32 research outputs found
To Infinity and Beyond: Scaling Economic Theories via Logical Compactness
Many economic-theoretic models incorporate finiteness assumptions that, while
introduced for simplicity, play a real role in the analysis. Such assumptions
introduce a conceptual problem, as results that rely on finiteness are often
implicitly nonrobust; for example, they may depend upon edge effects or
artificial boundary conditions. Here, we present a unified method that enables
us to remove finiteness assumptions, such as those on market sizes, time
horizons, and datasets. We then apply our approach to a variety of matching,
exchange economy, and revealed preference settings.
The key to our approach is Logical Compactness, a core result from
Propositional Logic. Building on Logical Compactness, in a matching setting, we
reprove large-market existence results implied by Fleiner's analysis, and
(newly) prove both the strategy-proofness of the man-optimal stable mechanism
in infinite markets and an infinite-market version of Nguyen and Vohra's
existence result for near-feasible stable matchings with couples. In a
trading-network setting, we prove that the Hatfield et al. result on existence
of Walrasian equilibria extends to infinite markets. In a dynamic matching
setting, we prove that Pereyra's existence result for dynamic two-sided
matching markets extends to a doubly infinite time horizon. Finally, beyond
existence and characterization of solutions, in a revealed-preference setting
we reprove Reny's infinite-data version of Afriat's theorem and (newly) prove
an infinite-data version of McFadden and Richter's characterization of
rationalizable stochastic datasets
"Almost-stable" matchings in the Hospitals / Residents problem with Couples
The Hospitals / Residents problem with Couples (hrc) models the allocation of intending junior doctors to hospitals where couples are allowed to submit joint preference lists over pairs of (typically geographically close) hospitals. It is known that a stable matching need not exist, so we consider min bp hrc, the problem of finding a matching that admits the minimum number of blocking pairs (i.e., is “as stable as possible”). We show that this problem is NP-hard and difficult to approximate even in the highly restricted case that each couple finds only one hospital pair acceptable. However if we further assume that the preference list of each single resident and hospital is of length at most 2, we give a polynomial-time algorithm for this case. We then present the first Integer Programming (IP) and Constraint Programming (CP) models for min bp hrc. Finally, we discuss an empirical evaluation of these models applied to randomly-generated instances of min bp hrc. We find that on average, the CP model is about 1.15 times faster than the IP model, and when presolving is applied to the CP model, it is on average 8.14 times faster. We further observe that the number of blocking pairs admitted by a solution is very small, i.e., usually at most 1, and never more than 2, for the (28,000) instances considered
On the Complexity of Stable Fractional Hypergraph Matching
In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system. Aharoni and Fleiner proved that there exists a stable fractional matching in every hypergraphic preference system. Furthermore, Kintali, Poplawski, Rajaraman, Sundaram, and Teng proved that the problem of finding a stable fractional matching in a hypergraphic preference system is PPAD-complete. In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system whose maximum degree is bounded by some constant. The proof by Kintali, Poplawski, Rajaraman, Sundaram, and Teng implies the PPAD-completeness of the problem of finding a stable fractional matching in a hypergraphic preference system whose maximum degree is 5. In this paper, we prove that (i) this problem is PPAD-complete even if the maximum degree is 3, and (ii) if the maximum degree is 2, then this problem can be solved in polynomial time. Furthermore, we prove that the problem of finding an approximate stable fractional matching in a hypergraphic preference system is PPAD-complete
Inducing stability in hedonic games
In many applications of coalition formation games, a key issue is that some desirable coalition structures are not elements of the core of these games. In these cases, it would be useful for an authority which aims to implement a certain outcome to know how far from the original game is the nearest game where the desirable outcome is part of the core. This question is at the center of this study. Focusing on hedonic games, we uncover previously unexplored links between such games and transferrable utility games, and develop a tailor-made solution concept for the transferrable utility game, the implementation core, to provide an answer to our question
Matching couples with Scarf’s algorithm
Scarf's algorithm [20] provides fractional core elements for NTU-games. Bir�o and
Fleiner [3] showed that Scarf's algorithm can be extended for capacitated NTU-games. In
this setting agents can be involved in more than one coalition at a time, cooperations may be
performed with di�erent intensities up to some limits, and the contribution of the agents can
also di�er in a coalition. The fractional stable solutions for the above model, produced by the
extended Scarf algorithm, are called stable allocations. In this paper we apply this solution
concept for the Hospitals / Residents problem with Couples (HRC). This is one of the most
important general stable matching problems due to its relevant applications, also well-known
to be NP-hard. We show that if a stable allocation yielded by the Scarf algorithm turns out
to be integral then it provides a stable matching for an instance of HRC, so this method
can be used as a heuristic. In an experimental study, we compare this method with other
heuristics constructed for HRC that are applied in practice in the American and Scottish
resident allocation programs, respectively. Our main �nding is that the Scarf algorithm
outperforms all the other known heuristics when the proportion of couples is high