Many centralized two-sided markets form a matching between participants by running a stable marriage algorithm. It is a well-known fact that no matching mechanism based on a stable marriage algorithm can guarantee truthfulness as a dominant strategy for participants. However, as we will show in this paper, in a probabilistic setting where the preference lists of one side of the market are composed of only a constant (independent of the the size of the market) number of entries, each drawn from an arbitrary distribution, the number of participants that have more than one stable partner is vanishingly small. This proves (and generalizes) a conjecture of Roth and Peranson [23]. As a corollary Ó of this result, we show that, with high probability, the truthful strategy is the best response for a given player when the other players are truthful. We also analyze equilibria of the deferred acceptance stable marriage game. We show that the game with complete information has an equilibrium in which a fraction of the strategies are truthful in expectation. In the more realistic setting of a game of incomplete information, we will show that the set of truthful strate-gies form a Ó

This dissertation studies mechanism design for various combinatorial problems in the presence of strategic agents. A mechanism is an algorithm for allocating a resource among a group of participants, each of which has a privately-known value for any particular allocation. A mechanism is truthful if it is in each participant’s best interest to reveal his private information truthfully regardless of the strategies of the other participants. First, we explore a competitive auction framework for truthful mechanism design in the setting of multi-unit auctions, or auctions which sell multiple identical copies of a good. In this framework, the goal is to design a truthful auction whose revenue approximates that of an omniscient auction for any set of bids. We focus on two natural settings — the limited demand setting where bidders desire at most a fixed number of copies and the limited budget setting where bidders can spend at most a fixed amount of money. In the limit demand setting, all prior auctions employed the use of randomization in the computation of the allocation and prices. Randomization

Abstract. We study the Student-Project Allocation problem (SPA), a generalisation of the classical Hospitals / Residents problem (HR). An instance of SPA involves a set of students, projects and lecturers. Each project is offered by a unique lecturer, and both projects and lecturers have capacity constraints. Students have preferences over projects, whilst lecturers have preferences over students. We present two optimal linear-time algorithms for allocating students to projects, subject to the preference and capacity constraints. In particular, each algorithm finds a stable matching of students to projects. Here, the concept of stability generalises the stability definition in the HR context. The stable matching produced by the first algorithm is simultaneously best-possible for all students, whilst the one produced by the second algorithm is simultaneously best-possible for all lecturers. We also prove some structural results concerning the set of stable matchings in a given instance of SPA. The SPA problem model that we consider is very general and has applications to a range of different contexts besides student-project allocation. 1

procedures

Abstract. This paper addresses strategies for the stable marriage problem. For the Gale-Shapley algorithm with men proposing, a classical theorem states that it is impossible for every cheating man to get a better partner than the one he gets if everyone is truthful. We study how to circumvent this theorem and incite men to cheat. First we devise coalitions in which a non-empty subset of the liars get better partners and no man is worse off than before. This strategy is limited in that not everyone in the coalition has the incentive to falsify his list. In an attempt to rectify this situation we introduce the element of randomness, but the theorem shows surprising robustness: it is impossible that every liar has a chance to improve the rank of his partner while no one gets hurt. To overcome the problem that some men lack the motivation to lie, we exhibit another randomized lying strategy in which every liar can expect to get a better partner on average, though with a chance of getting a worse one. Finally, we consider a variant scenario: instead of using the Gale-Shapley algorithm, suppose the stable matching is chosen at random. We present a modified form of the coalition strategy ensuring that every man in the coalition has a new probability distribution over partners which majorizes the original one. 1

We study strategy issues surrounding the stable marriage problem. Under the Gale-Shapley algorithm (with men proposing), a classical theorem says that it is impossible for every liar to get a better partner. We try to challenge this theorem. First, observing a loophole in the statement of the theorem, we devise a coalition strategy in which a non-empty subset of the liars gets a better partner and no man is worse off than before. This strategy is restricted in that not everyone has the incentive to cheat. We attack the classical theorem further by means of randomization. However, this theorem shows surprising robustness: it is impossible that every liar has the chance to improve while no one gets hurt. Hence, this impossibility result indicates that it is always hard to induce some people to falsify their lists. Finally, to overcome the problem of lacking motivation, we exhibit another randomized lying strategy in which every liar can expect to get a better partner, though with a chance of getting a worse one. 1

Abstract. The stable marriage problem has recently been studied in its general setting, where both ties and incomplete lists are allowed. It is NP-hard to find a stable matching of maximum size, while any stable matching is a maximal matching and thus trivially a factor two approximation. In this paper, we give the first nontrivial result for approximation of factor less than two. Our algorithm achieves an approximation ratio of 2/(1+L −2) for instances in which only men have ties of length at most L. When both men and women are allowed to have ties, we show a ratio of 13/7(< 1.858) for the case when ties are of length two. We also improve the lower bound on the approximation ratio to 21

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Abstract. Modellers would like to automatically translate a declarative conceptual model of a constrained combinatorial problem into an efficient design model. However, for efficiency, most solving techniques require a model-specific search strategy, which is inherently procedural and, thus, difficult to integrate with the declarative conceptual model. This paper describes a small, non-intrusive extension to the declarative modelling language Zinc that achieves such an integration by providing a number of generic search patterns that take Zinc user-defined functions as parameters. We show the generality of the approach by using it to implement three very different kinds of search: backtracking search, branchand-bound search, and local search. In each case, the work done by the modeller is small, while the performance is competitive with hand-coded search strategies. This demonstrates that search can be implemented in a manner that maintains the benefits of declarative modelling languages, while still being reasonably efficient. 1