Ordinal and cardinal solution concepts for two-sided matching

We characterize solutions for two-sided matching, both in the transferable- and in the nontransferable-utility frameworks, using a cardinal formulation. Our approach makes the comparison of the matching models with and without transfers particularly transparent. We introduce the concept of a no-trade stable matching to study the role of transfers in matching. A no-trade stable matching is one in which the availability of transfers does not affect the outcome

Social Welfare in One-Sided Matching Mechanisms

We study the Price of Anarchy of mechanisms for the well-known problem of one-sided matching, or house allocation, with respect to the social welfare objective. We consider both ordinal mechanisms, where agents submit preference lists over the items, and cardinal mechanisms, where agents may submit numerical values for the items being allocated. We present a general lower bound of $\Omega(\sqrt{n})$ on the Price of Anarchy, which applies to all mechanisms. We show that two well-known mechanisms, Probabilistic Serial, and Random Priority, achieve a matching upper bound. We extend our lower bound to the Price of Stability of a large class of mechanisms that satisfy a common proportionality property, and show stronger bounds on the Price of Anarchy of all deterministic mechanisms

Truthful approximations to range voting

We consider the fundamental mechanism design problem of approximate social welfare maximization under general cardinal preferences on a finite number of alternatives and without money. The well-known range voting scheme can be thought of as a non-truthful mechanism for exact social welfare maximization in this setting. With m being the number of alternatives, we exhibit a randomized truthful-in-expectation ordinal mechanism implementing an outcome whose expected social welfare is at least an Omega(m^{-3/4}) fraction of the social welfare of the socially optimal alternative. On the other hand, we show that for sufficiently many agents and any truthful-in-expectation ordinal mechanism, there is a valuation profile where the mechanism achieves at most an O(m^{-{2/3}) fraction of the optimal social welfare in expectation. We get tighter bounds for the natural special case of m = 3, and in that case furthermore obtain separation results concerning the approximation ratios achievable by natural restricted classes of truthful-in-expectation mechanisms. In particular, we show that for m = 3 and a sufficiently large number of agents, the best mechanism that is ordinal as well as mixed-unilateral has an approximation ratio between 0.610 and 0.611, the best ordinal mechanism has an approximation ratio between 0.616 and 0.641, while the best mixed-unilateral mechanism has an approximation ratio bigger than 0.660. In particular, the best mixed-unilateral non-ordinal (i.e., cardinal) mechanism strictly outperforms all ordinal ones, even the non-mixed-unilateral ordinal ones

Dominant and Nash Strategy Mechanisms for the Assignment Problem

In this paper, I examine the problem of matching or assigning a fixed set of goods or services to a fixed set of agents. I characterize the social choice correspondences that can be implemented in dominant and Nash strategies when transfers are not allowed. This is an extension of the literature that was begun by Gibbard (1973) and Satterthwaite (1975), who independently proved that if a mechanism is nonmanipulable it is dictatorial. For the classes of mechanisms that are described in the paper, the results imply that the only mechanisms that are implementable in dominant and Nash strategies are choice mechanisms that rely only on ordinal rankings. I also describe a subclass of mechanisms that are Pareto optimal. In addition, the results explain the modeling conventions found in the literature - that when nontransfer mechanisms are studied individuals are endowed with ordinal preferences, and when transfer mechanisms are studied individuals are endowed with cardinal preferences

Matching under Preferences

Matching theory studies how agents and/or objects from different sets can be matched with each other while taking agents\u2019 preferences into account. The theory originated in 1962 with a celebrated paper by David Gale and Lloyd Shapley (1962), in which they proposed the Stable Marriage Algorithm as a solution to the problem of two-sided matching. Since then, this theory has been successfully applied to many real-world problems such as matching students to universities, doctors to hospitals, kidney transplant patients to donors, and tenants to houses. This chapter will focus on algorithmic as well as strategic issues of matching theory. Many large-scale centralized allocation processes can be modelled by matching problems where agents have preferences over one another. For example, in China, over 10 million students apply for admission to higher education annually through a centralized process. The inputs to the matching scheme include the students\u2019 preferences over universities, and vice versa, and the capacities of each university. The task is to construct a matching that is in some sense optimal with respect to these inputs. Economists have long understood the problems with decentralized matching markets, which can suffer from such undesirable properties as unravelling, congestion and exploding offers (see Roth and Xing, 1994, for details). For centralized markets, constructing allocations by hand for large problem instances is clearly infeasible. Thus centralized mechanisms are required for automating the allocation process. Given the large number of agents typically involved, the computational efficiency of a mechanism's underlying algorithm is of paramount importance. Thus we seek polynomial-time algorithms for the underlying matching problems. Equally important are considerations of strategy: an agent (or a coalition of agents) may manipulate their input to the matching scheme (e.g., by misrepresenting their true preferences or underreporting their capacity) in order to try to improve their outcome. A desirable property of a mechanism is strategyproofness, which ensures that it is in the best interests of an agent to behave truthfully

A Pseudo-Market Approach to Allocation with Priorities

We propose a pseudo-market mechanism for no-monetary-transfer allocation of indivisible objects based on priorities such as those in school choice. Agents are given token money, face priority-specific prices, and buy utility-maximizing random assignments. The mechanism is asymptotically incentive compatible, and the resulting assignments are fair and constrained Pareto efficient. Hylland and Zeckhauser's (1979) position-allocation problem is a special case of our framework, and our results on incentives and fairness are also new in their classical setting

A Pseudo-Market Approach to Allocation with Priorities

We propose a pseudo-market mechanism for no-monetary-transfer allocation of indivisible objects based on priorities such as those in school choice. Agents are given token money, face priority-specific prices, and buy utility-maximizing random assignments. The mechanism is asymptotically incentive compatible, and the resulting assignments are fair and constrained Pareto efficient. Hylland and Zeckhauser's (1979) position-allocation problem is a special case of our framework, and our results on incentives and fairness are also new in their classical setting