Optimization of an Evaluation Function of the 4-sided Dominoes Game Using a Genetic Algorithm

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1 o Otimization of an Evaluation Function of the 4-sided Dominoes Game Using a Genetic Algorithm Nirvana S. Antonio, Cícero F. F. Costa Filho, Marly G. F. Costa, Rafael Padilla Abstract In 4-sided dominoes, the oular way of laying dominoes in Amazonas State, the strategies used for the game are more comlex than those adoted in the more traditional 2- sided dominoes, the most oular dominoes game layed in Brazil. This work resents the otimization of an evaluation function for the best move in 4-sided dominoes using a genetic algorithm. The evaluation function is comosed of terms that incororate the game s strategies and are defined as: unctuating, facilitating future moves and comlicating oonents moves. Coefficients were defined to determine the imortance of each term of the evaluation function and a set of arameters and oerators for imlementation of the genetic algorithm. The layers ability was calculated by the number of wins in 5,000 matches. The results obtained during the simulations showed that the team (comosed of 2 layers) that used the evaluation function with its coefficients otimized by the genetic algorithm won in more than 70% of the total matches. I. INTRODUCTION Dominoes is a game consisting of rectangular ieces divided in 2 halves, which are marked with black dots indicating a numeric value. Possibly having originated in China, dominoes is a oular game all over the world and can be layed in many different ways, deending on the region. In Brazil, the most oular way is called 2-sided dominoes. However, in Amazonas State, another variation of the game is more oular among the locals, dominoes layed on four sides. 4-sided dominoes is characterized by being a multi-agent roblem [1], because it is not a zero sum game, layed by four layers divided in two airs, which have imerfect information about the ossible moves of the other. When layed by 2 airs, it is considered a two-layer game, as there are only two scores. Therefore, each layer must develo strategies based on incomlete information in order to win the game with his/her artner. N.S. Antonio is with Universidade Federal do Amazonas/Centro de P&D em Tecnologia Eletrônica e da Informação UFAM/CETELI. Manaus. Amazonas. Brasil (tel: ; nirvana.sa@gmail.com). C. F. F. Costa Filho is with Universidade Federal do Amazonas/Centro de P&D em Tecnologia Eletrônica e da Informação UFAM/CETELI. Manaus. Amazonas. Brasil (tel:: ; ccosta@ufam.edu.br). M. G. F. Costa is with Universidade Federal do Amazonas/Centro de P&D em Tecnologia Eletrônica e da Informação UFAM/CETELI. Manaus. Amazonas. Brazil (tel: ; mcosta@ufam.edu.br). R. Padilla is with Universidade Federal do Amazonas/Centro de P&D em Tecnologia Eletrônica e da Informação UFAM/CETELI. Manaus. Amazonas. Brasil (tel: ; eng.rafaeladilla@gmail.com). The use of an evaluation function associated with the alha-beta algorithm for choosing the best move in games of erfect information was first roosed by Shannon [3]. This roosal was imlemented for chess [4] and checkers [5]. For games of imerfect information, the use of Shannon s roosal involves comlex reasoning about game strategy. It is mentioned for examle in the Bridge Baron rogram [6], develoed for Bridge. For 2-sided dominoes, some studies can be found in the literature seeking the best game strategy [7] [9]. In [2], we develoed a methodology for choosing the best move based on an evaluation function that combines in its terms information about the resent state of the game. To evaluate the develoed evaluation function, simulations were conducted for 4-sided dominoes games, in which a air used the evaluation function for choosing the best move while the other air erformed their moves based only on the dots of the dominoes on the table. The choice of the arameters of the evaluation function was done manually and as result, the first air won the game in more than 66% of the simulations. In this study, we will erform the otimization of the arameters that make u the evaluation function alied to some strategies that can be adoted by the layer. The otimization will be achieved by the otimization and search technique of the genetic algorithms (GA). GA is a very efficient technique for searching for otimal or near-otimal solutions. Moreover, it is easy to imlement and allows for arallel comuting. With the otimization rocess, we hoe that the air that uses the evaluation function to choose their moves erforms better than their oonents that use only the unctuations of the ieces on the table as a criterion to choose their moves, considered a basic strategy used by beginners. II. DOMINOES GAME Dominoes is a game consisting of 28 rectangular and flat ieces (or tiles). These ieces are divided in two halves that, in the classic form of the game, contain numbers ranging from zero to six dots, marking all combinations between these numbers. It can be layed by four layers, in airs or individually, where each layer receives seven ieces or, alternatively, can be layed by two layers, where the 14 remaining tiles are bought during the match. There are several ways to lay dominoes and the most common form in Brazil is called 2-sided dominoes. However, in Amazonas State, it is layed in another less oular way, 4-sided dominoes /11/$ IEEE 24

2 In 4-sided dominoes, the games are layed by two airs. Each layer must have seven tiles in hand at the beginning of each turn. In the first round, the first iece to be dislayed is the 6-6, called double six, having two halves numbered with 6 dots each. The layers have the ossibility to lay off the 4 sides of this first iece. In Figure 1, we can see a game where 3 oints have been oened from the double six. less than or equal to this sum. Figure 3 shows an examle where the garage is worth 10 oints since the score of the oonents is 12. Fig. 3. Garage examle of 10 oints P5) The last iece discarded by the layer who goes out is a double. The score of this move is equal to 20 oints. Fig. 1. Examle of a move that is scored 15 oints In the following rounds, the layer who hit (used u all their stones (tile ieces) before the other layers) in the revious round will start with the double of his choice. The main objective of 4-sided dominoes is to achieve a score of 200 or more oints, in one or more rounds of lay. During the game, there is the ossibility of scoring 5, 10,, 50 oints, in other words, multiles of 5. The chances of scoring can be defined by the following conditions: P1) The sum of the dots on the ends is a multile of 5 and the score recorded for the air is equal to this sum. Figure 1 exemlifies a move equivalent to 15 oints, because the sum of the dots on the ends of the double that started the game (2+5+8) is a multile of 5. However, Figure 2 shows an examle of a move where there is no score, because the sum of the dots on the ends is 14 (1+5+8). The game of 4-sided dominoes, having more elaborated goals and rules, adots more comlex strategies than those used in 2-sided dominoes, because in addition to scoring, the layer should also try to facilitate his (or his artner s) future moves and imede the moves of their oonents. To illustrate the strategies adoted during the game, it is considered that at the beginning of a round, a layer has in hand four or more ieces with the same number (Figure 4). In this scenario, the strategy is to try to make this same number resent on the most ossible sides, since this tye of move could force his oonent to ski his turn or may facilitate future moves. Fig. 4. Examle of a set of initial tiles of a layer where four of them have the number 5 Fig. 2. Examle of a move with no score P2) The oonent layer has no iece to fit in any of the 4 ends, so he skis his turn instead of laying, imeding an oonents lay is equivalent to a score of 20 oints for the air who caused the skiing. P3) When the layer causes all other layers to ass (including his artner). This ass, called a rooster is equivalent to a score of 50 oints. P4) When a layer hits, the dots of the ieces left in the hands of the oonents are totaled. This sum is called the garage. The score corresonds to the highest multile of 5 III. EVALUATION FUNCTION FOR SELECTING THE BEST MOVE Previously we develoed an intelligent agent for 4-sided dominoes that chooses the best move by an evaluation function [2]. The evaluation function consists of the sum of two terms, T n1 and T n2, as defined by equation (1). = + (1) In the exression above, the variable n identifies a move for which the evaluation function is calculated. The term T n1 corresonds to oints obtained by erforming move n. The term T n2 incororates the strategy of the game, corresonding to values that reresent how choosing move n can facilitate future actions of the layer and make future moves of the oonents even more difficult IEEE Conference on Comutational Intelligence and Games (CIG 11)

3 To calculate the terms T n1 and T n2 of the evaluation function, the current state of the game was used. The state of the game was defined as a set of 7 vectors, V i = {a i, b i, c i, d i, e i, f i, g i }, where a i corresonds to the number of ieces (tiles) layed with numbers 0; b i corresonds to the number of ieces layed with numbers 1, and so on. These vectors exress statistics udated every move. The definition of states of 4-sided dominoes is shown below: V 0 : quantity of ieces in the game for each number. V 1 : quantity of ieces in the hands of the layers for each number V 2 : quantity of ieces for each number at the tile ends. V 3 : quantity of ieces already layed by the air for each number. V 4 : indicates the numbers where the following oonent has already assed. If a 4 is equal to 1, the following oonent has already assed the numbering 0. If a 4 is equal to 0, the following oonent still has not assed the numbering 0, and so on. V 5 : indicates the numbers where the revious oonent has already assed. V 6 : indicates the numberings where the artner has already assed. The term T n1 incororates in the calculation of the evaluation function the oints obtained when the move otion n is made through situations P 1, P 2, P 3 and P 5, as shown in equation (2). = (2) It is noteworthy that the T n1 does not incororate situation P 4, which corresonds to the oints of the garage, because they cannot be counted while the round is not comlete. To understand the text that follows, let us define L as a ossible otion to lay, which means, a tile in the layer s hand. Each tile has numbers in its two halves. L 1 corresonds to the numbering of the half of the stone that matches one of the values in any tile ends of the game, and L 2 corresonds to the other half that does not match the values at the ends. Thus, if a layer has the otion to lace a iece numbered 5-3 and the game on the table is reared as illustrated in Figure 1, L 1 will receive the number 5, as it matches one side and L 2 will receive the number 3, which does not match the side. The term T n2 includes in its calculation two distinct ortions related to the game s strategy, as shown in equation (3): T = + (3) Portion E 1 considers the ossibility of forcing the subsequent oonent to ski his turn. To understand ortion E 1, it will be used in the following hyothetical situation: let us suose that in two of the tile ends there is the number n 1 and the layer taking a turn has three ieces in his hands with the same numbering. So, five of the seven ossible ieces with numbering n 1 do not belong to the oonent laying next; therefore, the robability of the oonent skiing his turn is higher if all the ends have this numbering. Given the ossibility of making the next oonent ski his turn and, thus score 20 oints, it is desirable that no iece with numbering L 1 = n 1 is discarded. In this case, the otion to lay a tile in which L 1 = n 1 is not desired from a strategic ersective, the negative sign for E 1 is given in the exression (3). The calculation of E 1 is erformed according to exression (4). = (4) The more ieces with numbering L 1 on the ends (reresented by vector V 2 ) and in the hands of the layer (reresented by the vector V 1 ), the greater the value of E 1 is. On the other hand, the value of E 1 should also be directly roortional to the number of ieces with the numbering L 1 already layed (vector V 0 ). The coefficients α 1, α 2 and α 3 control the imortance of each term of E 1 in relation to other arcels comromising the evaluation function. The ortion E 2 aims to facilitate future actions of the layer. The calculation of E 2 is erformed according to equation (5). = (5) This exression is very similar to the one roosed for E 1, but L 2 is used instead of L 1 as indeendent argument, which is the number the layer wants to add on the table. There is also a term related to facilitating his artner s lay, reresented by the vector V 3. In this case, the term examines the ieces layed by his artner to the numbering L 2 resented at the tile ends. The coefficients α 4, α 5, α 6 and α 7 control the imortance of each term in E 2 in relation to other arcels that comose the evaluation function. The evaluation function described in equations (1) - (5) involves the main cases considered by a layer while deciding the best move at each round. In other words, it rooses a ossible strategy to be adoted in 4-sided dominoes. In this work, this strategy will be called "Strategy 1" and its evaluation function will be used to calculate the fitness of the chromosomes during the execution of the genetic algorithm. At the end of the otimization, the genetic algorithm will find the coefficients α 1, α 2, α 3, α 4, α 5, α 6 and α 7 otimized for this strategy. Besides the strategy outlined above, other strategies were analyzed in this work. In the second strategy, the layer gives higher riority only to moves that can romote his own game; in other words, the coefficients α 1, α 2, α 3 (makes the oonents' moves more difficult) and α 7 (analyzes the moves that make his artner's game easier) are set to zero. Therefore, the evaluation function to be otimized by the genetic algorithm for Strategy 2 is made only by the terms of 2011 IEEE Conference on Comutational Intelligence and Games (CIG 11) 26

4 equation (6). At the end of otimization, the genetic algorithm will find the coefficients α 4, α 5 and α 6 otimized for this strategy. = (6) In the third strategy, the layer only gives higher riority to moves that can romote his artner's game, in other words, the coefficients α 1, α 2, α 3 (makes his oonents' moves harder) and α 5 (analyzes moves that make his own game easier) are set to zero. Therefore, the evaluation function otimized by the genetic algorithm for Strategy 3 is formed only by the terms of equation (7). At the end of otimization, the genetic algorithm will find the coefficients α 4, α 6 and α 7 otimized for this strategy. = (7) The fourth strategy gives higher riority to moves that make the oonents actions more difficult, in other words, coefficients α 4, α 5, α 6 and α 7 (facilitates the moves of the air) are set to zero. Therefore, the evaluation function to be otimized by genetic algorithm for Strategy 4 is formed only by the terms of equation (8). At the end of otimization, the genetic algorithm will find coefficients α 1, α 2 and α 3 otimized for this strategy. = (8) The fifth strategy to be considered in this work is the basic strategy of 4-sided dominoes, being the strategy used by novice layers. This strategy considers only the term T n1 of the evaluation function described in equation (1); in other words, only those oints are considered for choosing the best move. IV. OPTIMIZATION OF THE EVALUATION FUNCTION A. Parameters of the Genetic Algorithm In order to otimize the evaluation function using GA, we use a combination of arameters for better results, as described below. Coding: the chromosomes are encoded as a vector of tye double, which reresent arameters α 1, α 2, α 3, α 4, α 5, α 6 and α 7 of the evaluation function. Fitness function (fitness): the fitness function is resonsible for measuring the erformance of a chromosome during the simulation of domino matches. The fitness value of a chromosome is given by the number of wins resulting from the erformed matches. Poulation size: the size of the oulation directly affects the erformance of the algorithm. Very small oulations have less genetic diversity and can lead to a faster convergence of the algorithm to a local minimum. On the other hand, very large oulations make the algorithm too slow [10]. For the otimization erformed in this work, we use the following oulation sizes: 40, 60, 80 and 100. for the otimization rocess, two techniques of crossing were used: Two-oint and Uniform. The rate of crossover c, which defines the robability that the selected chromosomes ass by the rocess of crossing, will be 0.8 in this work. Mutation: To imlement the mutation rocess, the oerator we chose was the Uniform Mutation and the value of mutation robability, m, takes the values 0.1 and 0.5. Stoing criterion: as sto criterion of the genetic algorithm, we used the total number of generations in the following sizes: 50, 100, 150 and 200. B. Imlementation For the simulation of 4-sided dominoes, a rogram was develoed in Java with four agents acting as the four layers required for the match. In this rogram, the first air uses a basic strategy for choosing the best move and the second air may have its arameters α 1, α 2, α 3, α 4, α 5, α 6 and α 7 adjusted for each simulation. Thus, the simulator of 4-sided dominoes has as inut arameters the total number of desired matches and the set of coefficients α of the second air, returning the number of wins of this air. To imlement the otimization by genetic algorithm, we use the Genetic Algorithm toolbox of MATLAB, the GAtool. A function was created in MATLAB (dfitness.m) resonsible for calculating the fitness of each tested chromosome, by adjusting the inut arameters of the domino simulator in Java. Besides the fitness function (or fitness), the oulation tye was defined as a Double Vector, receiving values in the range of -10 to 10. In the selection rocess, we used the Roulette method, which simulates a roulette wheel with the area corresonding to each chromosome roortional to its fitness value; in other words, the robability of a chromosome being selected is directly roortional to its area on the roulette wheel. For the configuration of the reroduction rocess, the arameter Elite count, which indicates how many individuals with the best fitness values will be transmitted directly to the next generation without going through the reroduction, receives values corresonding to 10% of the oulation size. We used two crossover oerators (or crossover): Two-oint and Scattered (also known as Uniform crossover). The urose of erforming various otimizations is to find a combination of the genetic algorithm arameters required to achieve the otimal or near-otimal values for the coefficients α 1, α 2, α 3, α 4, α 5, α 6 and α 7 of the evaluation function, in a way that they allow the air that uses this function to choose the best move, has a better erformance than the air that only uses the strategy of adding oints during the match (coefficient α equal to zero). The set of arameters defined reviously gives us a combination of 64 executions of the genetic algorithm otimization for each strategy, in a total of 256 runs of the genetic algorithm. A summary of all variations of these IEEE Conference on Comutational Intelligence and Games (CIG 11)

5 arameters is shown in Figure 5: Fig. 5. Parameters used for the otimization of genetic algorithm. V. RESULTS AND DISCUSSION The following results show the otimizations obtained for Strategies 1, 2, 3 and 4 laying against the "basic strategy" game. For Strategy 1, the genetic algorithm erformed the otimization of the seven coefficients α. On the other hand, for Strategies 2, 3 and 4, the genetic algorithm erformed the otimization of only three coefficients α i, because the others are set to zero, deending on the tye of strategy. A. Otimization of the evaluation function of Strategy 1 The otimization of the evaluation function corresonding to the first strategy was divided into four grous according to the size of the oulation, receiving the values 40, 60, 80 and 100. For each arameter combination, the amount of wins was registered for air 2 and the coefficients resulting from the otimization rocess. Table 1 and Figure 6 show the best results for each otimization grou of the evaluation function of Strategy 1 together with the arameters used in the genetic algorithm during the otimization rocess. According to the simulations, the best results were obtained in the otimizations with a total of 100 generations, regardless of oulation size. On the other hand, an\ increase in oulation size led to an imrovement of the results. TABLE 1 Otimization results of the evaluation function of Strategy 1 Poulation Generations Two-oint % Scattered % Scattered % Two-oint % The best results obtained by air 2, which uses Strategy 1, are dislayed with the arameters used in Genetic Algorithm. The first grou of the otimizations corresonds to the oulation of 40 chromosomes. In this grou, a total of matches were held: air 2 won 3,462 matches, which reresents 69.24% of wins. The second grou corresonds to the combinations with oulation size equals to 60. In this grou, the best result is 3,484 winnings in 5,000 matches, which corresonds to a 69.68% yield. In the third grou of otimizations for the oulation of 80 chromosomes, the best fitness value achieved was 3,488 winnings (or 69.76%) for the air that used the evaluation function with the arameters otimized by the genetic algorithm. The last grou of otimizations of the evaluation function erformed tests for a oulation of 100 chromosomes. The best result is 3,503 winnings in 5,000 matches, which corresonds to a 70.06% yield for the air that used the evaluation function to choose the best move. This was also the best overall result obtained for Strategy 1 and the coefficients α 1, α 2, α 3, α 4, α 5, α 6 and α 7 of the evaluation function are shown in Table 5, in the end of the section. Fig. 6. Overall results of the otimization of the evaluation function of Strategy 1 B. Otimization of the evaluation function of Strategy 2 To otimize the coefficients of the evaluation function of Strategy 2, the values for a α 1, α 2, α 3 and α 7 were defined as 0. Once more, the otimization of the evaluation function was divided into grous according to the oulation size. Table 2 and Figure 7 show the best results obtained in the otimization of the evaluation function of Strategy 2. We can see that this strategy, where the layer focuses only on his own game, the results were worse than those in Strategy 1. The best overall result was 3,355 victories in 5,000 games, which reresents 67.1% success for the air that used the evaluation function to choose the best move. The coefficients α 4, α 5 and α 6 otimized in the evaluation function are shown in Table 5, at the end of this section. TABLE 2 Otimization results of the evaluation function of Strategy 2 Poulation Generations Two-oint % Scattered % Scattered % Two-oint % The best results obtained by air 2, which uses Strategy 2, dislayed along with arameters used in the Genetic Algorithm. C. Otimization of the evaluation function of Strategy 3 To otimize the coefficients of the evaluation function of Strategy 3, the values for α 1, α 2, α 3 and α 5 were defined as 0. As in the revious otimizations, the results were divided into grous according to the oulation size. Table 3 shows 2011 IEEE Conference on Comutational Intelligence and Games (CIG 11) 28

6 the best results for each grou of the otimization of the evaluation of Strategy 3 and the arameters used in the Genetic Algorithm. TABLE 4 Otimization results of the evaluation function of Strategy 4 Poulation Generations Two-oint % Scattered % Scattered % Scattered % The best results obtained by air 2, which uses Strategy 4, are dislayed along with the arameters used in the Genetic Algorithm. Fig. 7. Overall results of the otimization of the evaluation function of Strategy 2 TABLE 3 Otimization results of the evaluation function of Strategy 3 Poulation Generations Two-oint % Scattered % Two-oint % Scattered % The best results obtained by air 2, which uses Strategy 3, dislayed along with arameters used in the Genetic Algorithm. Fig. 7. Overall results of the otimization of the evaluation function of Strategy 3 D. Otimization of the evaluation function of Strategy 4 In Strategy 4, the layer wants only to imede the moves of his oonents, so the values for the α 4, α 5, α 6 and α 7 were defined as 0. Table 4 and Figure 9 resent the best results for each grou of the otimization of the evaluation of Strategy 4 and the arameters of the genetic algorithm. Fig. 8. Overall results of the otimization of the evaluation function of Strategy 4 Observing the above results, we note that Strategy 4 showed the worst results among all tested strategies, whereas the best result was the 61.04% of wins of air 2, or 3,052 victories in 5,000 matches. The otimized coefficients α 1, α 2 and α 3 in the evaluation function are shown in Table 5, as well as the otimized coefficients for the other strategies. All of the tested strategies showed significant results comared to the basic strategy, used by beginner layers. However, the evaluation function roosed initially reresented by Strategy 1, achieved the best results because it combines the three strategies in its equation. The best result obtained in this study, 70.06% yield, was also higher than the results obtained in [2] and [9]. In [2], the same intelligent agent was used to choose the best move, but the definition of the coefficients α i was done manually. His best result was 66% wins. In [9], where the research was based on 2-sided dominoes, an intelligent agent was also develoed to choose the moves. The agents conducted the decision of the move by inferences and assumtions based on revious moves of other layers. However, the stored data was only related to the numbers the layers layed. Seven different strategies were imlemented, including a traditional strategy based on the strategy adoted by real layers. However, to validate the intelligent agent, only one of the two layers varied his strategy in each simulation; the others layed with the traditional strategy. The best result obtained in this study was 54% wins and is not considered a significant result because it was within the tolerance limits set by the author due to the random characteristics of the game IEEE Conference on Comutational Intelligence and Games (CIG 11)

7 TABLE 5 Otimized coefficients for Strategies 1, 2, 3 and 4 Est. α 1 α 2 α 3 α 4 α 5 α 6 α VI. CONCLUSIONS This work resented the results of the coefficients otimization rocess of the evaluation function to choose the best move in 4-sided dominoes. To adjust the coefficients, we used the technique of searching and otimization of Genetic Algorithms alied to find the best combination that allowed the largest ossible number of wins. Four different strategies were evaluated and to obtain the best combination of coefficients of the evaluation functions, the genetic algorithm was run several times with different settings. The otimization results of the evaluation function were suerior to the results obtained in revious work, where 66% was obtained. Among the tested strategies, Strategy 1, which is a combination of the other three strategies, had the best results. Given these results, we conclude that the erformance of the genetic algorithm was excellent, being suerior to the others. Analyzing and comaring our results with results from other studies, ours roved to be more effective in the choice of the best moves in 4-sided dominoes. It is believed that the methodology alied to choose the best move can also be alied to other games where layers have imerfect information. REFERENCES [1] K. P. Sycara, Multiagent systems. AI Magazine, 1998, v. 19, n. 2, [2] N. S. Antonio, C. F. F. Costa Filho and M. G. F. Costa, Proosta de uma heurística ara o jogo de domino de 4 ontas, in Proc. VII Brazilian Symosium on Comuter Games and Digital Entertainment Comuting Track, Belo Horizonte, 2008, [3] C. E. Shannon, Programming a comuter for laying chess. Philosohical Magazine, 1950, 41(4), [4] M. S. Cambell, A. J. Hoane, F. H. Hus, Dee Blue, Artificial Intelligence, 2002, 134(1-2), [5] J. Shaeffer, One Jum Ahead: Challenging Human Suremacy in Checkers. Berlin: Sringer-Verlag, [6] S. J. J. Smith, D. S. Nau, T. A. Throo, Success in sades: Using AI lanning techniques to win the world chamionshi of comuter bridge, in Proceedings of the Fifteenth National Conference on Artificial Intelligence, Madison, Wisconsin, AAAI Press, 1998, [7] B. S. Chlebus, Domino-tiling games, Journal of Comuter and System Sciences, v. 32, n. 3, 1986, [8] H. C. Yen, A multiarameter analysis of domino tiling with an alication to concurrent systems. Theoretical Comuter Science, 2002, v. 98, n. 2, [9] A. G. de S. Garza, Evaluating Individual Player Strategies in a Collaborative Incomlete-Information Agent-Based Game Playing Environment, in IEEE Symosium on Comutational Intelligence and Games, 2006, [10] D. Ashlock, Evolutionary Comutation for Modeling and Otimization, Sringer, IEEE Conference on Comutational Intelligence and Games (CIG 11) 30