Let’s face the strong wind
Hasha Before reading the article to
, Xiaotian would like to ask if there are any RNG fans among the fans. Xiaotian is now recycling heartbroken RNG male fans, age requirements are 18-20, height 175+, body shape Come on if you look good, RNG will make you sad, I don’t know .
Okay, let’s get back to the topic. Recently, the topic “RNG lost to ” has topped the Weibo hot search. I believe that for many RNG fans, this is undoubtedly not a difficult fact to accept.
Everyone is asking: Why did we lose! Why did you lose! How could it be possible to lose! After all,
had a joke like this before: " I don't know who RNG is, but I know that he will be awesome every once in a while. "
was eliminated by G2 because RNG underestimated the enemy. After all, when the RNG players saw the G2 draw, every one of them burst into laughter. Looking at this scene now after losing, it is really ironic. It is no exaggeration to say that RNG lost to the team that least deserved to lose today. After the
competition ended, many fans also went to the official WeChat account to leave messages, some of which contained some fierce language.
In Xiaotian's opinion, this should also be a manifestation of "deep love and deep responsibility". As a fan, he must also hope to win the game. But victory or defeat is a common thing in military affairs. If you don't perform well this time, it doesn't mean next time.
As a rookie, Xiaotian, although I don’t watch competitive games on time, I still get excited when I see related videos. After all, LOL was also my youth.
still remembers that sophomore year, it was a hot summer. There were no classes in the afternoon, so several female students in the class discussed masturbating in the afternoon. Unfortunately, they were spanked three times in a row and they knelt down. So the five of us started discussing the reasons.
A: I was so angry that I was beaten up by the opponent. Me: It’s just that the tower was pushed too fast, and even though I asked you to join us, you never came. C: At that time, I had to play jungle and develop, so I didn’t have time. D: Let’s go to the next group. Me: I’ve told you several times but you haven’t come yet. A: We have to stick to other paths. D: How about we start now and the five of us go together. Me: Yes, ADC can demolish towers quickly. Let’s use all five of us as ADCs. In this way, we go to the middle lane and demolish the middle lane. There is usually only one person in the middle lane. If we fight five against one, we are not guaranteed to win. Together: Oops, what a great idea.
The results can be imagined. Five people walked all the way, and none of them had a chance to develop. We didn’t even have to wait for 20 shots, and we were all GG. Unwilling to admit defeat, we continued to experiment with five mages, five top laners, and five were successful. Alone, five junglers, and five support queens, I was so kneeling that afternoon that I couldn't stand up.
But we are still not convinced, and we all think that our tactics are fine. This is obviously a new tactical discovery!
So Xiaotian I ran to ask my senior LOL player cousin - mathematical modeling veteran .
After hearing this, my cousin sneered and said that you really asked the right person. But according to your approach, why are there still roles such as top laner, mid laner, jungler, ADC and support in League of Legends?
First of all, I think this is a mathematical problem that can be solved using a model. Mathematical Modeling Algorithms and Applications 2nd Edition
has two research ideas. One is based on data models and proves whether there are differences in the winning rates of different lineups through a large amount of battle data analysis.
The other method is to establish a mechanism analysis model based on certain assumptions, treat the system of both parties as a differential dynamics system, and solve the model. The former is relatively simple, and the conclusions obtained are real and intuitive; the latter is relatively complex, but can go deeper into the essence of the problem.
League of Legends is a team game. Obviously, the essence of this problem is to explore the efficiency of different collaborative combat combinations and select the optimal combination based on the results of model solution. This allows the establishment of an operations optimization model. However, because this problem is too complex, appropriate simplification is also necessary.
Research on multiplayer online competitive games based on Lanchester equation
Abstract : In order to solve the problem that current multiplayer online competitive games generally lack quantitatively described mathematical models, we first use the classic Lanchester equation to calculate the winning probability of both sides of the game. Conduct preliminary discussions. Based on the multivariate Lanchester equation and considering the coordination factors between game players, a differential equation model describing the battle process of multiplayer online competitive games is established. By using Vensim software to perform system dynamics simulations on the multiplayer online competitive game "League of Legends", the game process can be visually displayed and the results can be predicted.
Keywords : Lanchester equation; multiplayer online competitive games; synergy; system dynamics
Among many online games, multiplayer online competitive games occupy a considerable proportion of the online game market. Among them, Riot Games develops The online game "League of Legends" has an average of more than 27 million players per day, making it the online game with the most players currently[3]. Establishing a mathematical model to discuss the offensive and defensive strategies and the winning-lose relationship of this type of game will not only enable players to enhance their understanding of the game, but also promote the continuous development and upgrading of the game. Since the purpose of multiplayer online competitive games (especially tower pushing games) is to kill enemy characters and destroy enemy buildings (such as defense towers), based on the Lanchester equation commonly used in traditional war analysis, Combining the characteristics of different games to establish mathematical models is undoubtedly one of the best ways to solve such problems.
. Introduction to the classic Lanchester equation
The Lanchester equation was first proposed by the British engineer F.W. Lanchester during World War I to study the problem of deterministic solutions to war models. The assumptions of this model are very simple and only consider the number of troops and combat effectiveness of both combatants. Since the Lanchester equation does not take into account the political, economic and other factors of both combatants, it is only meaningful for the discussion of a single local battle. Using the Lanchester equation, some famous battles and wars in history were successfully analyzed and predicted, including the Vietnam War and the Battle of Iwo Jima between the United States and Japan [4]. The classic Lanchester equation studies two situations in modern wars. The first model is called the square rate model. This model assumes that both parties in the battle are completely exposed to the other side's field of vision and can transfer firepower from the destroyed target in time. For targets that have not been destroyed, this model is generally suitable for analyzing regular warfare, and the square rate equation is [5].
Through analysis, it can be obtained that the draw condition for both parties b(x2-x02)=a(y2-y02), where x and y respectively represent the number of troops of Party A and B respectively, and a represents each combat unit of Party B fighting against Party A. The kill rate of a unit is called the combat effectiveness of Party B's unit (or combat effectiveness coefficient, attrition rate coefficient, damage coefficient, etc.). Similarly, b is the combat effectiveness of Party A's unit.
The second model studied by the Lanchester equation is the linear rate model. This model assumes that both combatants cannot obtain information about the enemy target in time, so the attack has a certain degree of randomness. The loss of one's own forces is not only related to the opponent's forces, but also to the number of one's own forces per unit area. Obviously, this model is more suitable for guerrilla warfare. The linear rate equation is:
Through analysis, it can be found that the draw condition between the two sides is, x and y are defined as above, representing the combat effectiveness of Party B's unit, and representing the combat effectiveness of Party A's unit.
. Multiplayer online competitive game battle model based on the Lanchester equation
.1 A brief discussion of the multiplayer online competitive game battle model
The above is a brief introduction to the classic Lanchester equation, which is a model for modern warfare. Two different combat forms are established separately. Multiplayer online competitive games have many similarities with the war scenes described by the Lanchester equation. In order to discuss them in more detail, we might as well take the most popular tower-pushing online game "League of Legends" as an example to discuss the two. similarities between.The two sides of the
"League of Legends" game are the blue side and the purple side. Each side has 11 defense towers, 3 summoning crystals and 1 crystal hub. The winning condition of the game is to destroy the opponent's crystal hub. The
"League of Legends" game map is shown in Figure 1 (picture from Wikipedia). Each side has 5 players controlling 5 different "heroes". During the game, the 5 players on our side need to work together to gain money by killing enemy heroes, minions and wild monsters, constantly update equipment and destroy enemy buildings. object, and finally won the game. In
competitions, the key to one side's victory often lies in the victory of a "team battle" (that is, a battle involving multiple "heroes" from both sides). The outcome of a "team battle" is related to factors such as the equipment of the "heroes" on both sides, the player's operational proficiency, and the field of vision of the game map they control. Among them, the first two factors can correspond to the combat effectiveness coefficient in the Lanchester equation, and the latter one can correspond to different combat types (regular warfare and guerrilla warfare) in the Lanchester model.
"League of Legends" battle map
The following is an analysis of the "team battle" in the "League of Legends" game based on the Lanchester equation. In order to simplify the model, the individual differences between the five "heroes" on a certain side are ignored, and each Each "hero" represents the average level of players on that side. In games, team battles often take place in areas with bright visibility on the map. Both sides can see the movements of enemy units and can promptly transfer firepower from destroyed enemy units to undestroyed enemy units. Therefore, , the "team battle" in the game is obviously closer to the Lanchester square rate model. In a certain "team battle", if both sides A and B invest different numbers of troops, under such conditions, according to equation (1), by analyzing the phase trajectory of the Lanchester square rate model [4], we can get The conditions for Party B to win are:
According to formula (3), if in a certain "team battle" of the game, Party B only has 4 "heroes" participating in the battle due to various reasons, and Party A has 5, then this y0/x0 =0.8, if Party B wants to win, it must meet the condition a1.5625b.
In other words, at this time, the combat effectiveness of each hero of Party B per unit time must be 1.5625 times that of Party A. This is a big gap, which requires each player of Party B to gain huge equipment advantages and skilled operation proficiency. This is possible only under the circumstances.
In actual game battles, we often encounter situations where one player on a certain side is disconnected. In this case, through mathematical model analysis, it can be seen that it will be difficult for the other four players to win the game. In addition, the above analysis also shows that when a certain party's equipment and operating technology are backward, it is often easier for five players to adopt a "group" approach to gain an advantage in the battle.
.2 Application of multivariate Lanchester equation in multiplayer online competitive game battle model
.2.1 Multivariate Lanchester model
In the discussion of 2.1, this article ignores the individual differences between the five "heroes" of each party. , this assumption appears rough when analyzing general multiplayer online competitive games. Since under normal circumstances, different characters selected by each party have different characteristics and division of labor during battle, the individual differences between each "hero" must be considered.
In order to solve the above problems, the 10 "heroes" on both sides of the game can be regarded as 10 combat units, among which the 5 "heroes" on the own side have a cooperative combat relationship.The five combat units on the purple side are represented by P1, P2, P3, P4, P5, and the combat effectiveness of are represented by a1, a2, a3, a4, a5. The five combat units on the blue side are represented by B1, B. 2, B3, B4, B5, the combat effectiveness is β1, β2, β3, β4, β5, . Considering that each combat unit does not attack every enemy target equally, nor does it only attack a certain enemy target, therefore, it can be assumed that The attack power distribution coefficient of the j-th "hero" of the blue side to the i-th "hero" of the purple side is Φji, and the attack power distribution coefficient of the i-th "hero" of the purple side to the j-th "hero" of the blue side is φij .
Based on the above assumptions and definitions, the multivariate Lanchester equation [6] can be established as:
Among them, Pi can represent the blood volume of the i-th "hero" on the purple side. When the blood volume drops to 0, the "hero" "Killed. In the same way, Bj represents the blood volume of the j-th hero on the blue side. Through the multivariate Lanchester equation, we can more scientifically analyze the relationship between various combat units in multiplayer online competitive games, thereby establishing a more accurate model.
.2.2 Application of multivariate Lanchester equation based on coordination in multiplayer online competitive games
In multiplayer online competitive games, the different combat units of each party belong to a cooperative relationship. Due to the collaborative operations, In the process, one's own side can consume the other's combat effectiveness coefficient through mutual cooperation, and the resulting reduction in one's own losses can be equivalent to an increase in one's own blood volume [7]. Therefore, this part of the synergistic effect can be reflected in the multivariate Lanchester equation in the form of summation, and the multivariate Lanchester equation reflecting the synergistic effect can be established as:
In the above formula, bzj and Pki represent respectively The synergy coefficient between the "heroes" of the purple side and the blue side. The higher the coefficient, the better the coordination between different players in the team. It can be seen from this equation that in multiplayer online competitive games, the cooperation between players plays a very important role in winning the game.
.2.3 Preliminary discussion on the method of determining model parameters
The above discussed the multivariate Lanchester equation considering synergy, and applied it to the multiplayer online competitive game "League of Legends", establishing a relatively complete mathematical model. However, this model contains multiple unknown parameters, and the values of these parameters will be different for different "heroes" and different game matches. As of October 1, 2015, "League of Legends" has developed 126 "hero" characters. These "heroes" have different characteristics and skills, and the number of combinations of synergy between different "heroes" needs to be considered. Since each side needs to choose 5 different "heroes" in each game, either side will generate 126! /121! It is very difficult to accurately determine the parameter values between each combination of different selection combinations, and it is not necessary in practical applications. In view of the common game combinations and scenarios in multiplayer online competitive games, this article discusses the specific parameters of a relatively simplified game battle situation.
The five heroes in the "League of Legends" game can generally be divided into five types of roles, namely top laner, jungler, mid laner, ADC and support. Each of these roles corresponds to a number of relatively fixed hero choices. For the sake of simplicity, this article considers a team battle in which only two "heroes" participate in each side: Assume that in a certain team battle, both sides only participate in ADC and auxiliary, and other game characters are far apart and do not affect the outcome of the team battle. Influence. In the "League of Legends" game, the role of the ADC is equivalent to the core of fire output. It has a long range and causes high damage to enemy units. However, due to its low blood volume, it is easy to be killed by the opponent; its auxiliary role is mainly It protects the ADC from being killed by enemy units and controls the game's vision. It has low attack power but relatively high blood volume, and generally has control skills.
Let’s face the strong wind
Hasha Before reading the article to
, Xiaotian would like to ask if there are any RNG fans among the fans. Xiaotian is now recycling heartbroken RNG male fans, age requirements are 18-20, height 175+, body shape Come on if you look good, RNG will make you sad, I don’t know .
Okay, let’s get back to the topic. Recently, the topic “RNG lost to ” has topped the Weibo hot search. I believe that for many RNG fans, this is undoubtedly not a difficult fact to accept.
Everyone is asking: Why did we lose! Why did you lose! How could it be possible to lose! After all,
had a joke like this before: " I don't know who RNG is, but I know that he will be awesome every once in a while. "
was eliminated by G2 because RNG underestimated the enemy. After all, when the RNG players saw the G2 draw, every one of them burst into laughter. Looking at this scene now after losing, it is really ironic. It is no exaggeration to say that RNG lost to the team that least deserved to lose today. After the
competition ended, many fans also went to the official WeChat account to leave messages, some of which contained some fierce language.
In Xiaotian's opinion, this should also be a manifestation of "deep love and deep responsibility". As a fan, he must also hope to win the game. But victory or defeat is a common thing in military affairs. If you don't perform well this time, it doesn't mean next time.
As a rookie, Xiaotian, although I don’t watch competitive games on time, I still get excited when I see related videos. After all, LOL was also my youth.
still remembers that sophomore year, it was a hot summer. There were no classes in the afternoon, so several female students in the class discussed masturbating in the afternoon. Unfortunately, they were spanked three times in a row and they knelt down. So the five of us started discussing the reasons.
A: I was so angry that I was beaten up by the opponent. Me: It’s just that the tower was pushed too fast, and even though I asked you to join us, you never came. C: At that time, I had to play jungle and develop, so I didn’t have time. D: Let’s go to the next group. Me: I’ve told you several times but you haven’t come yet. A: We have to stick to other paths. D: How about we start now and the five of us go together. Me: Yes, ADC can demolish towers quickly. Let’s use all five of us as ADCs. In this way, we go to the middle lane and demolish the middle lane. There is usually only one person in the middle lane. If we fight five against one, we are not guaranteed to win. Together: Oops, what a great idea.
The results can be imagined. Five people walked all the way, and none of them had a chance to develop. We didn’t even have to wait for 20 shots, and we were all GG. Unwilling to admit defeat, we continued to experiment with five mages, five top laners, and five were successful. Alone, five junglers, and five support queens, I was so kneeling that afternoon that I couldn't stand up.
But we are still not convinced, and we all think that our tactics are fine. This is obviously a new tactical discovery!
So Xiaotian I ran to ask my senior LOL player cousin - mathematical modeling veteran .
After hearing this, my cousin sneered and said that you really asked the right person. But according to your approach, why are there still roles such as top laner, mid laner, jungler, ADC and support in League of Legends?
First of all, I think this is a mathematical problem that can be solved using a model. Mathematical Modeling Algorithms and Applications 2nd Edition
has two research ideas. One is based on data models and proves whether there are differences in the winning rates of different lineups through a large amount of battle data analysis.
The other method is to establish a mechanism analysis model based on certain assumptions, treat the system of both parties as a differential dynamics system, and solve the model. The former is relatively simple, and the conclusions obtained are real and intuitive; the latter is relatively complex, but can go deeper into the essence of the problem.
League of Legends is a team game. Obviously, the essence of this problem is to explore the efficiency of different collaborative combat combinations and select the optimal combination based on the results of model solution. This allows the establishment of an operations optimization model. However, because this problem is too complex, appropriate simplification is also necessary.
Research on multiplayer online competitive games based on Lanchester equation
Abstract : In order to solve the problem that current multiplayer online competitive games generally lack quantitatively described mathematical models, we first use the classic Lanchester equation to calculate the winning probability of both sides of the game. Conduct preliminary discussions. Based on the multivariate Lanchester equation and considering the coordination factors between game players, a differential equation model describing the battle process of multiplayer online competitive games is established. By using Vensim software to perform system dynamics simulations on the multiplayer online competitive game "League of Legends", the game process can be visually displayed and the results can be predicted.
Keywords : Lanchester equation; multiplayer online competitive games; synergy; system dynamics
Among many online games, multiplayer online competitive games occupy a considerable proportion of the online game market. Among them, Riot Games develops The online game "League of Legends" has an average of more than 27 million players per day, making it the online game with the most players currently[3]. Establishing a mathematical model to discuss the offensive and defensive strategies and the winning-lose relationship of this type of game will not only enable players to enhance their understanding of the game, but also promote the continuous development and upgrading of the game. Since the purpose of multiplayer online competitive games (especially tower pushing games) is to kill enemy characters and destroy enemy buildings (such as defense towers), based on the Lanchester equation commonly used in traditional war analysis, Combining the characteristics of different games to establish mathematical models is undoubtedly one of the best ways to solve such problems.
. Introduction to the classic Lanchester equation
The Lanchester equation was first proposed by the British engineer F.W. Lanchester during World War I to study the problem of deterministic solutions to war models. The assumptions of this model are very simple and only consider the number of troops and combat effectiveness of both combatants. Since the Lanchester equation does not take into account the political, economic and other factors of both combatants, it is only meaningful for the discussion of a single local battle. Using the Lanchester equation, some famous battles and wars in history were successfully analyzed and predicted, including the Vietnam War and the Battle of Iwo Jima between the United States and Japan [4]. The classic Lanchester equation studies two situations in modern wars. The first model is called the square rate model. This model assumes that both parties in the battle are completely exposed to the other side's field of vision and can transfer firepower from the destroyed target in time. For targets that have not been destroyed, this model is generally suitable for analyzing regular warfare, and the square rate equation is [5].
Through analysis, it can be obtained that the draw condition for both parties b(x2-x02)=a(y2-y02), where x and y respectively represent the number of troops of Party A and B respectively, and a represents each combat unit of Party B fighting against Party A. The kill rate of a unit is called the combat effectiveness of Party B's unit (or combat effectiveness coefficient, attrition rate coefficient, damage coefficient, etc.). Similarly, b is the combat effectiveness of Party A's unit.
The second model studied by the Lanchester equation is the linear rate model. This model assumes that both combatants cannot obtain information about the enemy target in time, so the attack has a certain degree of randomness. The loss of one's own forces is not only related to the opponent's forces, but also to the number of one's own forces per unit area. Obviously, this model is more suitable for guerrilla warfare. The linear rate equation is:
Through analysis, it can be found that the draw condition between the two sides is, x and y are defined as above, representing the combat effectiveness of Party B's unit, and representing the combat effectiveness of Party A's unit.
. Multiplayer online competitive game battle model based on the Lanchester equation
.1 A brief discussion of the multiplayer online competitive game battle model
The above is a brief introduction to the classic Lanchester equation, which is a model for modern warfare. Two different combat forms are established separately. Multiplayer online competitive games have many similarities with the war scenes described by the Lanchester equation. In order to discuss them in more detail, we might as well take the most popular tower-pushing online game "League of Legends" as an example to discuss the two. similarities between.The two sides of the
"League of Legends" game are the blue side and the purple side. Each side has 11 defense towers, 3 summoning crystals and 1 crystal hub. The winning condition of the game is to destroy the opponent's crystal hub. The
"League of Legends" game map is shown in Figure 1 (picture from Wikipedia). Each side has 5 players controlling 5 different "heroes". During the game, the 5 players on our side need to work together to gain money by killing enemy heroes, minions and wild monsters, constantly update equipment and destroy enemy buildings. object, and finally won the game. In
competitions, the key to one side's victory often lies in the victory of a "team battle" (that is, a battle involving multiple "heroes" from both sides). The outcome of a "team battle" is related to factors such as the equipment of the "heroes" on both sides, the player's operational proficiency, and the field of vision of the game map they control. Among them, the first two factors can correspond to the combat effectiveness coefficient in the Lanchester equation, and the latter one can correspond to different combat types (regular warfare and guerrilla warfare) in the Lanchester model.
"League of Legends" battle map
The following is an analysis of the "team battle" in the "League of Legends" game based on the Lanchester equation. In order to simplify the model, the individual differences between the five "heroes" on a certain side are ignored, and each Each "hero" represents the average level of players on that side. In games, team battles often take place in areas with bright visibility on the map. Both sides can see the movements of enemy units and can promptly transfer firepower from destroyed enemy units to undestroyed enemy units. Therefore, , the "team battle" in the game is obviously closer to the Lanchester square rate model. In a certain "team battle", if both sides A and B invest different numbers of troops, under such conditions, according to equation (1), by analyzing the phase trajectory of the Lanchester square rate model [4], we can get The conditions for Party B to win are:
According to formula (3), if in a certain "team battle" of the game, Party B only has 4 "heroes" participating in the battle due to various reasons, and Party A has 5, then this y0/x0 =0.8, if Party B wants to win, it must meet the condition a1.5625b.
In other words, at this time, the combat effectiveness of each hero of Party B per unit time must be 1.5625 times that of Party A. This is a big gap, which requires each player of Party B to gain huge equipment advantages and skilled operation proficiency. This is possible only under the circumstances.
In actual game battles, we often encounter situations where one player on a certain side is disconnected. In this case, through mathematical model analysis, it can be seen that it will be difficult for the other four players to win the game. In addition, the above analysis also shows that when a certain party's equipment and operating technology are backward, it is often easier for five players to adopt a "group" approach to gain an advantage in the battle.
.2 Application of multivariate Lanchester equation in multiplayer online competitive game battle model
.2.1 Multivariate Lanchester model
In the discussion of 2.1, this article ignores the individual differences between the five "heroes" of each party. , this assumption appears rough when analyzing general multiplayer online competitive games. Since under normal circumstances, different characters selected by each party have different characteristics and division of labor during battle, the individual differences between each "hero" must be considered.
In order to solve the above problems, the 10 "heroes" on both sides of the game can be regarded as 10 combat units, among which the 5 "heroes" on the own side have a cooperative combat relationship.The five combat units on the purple side are represented by P1, P2, P3, P4, P5, and the combat effectiveness of are represented by a1, a2, a3, a4, a5. The five combat units on the blue side are represented by B1, B. 2, B3, B4, B5, the combat effectiveness is β1, β2, β3, β4, β5, . Considering that each combat unit does not attack every enemy target equally, nor does it only attack a certain enemy target, therefore, it can be assumed that The attack power distribution coefficient of the j-th "hero" of the blue side to the i-th "hero" of the purple side is Φji, and the attack power distribution coefficient of the i-th "hero" of the purple side to the j-th "hero" of the blue side is φij .
Based on the above assumptions and definitions, the multivariate Lanchester equation [6] can be established as:
Among them, Pi can represent the blood volume of the i-th "hero" on the purple side. When the blood volume drops to 0, the "hero" "Killed. In the same way, Bj represents the blood volume of the j-th hero on the blue side. Through the multivariate Lanchester equation, we can more scientifically analyze the relationship between various combat units in multiplayer online competitive games, thereby establishing a more accurate model.
.2.2 Application of multivariate Lanchester equation based on coordination in multiplayer online competitive games
In multiplayer online competitive games, the different combat units of each party belong to a cooperative relationship. Due to the collaborative operations, In the process, one's own side can consume the other's combat effectiveness coefficient through mutual cooperation, and the resulting reduction in one's own losses can be equivalent to an increase in one's own blood volume [7]. Therefore, this part of the synergistic effect can be reflected in the multivariate Lanchester equation in the form of summation, and the multivariate Lanchester equation reflecting the synergistic effect can be established as:
In the above formula, bzj and Pki represent respectively The synergy coefficient between the "heroes" of the purple side and the blue side. The higher the coefficient, the better the coordination between different players in the team. It can be seen from this equation that in multiplayer online competitive games, the cooperation between players plays a very important role in winning the game.
.2.3 Preliminary discussion on the method of determining model parameters
The above discussed the multivariate Lanchester equation considering synergy, and applied it to the multiplayer online competitive game "League of Legends", establishing a relatively complete mathematical model. However, this model contains multiple unknown parameters, and the values of these parameters will be different for different "heroes" and different game matches. As of October 1, 2015, "League of Legends" has developed 126 "hero" characters. These "heroes" have different characteristics and skills, and the number of combinations of synergy between different "heroes" needs to be considered. Since each side needs to choose 5 different "heroes" in each game, either side will generate 126! /121! It is very difficult to accurately determine the parameter values between each combination of different selection combinations, and it is not necessary in practical applications. In view of the common game combinations and scenarios in multiplayer online competitive games, this article discusses the specific parameters of a relatively simplified game battle situation.
The five heroes in the "League of Legends" game can generally be divided into five types of roles, namely top laner, jungler, mid laner, ADC and support. Each of these roles corresponds to a number of relatively fixed hero choices. For the sake of simplicity, this article considers a team battle in which only two "heroes" participate in each side: Assume that in a certain team battle, both sides only participate in ADC and auxiliary, and other game characters are far apart and do not affect the outcome of the team battle. Influence. In the "League of Legends" game, the role of the ADC is equivalent to the core of fire output. It has a long range and causes high damage to enemy units. However, due to its low blood volume, it is easy to be killed by the opponent; its auxiliary role is mainly It protects the ADC from being killed by enemy units and controls the game's vision. It has low attack power but relatively high blood volume, and generally has control skills.
Assume that P1, , P2 and in equation (5) represent the purple side ADC and auxiliary respectively, and B1 and B2 respectively represent the blue side ADC and auxiliary. In actual game battles, the combat effectiveness coefficient can be considered as the damage value caused by one unit of blood of one "hero" to the other "hero" per unit time. The value of this value is related to the attack power and attack speed of the "hero". There are relationships. It can be seen from equation (5) that the damage caused by an ADC to the enemy "hero" per unit time is proportional to the ADC's blood volume. This assumption seems unscientific, but considering the actual battle situation, If the HP of one ADC decreases, the ADC will pay great attention to self-protection and output damage carefully, which will definitely affect its attack power. Therefore, the damage caused by the ADC and the HP of the ADC per unit time reflected in (5) The assumption of direct proportion is more reasonable. By querying the basic data of the relevant "heroes", the corresponding parameter values can be determined based on the above analysis.
. System dynamics modeling and simulation of multiplayer online competitive game battle model
System dynamics (SD) was first taught by J.W. Forrester of the Massachusetts Institute of Technology in 1956. Established to solve problems such as production management and inventory management, it was originally called Industrial Dynamics. After decades of development, this discipline has developed into an important transversal discipline integrating natural sciences and social sciences, with a wide range of applications [8]. The system dynamics method can clearly reflect the internal mechanism of the system and can perform dynamic simulation under small data conditions to complete the solution of specific complex system problems [9]. In this article, a multiplayer online competitive game battle model is established based on the multivariate Lanchester equation. This model involves the interaction between multiple units, and the relationship between the two units is described using differential equations. Therefore, the analysis and solution using traditional analytical solutions will be very complicated and the results obtained are not intuitive. Using system dynamics software to numerically simulate the game battle model can display the results more vividly.
Common system dynamics simulation software includes Vensim, DYNAMO, STELLA, etc. Among them, the Vensim software developed by the American Ventana Systems Company has the characteristics of friendly interface, powerful functions and relatively simple operation [10], so this article uses this software for game simulation. Bureau simulation.
.1 multiplayer online competitive game example simulation and analysis
conducted model establishment and parameter estimation based on the multivariate Lanchester equation in 2.2.2 and 2.2.3 respectively. It is still possible to discuss a method when conducting simulations and simulations. The simpler situation is the team battle described in 2.2.3 in which only two "heroes" participate in each side. In this case, the game battle model can be expressed as:
where P1, , P2, respectively represent the purple side ADC and auxiliary, B1, B2 respectively represent the blue side ADC and auxiliary. According to the battle model determined by equation (6), the flow diagram of the game battle system drawn using Vensim software is as follows.
Multiplayer online competitive game battle model system flow chart
According to the data analysis of different "heroes", in the following simulation, the selection of model parameter values are:
Φ11=Φ12=Φ21=Φ12=0.5, φ11=φ21
0=0.6 ,φ12=φ22=0.4, b12=b21=1.1x10 -5, p12=p21=3.7x10 -6, α1=0.103, α2=0. 025, β1=0.122, β2=0.027.For the sake of simplicity, in the parameter determination process of this article, it is assumed that the damage caused by the blue side's ADC and auxiliary attacks is evenly distributed to the opponent's two "heroes", while 60% of the damage caused by the purple side's ADC and auxiliary attacks is distributed to the opponent's ADC. . During the simulation, it is assumed that the initial health volumes of the blue side ADC and auxiliary are 1500 and 2300, and the initial health volumes of the purple side ADC and auxiliary are 1700 and 2000. It can be seen from the determined parameters that the purple side has a better distribution of output damage, while the blue side has stronger attack power and better coordination. Use Vensim to perform simulation, and the results are shown in the figure below.
Figure 3 The blood volume change results of each "hero" on both sides
From the simulation results, it can be seen that under the above parameter conditions, the purple side's ADC blood volume quickly dropped to below 300 within 20 seconds, and was on the verge of being "killed". As the core of the team's damage output, the purple side ADC has almost lost the ability to continue to output damage. Therefore, it can be determined that the blue team won this "team battle". Through Vensim simulation, different parameters can be adjusted at any time to predict the process and results of "team battles" under different conditions.
.2 model sensitivity analysis and game strategy research
In order to test the stability of the model system, the model parameter values can be adjusted for comparative simulation to study the impact of different initial conditions on the results of the game's "team battle". Under the parameters set in 3.1, the purple side lost the team battle. If the attack power and firepower distribution of the purple side and the blue side are not changed, the coordination between the purple side ADC and the support will become an important factor in determining the outcome of the "team battle". Under the condition that other parameters remain unchanged, the purple square synergy coefficient is enhanced, and the simulation results are shown in the figure below.
Figure 4 Comparison of blood volume changes of purple side ADC under the condition of enhanced purple side synergy coefficient
As can be seen from the above figure, when the purple side ADC and auxiliary increase the synergy coefficient by 30%, 100% and 200% respectively, its performance in the group The health value 20 seconds after the start of the battle increased from the original 250.9 to 303.7, 440.7 and 673.7 respectively. From this analysis, it can be seen that when the "hero" equipment is limited, strengthening the cooperation with teammates of the team can significantly improve The "team battle" effect increases the probability of winning the game.
. Conclusion
This article conducts a modeling analysis on the most popular multiplayer online competitive game "League of Legends" based on the multivariate Lanchester equation. Taking into account factors such as collaboration among game players, a relatively complete multiplayer online game is established. Competitive game battle model. Through the system dynamics simulation of the model, the progress of the game's "team battle" is vividly displayed, and the results can be predicted from this. Through sensitivity analysis of the model, the impact of different parameter values on the model running results can be discussed, thereby providing a reference for the formulation of more scientific game strategies.
I have to say that my cousin is indeed an academic. After reading this long string of words, emm... I still don’t seem to understand it. But it doesn't matter, it's enough to show off.
References
[1] iResearch. iResearch: China's online game market size in 2014 was 110.81 billion [EB/OL]. http://report.iresearch.cn/html/20150204/246130.shtml
[2] Baidu Encyclopedia. Esports_Baidu Encyclopedia [EB/OL]. http://baike.baidu.com/link?url=P7G4CwsU_dOHnPHZO_h1h339IdevH-s0OfF0y56dHpGi8r-zAa5-kKY4vBf0QRpIBwZsFwOrf0T-e_nEjPFtjq
[3] Zhongguancun Online. 》Daily active users breakthrough 27 million [EB/OL]. http://game.zol.com.cn/486/4869310.html
[4] Jiang Qiyuan, Xie Jinxing, Ye Jun. Mathematical Model (Fourth Edition) [M]. Beijing: Gao et al. Education Press, 2011.
[5] Wang Yong, Sun Tao, Li Xiaoyou, et al. Research on combat process modeling and simulation based on Lanchester equation[J]. Systems Engineering and Electronic Technology, 2009, Issue 07(7): 1677-1679.
[6] Zhang Xiaotian, Li Zhimeng, Deng Hongyan. Discussion on Lanchester Equation in Multidimensional Warfare[J]. Firepower and Command and Control, 2008, Issue 2:5-7.
[7] Peng Wencheng, Zhou Dian Jie, Zhang Wen. Lanchester equation of force loss based on coordinated operations[J]. Operations Research and Management, 2009, 18:128-131.
[8] Zhang Bo, Yu Zhaohui, Sun Qiang, et al. System Dynamics Introduction and review of related software [J]. Environment and Sustainable Development, 2010, Issue 2: 1-4.
[9] Tao Li, Yang Lei, Yan Bingzheng, et al. Quantification of social and individual factors in waste reduction classification Analysis[J]. Mathematical Modeling and Its Application, 2013, Issue Z1:23-26.
[10] Kong Hongshan, Zhang Mingqing, Tang Jun. Research on system dynamics model of Lanchester equation[J]. Computer Engineering and Design, 2011, 32:2789-2791.
Multiplayer Online Battle Arena Games Research Based on Lanchester Equation
Abstract: In view of the problem of current multiplayer online battle arena games generally lack of quantitative description of the mathematical model, this paper uses the classical Lanchester equation to discuss the winning probability of both side. Based on multi-dimensional Lanchester equation and considering factors in synergy among players of the game, the differential equation model is built to describe the war process of multiplayer online battle arena games. Through the system dynamics simulation for the multiplayer online battle arena game League of Legends by Vensim, it can show the game process and the results forecast.
Keywords: Lanchester equation; multiplayer online battle arena; coordination; system dynamics
Source: Zhihu Author: Mathematical modeling veteran
"Super mathematical modeling" (WeChat ID supermodeling), learn a little knowledge every day, easily understand various thinking, and become a fun rationalist. 600,000 mathematical elites are paying attention!
Figure 3 The blood volume change results of each "hero" on both sides
From the simulation results, it can be seen that under the above parameter conditions, the purple side's ADC blood volume quickly dropped to below 300 within 20 seconds, and was on the verge of being "killed". As the core of the team's damage output, the purple side ADC has almost lost the ability to continue to output damage. Therefore, it can be determined that the blue team won this "team battle". Through Vensim simulation, different parameters can be adjusted at any time to predict the process and results of "team battles" under different conditions.
.2 model sensitivity analysis and game strategy research
In order to test the stability of the model system, the model parameter values can be adjusted for comparative simulation to study the impact of different initial conditions on the results of the game's "team battle". Under the parameters set in 3.1, the purple side lost the team battle. If the attack power and firepower distribution of the purple side and the blue side are not changed, the coordination between the purple side ADC and the support will become an important factor in determining the outcome of the "team battle". Under the condition that other parameters remain unchanged, the purple square synergy coefficient is enhanced, and the simulation results are shown in the figure below.
Figure 4 Comparison of blood volume changes of purple side ADC under the condition of enhanced purple side synergy coefficient
As can be seen from the above figure, when the purple side ADC and auxiliary increase the synergy coefficient by 30%, 100% and 200% respectively, its performance in the group The health value 20 seconds after the start of the battle increased from the original 250.9 to 303.7, 440.7 and 673.7 respectively. From this analysis, it can be seen that when the "hero" equipment is limited, strengthening the cooperation with teammates of the team can significantly improve The "team battle" effect increases the probability of winning the game.
. Conclusion
This article conducts a modeling analysis on the most popular multiplayer online competitive game "League of Legends" based on the multivariate Lanchester equation. Taking into account factors such as collaboration among game players, a relatively complete multiplayer online game is established. Competitive game battle model. Through the system dynamics simulation of the model, the progress of the game's "team battle" is vividly displayed, and the results can be predicted from this. Through sensitivity analysis of the model, the impact of different parameter values on the model running results can be discussed, thereby providing a reference for the formulation of more scientific game strategies.
I have to say that my cousin is indeed an academic. After reading this long string of words, emm... I still don’t seem to understand it. But it doesn't matter, it's enough to show off.
References
[1] iResearch. iResearch: China's online game market size in 2014 was 110.81 billion [EB/OL]. http://report.iresearch.cn/html/20150204/246130.shtml
[2] Baidu Encyclopedia. Esports_Baidu Encyclopedia [EB/OL]. http://baike.baidu.com/link?url=P7G4CwsU_dOHnPHZO_h1h339IdevH-s0OfF0y56dHpGi8r-zAa5-kKY4vBf0QRpIBwZsFwOrf0T-e_nEjPFtjq
[3] Zhongguancun Online. 》Daily active users breakthrough 27 million [EB/OL]. http://game.zol.com.cn/486/4869310.html
[4] Jiang Qiyuan, Xie Jinxing, Ye Jun. Mathematical Model (Fourth Edition) [M]. Beijing: Gao et al. Education Press, 2011.
[5] Wang Yong, Sun Tao, Li Xiaoyou, et al. Research on combat process modeling and simulation based on Lanchester equation[J]. Systems Engineering and Electronic Technology, 2009, Issue 07(7): 1677-1679.
[6] Zhang Xiaotian, Li Zhimeng, Deng Hongyan. Discussion on Lanchester Equation in Multidimensional Warfare[J]. Firepower and Command and Control, 2008, Issue 2:5-7.
[7] Peng Wencheng, Zhou Dian Jie, Zhang Wen. Lanchester equation of force loss based on coordinated operations[J]. Operations Research and Management, 2009, 18:128-131.
[8] Zhang Bo, Yu Zhaohui, Sun Qiang, et al. System Dynamics Introduction and review of related software [J]. Environment and Sustainable Development, 2010, Issue 2: 1-4.
[9] Tao Li, Yang Lei, Yan Bingzheng, et al. Quantification of social and individual factors in waste reduction classification Analysis[J]. Mathematical Modeling and Its Application, 2013, Issue Z1:23-26.
[10] Kong Hongshan, Zhang Mingqing, Tang Jun. Research on system dynamics model of Lanchester equation[J]. Computer Engineering and Design, 2011, 32:2789-2791.
Multiplayer Online Battle Arena Games Research Based on Lanchester Equation
Abstract: In view of the problem of current multiplayer online battle arena games generally lack of quantitative description of the mathematical model, this paper uses the classical Lanchester equation to discuss the winning probability of both side. Based on multi-dimensional Lanchester equation and considering factors in synergy among players of the game, the differential equation model is built to describe the war process of multiplayer online battle arena games. Through the system dynamics simulation for the multiplayer online battle arena game League of Legends by Vensim, it can show the game process and the results forecast.
Keywords: Lanchester equation; multiplayer online battle arena; coordination; system dynamics
Source: Zhihu Author: Mathematical modeling veteran
"Super mathematical modeling" (WeChat ID supermodeling), learn a little knowledge every day, easily understand various thinking, and become a fun rationalist. 600,000 mathematical elites are paying attention!
