In this paper, a novel power auctioneers network (PAN) is proposed to enable the sharing of resources from licensed users in exchange for performance gains using the Cooperative Non-Orthogonal Multiple Access (C-NOMA) protocol. This system exploits a matching-based algorithm based on game theory, namely Pragmatic Distributed Algorithm (PDA), to handle the challenge of user pairing and power allocation in C-NOMA. PDA is a cooperative game that allows for the sharing of potential relays within a network through a round-robin approach, allowing for fairer resource distribution. This approach uses distributed methods to perform tasks on user devices to remove the strain on the base station’s resources. Monte Carlo simulations demonstrate that the addition of these games to a wireless network provides a significant performance increase compared to traditional orthogonal multiple access methods, the unoptimised C-NOMA and the Distributed Matching Algorithm (DMA) method. Lastly, the idea of instability in matching algorithms in the context of wireless communications is explored by altering the number of users within a network.

Abstract We consider the two-sided stable matching setting in which there may be uncertainty about the agents’ preferences due to limited information or communication. We consider three models of uncertainty: (1) lottery model—for each agent, there is a probability distribution over linear preferences, (2) compact indifference model—for each agent, a weak preference order is specified and each linear order compatible with the weak order is equally likely and (3) joint probability model—there is a lottery over preference profiles. For each of the models, we study the computational complexity of computing the stability probability of a given matching as well as finding a matching with the highest probability of being stable. We also examine more restricted problems such as deciding whether a certainly stable matching exists. We find a rich complexity landscape for these problems, indicating that the form uncertainty takes is significant.

The assignment problem is one of the most well-studied settings in multi-agent resource allocation. Aziz, de Haan, and Rastegari (2017) considered this problem with the additional feature that agents’ preferences involve uncertainty. In particular, they considered two uncertainty models neither of which is necessarily compact. In this paper, we focus on three uncertain preferences models whose size is polynomial in the number of agents and items. We consider several interesting computational questions with regard to Pareto optimal assignments. We also present some general characterization and algorithmic results that apply to large classes of uncertainty models.

We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomised mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomised mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of $$\frac{e}{e-1}$$ . The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of $$\frac{18}{13}$$ on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. By using a characterisation result of Bade, we show that any randomised mechanism that is a symmetrisation of a truthful, non-bossy and Pareto optimal mechanism has an improved lower bound of $$\frac{e}{e-1}$$ . Since our new mechanism is a symmetrisation of RSDM for strict preferences, it follows that this lower bound is tight. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomised strategy of the administrator who interviews the applicants.

The assignment problem is one of the most well-studied settings in social choice, matching, and discrete allocation. We consider this problem with the additional feature that agents' preferences involve uncertainty. The setting with uncertainty leads to a number of interesting questions including the following ones. How to compute an assignment with the highest probability of being Pareto optimal? What is the complexity of computing the probability that a given assignment is Pareto optimal? Does there exist an assignment that is Pareto optimal with probability one? We consider these problems under two natural uncertainty models: (1) the lottery model in which each agent has an independent probability distribution over linear orders and (2) the joint probability model that involves a joint probability distribution over preference profiles. For both of these models, we present a number of algorithmic and complexity results highlighting the difference and similarities in the complexity of the two models.

We consider Pareto optimal matchings (POMs) in a many-to-many market of applicants and courses where applicants have preferences, which may include ties, over individual courses and lexicographic preferences over sets of courses. Since this is the most general setting examined so far in the literature, our work unifies and generalizes several known results. Specifically, we characterize POMs and introduce the Generalized Serial Dictatorship Mechanism with Ties (GSDT) that effectively handles ties via properties of network flows. We show that GSDT can generate all POMs using different priority orderings over the applicants, but it satisfies truthfulness only for certain such orderings. This shortcoming is not specific to our mechanism; we show that any mechanism generating all POMs in our setting is prone to strategic manipulation. This is in contrast to the one-to-one case (with or without ties), for which truthful mechanisms generating all POMs do exist.

We study the House Allocation problem (also known as the Assignment problem), i.e., the problem of allocating a set of objects among a set of agents, where each agent has ordinal preferences (possibly involving ties) over a subset of the objects. We focus on truthful mechanisms without monetary transfers for finding large Pareto optimal matchings. It is straightforward to show that no deterministic truthful mechanism can approximate a maximum cardinality Pareto optimal matching with ratio better than 2. We thus consider randomized mechanisms. We give a natural and explicit extension of the classical Random Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation problem where preference lists can include ties. We thus obtain a universally truthful randomized mechanism for finding a Pareto optimal matching and show that it achieves an approximation ratio of eovere-1. The same bound holds even when agents have priorities (weights) and our goal is to find a maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On the other hand we give a lower bound of 18 over 13 on the approximation ratio of any universally truthful Pareto optimal mechanism in settings with strict preferences. In the case that the mechanism must additionally be non-bossy, an improved lower bound of eovere-1 holds. This lower bound is tight given that RSDM for strict preference lists is non-bossy. We moreover interpret our problem in terms of the classical secretary problem and prove that our mechanism provides the best randomized strategy of the administrator who interviews the applicants.

We study two-sided matching markets in which participants are initially endowed with partial preference orderings, lacking precise information about their true, strictly ordered list of preferences. We wish to reason about matchings that are stable with respect to agents' true preferences, and which are furthermore optimal for one given side of the market. We present three main results. First, one can decide in polynomial time whether there exists a matching that is stable and optimal under all strict preference orders that refine the given partial orders, and can construct this matching in polynomial time if it does exist. We show, however, that deciding whether a given pair of agents are matched in all or no such optimal stable matchings is co-NP-complete, even under quite severe restrictions on preferences. Finally, we describe a polynomial-time algorithm that decides, given a matching that is stable under the partial preference orderings, whether that matching is stable and optimal for one side of the market under some refinement of the partial orders.

The traditional model of two-sided matching assumes that all agents fully know their own preferences. As markets grow large, however, it becomes impractical for agents to precisely assess their rankings over all agents on the other side of the market. We propose a novel model of two-sided matching in which agents are endowed with known partially ordered preferences and unknown true preferences drawn from known distributions consistent with the partial order. The true preferences are learned through interviews, revealing the pairwise rankings among all interviewed agents, performed according to a centralized interview policy, i.e., an algorithm that adaptively schedules interviews. Our goal is for the policy to guarantee both stability and optimality for a given side of the market, with respect to the underlying true preferences of the agents. As interviews are costly, we seek a policy that minimizes the number of interviews. We introduce three minimization objectives: (very weak) dominance, which minimizes the number of interviews for any underlying true preference profile; Pareto optimality, which guarantees that no other policy dominates the given policy; and optimality in expectation with respect to the preference distribution. We formulate our problem as a Markov decision process, implying an algorithm for computing an optimal-in-expectation policy in time polynomial in the number of possible preference orderings (and thus exponential in the size of the input). We then derive structural properties of dominant policies which we call optimality certificates. We show that computing a minimum optimality certificate is NP-hard, suggesting that optimal-in-expectation and/or Pareto optimal policies could be NP-hard to compute. Finally, we restrict attention to a setting in which agents on one side of the market have the same partially ordered preferences (but potentially distinct underlying true preferences), and in which agents must interview before matching. In this restricted setting, we show how to leverage the idea of minimum optimality certificates to design a computationally efficient interview-minimizing policy. This policy works without knowledge of the distributions and is dominant (and so is also Pareto optimal and optimal-in-expectation).

In combinatorial auctions that use VCG, a seller can sometimes increase revenue by dropping bidders (see e.g. [5]). In our previous work [26], we showed that such failures of “revenue monotonicity” occur under an extremely broad range of deterministic strategyproof combinatorial auction mechanisms, even when bidders have “known single-minded” valuations. In this work we consider the question of whether revenue monotonic, strategyproof mechanisms for such bidders can be found in the broader class of randomized mechanisms. We demonstrate that—surprisingly—such mechanisms do exist, show how they can be constructed, and consider algorithmic techniques for implementing them in polynomial time. More formally, we characterize a class of randomized mechanisms defined for known single-minded bidders that are strategyproof and revenue monotonic, and furthermore satisfy some other desirable properties, namely participation, consumer sovereignty and maximality, representing the mechanism as a solution to a quadratically constrained linear program (QCLP). We prove that the QCLP is always feasible (i.e., for all bidder valuations) and give its solution analytically. Furthermore, we give an algorithm for running such a mechanism in time polynomial in the number of bidders and goods; this is interesting because constructing an instance of such mechanisms from our QCLP formulation in a naive way can require exponential time.