Nash bargaining in ordinal environments

We analyze the implications of Nash’s (1950) axioms in ordinal bargaining environments; there, the scale invariance axiom needs to be strenghtened to take into account all order-preserving transformations of the agents’ utilities. This axiom, called ordinal invariance, is a very demanding one. For two-agents, it is violated by every strongly individually rational bargaining rule. In general, no ordinally invariant bargaining rule satisfies the other three axioms of Nash. Parallel to Roth (1977), we introduce a weaker independence of irrelevant alternatives axiom that we argue is better suited for ordinally invariant bargaining rules. We show that the three-agent Shapley-Shubik bargaining rule uniquely satisfies ordinal invariance, Pareto optimality, symmetry, and this weaker independence of irrelevant alternatives axiom. We also analyze the implications of other independence axioms

Integer programming methods for special college admissions problems

We develop Integer Programming (IP) solutions for some special college admission problems arising from the Hungarian higher education admission scheme. We focus on four special features, namely the solution concept of stable score-limits, the presence of lower and common quotas, and paired applications. We note that each of the latter three special feature makes the college admissions problem NP-hard to solve. Currently, a heuristic based on the Gale-Shapley algorithm is being used in the application. The IP methods that we propose are not only interesting theoretically, but may also serve as an alternative solution concept for this practical application, and also for other ones

Group Strategyproof Pareto-Stable Marriage with Indifferences via the Generalized Assignment Game

We study the variant of the stable marriage problem in which the preferences of the agents are allowed to include indifferences. We present a mechanism for producing Pareto-stable matchings in stable marriage markets with indifferences that is group strategyproof for one side of the market. Our key technique involves modeling the stable marriage market as a generalized assignment game. We also show that our mechanism can be implemented efficiently. These results can be extended to the college admissions problem with indifferences

Manipulation Strategies for the Rank Maximal Matching Problem

We consider manipulation strategies for the rank-maximal matching problem. In the rank-maximal matching problem we are given a bipartite graph $G = (A \cup P, E)$ such that $A$ denotes a set of applicants and $P$ a set of posts. Each applicant $a \in A$ has a preference list over the set of his neighbours in $G$, possibly involving ties. Preference lists are represented by ranks on the edges - an edge $(a,p)$ has rank $i$, denoted as $rank(a,p)=i$, if post $p$ belongs to one of $a$'s $i$-th choices. A rank-maximal matching is one in which the maximum number of applicants is matched to their rank one posts and subject to this condition, the maximum number of applicants is matched to their rank two posts, and so on. A rank-maximal matching can be computed in $O(\min(c \sqrt{n},n) m)$ time, where $n$ denotes the number of applicants, $m$ the number of edges and $c$ the maximum rank of an edge in an optimal solution. A central authority matches applicants to posts. It does so using one of the rank-maximal matchings. Since there may be more than one rank- maximal matching of $G$, we assume that the central authority chooses any one of them randomly. Let $a_1$ be a manipulative applicant, who knows the preference lists of all the other applicants and wants to falsify his preference list so that he has a chance of getting better posts than if he were truthful. In the first problem addressed in this paper the manipulative applicant $a_1$ wants to ensure that he is never matched to any post worse than the most preferred among those of rank greater than one and obtainable when he is truthful. In the second problem the manipulator wants to construct such a preference list that the worst post he can become matched to by the central authority is best possible or in other words, $a_1$ wants to minimize the maximal rank of a post he can become matched to

Stable schedule matching under revealed preference

Baiou and Balinski (Math. Oper. Res., 27 (2002) 485) studied schedule matching where one determines the partnerships that form and how much time they spend together, under the assumption that each agent has a ranking on all potential partners. Here we study schedule matching under more general preferences that extend the substitutable preferences in Roth (Econometrica 52 (1984) 47) by an extension of the revealed preference approach in Alkan (Econom. Theory 19 (2002) 737). We give a generalization of the GaleShapley algorithm and show that some familiar properties of ordinary stable matchings continue to hold. Our main result is that, when preferences satisfy an additional property called size monotonicity, stable matchings are a lattice under the joint preferences of all agents on each side and have other interesting structural properties

Approximation algorithms for hard variants of the stable marriage and hospitals/residents problems

When ties and incomplete preference lists are permitted in the Stable Marriage and Hospitals/Residents problems, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is known to be NP-hard, even under very severe restrictions on the number, size and position of ties. In this paper, we describe polynomial-time 5/3-approximation algorithms for variants of these problems in which ties are on one side only and at the end of the preference lists. The particular variant is motivated by important applications in large scale centralised matching schemes

Bargaining over a finite set of alternatives

We analyze bilateral bargaining over a finite set of alternatives. We look for “good” ordinal solutions to such problems and show that Unanimity Compromise and Rational Compromise are the only bargaining rules that satisfy a basic set of properties. We then extend our analysis to admit problems with countably infinite alternatives. We show that, on this class, no bargaining rule choosing finite subsets of alternatives can be neutral. When rephrased in the utility framework of Nash (1950), this implies that there is no ordinal bargaining rule that is finite-valued

An Integer Programming Approach to the Student-Project Allocation Problem with Preferences over Projects

The Student-Project Allocation problem with preferences over Projects (SPA-P) involves sets of students, projects and lecturers, where the students and lecturers each have preferences over the projects. In this context, we typically seek a stable matching of students to projects (and lecturers). However, these stable matchings can have different sizes, and the problem of finding a maximum stable matching (MAX-SPA-P) is NP-hard. There are two known approximation algorithms for MAX-SPA-P, with performance guarantees of 2 and 32 . In this paper, we describe an Integer Programming (IP) model to enable MAX-SPA-P to be solved optimally. Following this, we present results arising from an empirical analysis that investigates how the solution produced by the approximation algorithms compares to the optimal solution obtained from the IP model, with respect to the size of the stable matchings constructed, on instances that are both randomly-generated and derived from real datasets. Our main finding is that the 32 -approximation algorithm finds stable matchings that are very close to having maximum cardinality

New and simple algorithms for stable flow problems

Stable flows generalize the well-known concept of stable matchings to markets in which transactions may involve several agents, forwarding flow from one to another. An instance of the problem consists of a capacitated directed network, in which vertices express their preferences over their incident edges. A network flow is stable if there is no group of vertices that all could benefit from rerouting the flow along a walk. Fleiner established that a stable flow always exists by reducing it to the stable allocation problem. We present an augmenting-path algorithm for computing a stable flow, the first algorithm that achieves polynomial running time for this problem without using stable allocation as a black-box subroutine. We further consider the problem of finding a stable flow such that the flow value on every edge is within a given interval. For this problem, we present an elegant graph transformation and based on this, we devise a simple and fast algorithm, which also can be used to find a solution to the stable marriage problem with forced and forbidden edges. Finally, we study the stable multicommodity flow model introduced by Kir\'{a}ly and Pap. The original model is highly involved and allows for commodity-dependent preference lists at the vertices and commodity-specific edge capacities. We present several graph-based reductions that show equivalence to a significantly simpler model. We further show that it is NP-complete to decide whether an integral solution exists

Sequential Deliberation for Social Choice

In large scale collective decision making, social choice is a normative study of how one ought to design a protocol for reaching consensus. However, in instances where the underlying decision space is too large or complex for ordinal voting, standard voting methods of social choice may be impractical. How then can we design a mechanism - preferably decentralized, simple, scalable, and not requiring any special knowledge of the decision space - to reach consensus? We propose sequential deliberation as a natural solution to this problem. In this iterative method, successive pairs of agents bargain over the decision space using the previous decision as a disagreement alternative. We describe the general method and analyze the quality of its outcome when the space of preferences define a median graph. We show that sequential deliberation finds a 1.208- approximation to the optimal social cost on such graphs, coming very close to this value with only a small constant number of agents sampled from the population. We also show lower bounds on simpler classes of mechanisms to justify our design choices. We further show that sequential deliberation is ex-post Pareto efficient and has truthful reporting as an equilibrium of the induced extensive form game. We finally show that for general metric spaces, the second moment of of the distribution of social cost of the outcomes produced by sequential deliberation is also bounded