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We highlight the tension between stability and equality in non transferable utility matching. We consider many to one matchings and refer to the two sides of the market as students and schools. The latter have aligned preferences, which in this context means that a schools utility is the sum of its students utilities. We show that the unique stable allocation displays extreme inequality between matched pairs.
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We study conditions for the existence of stable and group-strategy-proof mechanisms in a many-to-one matching model with contracts if students preferences are monotone in contract terms. We show that equivalence, properly defined, to a choice profile under which contracts are substitutes and the law of aggregate holds is a necessary and sufficient condition for the existence of a stable and group-strategy-proof mechanism. Our result can be interpreted as a (weak) embedding result for choice functions under which contracts are observable substitutes and the observable law of aggregate demand holds.
We provide necessary and sufficient conditions on the preferences of market participants for a unique stable matching in models of two-sided matching with non-transferable utility. We use the process of iterated deletion of unattractive alternatives (IDUA), a formalisation of the reduction procedure in Balinski and Ratier (1997), and we show that an instance of the matching problem possesses a unique stable matching if and only if IDUA collapses each participant preference list to a singleton. (This is in a sense the matching problem analog of a strategic game being dominance solvable.)
In a many-to-one matching model in which firms preferences satisfy substitutability, we study the set of worker-quasi-stable matchings. Worker-quasi-stability is a relaxation of stability that allows blocking pairs involving a firm and an unemployed worker. We show that this set has a lattice structure and define a Tarski operator on this lattice that models a re-equilibration process and has the set of stable matchings as its fixed points.
In discrete matching markets, substitutes and complements can be unidirectional between two groups of workers when members of one group are more important or competent than those of the other group for firms. We show that a stable matching exists and can be found by a two-stage Deferred Acceptance mechanism when firms preferences satisfy a unidirectional substitutes and complements condition. This result applies to both firm-worker matching and controlled school choice. Under the framework of matching with continuous monetary transfers and quasi-linear utilities, we show that substitutes and complements are bidirectional for a pair of workers.
We provide two characterizations, one axiomatic and the other neuro-computational, of the dependence of choice probabilities on deadlines, within the widely used softmax representation [ p_{t}left( a,Aright) =dfrac{e^{frac{uleft( aright) }{lambda left( tright) }+alpha left( aright) }}{sum_{bin A}e^{frac{uleft( bright) }{lambda left( tright) }+alpha left( bright) }}% ] where $p_{t}left( a,Aright) $ is the probability that alternative $a$ is selected from the set $A$ of feasible alternatives if $t$ is the time available to decide, $lambda$ is a time dependent noise parameter measuring the unit cost of information, $u$ is a time independent utility function, and $alpha$ is an alternative-specific bias that determines the initial choice probabilities reflecting prior information and memory anchoring. Our axiomatic analysis provides a behavioral foundation of softmax (also known as Multinomial Logit Model when $alpha$ is constant). Our neuro-computational derivation provides a biologically inspired algorithm that may explain the emergence of softmax in choice behavior. Jointly, the two approaches provide a thorough understanding of soft-maximization in terms of internal causes (neurophysiological mechanisms) and external effects (testable implications).
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