# Payoff Distribution in a Multi-Company Extraction Game with Uncertain Duration

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## Abstract

**:**

## 1. Introduction

## 2. Notation and Non-Cooperative Setup

#### 2.1. Problem Statement

- ${u}_{11}\in {U}_{11},{u}_{12}\in {U}_{12},\dots ,{u}_{1M}\in {U}_{1M},\dots ,{u}_{N1}\in {U}_{N1},\dots ,{u}_{NM}\in {U}_{NM}$ are the extraction effort levels of the N companies involved in pulling out M exhaustible resources. More precisely, ${u}_{ij}$ is the effort exerted by firm i to extract resource j. The only requirement for the control sets ${U}_{ij}$, for $i=1,\dots ,N$, $j=1,\dots ,M$, concerns the non-negativity of effort levels, so we can assume ${U}_{ij}\subseteq {\mathbb{R}}_{+}$, for all $i,j$. (We do not impose any other constraint both on the control sets and on the state set, thus admitting any possible level. Because such sets are not compact in principle, maximum points may fail to exist, hence the choice of the payoff functions is crucial to have an equilibrium structure.)
- $x(t)=({x}_{1}(t),\dots ,{x}_{M}(t))$ is the state vector indicating the quantities of the exhaustible resources available to be extracted by the companies. We assume $x\in X\subseteq {\mathbb{R}}_{+}^{M}$.
- The M dynamic constraints of the game are given by:$$\left(\right)open="\{"\; close>\begin{array}{c}\dot{x}(t)=g(x(t),{u}_{11}(t),\dots ,{u}_{NM}(t))\hfill \\ x({t}_{0})={x}_{0}\in {\mathbb{R}}_{+}^{M}\hfill \end{array}$$
- The interval over which the game is played is $[{t}_{0},\phantom{\rule{4pt}{0ex}}T]\subset {\mathbb{R}}_{+}$, where ${t}_{0}\ge 0$ and $T<\infty $.
- The final instant of the game, i.e., the exact time at which all companies stop the extraction, is described by the random variable $\widehat{t}\in [{t}_{0},\phantom{\rule{4pt}{0ex}}T]$. The cumulative distribution function (c.d.f.) of $\widehat{t}$ is given by ${F}^{p}(t)$, which is assumed to have a break (jump) of length $p>0$. The jump occurs at instant ${t}_{1}\in [{t}_{0},\phantom{\rule{4pt}{0ex}}T]$, i.e., it can be described as follows (Figure 1):$${F}^{p}(t)=\left(\right)open="\{"\; close>\begin{array}{c}F(t),\phantom{\rule{2.em}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\in [{t}_{0},{t}_{1})\hfill \\ F(t)+p,\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\in [{t}_{1},T]\hfill \end{array}$$
- The instantaneous payoff of the i-th player at the moment $\tau \in [{t}_{0},T]$ is defined as ${h}_{i}(x(\tau ),{u}_{i1}(\tau ),\dots ,{u}_{iM}(\tau ))$. To shorten the notation, we write$${h}_{i}(x(\tau ),{u}_{i1}(\tau ),\dots ,{u}_{iM}(\tau ))={h}_{i}(\tau ).$$The i-th related integral function is:$${H}_{i}(t)={\int}_{{t}_{0}}^{t}{h}_{i}(\tau )d\tau .$$
- The i-th objective function is represented by the following integral payoff to be maximized:$${K}_{i}({t}_{0},{x}_{0},{u}_{11},\dots ,{u}_{NM})=\underset{{t}_{0}}{\overset{T}{\int}}\left(\right)open="("\; close=")">\underset{{t}_{0}}{\overset{t}{\int}}{h}_{i}(x(\tau ))d\tau $$

**Proposition**

**1.**

**Proof.**

#### 2.2. Problem Statement for a Subgame

**Subgame starting at $\theta <{t}_{1}$:**Consider a subgame ${\Gamma}^{T}(\theta ,\tilde{x})$ such that $\theta \in [{t}_{0};{t}_{1})$. The conditional c.d.f. in the considered subgame takes the following form:

**Subgame starting at $\widehat{\theta}\ge {t}_{1}$:**Consider a subgame ${\Gamma}^{T}(\widehat{\theta},\tilde{x})$ such that $\widehat{\theta}\in [{t}_{1},\phantom{\rule{4pt}{0ex}}T]$. The conditional cumulative distribution function in the considered subgame takes the following form:

**Proposition**

**2.**

#### 2.3. Open-loop Nash equilibrium

**Definition**

**1.**

## 3. Main Results in the Cooperative Setup

**Definition**

**2.**

**Definition**

**3.**

- 1.
- for all $\theta \in [{t}_{0},{t}_{1})$ the vector ${\xi}^{\theta}=({\xi}_{1}^{\theta},\dots ,{\xi}_{N}^{\theta})$, where$${\xi}_{i}^{\theta}={\displaystyle \frac{1}{1-F(\theta )}}\left(\right)open="["\; close="]">{\int}_{\theta}^{T}{\beta}_{i}(t)(1-F(t))dt-p{\int}_{{t}_{1}}^{T}{\beta}_{i}(t)dt$$
- 2.
- for all $\widehat{\theta}\in [{t}_{1},\phantom{\rule{4pt}{0ex}}T]$ the vector ${\widehat{\xi}}^{\widehat{\theta}}=\left(\right)open="("\; close=")">{\widehat{\xi}}_{1}^{\widehat{\theta}},\dots ,{\widehat{\xi}}_{N}^{\widehat{\theta}}$ where$${\widehat{\xi}}_{i}^{\widehat{\theta}}={\displaystyle \frac{1}{1-p-F(\widehat{\theta})}}{\int}_{\widehat{\theta}}^{T}{\beta}_{i}(t)(1-p-F(t))dt,$$

**Lemma**

**1.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Theorem**

**1.**

- 1.
- if $\tau \in [{t}_{0},\phantom{\rule{4pt}{0ex}}{t}_{1})$,$${\beta}_{i}(\tau )={\displaystyle \frac{f(\tau )}{1-F(\tau )}}{\xi}_{i}(\tau ,{x}^{*}(\tau ),T)-{\xi}_{i}^{\prime}(\tau ,{x}^{*}(\tau ),T),$$
- 1.
- if $\tau \in [{t}_{1},\phantom{\rule{4pt}{0ex}}T]$$${\beta}_{i}(\tau )={\displaystyle \frac{f(\tau )}{1-p-F(\tau )}}{\xi}_{i}(\tau ,{x}^{*}(\tau ),T)-{\xi}_{i}^{\prime}(\tau ,{x}^{*}(\tau ),T),$$

## 4. An Example

**noncooperative**open-loop optimal trajectories of state and controls in relation to the noncooperative form of the game using Pontryagin’s maximum principle, which is one of the two major procedures for equilibrium structure in differential games [31]. In this model, this method is suitable, because the open-loop trajectories are easily visualized in ${K}_{i}(\xb7)$. Each company aims to solve the following problem:

## 5. Conclusions and Further Developments

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**An example of a c.d.f. ${F}^{p}(t)$ in the interval $[{t}_{0},\phantom{\rule{4pt}{0ex}}T]$.

**Figure 2.**The exponential c.d.f. $F(t)=1-{e}^{-(t-{t}_{0})}$ in the interval $[{t}_{0},\phantom{\rule{4pt}{0ex}}{t}_{1}]$.

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**MDPI and ACS Style**

Gromova, E.; Malakhova, A.; Palestini, A.
Payoff Distribution in a Multi-Company Extraction Game with Uncertain Duration. *Mathematics* **2018**, *6*, 165.
https://doi.org/10.3390/math6090165

**AMA Style**

Gromova E, Malakhova A, Palestini A.
Payoff Distribution in a Multi-Company Extraction Game with Uncertain Duration. *Mathematics*. 2018; 6(9):165.
https://doi.org/10.3390/math6090165

**Chicago/Turabian Style**

Gromova, Ekaterina, Anastasiya Malakhova, and Arsen Palestini.
2018. "Payoff Distribution in a Multi-Company Extraction Game with Uncertain Duration" *Mathematics* 6, no. 9: 165.
https://doi.org/10.3390/math6090165