A many-to-one job market: more about the core and the competitive salaries

TL;DR

The paper analyzes many-to-one transfer‑valuation markets with firms and unit-capacity workers, focusing on the core and its relation to competitive salaries. It provides both axiom‑based and graph‑theoretic characterizations of maximum and minimum competitive salaries and introduces tight digraphs to characterize extreme core allocations. A key result is that the kernel need not be contained in the core and that the core and bargaining set may fail to coincide in this setting, highlighting robustness concerns for stable allocations. The authors also present a lexicographic max–min procedure that generates all extreme competitive salary vectors, and extend the analysis to Kaneko’s buyer–seller market, illustrating broad applicability and implications for stability and non-manipulability in two-sided markets.

Abstract

This paper studies many-to-one assignment markets, or matching markets with wages. Although it is well-known that the core of this model is non-empty, the structure of the core has not been fully investigated. To the known dissimilarities with the one-to-one assignment game, we add that the bargaining set does not coincide with the core and the kernel may not be included in the core. Besides, not all extreme core allocations can be obtained by means of a lexicographic maximization or a lexicographic minimization procedure, as it is the case in the one-to-one assignment game. The maximum and minimum competitive salaries are characterized in two ways: axiomatically and by means of easily verifiable properties of an associated directed graph. Regarding the remaining extreme core allocations of the many-to-one assignment game, we propose a lexicographic procedure that, for each order on the set of workers, sequentially maximizes or minimizes each worker's competitive salary. This procedure provides all extreme vectors of competitive salaries, that is all extreme core allocations.

Figures (4)

• Figure 1: $C(W)$ for the many-to-one market of Example \ref{['example:core_workers']}
• Figure 2: digraph of minimum vector $(3,2,0)$, digraph of maximum vector $(8,6,4)$
• Figure 3: digraph of extreme vector $(3,3,0)$ digraph of extreme vector $(7,6,4)$
• Figure 4: Extended tight digraph of the minimum CE payoff vector (4,2,0).

Theorems & Definitions (32)

• Proposition 1
• Proposition 2
• Example 3
• Example 4
• Corollary 5
• Proposition 6
• proof
• Definition 7
• Theorem 8
• proof