Game-theoretic analysis of the Restricted Play principle
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Introduction 3
Literature review 5
Chapter 1. Problem formulation 7
1.1. Game-theoretic description 7
1.2. Aspects of a game 11
1.3. Criteria definitions 12
Chapter 2. Example: Continuous Colonel Blotto 16
2.1. Utilization case 18
2.1.1 Rational behaviour 18
2.1.2 Sub-optimal behaviour 20
2.2. Modified utilization case 20
2.3. Capture case 21
Conclusion 24
Appendix 25
References 39
Literature review 5
Chapter 1. Problem formulation 7
1.1. Game-theoretic description 7
1.2. Aspects of a game 11
1.3. Criteria definitions 12
Chapter 2. Example: Continuous Colonel Blotto 16
2.1. Utilization case 18
2.1.1 Rational behaviour 18
2.1.2 Sub-optimal behaviour 20
2.2. Modified utilization case 20
2.3. Capture case 21
Conclusion 24
Appendix 25
References 39
Game balance is the fine-tuning phase of the game development, involving the slight adjustment of the game rules to make the game closer to its intended form. This process may often be costly, but it is crucial for how the game is perceived by its target audience. Therefore, game balance is a key for the game’s success. Moreover, the applications are not limited solely to the game development. As a consequence, there is a demand for the high-efficient methods of the game balance.
This thesis is an attempt to formulate some empiric game balance criteria mathematically. It uses the concept of restricted play first introduced by A. Jaffe to define certain game imbalance criteria formally. In it is described in the following way: "...to understand the balance of some dynamic, we may frame it as the fairness of a match between two players, one of whom is restricted in a way that highlights that dynamic. In other words, we hallucinate a player (realistic or otherwise) whose behavior captures that dynamic or the lack thereof".
provides some examples on the applications of this technique: for example, to evaluate the fairness of starting conditions, a fairness of a match between a restricted player always choosing a specific starting condition and an unrestricted player may be assessed. Similarly, the importance of playing unpre- dictably may be derived from the observations of the match between a normal player and a player who implements the low-entropy mixed strategies. The same logic is applied to a diversity of other balance features.
It should be noted that the original formulation assumes the (at least theoretical) possibility of holding a match between two players. Therefore, in this form, this principle cannot be directly applied to, for instance, single-player games, for which the problem of obtaining the effective balancing techniques (for example, difficulty evaluation, as in remains relevant. On the other hand, the notion of fairness is not well-defined and thus, before mathematical reasoning may be applied, it is necessary to introduce the formal definition for this concept.
Due to the reasons outlined above, for the purposes of this thesis, we introduce a more formal, albeit less general, formulation of the restricted play principle. The main difference between this and the original version is that, in our case, we compare the performance of two imaginary versions of the same player, one of which is restricted in its choices. The performance is defined in two different ways, emerging from the game theory and probability theory correspondingly: through worst-case and average-case payoff values. While it is natural to apply the first estimate of a payoff when a player has no prior information about his/her possible opponent, the second estimate is more suited to the cases when such information is widely available, for example, when there is some statistical data on the preferences of the game’s audience.
Using the game-theoretic concept of a normal-form game as the modelling framework, we provide formal definitions for some game balance criteria through the notions of aspect set and aspect mapping. Defined arbitrarily by a game designer according to his or her goals, the aspects can be any kind of mathematical objects corresponding to the important qualities of the player strategies, which, for any concrete strategy of a concrete player, may be either implemented by this strategy or not. This correspondence is strictly defined by the introduction of the aspect mapping, which maps any element of the aspect set to the set of strategies implementing it. Then, the formulations of the considered criteria emerge naturally from the single question, namely, whether there exists a rational strategy that doesn’t implement a certain aspect. There, rational strategy is defined as a strategy yielding a non-negative (worst-case or average-case) payoff of a player implementing it.
The analogous questions regarding the existence of a rational strategy that does implement an aspect or about whether all the rational strategies do/don’t implement an aspect can be easily reduced to that form by introducing the complementary aspect, which is implemented by a strategy if and only if this strategy doesn’t implement the given one. This makes the corresponding criteria applicable to both the situations where we want to give a player the freedom to avoid the aspect without sacrificing some payoff and the cases when we implicitly try to force the players to use only the strategies implementing a certain aspect, crucial for delivering the right gameplay experience.
Alternatively, instead of a rational strategy one may use a sub-optimal strategy, which yields a payoff lying within the given range of a maximum possible value. For this modification, a separate set of criteria is also constructed.
The whole proposed approach heavily relies on the assumption that players tend to play the game as rationally as possible given their knowledge and skills. This assumption doesn’t hold for every real-world gameplay scenario. Normally, the goal of each player is to have a fun time playing. This doesn’t always correspond to getting the non-negative or maximum (depending on the rationality definition) payoff. Such discrepancy often leads to the behaviour of the players becoming more uncontrollable than it is expected to be. A so-called griefing, i.e., a deliberate act of irritating and harassing other players, may serve as an example of such disruptive behaviour.
This thesis is an attempt to formulate some empiric game balance criteria mathematically. It uses the concept of restricted play first introduced by A. Jaffe to define certain game imbalance criteria formally. In it is described in the following way: "...to understand the balance of some dynamic, we may frame it as the fairness of a match between two players, one of whom is restricted in a way that highlights that dynamic. In other words, we hallucinate a player (realistic or otherwise) whose behavior captures that dynamic or the lack thereof".
provides some examples on the applications of this technique: for example, to evaluate the fairness of starting conditions, a fairness of a match between a restricted player always choosing a specific starting condition and an unrestricted player may be assessed. Similarly, the importance of playing unpre- dictably may be derived from the observations of the match between a normal player and a player who implements the low-entropy mixed strategies. The same logic is applied to a diversity of other balance features.
It should be noted that the original formulation assumes the (at least theoretical) possibility of holding a match between two players. Therefore, in this form, this principle cannot be directly applied to, for instance, single-player games, for which the problem of obtaining the effective balancing techniques (for example, difficulty evaluation, as in remains relevant. On the other hand, the notion of fairness is not well-defined and thus, before mathematical reasoning may be applied, it is necessary to introduce the formal definition for this concept.
Due to the reasons outlined above, for the purposes of this thesis, we introduce a more formal, albeit less general, formulation of the restricted play principle. The main difference between this and the original version is that, in our case, we compare the performance of two imaginary versions of the same player, one of which is restricted in its choices. The performance is defined in two different ways, emerging from the game theory and probability theory correspondingly: through worst-case and average-case payoff values. While it is natural to apply the first estimate of a payoff when a player has no prior information about his/her possible opponent, the second estimate is more suited to the cases when such information is widely available, for example, when there is some statistical data on the preferences of the game’s audience.
Using the game-theoretic concept of a normal-form game as the modelling framework, we provide formal definitions for some game balance criteria through the notions of aspect set and aspect mapping. Defined arbitrarily by a game designer according to his or her goals, the aspects can be any kind of mathematical objects corresponding to the important qualities of the player strategies, which, for any concrete strategy of a concrete player, may be either implemented by this strategy or not. This correspondence is strictly defined by the introduction of the aspect mapping, which maps any element of the aspect set to the set of strategies implementing it. Then, the formulations of the considered criteria emerge naturally from the single question, namely, whether there exists a rational strategy that doesn’t implement a certain aspect. There, rational strategy is defined as a strategy yielding a non-negative (worst-case or average-case) payoff of a player implementing it.
The analogous questions regarding the existence of a rational strategy that does implement an aspect or about whether all the rational strategies do/don’t implement an aspect can be easily reduced to that form by introducing the complementary aspect, which is implemented by a strategy if and only if this strategy doesn’t implement the given one. This makes the corresponding criteria applicable to both the situations where we want to give a player the freedom to avoid the aspect without sacrificing some payoff and the cases when we implicitly try to force the players to use only the strategies implementing a certain aspect, crucial for delivering the right gameplay experience.
Alternatively, instead of a rational strategy one may use a sub-optimal strategy, which yields a payoff lying within the given range of a maximum possible value. For this modification, a separate set of criteria is also constructed.
The whole proposed approach heavily relies on the assumption that players tend to play the game as rationally as possible given their knowledge and skills. This assumption doesn’t hold for every real-world gameplay scenario. Normally, the goal of each player is to have a fun time playing. This doesn’t always correspond to getting the non-negative or maximum (depending on the rationality definition) payoff. Such discrepancy often leads to the behaviour of the players becoming more uncontrollable than it is expected to be. A so-called griefing, i.e., a deliberate act of irritating and harassing other players, may serve as an example of such disruptive behaviour.
We have shown how some of the game balance problems can be reformulated as rigorous mathematical criteria. To do this, we introduced an improved restricted play principle, applicable for games with any number of players and not requiring simulation of the game instances. After that, using the flexible framework of normal-form games, we proposed two measures of strategy effectiveness: the worst-case and the average-case payoffs. To be able to work with the realistic behaviour of the players, who do not always play optimally, we introduced two reasonable behaviour schemes: rational and suboptimal behaviour. After that, we provided formal criteria that can be used to assess the balance of a game. These criteria were then illustrated using the concrete example: a continuous version of the Colonel Blotto game.
To the best of our knowledge, this thesis is novel in the application of formal methods to game balance problems. Other papers focused mainly on two subjects. The first one is performing manual statistical analysis on either real data collected using various methods (telemetry, public APIs, etc.) or data derived from the
matches between AI players. The second subject is automatic adjustment of the difficulty in single-player games. On the contrary, this thesis proposes an approach that doesn’t require manual analysis and is applicable to multiplayer games.
We believe that further work on the formalization of game balance problems may lead to significant improvements in the field of playtesting. The criteria proposed in this thesis allow for the development of automatic testing systems, which will both reduce the amount of manual playtesting needed and provide a way to monitor matches between real players to be able to react in time in case some design flaws went unnoticed. This will allow designers to focus on high-level goals while parameter tweaking is being done automatically.
To the best of our knowledge, this thesis is novel in the application of formal methods to game balance problems. Other papers focused mainly on two subjects. The first one is performing manual statistical analysis on either real data collected using various methods (telemetry, public APIs, etc.) or data derived from the
matches between AI players. The second subject is automatic adjustment of the difficulty in single-player games. On the contrary, this thesis proposes an approach that doesn’t require manual analysis and is applicable to multiplayer games.
We believe that further work on the formalization of game balance problems may lead to significant improvements in the field of playtesting. The criteria proposed in this thesis allow for the development of automatic testing systems, which will both reduce the amount of manual playtesting needed and provide a way to monitor matches between real players to be able to react in time in case some design flaws went unnoticed. This will allow designers to focus on high-level goals while parameter tweaking is being done automatically.
