# Fairness Is an Emergent Self-Organized Property of the Free Market for Labor

## Abstract

**:**

## 1. Introduction

## 2. Emergence of Fairness through Self-organizing Free Market Dynamics

_{ave}= M/N. Let us assume that there are k categories of employees -- ranging from secretaries to the CEO, contributing in different ways towards the company’s overall success and value creation. All employees in category i contribute value V

_{i}, i ∈ {1,2, … k}, such that V

_{1}< V

_{2}<….< V

_{k}. Let the corresponding value at S

_{ave}be V

_{ave}, occurring at category s. Since all employees are contributing unequally, some more some less, they all need to be compensated differently, commensurate with their relative contributions towards the overall value created by the company. Instead, A has an egalitarian policy that all employees are equal and therefore pays all of them the same salary, S

_{ave}, irrespective of their contributions. The salary of the CEO is the same as that of an administrative assistant in the mail room. This salary distribution is a sharp vertical line at S

_{ave}, as seen in Figure 1(a), the Kronecker delta function of magnitude N, given by:

_{is}, where δ

_{is}= 1, if i = s and δ

_{is}= 0, if i ≠ s

_{j}such that V

_{j}> V

_{ave}(e.g., senior engineers, vice presidents, CEO) would feel that their contributions are not fairly valued and compensated for by A, and will therefore be motivated to leave for other companies where they are offered higher salaries. Hence, in order to survive A will be forced to match the salaries offered by others to retain these employees, thereby forcing the distribution to spread to the right of S

_{ave}, as seen in Figure 1(b).

_{j}such that V

_{j}< V

_{ave}, will motivate candidates with the relevant skill sets (e.g., low-level administration, sales and marketing staff) from other companies to compete for these higher paying positions in A. This competition will eventually drive the compensation down for these overpaid employees forcing the distribution to spread to the left of S

_{ave}, as seen in Figure 1(c). Eventually, we will have a distribution that is not a delta function, but a broader one where different employees make different salaries depending on the values of their contributions. The funds for the higher salaries now paid to the formerly underpaid employees (i.e., those who satisfy V

_{j}> V

_{ave}) come out of the savings resulting from the reduced salaries of the formerly overpaid group (i.e., those who satisfy V

_{j}< V

_{ave}), thereby conserving the total salary budget M.

## 3. Entropy as the Appropriate Measure of Fairness in an Ideal Free Market System

#### 3.1. Entropy as a Measure of Fairness: Statistical Thermodynamics Approach

_{ave}= E/N. In such an isolated system, both N and E are conserved. Now imagine starting this system at t=0 in the initial state where all molecules are assigned to have exactly the same energy E

_{ave}. In time, the system will evolve from this initial state of a Kronecker delta distribution in energy to one where the energy distribution is more spread out as a result of intermolecular collisions which cause varied energy exchanges among the molecules.

_{i}of an employee i. In reality, this space will be more complicated with additional dimensions that capture other features such as titles, awards, education, experience, etc., but let us restrict ourselves to salary as it is the dominant feature. A point in this ideal phase space represents the state of the company at some time t by specifying the salaries of all the employees. Company A is an isolated system, in the thermodynamic sense, with respect N and M (i.e., these are conserved) but an open system with respect to information exchanges regarding salaries and job openings elsewhere in the free market economic environment.

_{1}, S

_{2}, …,S

_{1000}}, at any given time t corresponds to defining the microstate of A. Specifying how many employees are in each of the k salary categories, at any given time, corresponds to defining the macrostate of A. Generally, for a large N, there will be an extremely large number of macrostates. Most macrostates, in turn, would contain an unimaginably large number of different microstates. This property is known as multiplicity (W) in statistical mechanics. The Kronecker delta function configuration presented above is a unique exception, a macrostate that has only one microstate. As the company adapts and evolves over time, it will wander from one microstate to another. This does not mean, however, it will necessarily move from one macrostate to another as many different microstates could correspond to the same macrostate due to multiplicity.

^{220}ways, through the various permutations of the employees, thus resulting in a multiplicity of 10

^{220}microstates (we show below how multiplicity is calculated). Let us further assume that there are three classes of employees: low skilled (contributing value V

_{low}), medium skilled (V

_{med}), and high skilled (V

_{high}), such that V

_{low}< V

_{med}< V

_{high}, implying that S

_{low}< S

_{med}< S

_{high}. Let us further stipulate that the number of low skilled employees is 500, medium skilled 300, and high skilled 200.

_{i}is the number of employees in the value category i , subject to the constraints:

#### 3.2. Entropy as a Measure of Fairness: Axiomatic Approach

- (i)
- Axiom of Continuity: Fairness measure f(S) is continuous. This states that a small change in salary results in a small change in the fairness measure.
- (ii)
- Axiom of Homogeneity: Fairness measure f(S) is a homogeneous function of degree zero. This axiom states that the fairness measure is independent of the unit of measurement or absolute magnitude of the salaries.
- (iii)
- Axiom of Asymptotic Saturation: Fairness measure f(S) of equal resource allocations eventually becomes independent of the number of employees. This is needed to ensure uniqueness of the fairness measure and invariance under change of variable due to scaling.
- (iv)
- Axiom of Irrelevance of Partition: If one partitions the elements of S into two parts S = [ S
^{1}, S^{2}], then the fairness index f(S^{1}, S^{2}) can be computed recursively and is independent of the partition. - (v)
- Axiom of Monotonicity: For n = 2 employees who are contributing equally, the fairness measure f(α, 1− α) is monotonically increasing as the absolute difference between the two elements (i.e., |1 − 2α|) shrinks to zero. This axiom involves a value statement on fairness–i.e., when there are just two equal employees, more equalized is more fair. This axiom specifies an increasing direction of fairness and ensures uniqueness of f(S).

## 4. Essence of Entropy is Fairness

_{i}is the salary of ith employee. It is related to Shannon’s entropy H by the following equation:

“{The Theil} index can be interpreted as the expected information content of the indirect message which transforms the population shares as prior probabilities into the income shares as posterior probabilities.”

“But the fact remains that {The Theil index} is an arbitrary formula, and the average of the logarithms of the reciprocals of income shares weighted by income is not a measure that is exactly overflowing with intuitive sense.”“Given the association of doom with entropy in the context of thermodynamics it may take a little time to get used to entropy as a good thing (‘How grand, entropy is on the increase!’), …”

## 5. Maximally Fair Salary Distribution at Equilibrium

## 6. Discussion

## Acknowledgments

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Venkatasubramanian, V.
Fairness Is an Emergent Self-Organized Property of the Free Market for Labor. *Entropy* **2010**, *12*, 1514-1531.
https://doi.org/10.3390/e12061514

**AMA Style**

Venkatasubramanian V.
Fairness Is an Emergent Self-Organized Property of the Free Market for Labor. *Entropy*. 2010; 12(6):1514-1531.
https://doi.org/10.3390/e12061514

**Chicago/Turabian Style**

Venkatasubramanian, Venkat.
2010. "Fairness Is an Emergent Self-Organized Property of the Free Market for Labor" *Entropy* 12, no. 6: 1514-1531.
https://doi.org/10.3390/e12061514