Testing substitutability

Japan Journal of Industrial and Applied Mathematics, 2012

In this paper, we study a many-to-many matching model with contracts. We extend the economic model of Hatfield and Milgrom by allowing a partial order on the possible bilateral contracts of the agents in a two-sided market economy. To prove that a generalized stable allocation exists, we use generalized form of properties like path independence and substitutability. The key to our results is the well-known lattice theoretical fixed point theorem of Tarski. The constructive proof of this fixed point theorem for finite sets turns out to be the appropriate generalization of the Gale-Shapley algorithm also in our general setting.

2019

A major goal in algorithmic game theory is to understand the communication complexity of truthful combinatorial auctions. There were few known lower bounds on the number of value queries required, and no known lower bounds on the number of demand queries or amount of general communication required. This paper shows that for rich enough domains of valuation functions, the communication complexity of a truthful mechanism on that domain is characterized by the “taxation complexity” of that mechanism. A similar characterization is developed for the number of value queries and demand queries in terms of “menu complexity” and “affinity”. This provides a new technique to lower bound the communication complexity of truthful mechanisms. The author also shows that truthful two-player mechanisms give rise to simultaneous mechanisms achieving the same welfare, suggesting the possibly easier approach of proving lower bounds for simultaneous communication problems.

American Economic Review, 2012

We consider the e¢ cient allocation of a single good with interdependent values in a quasi-linear environment. We present an approach to modelling interdependent preferences distinguishing between "payo¤ types" and "belief types" and report a characterization of when the e¢ cient allocation can be partially Bayesian implemented on a …nite type space. The characterization can be used to unify a number of su¢ cient conditions for e¢ cient partial implementation in this classical auction setting.

Arxiv preprint math/0401199, 2004

In this paper, we consider the problem of choosing a set of multi-party contracts, where each coalition of agents has a non-empty finite set of feasible contracts to choose from. We call such problems, contract choice problems. We provide conditions under which a contract ...

SSRN Electronic Journal, 2017

Each acceptant and substitutable choice rule is known to have a maximizer-collecting representation: there exists a list of priority orderings such that from each choice set that includes more elements than the capacity, the choice is the union of the priority orderings' maximizers (Aizerman and Malishevski, 1981). We introduce the notion of a prime atom and constructively prove that the number of prime atoms of a choice rule determines its smallest size maximizer-collecting representation. We show that responsive choice rules require the maximal number of priority orderings in their maximizer-collecting representations among all acceptant and substitutable choice rules. We characterize maximizer-collecting choice rules in which the number of priorities equals the capacity. We also show that if the capacity is greater than three and the number of elements exceeds the capacity by at least two, then no acceptant and substitutable choice rule has a maximizer-collecting representation of the size equal to the capacity.

2012

In this appendix, we provide a proof of the maximal domain result (Theorem 7) of "Stability and Competitive Equilibrium in Trading Networks." We also provide the counterexamples referenced in Section 4.2 of "Stability and Competitive Equilibrium in Trading Networks."

Journal of Economic Theory, 2013

In a setting where agents have quasi-linear utilities over social alternatives and a transferable commodity, we consider three properties that a social choice function may possess: truthful implementation (in dominant strategies); monotonicity in differences; lexicographic affine maximization. We introduce the notion of a flexible domain of preferences that allows elevation of pairs and study which of these conditions implies which others when the domain is flexible. We provide a generalization of the theorem of Roberts (1979) in restricted valuation domains. Flexibility holds (and the theorem is not vacuous) if the domain of valuation profiles is restricted to the space of continuous functions defined on a topological space, or the space of piecewise linear functions defined on an affine space, or the space of smooth functions defined on a compact differentiable manifold. Our results can be applied in both public and private goods allocation settings, with finite or infinite alternative sets.

2008

Skyline Queries have recently received a lot of attention due to their intuitive query capabilities. Following the concept of Pareto optimality all 'best' database objects are returned to the user. However, this often results in unmanageable large result set sizes hampering the success of this innovative paradigm. As an effective remedy for this problem, trade-offs provide a natural concept for dealing with incomparable choices. But such trade-offs are not reflected by the Pareto paradigm. Thus, incorporating them into the users' preference orders and adjusting skyline results accordingly needs special algorithms beyond traditional skylining. For the actual integration of trade-offs into skylines, the problem of ensuring the consistency of arbitrary trade-off sets poses a demanding challenge. Consistency is a crucial aspect when dealing with multi-dimensional trade-offs spanning over several attributes. If the consistency should be violated, cyclic preferences may occur in the result set. But such cyclic preferences cannot be handled by information systems in a sensible way. Often, this problem is circumvented by restricting the trade-offs' expressiveness, e.g. by altogether ignoring some classes of possibly inconsistent tradeoffs. In this paper, we will present a new algorithm capable of efficiently verifying the consistency of any arbitrary set of trade-offs. After motivating its basic concepts and introducing the algorithm itself, we will also show that it exhibits superior average-case performance. The benefits of our approach promise to pave the way towards personalized and cooperative information systems.

2009

The existence of incentive-compatible computationally-efficient protocols for combinatorial auctions with decent approximation ratios is the paradigmatic problem in computational mechanism design. It is believed that in many cases good approximations for combinatorial auctions may be unattainable due to an inherent clash between truthfulness and computational efficiency. However, to date, researchers lack the machinery to prove such results. In this paper, we present a new approach: We take the first steps towards the development of new technologies for lower bounding the VC-dimension of k-tuples of disjoint sets. We apply this machinery to prove the first computational-complexity inapproximability results for incentive-compatible mechanisms for combinatorial auctions. These results hold for the important class of VCG-based mechanisms, and are based on the complexity assumption that NP has no polynomial-size circuits. We believe that our approach holds great promise. Indeed, subsequently to our work, Buchfuhrer and Umans , and, independently, Dughmi et al. , strengthened our results via extensions of our techniques, thus proving conjectures we presented in

Economics Letters, 1984

It is shown that if there are three commodities or less, an excess demand function that fulfills the property of Gross Substitutability satisfies also the Weak Axiom of Revealed Preference.