# Core Stability and Core Selection in a Decentralized Labor Matching Market

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Motivation

## 2. Related Literature

## 3. Model

#### 3.1. Generalized Many-to-One Matching

#### Consequence of Linearly Separable Utilities

**Proposition 1.**

**Proof.**

#### 3.2. Re-Formulation

#### 3.3. Solution Concepts

**Remark 2.**

#### 3.4. Dynamics

#### 3.4.1. Inter-Company Bargaining

#### 3.4.2. Intra-Company Bargaining

## 4. Analysis

**Theorem 3.**

- the process is absorbed into the core with probability 1 in finite time,
- for $\beta \to \infty $ and $\epsilon >0$, the process converges to $fw$-BS in the core.

#### 4.1. Optimality and Stability

**Lemma 4.**

**Proof.**

**Lemma 5.**

- 1a.
- there exists a positive probability transition such that the number of unemployed workers increases and no previously unemployed worker is employed,and/or
- 1b.
- there exists a positive probability transition such that the sum of aspiration levels decreases,or
- 2.
- $[{\mathbf{M}}^{t},{\mathbf{a}}^{t},{\mathbf{\alpha}}^{t}]$ is such that any worker either has aspiration level zero and is unemployed or he is employed in an optimal matching with stable matching and aspiration levels.

**Proof.**

**Lemma 6.**

**Proof.**

**Lemma 7.**

**Proof.**

#### 4.2. Equity

**Lemma 8.**

**Proof.**

**Lemma 9.**

**Proof.**

**Lemma 10.**

**Proof.**

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

^{1.}Alternatively, the auction may begin with maximum wages, and with reductions being made by workers without offers.^{3.}Two workers are gross substitutes for one another if the demand of any firm for one does not go down if the wage of the other goes up.^{4.}Nax et al. [29] generalize this assumption of a one-dimensional aspiration vector to multi-dimensional demand vectors. A similar generalization may be applied in this framework.^{5.}The equality in Equation (8) stems from the following tie-breaking rule: Firms (and workers) prefer to be active (employed) over being inactive (unemployed). If a worker is currently employed with a payoff of zero they prefer to remain with their current company over joining another company where their payoff will be zero.^{6.}If ${\mathcal{A}}_{\epsilon}({({\mu}_{j}^{t+1})}_{j\in C})=0$, we set set $\nu ({({\mu}_{j}^{t+1})}_{j\in C})=\infty $.^{7.}Rochford [44] first introduced such a solution as a pairwise-bargained solution for the one-to-one assignment game.^{8.}Note that this function is often used for smooth-perturbed best-response modeling; other functions, such as probit, would yield qualitatively the same result.

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Nax, H.H.; Pradelski, B.S.R.
Core Stability and Core Selection in a Decentralized Labor Matching Market. *Games* **2016**, *7*, 10.
https://doi.org/10.3390/g7020010

**AMA Style**

Nax HH, Pradelski BSR.
Core Stability and Core Selection in a Decentralized Labor Matching Market. *Games*. 2016; 7(2):10.
https://doi.org/10.3390/g7020010

**Chicago/Turabian Style**

Nax, Heinrich H., and Bary S. R. Pradelski.
2016. "Core Stability and Core Selection in a Decentralized Labor Matching Market" *Games* 7, no. 2: 10.
https://doi.org/10.3390/g7020010