# Long-Run Equilibrium in the Market of Mobile Services in the USA

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Coupled Fixed Points of Semi-Cyclic Maps

**Definition 1**

**Definition 2**

**.**Let A and B be nonempty subsets of a metric space $(X,\rho )$. The ordered pair of maps $(F,G)$, $F:A\times A\to B$ and $G:B\times B\to A$ is called a cyclic ordered pair of maps.

**Definition 3**

**.**Let A and B be nonempty subsets of a metric space $(X,\rho )$ and $(F,G)$ be a cyclic ordered pair of maps. An ordered pair $(x,y)\in A\times A$ is said to be a coupled fixed point of F in A if $x=F(x,y)$ and $y=F(y,x)$.

**Definition 4**

**.**Let A and B be nonempty subsets of a metric space $(X,\rho )$ and $(F,G)$ be a cyclic ordered pair of maps. We say that the cyclic ordered pair of maps $(F,G)$ is a cyclic contraction if there exists $\alpha \in (0,1/2)$ such that

**Theorem 1**

**.**Let A and B be nonempty subsets of a metric space $(X,\rho )$ and $(F,G)$ be a cyclic contraction. Then, F and G have a unique common coupled fixed point $({x}_{0},{y}_{0})\in A\times A\cup B\times B$, i.e., ${x}_{0}=F({x}_{0},{y}_{0})=G({x}_{0},{y}_{0})$ and ${y}_{0}=F({y}_{0},{x}_{0})=G({y}_{0},{x}_{0})$.

**Definition 5**

**Definition 6**

**Theorem 2**

- 1.
- The two players are producing homogeneous goods that are perfect substitutes.
- 2.
- The first player can produce quantities from the set A, and the second one can produce quantities from the set B, where A and B are closed, nonempty subsets of a complete metric space $(X,\rho )$.
- 3.
- Let there be a closed subset $D\subseteq A\times B$ and maps $F:D\to A$, $G:D\to B$ such that$$\left(F\right(x,y),G(x,y\left)\right)\subseteq D$$
- 4.
- Let $\alpha <1$, such that the inequality:$$\rho \left(F\right(x,y),F(u,v\left)\right)+\rho \left(G\right(x,y),G(u,v\left)\right)\le \alpha \left(\rho \right(x,u)+\rho (y,v\left)\right)$$

#### 2.2. Approximation with Sigmoid Functions

- Nonlinearity in an intuitive way with respect to a small number of parameters.
- Switching agents’ behavior, e.g., between different suppliers, products, etc.
- As a robust technical tool capable of modeling changes in a modular way by scaling these changes down to a predefined range (a to b), thus allowing us to incorporate strategies that are built on incremental changes of output.

#### 2.3. Testing Model’s Fit

## 3. Empirical Response Functions

#### 3.1. Data and Empirical Strategy

#### 3.2. Approximation of the Evolution of Smartphone Subscriptions over Time

#### 3.3. Construction of the Response Function

#### 3.4. Alternative Model with Consumer Shares

#### 3.5. Constructing the Response Functions for the Alternative Model

## 4. Model Fit

#### 4.1. The Least Squares Model

#### 4.2. The Sigmoid Model

#### 4.3. The Linear Approximation of the Sigmoid Model

## 5. Existence, Uniqueness and Stability of Market Equilibrium

**Proposition 1.**

**Proof.**

## 6. Using the Empirical Response Functions to Simulate Convergence to the Equilibrium

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**A few basic sigmoid functions: $\frac{x}{1+\left|x\right|}$, $\frac{2}{\pi}\mathrm{arctan}\left(\right)open="("\; close=")">\frac{\pi}{2}x$, $\frac{x}{\sqrt{1+{x}^{2}}}$ and $\mathrm{erf}\left(\right)open="("\; close=")">\frac{\sqrt{\pi}}{2}x$.

**Figure 3.**Percentage shares in the mobile market in the USA. (

**a**) Graphic data for the seven mobile operators in the USA (2009–2020). (

**b**) Graphic data for the two biggest operators, AT&T in blue and Verizon in red (2009–2020).

**Figure 4.**Total number of mobile users (2009–2020) and their approximation by the sigmoid function $S\left(y\right)$.

**Figure 6.**An approximation from using the ordered pair of response functions $(F,G)$. (

**a**) AT&T: blue—the real data, red—an approximation using ${u}_{n+1}=F({u}_{n},{v}_{n})$, green—an approximation using ${u}_{n+1}={F}^{n+1}({u}_{0},{v}_{0})$. (

**b**) Verizon: blue—the real data, red—an approximation using ${u}_{n+1}=F({u}_{n},{v}_{n})$, green—an approximation using ${u}_{n+1}={G}^{n+1}({u}_{0},{v}_{0})$.

**Figure 7.**An approximation from using the ordered pair of response functions $({F}_{1},{G}_{1})$. (

**a**) AT&T: blue—the real data, red—an approximation using ${u}_{n+1}={F}_{1}({u}_{n},{v}_{n})$, green—an approximation using ${u}_{n+1}={F}_{1}^{n+1}({u}_{0},{v}_{0})$. (

**b**) Verizon: blue—the real data, red—an approximation using ${u}_{n+1}={F}_{1}({u}_{n},{v}_{n})$, green—an approximation using ${u}_{n+1}={G}_{1}^{n+1}({u}_{0},{v}_{0})$.

**Figure 8.**An approximation from using the ordered pair of response functions $({F}_{2},{G}_{2})$. (

**a**) $AT\&T$: blue—the real data, red—an approximation using ${u}_{n+1}={F}_{2}({u}_{n},{v}_{n})$, green—an approximation using ${u}_{n+1}={F}_{2}^{n+1}({u}_{0},{v}_{0})$. (

**b**) $Verizon$: blue—the real data, red—an approximation using ${u}_{n+1}={F}_{2}({u}_{n},{v}_{n})$, green—an approximation using ${u}_{n+1}={G}_{2}^{n+1}({u}_{0},{v}_{0})$.

**Figure 9.**A simulation with the ordered pair of response functions $({F}_{1},{G}_{1})$ (blue color for customers of AT&T and red color for Verizone ones) (

**a**) Evolution of the market with the sigmoid model if the initial start is $(20,70)$. (

**b**) Evolution of the market with the sigmoid model if the initial start is $(70,170)$.

**Figure 10.**A simulation with the ordered pair of response functions $({F}_{2},{G}_{2})$ (blue color for customers of AT&T and red color for Verizone ones) (

**a**) Evolution of the market with the linear approximation of the sigmoid model if the initial start is $(20,70)$. (

**b**) Evolution of the market with linear approximation of the sigmoid model if the initial start is $(70,170)$.

**Figure 11.**A simulation with the ordered pair of response functions $(F,G)$ (blue color for customers of AT&T and red color for Verizone ones) (

**a**) Evolution of the market with the least square model if the initial start is $(20,70)$. (

**b**) Evolution of the market with the least square model if the initial start is $(70,170)$.

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**MDPI and ACS Style**

Badev, A.; Kabaivanov, S.; Kopanov, P.; Zhelinski, V.; Zlatanov, B.
Long-Run Equilibrium in the Market of Mobile Services in the USA. *Mathematics* **2024**, *12*, 724.
https://doi.org/10.3390/math12050724

**AMA Style**

Badev A, Kabaivanov S, Kopanov P, Zhelinski V, Zlatanov B.
Long-Run Equilibrium in the Market of Mobile Services in the USA. *Mathematics*. 2024; 12(5):724.
https://doi.org/10.3390/math12050724

**Chicago/Turabian Style**

Badev, Anton, Stanimir Kabaivanov, Petar Kopanov, Vasil Zhelinski, and Boyan Zlatanov.
2024. "Long-Run Equilibrium in the Market of Mobile Services in the USA" *Mathematics* 12, no. 5: 724.
https://doi.org/10.3390/math12050724