• Abstract. Suppose agents are to be matched to objects and arrive over time without a definite terminal date. Although the set of core matchings can then be empty, an extended version of the top trading cycles algorithm shows that a Pareto-optimal weak-core matching always exists. Optimal matchings face a difficulty however: some of the agents linked by chains of trades may have lifespans that do not overlap, thus obstructing their trades. To address this problem, we let matchings be implemented via competitive markets. Competitive equilibria always exist and any matching in the core can be competitively implemented. Moreover full core equivalence, where allocations are in the core iff they can be competitively implemented, holds for a dense set of models. The extended algorithm also yields a strategyproof mechanism, comparably to the finite model.

Sophie Bade (Royal Holloway)

19 January '22

• Bio. Sophie Bade is a Professor of Economics at Royal Holloway. She received her PhD from NYU under the supervision of Efe Ok. In her PhD, Sophie worked on game theoretic models where the players' behavior deviates from standard rationality assumptions. While she continues to be interested in how to integrate novel decision theoretic models into interactive settings, she developed a second strand of research on mechanism design. As a first step in that direction, Sophie worked on house matching markets with endogenous information acquisition. More recent interests include random matching models, the notion of obvious strategyproofness, and matching problems with convex preferences.

Historical Notes and Further Readings

• Classic matching theory concerns problems with finitely many agents and objects. Shapley and Scarf's (1974) housing markets are a model of finitely many agents and equally many houses. Gale and Shapley's (1962) marriage markets are defined for finitely many men and women. We start with the observation that these models are in practice always applied to sets with arbitrary boundaries. Al Roth famously suggests using matching models for patients with end stage renal disease and donor kidneys. To apply existing models we then must use - somewhat arbitrary - spatial and temporal criteria to limit the patients and kidneys who take part. In our paper we remove these limits from the model and consider the case of countably many agents and objects. Otherwise, we depart as parsimoniously as possible from Shapley and Scarf (1974).
  
  Our results that the Pareto optimal weak core of is always non-empty and that there exist strategyproof Pareto optimal matching mechanisms rely on an extension of Gale's top trading cycles, introduced by Shapely and Scarf (1974). In an interesting contrast, Jagadeesan (2018) shows that the Gale-Shapley (1962) deferred acceptance algorithm for bilateral matching applies without change to the case of countably many agents. The extension of Gale's top trading cycles is not sufficient to show the existence of market equilibrium. For that existence proof we use a Cantor diagonalisation argument that bears some resemblance to arguments for the existence of equilibria in the overlapping generations model of general equilibrium theory (e.g., Balasko, Cass and Shell (1980)).

• Related Work

1. Balasko Y., D. Cass, and K. Shell. "Existence of competitive equilibrium in a general overlapping-generations model." Journal of Economic Theory (1980) 23: 307-322.
2. Gale D. and L. Shapley. "College admissions and the stability of marriage." American Mathematical Monthly (1962) 69: 9-14.
3. Jagadeesana R. "Lone wolves in infinite, discrete matching markets." Games and Economic Behavior (2018) 108: 275-286.
4. Shapley L. and H. Scarf. "On Cores and Indivisibility." Journal of Mathematical Economics (1974) 1: 23-37.