Game Theory and Social Interaction for Selection and Crossover Pressure Control in Genetic Algorithms: An Empirical Analysis to Real-Valued Constrained Optimization

Game Theory (GT) formalizes dispute scenarios between two or more players where each one makes a move following their strategy profiles. The following paper introduces the integration of GT to selection and crossover steps of Genetic Algorithms as an evolutionary model of the representation of population in a similar way to human social evolution. Two ideas are proposed to be incorporated into the GA. First, the Genetic Algorithm with Social Interaction (GASI), a family of GAs that uses GT in selection phase to increase the diversification of the solutions. Second, the (Game-Based Crossover) GBX and GBX2 crossover operators, competition-based tournament selection methods that employ social dispute to generate more diverse offspring. Performance and robustness of the new approaches were assessed by ten continuous and constrained engineering design optimization problems and compared against variants of the canonical GA, as well as well-known heuristics from the literature. Results indicate significant performance relevance in most instances compared to other algorithms and highlight the benefits of combining GT and GA.


I. INTRODUCTION
Genetic Algorithms (GA) [1] are considered as a well-known population-based meta-heuristic for solving non-continuous and non-linear problems, where exact algorithms do not solve [2]. GA design is completely based in the Darwinian theory of natural selection. The theory states that the fittest individuals are more likely to produce offspring than less able individuals who are eliminated if they do not adapt or mutate in order to improve their capabilities [3].
The associate editor coordinating the review of this manuscript and approving it for publication was Rashid Mehmood .
Evolutionary algorithms have been extensively studied by researchers over the years. A common practice is combining GA's with other heuristics or exact methods, such as local or global search procedures to enhance features of GA's to solve problems of more complex domains [4]- [9]. A promising direction for the hybridization of GA is the incorporation of aspects of Game Theory (GT) combining them to new approaches together with pre-established canonical GA mechanisms [10].
With the emergence of the prisoner (DP) dilemma in the middle of the last century, an approach based on GT aroused great interest in the academic community [11], with applications in different areas of knowledge, such as [12], VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ international relations [13], economics [14], psychology [15], biology [16], applied mathematics [17], moral philosophy [11] and interdisciplinary research [18], [19]. One of the reasons behind this is related to the analysis of game theory under the rules of PD, which tends to stimulate non-cooperative behavior for all participants when compared to a cooperation strategy between the two rational agents. As a result, the prisoner's dilemma can be extrapolated to various real-world applications of multi-agent interaction, where failure to rationally achieve a cooperative outcome implies assessments and approaches focused on reducing losses or decisions based on lower gains [11]. The integration of the GT with the GA opens several possibilities to improve GA genetic operators. Game Theory provides the means to analyze situations in which players perform interdependent actions, when pondering and anticipating your opponents' next steps. Any system involving any decision-making processes, whether in a dispute between opponents (for example, board games, competition or cooperation between intelligent agents [20]) or in environments involving risk control analysis (for example, economics [21]) fit in the category of problems that can be modeled as a typical Game Theory Problem.

A. PROBLEM DISCUSSION
The classic structure of GA does not effectively implement the concept of phenotype when describing it as just the individual's fitness value. It, the phenotype, can be defined by morphological, physiological and behavioral characteristics, as described by Evolutionary Biology, when stating that it contributes a lot to the evolutionary process of individuals in a population. Futhermore, its concept is also closely related to the behavior of individuals. And it is a determining factor for the survival of your genes when considering the influence exerted by the environment on behavior. Consequently it reflects on the following generations, with the maintenance of certain genes, through the biological inheritance of behavioral characteristics, whether in human populations or other animal species [22]. Thus, Teixeira [22] introduced the GT as a phenotypic component in the context of GA with the objective of making its mechanism closer to that observed in nature. Thus, a conflict of interest environment was introduced in which rational agents must make decisions to maximize their fitness value during the selection process, receiving a payoff value at the end of the dispute. Each one of them stores the information of their strategy profile as a phenotypic encoding. GT-based competition can also be used to adjust their fitness values during the selection phase and, thus, allow individuals not so apt to evolve and generate offspring, when passing on their genetic material to the next generations. Individual payoff information from the disputes are also used to move the population within a certain region of the search space towards when expanding the power of exploration the algorithm. The basic scheme of the proposed algorithms are described in Fig.1. In other words, a social interaction scheme based on GT is proposed which offers a set of tools by allowing the design and implementation of a new class of crossover operators. In this way, a solution search feature is encoded in the strategy profile of each candidate solution and the maximum individual payment allowed is adaptively estimated based on the state of set of solutions. This article extends the study shown in [10], where it was presented a new mechanism of selective pressure's self-adaptive control using GT in the selection phase (GASI), as well as two new crossover methods, named GBX and GBX2. The crossovers' dispute environment is modeled from the Prisoner's Dilemma paradigm, the most famous and studied game from the field of GT. Candidate solutions employ strategies based on their behavior strategies against their opponents to improve their individual adaptability (in this context, their fitness values).
The paper is divided into the following sections: related works and State-of-the-Art in GA and GT (Section II); a brief theoretical foundation of Genetic Algorithms and Game Theory (Section III); introduction to GASI algorithms (Section IV) and GBX/GBX2 crossover (Section V); the experiments' setup and parameters, describing each problems instances and their results analyses (Section VI); and then, final remarks based on the results and future works (Section VII).

II. RELATED WORKS
As said early, this paper is a continuation of the study published by Pereira et al. [10], who conducted an in-depth analysis from the influence of four dispute configurations (Hawk-Dove, Stag Hunt, Harmonic Games and Prisoner's Dilemma) to the GA's selection operator aiming to solve problems of the following form: Although the results indicate results and positive influence of  GT in the control of selective pressure and diversity, insignificant improvements of objective function values were reported  compared to the original GA. Several works in the scientific literature incorporate GT to EA's, particularly in Genetic Algorithms context. Leher [3] proposed an implementation of GA to the Traveling Salesman Problem (TSP) whose selection operator is based on the Hawk-Dove game. In Periaux et al. [24], a new evolutionary strategy was discussed to GA's using Nash equilibrium for multiobjective optimization problems involving aerodynamic applications.
Teixeira et al. [22], [25] developed a series of hybrids metaheuristics where Game Theory plays an important role as a step of Social Interaction of these algorithms. They emphasize reducing selective pressure, maintenance of diversity and introduction of the social aspect through the codification of a behavior strategy added to the solution of the problem. From those, we cite the Fuzzy Social Interaction Genetic Algorithm (F-SIGA) that uses fuzzy logic to model the payoff tables in the disputes and the behavior strategies [26], the Ant Colony Algorithm using Social Interaction (ACO-GT) [27] and the Artificial Immune Systems Optimization that uses GT (AISO-GT) to select the best clones based on the Prisioner's Dillema [28]. Also, following the same line of thought of Teixeira et al. [29], Lahoz-Beltra et al. [30] developed a Genetic Algorithm that uses GT to choose selection operators applied Knapsack Problem. One of the main features of this work is the variety of Social Interaction (SI) game models: Prisoner's Dilemma (PD), Chicken Game (CG), Mixed Polymorphism (MP), Friend or Foe (FOF), Facultative Defection (FD) and Battle of Sexes (BS).

A. COMPARISON OF PROPOSITIONS
The GASI family of algorithms introduces the following improvements and mechanisms based on Game Theory: Values from payoff table are based on the best and worst fitness in the actual iteration, allowing the gains obtained by each solution to be automatically adapted to each problem; • Parent competition crossover (GBX/GBX2): Individuals selected for crossover compete among themselves in a iterated social game environment, where maximum payoff values are dynamically assigned. This establishes a balance between exploitation and exploitation. In this scenario, payoff accumulated during the rounds of a game are used to generate the new offspring. Furthermore, we analyze the feasibility of the individuals selection (GASI) as well as the crossover scheme (GBX/GBX2) to any bioinspired metaheuristic. The performance of the proposed mechanisms is assessed by experiments using benchmark functions of constrained optimization. The GBX and GASI are compared against GA's with alternative crossover methods and some of the state-ofthe-art algorithms for the given problems.

III. THEORETICAL BACKGROUND
As a way of adapting the Social Interaction model to GA, each solution stores information related to its individual genotypic characteristics, their behavior strategiy profile. The strategy profile represents the probability of a player, that is, a solution to opt for a specified change in the dispute environment. The new encoding scheme allows a diversification of interaction between solutions of the solution set beyond the basic rules of natural selection. Competition based on social interactions (encoded in each individual's strategies) affects the selection and crossover process. The social competition is mathematically formalized by Game Theory as a conflict of interests. The social interaction environment is defined [22] as follows: each agent (solutions of the solution set in the GA) compete among each other for available resources that can only be attained in a dispute (Game), so that they may evolve along their existence using these resources [22]. Fig.2 depicts the encoding of a solution of the Genetic Algorithm with Social Interaction (GASI), a combination of the theoretical foundation of GA's together with the Game Theory structure. The genotypic aspect of the algorithm is the real number encoding of the solutions, while the phenotypic aspect is the strategy profile that describes their behavior and their payoff value obtained from adopting such strategy.  The following data structures are sets of each index: • Genotype: 3D matrix of decision variables associated to a particular solution (X ); 2D matrix of real valued fitness (f (x)); 2D matrix of normalized fitness (f (n) (x)); 2D matrix with the number of infeasible constraints ((Xv)); 3D matrix of the value of each constraint c(x) and h(x) (Xr): • Phenotype: 3D matrix associated to each individual probability of doing a decision in a Game (Xs); 2D matrix related to the strategy profile (Xb); 3D matrix of the game payoff obtained for each individual (Xga); 2D Matrix of social interaction fitness(f (s) (x)); • Total Value: 2D matrix with the total fitness (f (t) (x)). As mentioned before, solutions of the same population in the GASI participating in a dispute-based process where elements of each individual phenotype provide the necessary information to take part in the social environment framework. The social interaction dispute environment is composed of four elements [22], [31]: • Players: Any solution or group of solutions with decision-making capabilities that has influence over other competitors; • Rationality: All players are presumed to have mechanisms that provide logical capability. They use the most appropriate means to achieve their goals, i.e., maximizing gains; • Strategy Profile: Each player takes into account the fact that they interact against rational opponents, so decisions made by opponents reflects upon the gains and the behavior of the player. There are three types of behavioral strategies: A) Pure strategies where players always choose the same behavior; B) Mixed strategies where players have fixed probabilities to do any move at their convenience C) Random strategies. See behavior matrix (Xb [i] ) in Fig.2; • Game: Consists of a formal model of the conflict of interests between players. The game environment present in the GASI and the GBX and GBX2 crossover operators are modeled for two-player iterated games.

A. GAMING AND THE PRISONER'S DILEMMA
Proposed by researchers of the RAND Corporation in the 1950's and formalized by Albert William Tucker, the Prisoner's Dilemma is one of the most well-known nonzero-sum, 1 non-cooperative game from Game Theory. The context of the game is as follows: two individuals are arrested, accused of the same crime and put into different cells without any communication between each other. Each individual has one option, either confess its crime (Cooperate, C) or deny participation in the crime (Defect, D). In this scenario, we have the following consequences to each individual based on their choice [22], [32], [33]: 1) If one confesses and the other defects, the first is arrested for three months while the latter for two years; 2) If both confess, they are arrested for one year; 3) If none confess, they are arrested for six months.  A payoff value is associated to the decision each player has taken. They are as follows: • T (temptation): Given to the player that defected while the opponent cooperated.
• R (reward): Given to both players that cooperated.
• P (punishment): Given to both players that defected. • S (sucker): Given to the player that cooperated while the opponent defected. In the classical instance of the Prisoner's Dilemma the values of T , R, P, S shown in Table 1 must adhere to the following rule: The available actions, cooperate (C) and defect (D) are encoded in the behavior matrix (Xb) that determines the strategy that a solution will employ during the social interaction phase. Four distinct strategies based on [30], [34], [35] are employed in GASI: 1) Pure strategies: a) ALL-C (Always cooperate):The player will always cooperate regardless of the opponent last move; b) ALL-D (Always defect): The player will always defect regardless of the opponent last move. 2) Mixed strategies: a) TFT (Tit-for-Tat): The player will start cooperating and then mirror the last action of his opponent; b) RAND (Random behavior): The player will take random actions regardless of his opponent's last move. The strategy encoding of each solution (Xs) allows for new strategies to be generated by crossover and/or mutation so they can be inherited by their offspring. Table 2 describes the encoding of each strategy represented by two positive ternary numbers that can assume values between 0 and 2.
In the following section, we present two applications of the Prisoner's Dilemma-based game. First, a selective pressure control system where less fit solutions have the opportunity to be chosen for reproduction after a second assessment based on the payoff obtained from the social interaction dispute (i.e. GASI-S and GASI-POP). Second, two crossover methods where offspring are generated based on the payoff of the parents from a R-rounds game (i.e. GBX, GBX2).

IV. SOCIAL INTERACTION AS A SELECTIVE PRESSURE CONTROLLER FOR GENETIC ALGORITHMS
The GASI algorithm (Genetic Algorithms with Social Interaction) adds a mechanism based on Game Theory to control the selective pressure of the selection phase, invariant to the selection method that was used (e.g. tournament, roulette). The fitness assessment of each solution now consists of the weighted sum of the objective function value (real fitness) and the payoff value obtained at the social dispute phase. Individuals with lower fitness values will be able to increase their chances of surviving and generating offspring by participating in the social dispute in order to maximize their individual winnings.
The selective pressure is controlled by a parameter that defines the maximum percentage of solutions or competitions (roulette and tournament, respectively) that will participate in the social dispute step. We present two different implementations of the selective pressure control where the social dispute takes place at different steps of the algorithm and employs distinct solution selection scheme. GASI-POP and GASI-S are summarized in Table 3 along the canonical GA in order to highlight their differences.
The first, GASI-POP runs the social dispute step to all solutions of the population prior to the selection step. Here, players are previously selected based on the arrangement of the list of individuals of X (see Fig.2). Fig.3 illustrates the GASI-POP method of choosing players for the social dispute phase, as well as its selection scheme.
While in the GASI-POP every solution of the population undergoes the social dispute step, the GASI-S only includes solutions that have been chosen in the tournament table selection. Players are allocated to games based on the arrangement of elements in the list of individuals of each competition. 2 The game step takes place during the selection 2 There is a noticeable difference between competition and game in the context of the GASI-S algorithm. Game is the process where two solutions from the list of competitors (tournament) will compete for earnings in accordance to a payoff table. Competition is the process of choosing the best individual in a tournament selection.   For each algorithm, computation of the social fitness at distinct steps of the execution results in major changes to the total fitness value. In the GASI-POP algorithm, the social fitness (f (s) (x)) values (f (t) (x)) is obtained before the selection phase, so the each opponent is fixed for each player. Whereas in GASI-S the payoff is obtained during the elimination process at the tournament selection with random opponents, generating payoff values that vary according to the behavioral combinations between players.
The following sections covers in detail the following points: the social dispute process based on variants of the Prisoner's Dilemma; the fitness normalization computation changing the T , R, P, S payoff rules; and the computation of the total fitness (f (t) (x)) based on social fitness values.

A. THE RULES OF THE GAME
As opposed to algorithms that include the social interaction step with fixed (constant and user-defined) payoff values [22], [23], [25], [27], [36], GASI-POP and GASI-S features a self-adaptive payoff table based on the worst (Pf w (n) ) and best (Pf b (n) ) normalized fitness values. The GASI-POP and GASI-S payoff table values are adjusted according to the values of the objective function fitness.
The fitness normalization follows Deb [37] rules for constrained optimization problems. In it, solutions that violated either equality or inequality constraints have those values added to the fitness adjusted by a penalty function. The normalization is described by (3) and (4). Constraint values (C re ) must meet the condition that they are equal to or less than zero: (3) The variables in (3) are: normalized fitness (f (n) (x)); objective value fitness (f (x)); worst feasible fitness value 3 (Pfw(x)); constraint index re (G re (x)); Number of Restrictions (R); and solution set without violations (S fe ). The T > R > P > S payoff values of the Prisoner's Dilemma are adjusted in each generation to include the best and the worst normalized fitness of the population, preserving their respective ratios, 0.4 to T ; 0.3 to ; 0.2 to P; 0.1 to S. 4 The adjustment process of the T > R > P > S values is described in (5) to (8), where Pf w (n) (x) and Pf g (n) (x) corresponds to the worst and best overall normalized fitness values, respectively, and Ro is the number of rounds in a game.
A depiction of an iterated R-round game is shown in Fig.5. The game consists of a two-round game of four solutions (X = {A, B, E, F}). The best and worst value of the objective value fitness is 8 and 40, respectively. Payoff values are calculated based on (5) to (8). Parameters α and β (introduced below) have fixed values of 1 for the sake of understanding.
Two mechanisms control the selection pressure: • Number of rounds of tournament selection or game rate (Rt GAMES ): Low values of Rt GAMES influence the increase in selective pressure, decreasing the chances of the total fitness being used instead of the objective function fitness 5 ; • Scaling factors α and β: Both parameters denote the relevance the social and objective fitness in the computation of the total fitness. The relationship between them is inversely proportional; In the implementation of the GASI-POP and GASI-S α is gradually increased while β remains fixed. This increases the relevance of the social fitness at the starting generations, increasing the likelihood of solutions with less objective solution quality to be chosen for crossover. The continuous decrease in the importance of social fitness tends to increase the selective pressure of the population, which may lead premature convergence of solutions to local optima. Equations (9) and (10) illustrate the linear variation of α and β at each iteration, where the parameters α start , α end , β start and β end are the starting and ending limits (or values) for α and β, 5 In constrained function optimization problems the objective function fitness is not normalized. During the process of comparison of solutions, the objective function fitness is taken into account together with the number of violated constraints information following Deb's Rule [37].
while iter variable and the constant ITER MAX are the current iteration and the total number of iterations, respectively.
• Calculation of the objective function fitness (α) control parameter: • Calculation of the social fitness (β) control parameter: At the end of the social dispute step, the total fitness (f (t) (x)) is calculated based on the values of social fitness f (s) (x), normalized objective function fitness f n) (x) and the α and β weighting factors. Computation of the total fitness value used in the Genetic Algorithm selection stage is due to (11) 6 : Algorithm 1 describes the entire social interaction step of the GASI-POP and GASI-S. Changes made to the standard Genetic Algorithm versions proposed by [38] (simplified) and [1] are highlighted in red. Social Interaction can also take part in the crossover process.

V. GAME-BASED CROSSOVER (GBX)
The GBX crossover allows the offspring to perform a wider local search around the region of the parent solutions. It can be said that the proposed recombination method is in the same class of exploration and exploitation focused search algorithms such as Fuzzy Connectives Based Crossover (FCBX) [39], Blended Crossover Alpha (BLX -α), Parent-Centric Alpha Crossover (PBX-α) [40], Simulated Binary Crossover (SBX) [41], hybrid methods [42], among others 7

A. AN EXPLORATION-EXPLOITATION BASED APPROACH
Exploration and exploitation are equally important given their divergent nature, therefore control of both must be maintained throughout the iterations for any search algorithm [45]. The GBX crossover employs social dispute as a mechanism to place solutions in either a exploitative or explorative search region. Earnings from the social interaction phase will determine the thresholds between diversification and intensification regions around the parent solution. The decision variables of the offspring are influenced by a random variable sampled from a normal distribution (N (0, 1)) that shifts the decision variables of the offspring towards either exploration or exploitation search zones. Fig.6 shows the separation of the default search space for two offspring generated by the same pair of parents. The intervals determined by the ∇X [p1] [j] and ∇X [p2] [j] direction is the interval that the parent decision variables Initialize strategy profile values; Initialize decision variables; Compute objective function Fitness(f (x)); Find best objective function value (Pf g (r) (x)); Find worst objective function value (Pf w (r) (x)); Compute normalized Fitness(f (n) (x)); Find best normalized fitness(Pf g (n) (x)); Find worst normalized fitness(Pf w (n) (x)); if GASI-POP AND i is odd then Mutate offspring 2; Find global best feasible objective function fitness (Pf g (r) (x)); Find global best feasible objective function fitness (Pf w (r) (x)); Computed offspring normalized Fitness(f (n) (x)); Find best feasible normalized fitness (Pf g (n) (x)); Find worst feasible normalized fitness (Pf w (n) (x)); if GASI-POP then Game: P1 (i * 2) vs. P2 ((i * 2) − 1); Obtain P1 and P2 social Fitness(f (s) (x)); Obtain P1 and P2 total Fitness (f (t) (x)); The structure of the games in the social dispute step of the GBX is similar to the GASI-S and GASI-POP with a few modifications to the calculation of the T , R, P, S payoff values. In the GBX, the payoff values are estimated from the minimum and maximum search space limits of the X [i][j] direction index. Equations (12) to (15) describe the GBX T , R, P, S computation. The earnings are scaled by a random variable sampled according to U (0.8, 1.2). where: To better understand the solution generation scheme of the GBX, Tables 4 and 5    Although the T , R, P, S values follows relation (2), they may vary according to the search space associated to each decision variable j. That is, the T , R, P, S values of the offspring of P1 may not be the same as P2. The game environment in the social interaction scheme of Fig.7 is expressed by (20) and (21) The total T , R, P, S values obtained by each player is represented by (Z [i][j] ), a summation of all gains along the time horizon of iterations. The summation is described in (22) where r is the number of rounds, N Ro the maximum number of rounds and Z [i][j] the total payoff of the j-th decision variable of the i-th player: Offspring of p1 and p2 are generated by (23) based on Z (default mode). As stated above, under exploration and exploitation conditions (see Fig.7), the gains applied to the jth decision variable (Z [i] [j] ) determines the boundary between diversification and intensification. The choice of which region the offspring will be placed into is determined by the absolute value of a random number sampled by a standard normal distribution (a ∼ N (0, 1)).
Besides the default offspring generation scheme, the GBX crossover features an additional two schemes, Global best Exploitation and Scattering. Both are covered in the following subsections.

B. GLOBAL BEST EXPLOITATION MODE (GBEM)
The GBEM is a mechanism that intensifies local search around the best solution region. The main difference from the default mode is in determining the minimum and maximum search ranges applied to the calculation of the T R P S values. In this scenario, the boundaries are determined by the difference between the jth parent-related decision variable value (P1 or P2) and the best overall current iteration P g [j] . Several criteria must be followed in order to guarantee effectiveness of the GBEM: • GBEM can only be used when the amount of completed iterations reaches a certain user-defined threshold; • Based on empirical evidence, a proportion of 80%/20% of solutions of the population should employ the GBEM. This is so to prevent solutions from collapsing in the subspace of the overall best; The first stage of GBEM is the initialization of the search boundaries around the global best P G [j] . Two factors are verified, the difference in the distance between the jth decision  Regarding the GBEM initialization in Algorithm 2 and Fig.8, the calculation of the search space is as follows: • 1 norm between jth decision variables of global best (P g [j] ) and parent i (P1 or P2) due to (24)   , Following the calculation of the bounds, gains are computed in the social interaction step followed by the generation of offspring. Payoff values T , R, P, S are the same as those described in the previous subsection ((13) to (16)) with the inclusion of minimum and maximum bounds in (27) N (0, 1)), otherwise we use (17). The offspring generation process is modified in GBEM mode. The offspring use the payoff values of the Social Interaction multiplied by a linear growth function θ (31) based on the number of t iterations bounded by t max . This function is similar to the linear weighting functions (9) and (10). Equation (31) forces solutions to explore solutions that are closer to the global best in early executions of GBEM mode, increasing the intensification of the search in later stages.
Acceptable values for θ are recommended to be between θ start = 0.6 e θ end = 1.0. Low values of θ start create offspring that are closer the global best in early stages of the algorithm. Lastly, new offspring are generated by (32). Initial values for direction value ∇ are randomly generated following (32) and (33). where:

C. SCATTERING MODE (SM)
To avoid rapid degradation of diversity and further explore the search space, the GBX crossover controls the spread of solutions across the solution space. This mechanism applies a random probability that the offspring will be randomly placed within the search space. The amount of solutions to undergo Scattering is determined by a user-defined parameter ρ within the [0, 1] range. It is recommended to choose values close to 0 for ρ. Scattering mode is activated once every other iteration. Algorithm 3 illustrates the process, where ω represents the amount of iterations that compose a cycle, and cont is the iteration counter variable within each cycle.
Offspring in scattering is generated by (34), where U (X [j] MIN , X [j] MAX ) is a uniform random value between the lower (X [j] MIN ) and upper (X [j] MAX ) bounds of the jth decision variable:

1) AN ALTERNATIVE VERSION OF GBX (GBX2)
The GBX2 crossover simulates win and loss scenarios with each round of dispute between parents P1 and P2 by the use of random variables according to a normal distribution (N (0.1)) inputted along with gains of each round. The sum of gains obtained during the social dispute expressed in (22) N (0, 1). It is also possible to control the distance at which the new individual's decision variable j will be in relation to its predecessor by using a linear growth/decay function; The GBX2 crossover employs a scheme to change the T , R, P, S values where the φ [i][j] perturbation factor is a random number according to a normal distribution (N (0, 1)), described in (39).
Another essential modification is in the calculation of lower and upper bounds of the GBX2. A fixed value is due 9 The implementation of the normal Gaussian distribution-based random value generator used in this article (N (µ, σ 2 ), where µ = 0 and σ = 1) is based on the Box-Muller method, which produces a pair of standard and independent normal random numbers based on a pair of uniformly generated numbers (U (0, 1)). For more information, see [46]- [48]. by the subtraction of X [j] MAX and X [j] MIN . 10 Equations (35) to (38) show the T, R, P, S estimations for GBX2.
{0.4, 0.3, 0.2, 0.1}) define the maximum percentage of the difference of the lower and upper bounds. In the T , R, P, S computations presented in this article, the maximum value of the boundary difference is 40% for T , 30% for R, 20% for P and 10% for S. where: After the Social Interaction step, the total payoff (Z [i][j] ) is obtained from the sum described in (40) 11 where Pay [i][r] is the T, R, P, S value of the ith solution in the r round whose behavior (cooperation or defection) are described in (20) and (21). The displacement direction of the jth decision variable is determined by (41). 12 If Z [i][j] is negative, the new solution to be stored in the descendant j decision variable will be lower than that found in its parent (P1 or P2). Otherwise, it will be greater than or equal to it.
Prior to the creation of offspring, the GBX2 crossover uses a linear growth/decay function ( ) to compute a weight to scale (Z [i][j] ) after the iterated games. This gradually controls the distance of new individuals from their respective parents. A monotonically increasing value of distances offspring from their predecessor, whereas a monotonically decreasing value of move solutions closer to their parents (P1 or P2). The control function is described by (42): After updating , offspring are generated by (43): GBX2 has a limited exploitation capacity in order to favor exploitation around parents. If there is a need to generate offspring even farther from their parents, in (42) can be set so that the limits represented by constants start or end have values greater than 1. GBEM and Scattering can be integrated to the GBX2 without the need of any adjustments.

VI. EXPERIMENTS AND RESULTS
Behavior and performance of the proposed algorithms is assessed by 10 well-known benchmark constrained optimization functions from the engineering field ( [10], [49]- [55]). We implemented five crossover methods for means of comparison. Two are associated with the proposal (GBX, GBX2) and three for comparison (BLX-α, 13 UX, 14 AX 15 ). 11 The total payment rate applied to the GBX2 crossover is similar to that found in the GBX method (22). 12 Applicable only to the GBX2 crossover. For GBX crossover, see (12). 13 BLX: Blended Crossover. 14 UX: Uniform Crossover. 15 AX: Arithmetic Crossover. VOLUME 8, 2020 For each crossover three different selection strategies are used, namely: Classic Tournament selection GA, Population-based Social Tournament GA (GASI-POP); Selection restricted Social Interaction-based Tournament selection GA (GASI-S). Parameters the experiment are described in Table 6. Tests were conducted in a machine with the following hardware configuration: Apple MacBook Pro mid-2012 with NVIDIA GeForce GT 650M GPU 1024 MB GDDR5 VRAM with 900 MHz clock and 384 CUDA Colors; Intel Core i7 ''Ivy Bridge'' 2.9 GHz CPU; 16 GB DDR3L RAM clocked at 1600 MHz. The operating system used was macOS Sierra 10.12.6, with CUDA driver version 9.1.

A. THREE BAR TRUSS DESIGN PROBLEM: TBT
Optimization of a two-dimensional function (x j = {x 1 , x 2 }) representing the volume subject to stress constraints on each side of the lattice [49], [56], [57].
Subject to:   Described by Arora [67] and Belegundu [68], this problem consists of the minimization of the weight of a tension/compression spring. The problem features four constraints describing the minimal deflection, shearing tension, VOLUME 8, 2020 surge frequency and external diameter limits. Design variables are the average spool diameter (x 1 ), yarn diameter (x 2 ), and active spool number (x 3 ) Subject to: Bounds of the design variables {x 1 ,

F. TABULAR COLUMN DESIGN PROBLEM -TCD
The main objective is to design a uniform tabular section column with a length (L) equal to 250 cm at a minimum cost that includes material and construction costs that can withstand a compression load P of 2500 kgf . This is a two-dimensional problem (x j = {x 1 , x 2 }) whose design variables are: Average column diameter (d = x 1 ); column thickness (t = x 2 ). The column is made of a material with a yield stress (σ x ) of 500 kgf /cm 2 , a modulus of elasticity (E) of 0.85 × 106 kgf /cm 2 and a density (ρ) equal to 0.0025 kgf /cm 2 .
Subject to: In order to find the minimum cost of fabricating a welded beam subject to shear stress conditions (τ ), bending stress  on the beam (θ), bar buckling load (P c ), final beam deflection (δ), and side constraints, the WBD problem consists of four design variables (x j = {x 1 , x 2 , x 3 , x 4 }): Weld thickness (h = x 1 ), weld joint length (l = x 2 ); beam width (t = x 3 ); beam thickness b = x 4 . In this work, we use two of the WBD, the first (WBD1) has six constraints while the second (WBD2) adds a seventh constraint accompanied by changes in deflection (δ(x)), buckling calculations (P c (x)) and polar moment of inertia (J (x)).

H. RESULTS
Results were obtained based on the collection of 60 test runs for each algorithm. The implementation of other crossover methods (i.e. BLX-α, AX, UX) with those proposed in this article (i.e. GBX, GBX2) allows an explicit comparison, both in terms of convergence and diversity in the search space.
The algorithms implemented are also compared to other techniques published in the scientific literature.

1) COMPARATIVE TABLES (PERFORMANCE)
Tables 7 to 13 show the following statistics measured in the experiment (fitness) of all runs: Mean of the best; median of best; standard deviation, best and worst observed. Ranking of the best observed results for mean and best fitness statistic is displayed in Table 14. It is possible to observe that the GBX2 crossover method had the best results for five functions for the mean fitness statistic (DPV1, DPV2, MWTCS, WBD1, WBD2), as well as comparable fitness to the best fitness statistic in seven instances (DPV1, DPV2, HNO1, MWTCS, TBT, WBD1, WBD2).
No significant results were detected in terms of the mean of the population and global best between implementations that use Social Interaction in the selection process      (GASI-POP, GASI-S) and those that use only standard tournament (GA). Although the decrease in selective pressure based on the social dispute did not contribute significantly to the quality of the solutions, the best ranking values were obtained by the following algorithms: GA, GBX2, GASI-S GBX2 and GASI-POP GBX2, respectively. For algorithms based on the GBX crossover, no significant improvements in global best fitness was observed, with the best performance in only two functions (i.e. HNO1, HNO2), both having their design constraints bounded in a closed interval; it is important to notice that excluding the HNO1 and HNO2 instances, the proposed algorithms had better results than the ones published in the scientific literature.

2) STATISTICAL ANALYSIS
Statistical significance of the data is validated by the Mann-Whitney U non-parametric test. Statistical significance of one algorithm in relation to another is confirmed if p < 0.05, otherwise the compared solutions have no statistical difference. Data in Tables 15 and 16 was obtained from the mean of the best observed fitness of each run. Based on these results, it is possible to affirm that the algorithms based on the crossover GBX2 without Social Interaction in the selection has statistical relevance in relation to the BLX-α, AX and UX methods in the 10 instances. Whereas for the GBX, the null hypothesis was failed to be rejected in two instances (TBT, WBD2).
Regarding the GASI-POP and GASI-S algorithms, the p values suggests that the GASI family of algorithms had no statistical significance compared to their GA counterparts (see Table 17). In general, it is possible to conclude, based on the tests below that the use of Social Interaction in the selection process has significantly contributed to any difference to the improvement of results in relation to its counterparts.

3) ANALYSIS OF CONVERGENCE AND DIVERSITY
The following graphs illustrate the behavior of the mean of all runs for assessment of convergence and diversity throughout the iterations for each algorithm, exposed in Table 14. Data was normalized using (3) and (4) in order to adapt results with violated constraints to the plots. For both graphs (convergence and diversity), the following equations are used: • Mean of the global best (Fig.10): Mean of the best fitness of each run in iteration t. Variables of (98) are: global best fitness of run P g at time step t (Pfg(x) [ex] ); Total of runs (N ex ); current run index (ex).
• Population diversity (Fig.11): Population diversity estimation proposed by Ursem [108] is measured for all algorithms. Variables related to (99) In accordance to the graphs of Fig.10 and Fig.11, several conclusions can be made: 1) The algorithms based on the GBX2 crossover (GA GBX, GASI GBX2(S), GASI GBX2(P)) were shown to have the best fitness values in five problem instances (DPV1, DPV2, MWTCS, WBD1, WBD2). It is also verified that the GBX2 converged to an accumulation point faster than the others in six engineering problems (i.e., DPV1, DPV2, MWTCS, SRD11, WBD1, WBD2); 2) In box constrained problems (i.e., HNO1, HNO2), GBX algorithms achieved slightly higher performance when compared to competing methods, specifically against AX crossovers (i.e., GA AX, GASI-POP AX, GASI-S AX). Such performance is expressed both in the global fitness average and in the fast convergence; 3) A deceleration in the intensity of the convergence was detected in algorithms with Social Interaction during the selection phase in comparison to the versions with standard selection. For diversity, GASI algorithms showed a slight increase which may vary based on the Game parameters (see Table 6); 4) The inclusion of the GBEM (marked with black arrows) during the last generations has significantly influenced the convergence speed (Fig.10). A sudden drop is observed in the mean statistic, smoother for GBX2 and steeper for GBX. Regarding the diversity of the population (Fig.11), the GBX method showed a significant drop after using the GBEM scheme in all evaluated functions, whereas in GBX2, the loss of diversity occurs with less intensity in seven tested problems (DPV1, DPV2, TBT, TCD, MWTCS, WBD1, WBD2); 5) During the execution of the algorithms based on GBX and GBX2, it is possible to identify the periodic effect of the SM mode (marked with red arrows) in Fig.11, where the scattering process causes a sudden increase in the diversity of the population;

VII. FINAL REMARKS AND CONCLUSION
In this paper, we introduced a social interaction step rooted on concepts of game theory that runs at the selection or crossover steps of the GA. Running the social dispute at the selection step expands the ability to diversify the solutions available in the search space (GASI). While in the crossover, Game Theory schemes generate new offspring (GBX/GBX2). A combination of the GASI and GBX/GBX2 crossover resulted in six new metaheuristics. The performance of these new algorithms were evaluated by ten engineering optimization problems with equality and inequality constraints. For comparison means, previously published techniques (e.g. particle swarm optimization (PSO), differential evolution (DE), ant colony optimization (ACO), Lagrange multipliers, Branchand-Bound method) were chosen to validate the approach. Experiment data suggests viability and good performance in the use of mechanisms based on Social Interaction applied in the crossover. GBX2 presented the statistically relevant results in eight of the ten instances in spite of a gradual loss of diversity over the generations. The GBX crossover has not shown satisfactory convergence speed to accumulation points despite its good ranking score (inferior only to the GBX2 method) and highest degree of diversity. A possible solution to this deficiency is the implementation of other techniques aimed at the estimation of a non-random and focused on the use of information about the current state of the population direction vector. Another viable option would be the inclusion of local memory associated with previously found solutions.
Regarding the performance of the GASI algorithms, the data indicates a limited influence to the diversity and the search capacity of the algorithm. In short, the use of game-based dispute applied to the control of selective pressure in the process of selecting parents does not have a relevant impact based on the data obtained. It is possible to obtain a different performance by changing the parameters observed in Table 6, emphasizing the rate in which games take place. However, in a preliminary general analysis it is recommended to use additional approaches in order to improve the quality of optimization in the methods of the GASI family, such as the use of the diversity value to adjust parameters along the inclusion of other information to be used during selection criteria (i.e. local memory, spatial distance between individuals).
In general, the data obtained in the experiments suggests the inclusion of Social Interaction techniques for population based metaheuristics. For future work, we highlight the improvement of the mechanisms proposed in this article such as the assignment of behaviors for each decision variable instead of just a single strategy profile for the individual as a whole. It is also possible to emphasize the use of other payment rules available in addition to the Prisoner's Dilemma (e.g. Stag Hunt, Harmonic Games, Friend or Foe), regulation of payment through Fuzzy agents, as well as the expansion of the Social Interaction model used in the proposed algorithms so that they can house more rules of dispute and other combinations of strategy, thus making the game more complex and subject to a greater number of combination of results.