwave (W) frequencies. If properly designed, such dual-mode base stations can enhance mobility and handover in highly mobile wireless environments. In this paper, a novel approach for analyzing and managing mobility in joint W-mmW networks is proposed. The proposed approach leverages device-level caching along with the capabilities of dual-mode base stations to minimize handover failures, reduce inter-frequency measurement energy consumption, and provide seamless mobility in emerging dense heterogeneous networks. First, fundamental results on the caching capabilities, including caching probability and cache duration are derived for the proposed dual-mode network scenario. Second, the average achievable rate of caching is derived for mobile users. Third, the proposed cache-enabled mobility management problem is formulated as a dynamic matching game between mobile user equipments (MUEs) and small base stations (SBSs). The goal of this game is to find a distributed handover mechanism that subject to the network constraints on HOFs and limited cache sizes, allows each MUE to choose between executing an HO to a target SBS, being connected to the macrocell base station (MBS), or perform a transparent HO by using the cached content. The formulated matching game allows capturing the dynamics of the mobility management problem caused by HOFs. To solve this dynamic matching problem, a novel algorithm is proposed and its convergence to a two-sided dynamically stable HO policy is proved. Numerical results corroborate the analytical derivations and show that the proposed solution will provides significant reductions in both the HOF and energy consumption by MUEs.
... First, the UAMS problem objective is to solve the user association and mode selection, which is formulated as a matching game between users and SBSs assuming a fixed power allocation. Subsequently, a local matching algorithm inspired from the Gale and Shapley algorithm [29], [30] is proposed. To solve the problem locally, knowledge of inter-cell interference is needed. ...
... The detailed matching algorithm is presented in Algorithm 1. Remark 5. For a fixed preference profile, Algorithm 1 is guaranteed to find a two-sides stable matching between SBSs and users [29]. if |M j | = 1, 7: ...
... Then, the SINR expressions from (3) and (4) is rewritten as: By substituting (29) and (30) in (28), the objective function becomes: ...
In this paper, the problem of uplink (UL) and downlink (DL) resource optimization, mode selection and power allocation is studied for wireless cellular networks under the assumption of in-band full duplex (IBFD) base stations, non-orthogonal multiple access (NOMA) operation, and queue stability constraints. The problem is formulated as a network utility maximization problem for which a Lyapunov framework is used to decompose it into two disjoint subproblems of auxiliary variable selection and rate maximization. The latter is further decoupled into a user association and mode selection (UAMS) problem and a UL/DL power optimization (UDPO) problem that are solved concurrently. The UAMS problem is modeled as a many-to-one matching problem to associate users to small cell base stations (SBSs) and select transmission mode (half/full-duplex and orthogonal/non-orthogonal multiple access), and an algorithm is proposed to solve the problem converging to a pairwise stable matching. Subsequently, the UDPO problem is formulated as a sequence of convex problems and is solved using the concave-convex procedure. Simulation results demonstrate the effectiveness of the proposed scheme to allocate UL and DL power levels after dynamically selecting the operating mode and the served users, under different traffic intensity conditions, network density, and self-interference cancellation capability. The proposed scheme is shown to achieve up to 63% and 73% of gains in UL and DL packet throughput, and 21% and 17% in UL and DL cell edge throughput, respectively, compared to existing baseline schemes.
... Critically, the core in this game is non-empty [3]. Indeed, the core may contain an exponential number of stable matchings [7]. ...
... The celebrated deferred-acceptance algorithm by Gale and Shapley [3] outputs the doctoroptimal stable matching when the doctors make proposals (see Algorithm 1) and the hospitaloptimal stable matching when the hospitals propose. These results hold regardless of the specific order of proposals. ...
We study the stable matching problem under the random matching model where the preferences of the doctors and hospitals are sampled uniformly and independently at random. In a balanced market with n doctors and n hospitals, the doctor-proposal deferred-acceptance algorithm gives doctors an expected rank of order for their partners and hospitals an expected rank of order for their partners. This situation is reversed in an unbalanced market with n+1 doctors and n hospitals, a phenomenon known as the short-side advantage. The current proofs of this fact are indirect, counter-intuitively being based upon analyzing the hospital-proposal deferred-acceptance algorithm. In this paper we provide a direct proof of the short-side advantage, explicitly analyzing the doctor-proposal deferred-acceptance algorithm. Our proof sheds light on how and why the phenomenon arises.
... For the distributed approach, we propose a distributed game based on matching theory in which UEs apply to individual BSs and receive a response for association action which ensures load balancing of the responding BS. In this distributed approach, no central entity exists and there are no messages or information -Perform a DA matching game [30] as follows: 7 while R ≠ ∅ do 8 Each UE ∈ R applies to its th preferred BS in P UE ; 9 Each BS forms its current waiting list W from its new applicants and its previous waiting list W −1 ; ...
... We propose a matching game distributed load balancing (MG-DLB) algorithm. At each learning step , the UEs and BSs play a distributed deferred acceptance (DA) matching game [30] which leads to load-balanced association resulting from each BS controls how many UEs to admit based on its load balancing constraint. From the updated U-tables at each UE and each BS, the UEs and the BSs build their own preference list using the U-value vector corresponding to the current state of each UE. ...
As next generation cellular networks become denser, associating users with the optimal base stations at each time while ensuring no base station is overloaded becomes critical for achieving stable and high network performance. We propose multi-agent online Q-learning (QL) algorithms for performing real-time load balancing user association and handover in dense cellular networks. The load balancing constraints at all base stations couple the actions of user agents, and we propose two multi-agent action selection policies, one centralized and one distributed, to satisfy load balancing at every learning step. In the centralized policy, the actions of UEs are determined by a central load balancer (CLB) running an algorithm based on swapping the worst connection to maximize the total learning reward. In the distributed policy, each UE takes an action based on its local information by participating in a distributed matching game with the BSs to maximize the local reward. We then integrate these action selection policies into an online QL algorithm that adapts in real-time to network dynamics including channel variations and user mobility, using a reward function that considers a handover cost to reduce handover frequency. The proposed multi-agent QL algorithm features low-complexity and fast convergence, outperforming 3GPP max-SINR association. Both policies adapt well to network dynamics at various UE speed profiles from walking, running, to biking and suburban driving, illustrating their robustness and real-time adaptability.
... We focus on the existence and structure of stable outcomes in decentralized, real-world matching markets. In production networks that we consider in this paper, stable outcomes play the role of equilibrium and may serve as a reasonable prediction of the outcome of market interactions (Kelso and Crawford, 1982, Roth, 1984. 1 We obtain a general result: any trading network has an outcome that satisfies a natural extension of pairwise stability (Gale and Shapley, 1962). Our model of matching markets subsumes many previous models of matching with contracts, including many-to-one (Gale and Shapley, 1962, Crawford and Knoer, 1981, Kelso and Crawford, 1982, Hatfield and Milgrom, 2005 and many-to-many matching markets (Roth, 1984, Sotomayor, 1999, 2004, Echenique and Oviedo, 2006, Klaus and Walzl, 2009). ...
... In production networks that we consider in this paper, stable outcomes play the role of equilibrium and may serve as a reasonable prediction of the outcome of market interactions (Kelso and Crawford, 1982, Roth, 1984. 1 We obtain a general result: any trading network has an outcome that satisfies a natural extension of pairwise stability (Gale and Shapley, 1962). Our model of matching markets subsumes many previous models of matching with contracts, including many-to-one (Gale and Shapley, 1962, Crawford and Knoer, 1981, Kelso and Crawford, 1982, Hatfield and Milgrom, 2005 and many-to-many matching markets (Roth, 1984, Sotomayor, 1999, 2004, Echenique and Oviedo, 2006, Klaus and Walzl, 2009). ...
We consider a model of matching in trading networks in which firms can enter into bilateral contracts. In trading networks, stable outcomes, which are immune to deviations of arbitrary sets of firms, may not exist. We define a new solution concept called trail stability. Trail-stable outcomes are immune to consecutive, pairwise deviations between linked firms. We show that any trading network with bilateral contracts has a trail-stable outcome whenever firms' choice functions satisfy the full substitutability condition. For trail-stable outcomes, we prove results on the lattice structure, the rural hospitals theorem, strategy-proofness, and comparative statics of firm entry and exit. We also introduce weak trail stability which is implied by trail stability under full substitutability. We describe relationships between the solution concepts.
... Meanwhile, it minimizes the exposure of option insurer's position to any potential losses. The business strategy of matching is analogous to the generalized Tian Ji's horse racing strategy [18][19][20] and the Nobel prize-winning stable allocation theory [21,22]. In the end, the novelty of financial option insurance is elaborated in three aspects including market acceptance, risk profile and profitability, and positive market effect. ...
... After the completion of ranking matrix, by repeating acceptanceand-rejection procedure as mentioned above, it is capable of determining the most optimal matching in the combination of {A β, B α, C δ, D γ}, as shown in Figure 9(c). The developed business strategy of matching is analogous to the generalized Tian Ji's horse racing strategy [18][19][20] and the Nobel prize-winning stable allocation theory [21,22]. In our case, the preferences can easily be quantified in terms of moneyness of financial option. ...
The option is a financial derivative, which is regularly employed in reducing the risk of its underlying securities. However, investing in option is still risky. Such risk becomes much severer for speculators who utilize option as a means of leverage to increase their potential returns. In order to mitigate risk on their positions, the rudimentary concept of financial option insurance is introduced into practice. Two starkly-dissimilar concepts of insurance and financial option are integrated into the formation of financial option insurance. The proposed financial product insures investors option premiums when misfortune befalls on them. As a trade-off, they are likely to sacrifice a limited portion of their potential profits. The loopholes of prevailing financial market are addressed and the void is filled by introducing a stable three-entity framework. Moreover, a specifically designed mathematical model is proposed. It consists of two portions: the business strategy of matching and a verification-and-modification process. The proposed model enables the option investors with calls and puts of different moneyness to be protected by the issued option insurance. Meanwhile, it minimizes the exposure of option insurers position to any potential losses.
... College Admission and the stability of Marriage is a wellknown problem, introduced by Gale and Shapley [31]. In its most popular variants, there are two disjoint sets of cardinality, n. ...
... Gale and Shapley [31] provide the solution and the algorithm for stable pairing. They also proved that there always exists a stable match for such type of problem. ...
To balance the load and to discourage the free-riding in peer-to-peer (P2P) networks, many incentive mechanisms and policies have been proposed in recent years. Global peer ranking is one such mechanism. In this mechanism, peers are ranked based on a metric called contribution index. Contribution index is defined in such a manner that peers are motivated to share the resources in the network. Fairness in the terms of upload to download ratio in each peer can be achieved by this method. However, calculation of contribution index is not trivial. It is computed distributively and iteratively in the entire network and requires strict clock synchronization among the peers. A very small error in clock synchronization may lead to wrong results. Furthermore, iterative calculation requires a lot of message overhead and storage capacity, which makes its implementation more complex. In this paper, we are proposing a simple incentive mechanism based on the contributions of peers, which can balance the upload and download amount of resources in each peer. It does not require iterative calculation, therefore, can be implemented with lesser message overhead and storage capacity without requiring strict clock synchronization. This approach is efficient as there are very less rejections among the cooperative peers. It can be implemented in a truly distributed fashion with O(N) time complexity per peer.
... The game and its analysis are novel, so far as we know. Stable marriage [10] and its variants have been extensively studied, but the connection to games appears to be new. Many of the results that we use on matchings of random point sets are taken from [17]. ...
... Stable matching can be applied to a wide variety of settings involving agents each of which has preferences over the others. The concept was introduced in a celebrated paper of Gale and Shapley [10], who considered the setting of n heterosexual marriages between n girls and n boys, each of whom has an arbitrary preference order over those of the opposite sex. Gale and Shapley gave a beautiful algorithm proving the existence of a stable matching in this case. ...
We introduce a two-player game involving two tokens located at points of a fixed set. The players take turns to move a token to an unoccupied point in such a way that the distance between the two tokens is decreased. Optimal strategies for this game and its variants are intimately tied to Gale-Shapley stable marriage. We focus particularly on the case of random infinite sets, where we use invariance, ergodicity, mass transport, and deletion-tolerance to determine game outcomes.
... Since the seminal paper by Gale and Shapley [7], the stable matching and its generalizations have been widely studied in mathematics, economics, and computer science; see, e.g, [8,23,17] for more detail. As a noteworthy connection to another discrete structure, Conway [13] pointed out that the set of stable matchings forms a distributive lattice under a natural dominance relation. ...
... For sake of disambiguation, we adopt specific stable matchings defined algorithmically as follows. For a stable matching instance, a stable matching can be obtained by a simple algorithm, so-called the deferred acceptance algorithm [7,19] (see Algorithm 1). In each iteration, an unmatched left vertex u proposes to the most-preferred right vertex v in u's preference list to whom it hasn't yet proposed. ...
We explore novel connections between antimatroids and matchings in bipartite graphs. In particular, we prove that a combinatorial structure induced by stable matchings or maximum-weight matchings is an antimatroid. Moreover, we demonstrate that every antimatroid admits such a representation by stable matchings and maximum-weight matchings.
... To find the stable policy π j , the deferred acceptance (DA) algorithm, originally introduced in [49], can be adopted. Hence, we introduce Algorithm 1 based on the DA algorithm ...
... Since it is based on a variant of the DA process, Algorithm 1 is guaranteed to converge to a stable matching as shown in [49]. Moreover, among the set of all stable solutions, Algorithm 1 yields the solution that is PO for the D-BSs. ...
In this paper, a novel framework is proposed for optimizing the operation and performance of a large-scale, multi-hop millimeter wave (mmW) backhaul within a wireless small cell network (SCN) that encompasses multiple mobile network operators (MNOs). The proposed framework enables the small base stations (SBSs) to jointly decide on forming the multi-hop, mmW links over backhaul infrastructure that belongs to multiple, independent MNOs, while properly allocating resources across those links. In this regard, the problem is addressed using a novel framework based on matching theory that is composed to two, highly inter-related stages: a multi-hop network formation stage and a resource management stage. One unique feature of this framework is that it jointly accounts for both wireless channel characteristics and economic factors during both network formation and resource management. The multi-hop network formation stage is formulated as a one-to-many matching game which is solved using a novel algorithm, that builds on the so-called deferred acceptance algorithm and is shown to yield a stable and Pareto optimal multi-hop mmW backhaul network. Then, a one-to-many matching game is formulated to enable proper resource allocation across the formed multi-hop network. This game is then shown to exhibit peer effects and, as such, a novel algorithm is developed to find a stable and optimal resource management solution that can properly cope with these peer effects. Simulation results show that the proposed framework yields substantial gains, in terms of the average sum rate, reaching up to 27% and 54%, respectively, compared to a non-cooperative scheme in which inter-operator sharing is not allowed and a random allocation approach. The results also show that our framework provides insights on how to manage pricing and the cost of the cooperative mmW backhaul network for the MNOs.
... Matching games [7] is a field of game theory that have proved to be successful in explaining achievements and failures of matching and allocation mechanisms in decentralized markets. Gale and Shapley published one of the earliest and probably most successful paper on the subject [3] and solved the stable marriage and college admissions association problem with a polynomial time algorithm called DAA. ...
... Shapley's deferred acceptance algorithm in its college-admission form with APs preferences over groups of users and users preferences over individual APs is a stable matching mechanism Step 1: Initialization; 3 Step 1.a: All APs and users are marked unengaged. ...
In multi-rate IEEE 802.11 WLANs, the traditional user association based on the strongest received signal and the well known anomaly of the MAC protocol can lead to overloaded Access Points (APs), and poor or heterogeneous performance. Our goal is to propose an alternative game-theoretic approach for association. We model the joint resource allocation and user association as a matching game with complementarities and peer effects consisting of selfish players solely interested in their individual throughputs. Using recent game-theoretic results we first show that various resource sharing protocols actually fall in the scope of the set of stability-inducing resource allocation schemes. The game makes an extensive use of the Nash bargaining and some of its related properties that allow to control the incentives of the players. We show that the proposed mechanism can greatly improve the efficiency of 802.11 with heterogeneous nodes and reduce the negative impact of peer effects such as its MAC anomaly. The mechanism can be implemented as a virtual connectivity management layer to achieve efficient APs-user associations without modification of the MAC layer.
... In this work we provide a possible approach to the DP TPC event matching problem using an algorithm from Gale and Shapley [7], which gives a solution to the stable marriage problem. Matching, in mathematical sense, is selecting a set of independent edges in a graph without common vertices [8]. ...
... For multiple scatter events, other algorithms may also be considered as there could be additional secondary events belonging to the same primary event. Gale and Shapley also provided a solution to a similar problem, the so-called college admission problem [7]. In this paper, we study the application of the stable marriage and college admission algorithms both to single and multiple scatter events using data generated by a toy Monte Carlo. ...
The problem of matching primary and secondary light signals, belonging to the same event, is presented in the context of dual-phase time projection chambers. In large-scale detectors, the secondary light emission could be delayed up to order of milliseconds, which, combined with high signal rates, could make the matching of the signals challenging. A possible approach is offered in the framework of the Stable Marriage and the College Admission problem, for both of which solutions are given by the Gale-Shapley algorithm.
... 2) EV-Consumer-Oriented and EV-Provider-Oriented V2V Matching Algorithms: The Gale-Shapley algorithm [32] has been proposed as an efficient method to find a stable one-toone matching between men and women in the stable marriage problem. Similarly, in our investigated problem, EVs as energy consumers and EVs as energy providers can be regarded as men and women, respectively. ...
... The EV-consumer-oriented algorithm yields an EV-consumeroptimal stable matching, in which each EV as energy consumer has the best matched partner that it can have in any stable matching, whereas the EV-provider-oriented algorithm leads to an EV-consumer-optimal output. This property is referred to as the polarization of stable matchings [32]. ...
In this paper, we investigate flexible power transfer among electric vehicles (EVs) from a cooperative perspective in an EV system. First, the concept of cooperative EV-to-EV (V2V) charging is introduced, which enables active cooperation via charging/discharging operations between EVs as energy consumers and EVs as energy providers. Then, based on the cooperative V2V charging concept, a flexible energy management protocol with different V2V matching algorithms is proposed, which can help the EVs achieve more flexible and smarter charging/discharging behaviors. In the proposed energy management protocol, we define the utilities of the EVs based on the cost and profit through cooperative V2V charging and employ the bipartite graph to model the charging/discharging cooperation between EVs as energy consumers and EVs as energy providers. Based on the constructed bipartite graph, a max-weight V2V matching algorithm is proposed in order to optimize the network social welfare. Moreover, taking individual rationality into consideration, we further introduce the stable matching concepts and propose two stable V2V matching algorithms, which can yield the EV-consumer-optimal and EV-provider-optimal stable V2V matchings, respectively. Simulation results verify the efficiency of our proposed cooperative V2V charging based energy management protocol in improving the EV utilities and the network social welfare as well as reducing the energy consumption of the EVs.
... Nash Equilibria is a proposed solution of a non-cooperative game involving two or more players in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy [3][4][5]. In other words, Nash equilibrium [6][7][8] is a configuration of strategies in a game in which no player has an immediate incentive to change his own initial strategy unilaterally. If two players Alice and Bob choose strategies A and B, (A, B) is a Nash equilibrium if Alice has no other strategy available that does better than A at maximizing her payoff in response to Bob choosing B, and Bob has no other strategy available that does better than B at maximizing his payoff in response to Alice choosing A. Nash equilibria more often than not lead to collectively horrible outcomes: ...
... As in the seminal studies ofGale and Shapley (1962) orRoth and Sotomayor (1992).16 SeeIrving (1994),Abdulkadiroglu et al. (2009), Domaniç et al. (2017,Chen and Li (2019), or Bonifacio et al. (2023) for example. ...
From a normative viewpoint, there is no compelling reason for preferring the weak over the strong core, and vice versa. However, the situation changes significantly from a mechanism design perspective. We work in a rights structures environment, where the role of the social planner is to allocate rights to individuals or coalitions which allow them to change the status-quo state. While coalitions are irrelevant for implementation in weak core (Koray and Yildiz in J. Econ. Theory 176:479-502, 2018; Korpela et al. in J. Econ. Theory 185:104953, 2020), our results show that they are fundamental for implementation in strong core. We fully characterize the implementation of social choice rules in strong core to outline this distinction. For robustness, we also characterize double implementation in weak and strong core which we show to be equivalent to implementation in weak core. Finally, we show that this equivalence breaks down in the more realistic case of implementation by codes of rights, where the set of states coincides with the set of outcomes.
... Matching markets are increasingly recognized for their role in aligning varying preferences of agents and enabling decentralized resource distribution [18]. Nonetheless, the prevalent model for pairing agents as proposed in [23] shows inadequacies in managing uncertainties and achieving competitive equilibria. Consequently, it has become clear that bargaining processes are essential to reach a consensus on surplus distribution among matched agents [12]. ...
This study examines the dynamics of bargaining in a social system that incorporates risk sharing through exchange network models and stochastic matching between agents. The analysis explores three scenarios: convergent expectations, divergent expectations, and social preferences among model players. The study introduces stochastic shocks through a Poisson process, which can disrupt coordination within the decentralized exchange mechanism. Despite these shocks, agents can employ a risk-sharing protocol utilizing Pareto weights to mitigate their effects. The model outcomes do not align with the generalized Nash bargaining solutions across all scenarios. However, over a sufficiently long time frame, the dynamics consistently converge to a fixed point that slightly deviates from the balanced outcome or Nash equilibrium. This minor deviation represents the risk premium necessary for hedging against mutual risk. The risk premium is at its minimum in the scenario with convergent expectations and remains unchanged in the case involving social preferences.
... Intuitively, a choice function selects a subset of students from a pool of applicants. For example, in the classical deferred acceptance algorithm [Gale and Shapley, 1962], a school's choice function simply selects students based on its priority order, up to its capacity. In this paper, we aim to design a choice function that balances achieving diversity goals with respecting school priorities. ...
Student placements under diversity constraints are a common practice globally. This paper addresses the selection of students by a single school under a \emph{one-to-one convention}, where students can belong to multiple types but are counted only once based on one type. While existing algorithms in economics and computer science aim to help schools meet diversity goals and priorities, we demonstrate that these methods can result in significant imbalances among students with different type combinations. To address this issue, we introduce a new property called \emph{balanced representation}, which ensures fair representation across all types and type combinations. We propose a straightforward choice function that uniquely satisfies four fundamental properties: maximal diversity, non-wastefulness, justified envy-freeness, and balanced representation. While previous research has primarily focused on algorithms based on bipartite graphs, we take a different approach by utilizing flow networks. This method provides a more compact formalization of the problem and significantly improves computational efficiency. Additionally, we present efficient algorithms for implementing our choice function within both the bipartite graph and flow network frameworks.
... Sanjay Srivastava, Timofiy Mylovanov, and Rakesh Vohra consider a variation on Crawford and Sobel's model in which the sender is potentially a "bullshitter" who doesn't care about the truth at all (formally, the sender does not know their type when they send a message, and moreover has state-independent preferences). The authors show that depending on the bullshitter's preferences, more or less information can be transmitted in equilibrium and the (informed) sender and the receiver can be either made better or worse off. 2 The remaining papers in the issue are related to two of Vince's papers that beautifully linked general equilibrium theory with the theory of matching markets introduced by Gale & Shapley (1962). In response to these two papers coauthored with Elsie Knoer and Lex Kelso respectively, Al Roth inscribed Vince's copy of Roth & Sotomayor (1990), "For Vince, who discovered that this stuff was economics." ...
... This in turn implies that people differ in their opinions about what constitutes a desirable partner or at least about who is worth pursuing. At the other extreme, and more in line with biological studies of mate selection [2][3][4], lies the competition hypothesis, which assumes that there is consensus about what constitutes a desirable partner and that mate seekers, regardless of their own qualifications, pursue those partners who are universally recognized as most desirable [5][6][7][8]. Paradoxically this can also produce couples who resemble one another in terms of desirability, as the most desirable partners pair off with one another, followed by the next most desirable, and so on. To the extent that desirability correlates with attributes like age, physical attractiveness, and education, the matching and competition hypotheses can, as a result, produce similar equilibrium patterns of mixing [5,9,10]. ...
Romantic courtship is often described as taking place in a dating market where men and women compete for mates, but the detailed structure and dynamics of dating markets have historically been difficult to quantify for lack of suitable data. In recent years, however, the advent and vigorous growth of the online dating industry has provided a rich new source of information on mate pursuit. Here we present an empirical analysis of heterosexual dating markets in four large US cities using data from a popular, free online dating service. We show that competition for mates creates a pronounced hierarchy of desirability that correlates strongly with user demographics and is remarkably consistent across cities. We find that both men and women pursue partners who are on average about 25% more desirable than themselves by our measures and that they use different messaging strategies with partners of different desirability. We also find that the probability of receiving a response to an advance drops markedly with increasing difference in desirability between the pursuer and the pursued. Strategic behaviors can improve one's chances of attracting a more desirable mate, though the effects are modest.
... A marriage is called stable if there is no unmarried pair (a man, a woman) who prefer each other to their respective partners in the marriage. A classic theorem, due to Gale and Shapley [4], asserts that, given any system of preferences {ρ j , σ j } j∈ [n] , there exists at least one stable marriage M. ...
Following up a recent work by Ashlagi, Kanoria and Leshno, we study a stable matching problem with unequal numbers of men and women, and independent uniform preferences. The asymptotic formulas for the expected number of stable matchings, and for the probabilities of one point--concentration for the range of husbands' total ranks and for the range of wives' total ranks are obtained.
... The (stable) marriage problem (MP), proposed by Gale and Shapley (1962), combines a rich mathematical structure with important real-life applications. The MP consists of two groups of agents, men and women. ...
When several two-sided matching markets merge into one, it is inevitable that some agents will become worse off if the matching mechanism used is stable. I formalize this observation by defining the property of integration monotonicity, which requires that every agent becomes better off after any number of matching markets merge. Integration monotonicity is also incompatible with the weaker efficiency property of Pareto optimality. Nevertheless, I obtain two possibility results. First, stable matching mechanisms never hurt more than one-half of the society after the integration of several matching markets occurs. Second, in random matching markets there are positive expected gains from integration for both sides of the market, which I quantify.
This paper proposes an efficient depth-first search algorithm to solve the maximum stable marriage problem with ties and incomplete preference lists. The key idea of the algorithm is to initialize an empty matching and mark all men as unmatched. In each iteration, an unmatched man proposes to the most preferred woman on his rank list. If the woman is unmatched or prefers the proposing man over her current partner, she is assigned to the man, forming a new pair in the matching. Otherwise, she keeps her current partner and rejects the proposing man. When a man is rejected by a woman, he becomes unmatched. The algorithm then recursively processes the next unmatched man until it either finds a complete matching or reaches a predefined maximum number of iterations. Experimental results on randomly generated datasets show that our algorithm is effective in producing high- quality solutions to the problem. Keywords: Depth-first search; Gale-Shapley algorithm; stable matching; stable marriage problem; ties and incomplete lists
The vehicle-cargo matching (VCM) sector within the freight industry faces significant challenges, including substantial fluctuations in vehicle-cargo supply and demand quantities and suboptimal matching efficiency. This study addresses these issues by innovatively establishing a multi-objective VCM decision model and segmenting typical business scenarios based on varying vehicle-cargo supply–demand ratios, with objective functions scientifically set according to the diverse requirements of each participant. This model fully considers the characteristics of the VCM system and is solved using appropriate Kuhn–Munkres (an improved Hungarian algorithm) algorithms to ensure efficient and accurate results. Evaluation indicators are also innovatively set from both economic and social benefit perspectives, incorporating managerial assessments and corporate development requirements. The study compares the impacts of different objective functions on VCM outcomes across various scenarios. Results indicate that, in most cases, the platform can achieve maximum profits without explicitly targeting profit maximization, thus accommodating other managerial assessment requirements. Compared to previous single-objective function studies, this approach increases economic efficiency by 22.14 and decreases the empty driving rate by 12.1. The model is directly applicable to real-world logistics, offering a practical, comprehensive solution for maximizing resources and profitability within the evolving freight industry landscape.
In this paper, we conceptualize parking in urban areas as a matching problem. We begin by formulating street parking behavior as a strategic game and develop a matching framework for this scenario. We demonstrate that the set of Nash equilibria in the parking game directly corresponds to the set of stable outcomes in the associated matching problem. Subsequently, we propose a centralized mechanism designed to allocate available parking spaces to drivers in a stable manner, facilitated through a digital platform, such as a mobile application. This mechanism can accommodate various practical needs by establishing a priority structure that may include distance priority, price priority, or resident priority.
There is evidence that many people do not report their types truthfully in strategy-proof mechanisms. One of the leading explanations is mechanism complexity. We propose a novel way of describing any strategy-proof mechanism with the aim to reduce the ensuing cognitive load on players and, hence, to increase truthful reporting. The description highlights that a player is guaranteed to obtain a most preferred feasible allocation based on her reported type and that feasibility depends only on the other players’ reports. Optimality of truthful reporting should therefore become more “apparent.” We experimentally test this prediction using the top trading cycles mechanism for matching applicants to positions. The proposed description increases truth-telling. This increase positively interacts with subject numeracy.
This paper was accepted by Dorothea Kübler, behavioral economics and decision analysis.
Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2023.02531 .
Connected and automated vehicles (CAVs) provide the most intriguing opportunity for enabling users to significantly improve safety and transportation efficiency by monitoring network conditions and making better operating decisions. CAVs, however, could alter the tendency to travel, eventually leading to a high traffic demand and causing rebound effects (e.g., increasing vehicle miles traveled). This chapter provides a control framework to distribute travel demand in a given transportation network, resulting in a socially optimal mobility system that travelers would be willing to accept. A “socially optimal mobility system” implies a mobility system that (1) is efficient (in terms of energy consumption and travel time), (2) mitigates rebound effects, and (3) ensures equity in transportation.
The traditional MapReduce frameworks were originally designed for processing data within a single cluster and are not suitable for handling geo-distributed data. Consequently, alternative approaches such as Hierarchical and Geo-Hadoop have been proposed to address this limitation. However, these approaches still face challenges in efficiently managing inter-cluster data transfer, particularly considering the heterogeneity of clusters and varying bandwidth among them. Moreover, the need to transmit results to a central global reducer for geo-distributed MapReduce operations adds unnecessary complexity. To tackle these issues, we introduce Extended Cross-MapReduce (ECMR), a framework that integrates resource heterogeneity and network links in geo-distributed MapReduce workflows. ECMR optimizes data management and determines the necessary data volume for generating final results. To enhance performance, ECMR leverages the overlap between data transfer and execution time by utilizing multiple global reducers and grouping temporary results that require data transfer over the Internet. In ECMR, we propose a bipartite graph and extend the Gale-Shapley algorithm to determine the optimal number of clusters and select the most suitable locations for global reducers. Through extensive experimental evaluations conducted on a real testbed, we demonstrate the effectiveness of our proposed ECMR method. The results exhibit significant improvements over traditional Hierarchical and Geo-Hadoop approaches, achieving reductions of up to 81% and 85% in overall makespan, respectively.
In this paper, we establish conditions for the existence of an equilibrium in the equilibrium flow problem studied by Galichon et al. (2024). The problem nests several classical economic models such as bipartite matching models, hedonic pricing models, shortest-path and minimum-cost flow problems, and time-dependent routing problems.
We study deviations by a group of agents in the three main types of matching markets: the house allocation, the marriage, and the roommates models. For a given instance, we call a matching k -stable if no other matching exists that is more beneficial to at least k out of the n agents. The concept generalizes the recently studied majority stability [57]. We prove that whereas the verification of k -stability for a given matching is polynomial-time solvable in all three models, the complexity of deciding whether a k -stable matching exists depends on and is characteristic to each model.
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