# A Nikaido Isoda-Based Hybrid Genetic Algorithm and Relaxation Method for Finding Nash Equilibrium

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### 2.1. Definitions

**x**

_{i}in the Euclidean space ${\mathit{R}}^{{m}_{i}}$. All players take a collective action $x=\left({x}_{1},....,{x}_{n}\right)\in {R}^{{m}_{1}}\times \cdots \times {R}^{{m}_{n}}$. Let ${X}_{i}\subseteq {R}^{{m}_{i}}$ be an action set of player i, and ${\varphi}_{i}:{X}_{i}\to R$ be the corresponding payoff function. Let $X$ be the collective action set. By definition, $X\subseteq {X}_{1}\times \cdots \times {X}_{n}\subseteq {R}^{{m}_{1}}\times \cdots \times {R}^{{m}_{n}}={R}^{m}$. Let $x=\left({x}_{1},....,{x}_{n}\right)$ and $y=\left({y}_{1},....,{y}_{n}\right)$ be elements of the collective action set ${X}_{1}\times \cdots \times {X}_{n}$. An element $\left({y}_{i}|x\right)\equiv \left({x}_{1},....,{x}_{i-1},{y}_{i},{x}_{i+1},\dots ,{x}_{n}\right)$ of the collective action set is interpreted as a collection of actions when the i-th player tries

**y**

_{i}, while the remaining players still keep ${x}_{j},j=1,\dots ,i-1,i+1,\dots ,n$.

**Definition**

**1.**

**Definition**

**2.**

**x**

_{i}to

**y**

_{i}, while all other players continue playing according to

**x**. That means that one player changes the action, while the others do not. Thus, the function represents the sum of these improvements in the payoff. Note that the maximum value of this function is always nonnegative for a given

**x**. Furthermore, the function is nonpositive for all feasible when it is a NE, since no player can improve his payoff at equilibrium. In consequence, each summand can be at most zero at the NE [1].

**Definition**

**3.**

_{eq}and N

_{ieq}are the number of the equality and inequality constraints, respectively. Consequently, the NE needs to satisfy not only the condition (4), but also the constraints (5) and (6).

#### 2.2. NI-Based Relaxation Method

**x**):

**y**that is corresponding to Z(

**x**) be denoted by

**y**

^{max}(

**x**). In case of the generalized Nash Equilibrium problem,

**y**

^{max}(

**x**) must fulfill all constraint requirements (5) and (6). The optimum response function returns the set of players’ actions whereby the players all try to unilaterally maximize their respective payoffs. By playing actions

**y**

^{max}(

**x**) rather than

**x**, the players approach the equilibrium.

**x**

^{s}to the NI function (2), the optimum response function ${y}^{max}\left({x}^{s}\right)$ is obtained from solving the optimization problem (7). ${y}^{max}\left({x}^{s}\right)$ is then input into (8) and results in

**x**

^{s}

^{+1}, which is successively used in the NI function (2). The procedure repeats until the condition (4) is satisfied, i.e.,

**x**

^{t}is the NE. There are two remarks regarding the determination of the NE. First, it can be difficult to attain the zero value of the NI function. In practice, the procedure is allowed to stop when

**y**

^{max}(

**x**) is normally carried out in analytical manners. However, when the NI function is complicated, the analytical form of the optimum response function may not be possible to obtain. The next section explains an alternative method of obtaining ${y}^{max}\left({x}^{s}\right)$, from which its numerical value is computed instead of deriving the closed form solution.

## 3. NI-Based Hybrid Genetic Algorithm and Relaxation Method

#### 3.1. Methodological Concept

**y**

^{max}(

**x**) [2,3]. This section proposes a method for determining the numerical value

**y**

^{max}(

**x**). The so-called Genetic Algorithm (GA) is applied for this purpose. Let ${\Psi}_{min}$ be the minimum value of the NI function during the computational procedure, ${\Psi}_{int}$ be the intermediate value during the computation loop, count be the counter of computation loop, and L be the maximum number of computation loop. The NI-based hybrid GA and relaxation method has the following steps:

- Starting with s = 1, initialize
**x**^{s}which satisfies the constraints (5) and (6). Then initialize ${\Psi}_{min}$ and ${\Psi}_{int}$ to large values compared to ε, e.g., equal to 1. - While ${\Psi}_{int}>\epsilon $ and $count\le L$, perform steps 3 to 7.
- Using GA, compute ${y}^{max}\left({x}^{s}\right)$ from the constrained optimization problem:Maximize$$\Psi \left({x}^{s},{y}^{max}\left({x}^{s}\right)\right)$$subject to$${c}_{eq,r}\left({y}^{max}\left({x}^{s}\right)\right)=0\text{\hspace{1em}}(r=1,\dots ,{N}_{eq})$$$${c}_{ieq,q}\left({y}^{max}\left({x}^{s}\right)\right)\le 0\text{}(q=1,\dots ,{N}_{ieq})\text{}$$
- Compute $\Psi \left({x}^{s},{y}^{max}\left({x}^{s}\right)\right)$, if $\left|\Psi \left({x}^{s},{y}^{max}\left({x}^{s}\right)\right)\right|\le {\Psi}_{min}$:
- Assign $\left|\Psi \left({x}^{s},{y}^{max}\left({x}^{s}\right)\right)\right|$ to ${\Psi}_{min}$.
- Assign ${x}^{s}$ to
**x**^{*}and ${y}^{max}\left({x}^{s}\right)$ to**y**^{*}.

- Assign $\left|\Psi \left({x}^{s},{y}^{max}\left({x}^{s}\right)\right)\right|$ to ${\Psi}_{int}$.
- Compute ${x}^{s+1}=\left(1-{\alpha}_{s}\right){x}^{s}+{\alpha}_{s}{y}^{max}\left({x}^{s}\right)$ and assign ${x}^{s+1}$ to ${x}^{s}$.
- Compute count = count + 1.

**x**

^{*}. The procedure of the NI-based hybrid GA and relaxation method is summarized in Figure 1. The GA is described in the following section.

#### 3.2. Genetic Algorithm (GA)

**x**

^{s}, the fitness function $F\left({y}^{max}\left({x}^{s}\right)\right)$ of a chromosome is

_{Pop}is the population size. The fitness value F

_{j}is obtained from (20). Two-point crossover and binary mutation are then applied to the reproduced chromosomes to result in the new population generation [27]. The whole procedure is to be explained through the use of numerical examples in the following section.

## 4. Illustrative Examples

#### 4.1. River Basin Pollution Game

_{j}. The players must meet environmental conditions set by a local authority. Pollutants may be expelled into the river, where they disperse. Two monitoring stations l = 1, 2 are located along the river, at which the local authority has set maximum pollutant concentration levels.

_{1}and d

_{2}determine the inverse demand law and are equal to 3.0 and 0.1, respectively. The values of c

_{1j}and c

_{2j}are given in Table 1, and K

_{l}= 100 (l = 1, 2). u

_{j}

_{l}are the decay and transportation coefficients from player j to location l. e

_{j}is the emission coefficient of player j, which is also given in Table 2 [35].

^{−4}. The evolution of the NI function value is plotted in Figure 2.

#### 4.2. Electricity Market Game

_{g}and P

_{C}are the power generation of a unit and a company, respectively.

^{−4}. The evolution of NI function value is plotted in Figure 3. The search terminates after the maximum number of loops, 50, is reached. The values of the strategic variables that yield the minimum NI function, ${\psi}_{min}=1.3\times {10}^{-4}$, are taken as the NE. The obtained NE is compared to the literature [38] in Table 6.

#### 4.3. Nash–Cournot Aggregative Game of Power Generation Firms

_{i}is the output power of firm i in MW, d

_{i}is the local demand in MW, and P

_{imin}and P

_{imax}are the upper and lower bounds of the output power P

_{i}, respectively. $p\left(\sigma \right)$ is the electricity price and is defined by

^{−4}.

#### 4.4. Game with Local Optima

^{−8}.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NE | Nash Equilibrium |

GA | Genetic Algorithm |

DE | Differential Evolution |

NI | Nikaido–Isoda |

NSGAII | Non-dominated Sorting Genetic Algorithm II |

NDEMO | Nash Domination Evolutionary Multiplayer Optimization |

NIDE | Nikaido–Isoda Differential Evolution |

NESGA | Nash Equilibrium Sorting Genetic Algorithm |

GSA | Gravitational Search Algorithm |

IPO | Inclined Planes System Optimization Algorithm |

CSA | Capuchin Search Algorithm |

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Method | Check of Nash Equilibrium Condition | Creation of New Actions | Ability to Solve Generalized Nash Problems | Hardship in Implementation |
---|---|---|---|---|

Evolutionary Computation [8] | Nash Non-Dominated Sorting | Evolutionary Algorithm (NSGAII) | Not Addressed | High |

NDEMO [9] | Nash Dominance Sorting | Evolutionary Algorithm (DE) | Applicable | High |

NIDE [7] | Nikai–Isoda Function Computation | Evolutionary Algorithm (DE) | Not Addressed | Moderate |

Co-evolutionary Approach [10,21] | Nash Non-Dominated Sorting | Co-evolutionary Algorithm (GA and DE) | Applicable | High |

NESGA [13] | Sorting of Payoff Function Values | Evolutionary Algorithm (GA) | Not Addressed | High |

Distributed Algorithm [14] | Payoff Function Evaluation and Sorting | Dynamic Equations | Limited to An Equality Constraint | High |

NI-based Hybrid GA and Relaxation Method (Proposed Method) | Nikai–Isoda Function Computation | Relaxation Method | Applicable | Low |

Player j | c_{1j} | c_{2j} | e_{j} | u_{j}_{1} | u_{j}_{2} |
---|---|---|---|---|---|

1 | 0.10 | 0.01 | 0.50 | 6.5 | 4.853 |

2 | 0.12 | 0.05 | 0.25 | 5.0 | 6.250 |

3 | 0.15 | 0.01 | 0.75 | 5.5 | 3.750 |

Pulp Producing Level | The Proposed Method | [Krawczyk and Uryasev, 2000] [35] |
---|---|---|

x_{1} | 21.570 | 21.140 |

x_{2} | 15.860 | 16.030 |

x_{3} | 2.440 | 2.728 |

Company # | Generator # | ${\mathit{P}}_{\mathit{g}}^{\mathit{m}\mathit{i}\mathit{n}}$ (MW) | ${\mathit{P}}_{\mathit{g}}^{\mathit{m}\mathit{a}\mathit{x}}$ (MW) | ${\mathit{P}}_{\mathit{C}}^{\mathit{m}\mathit{i}\mathit{n}}$ (MW) | ${\mathit{P}}_{\mathit{C}}^{\mathit{m}\mathit{a}\mathit{x}}$ (MW) |
---|---|---|---|---|---|

1 | 1 | 0 | 80 | 0 | 80 |

2 | 2 | 0 | 80 | 0 | 130 |

3 | 0 | 50 | |||

3 | 4 | 0 | 55 | 0 | 125 |

5 | 0 | 30 | |||

6 | 0 | 40 |

Generator # | ${\mathit{c}}_{\mathit{i}}$ [$/MW^{2}h] | ${\mathit{d}}_{\mathit{i}}$ [$/MWh] | ${\mathit{e}}_{\mathit{i}}$ [$/h] |
---|---|---|---|

1 | 0.04 | 2 | 0 |

2 | 0.035 | 1.75 | 0 |

3 | 0.125 | 1 | 0 |

4 | 0.0166 | 3.25 | 0 |

5 | 0.05 | 3 | 0 |

6 | 0.05 | 3 | 0 |

Power Generation Unit | The Proposed Method (MW) | (Contreras et al., 2004) [38] (MW) |
---|---|---|

${P}_{g1}$ | 46.66 | 46.66 |

${P}_{g2}$ | 32.15 | 32.16 |

${P}_{g3}$ | 15.01 | 15 |

${P}_{g4}$ | 22.13 | 22.13 |

${P}_{g5}$ | 12.35 | 12.33 |

${P}_{g6}$ | 12.31 | 12.33 |

Parameter | Firm 1 | Firm 2 | Firm 3 | Firm 4 | Firm 5 |
---|---|---|---|---|---|

${\alpha}_{i}$ | 5 | 8 | 6 | 9 | 7 |

${\beta}_{i}$ | 12 | 10 | 11 | 11 | 13 |

${\gamma}_{i}$ | 2 | 1 | 3 | 1.5 | 0.5 |

${\varsigma}_{i}$ | 25 | 48 | 15 | 30 | 45 |

d_{i} | 45 | 35 | 25 | 40 | 30 |

P_{imin} | 20 | 45 | 10 | 20 | 25 |

P_{imax} | 37 | 55 | 25 | 33 | 40 |

Firm i | Output Power P_{i} (MW) |
---|---|

1 | 28.51 |

2 | 54.66 |

3 | 18.84 |

4 | 32.99 |

5 | 40.00 |

Parameter | Value |
---|---|

Population size | 50 |

Maximum number of iterations | 400 |

Probability of crossover | 0.35 |

Mutation factor | 0.45 |

Termination tolerance | 0.0001 |

Parameter | Value |
---|---|

Population size | 50 |

Maximum number of iterations | 1000 |

Number of runs | 30 |

G0 | 100 |

α | 20 |

K0 | 50 and is decreased linearly to 1 |

Parameter | Value |
---|---|

Population size | 50 |

Maximum number of iterations | 1000 |

Number of runs | 30 |

c_{1} | 0.72 |

c_{2} | 2.76 |

shift_{1} | 72.47 |

shift_{2} | 188.51 |

scale_{1} | 0.04 |

scale_{2} | 0.82 |

Parameter | Value |
---|---|

Population size | 30 |

Maximum number of iterations | 1000 |

Number of runs | 30 |

Velocity control constant | 1 |

Inertia parameter | 0.7 |

Balance factor | 0.7 |

Elasticity factor | 9 |

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**MDPI and ACS Style**

Harnpornchai, N.; Wonggattaleekam, W.
A Nikaido Isoda-Based Hybrid Genetic Algorithm and Relaxation Method for Finding Nash Equilibrium. *Mathematics* **2022**, *10*, 81.
https://doi.org/10.3390/math10010081

**AMA Style**

Harnpornchai N, Wonggattaleekam W.
A Nikaido Isoda-Based Hybrid Genetic Algorithm and Relaxation Method for Finding Nash Equilibrium. *Mathematics*. 2022; 10(1):81.
https://doi.org/10.3390/math10010081

**Chicago/Turabian Style**

Harnpornchai, Napat, and Wiriyaporn Wonggattaleekam.
2022. "A Nikaido Isoda-Based Hybrid Genetic Algorithm and Relaxation Method for Finding Nash Equilibrium" *Mathematics* 10, no. 1: 81.
https://doi.org/10.3390/math10010081