c Copyright 2013 Alexander Jaffe Understanding Game Balance with Quantitative Methods Alexander Jaffe A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2013 Reading Committee: James R. Lee, Chair Zoran Popovi´c,Chair Anna Karlin Program Authorized to Offer Degree: UW Computer Science & Engineering University of Washington Abstract Understanding Game Balance with Quantitative Methods Alexander Jaffe Co-Chairs of the Supervisory Committee: Professor James R. Lee CSE Professor Zoran Popovi´c CSE Game balancing is the fine-tuning phase in which a functioning game is adjusted to be deep, fair, and interesting. Balancing is difficult and time-consuming, as designers must repeatedly tweak parameters and run lengthy playtests to evaluate the effects of these changes. Only recently has computer science played a role in balancing, through quantitative balance analysis. Such methods take two forms: analytics for repositories of real gameplay, and the study of simulated players. In this work I rectify a deficiency of prior work: largely ignoring the players themselves. I argue that variety among players is the main source of depth in many games, and that analysis should be contextualized by the behavioral properties of players. Concretely, I present a formalization of diverse forms of game balance. This formulation, called `restricted play', reveals the connection between balancing concerns, by effectively reducing them to the fairness of games with restricted players. Using restricted play as a foundation, I contribute four novel methods of quantitative balance analysis. I first show how game balance be estimated without players, using sim- ulated agents under algorithmic restrictions. I then present a set of guidelines for using domain-specific models to guide data exploration, with a case study of my design work on a major competitive video game. I extend my work on this game with novel data visualization techniques, which overcome limitations of existing work by decomposing data in terms of player skill. I finally present an advanced formulation of fairness in games - the first to take into account a game's metagame, or player community. These contributions are supported by a detailed exploration of common understandings of game balance, a survey of prior work in quantitative balance analysis, a discussion of the social benefit of this work, and a vision of future games that quantitative balance analysis might one day make possible. TABLE OF CONTENTS Page List of Figures . iii Chapter 1: Introduction . 1 1.1 The Problem of Balancing . 1 1.2 Introducing Quantitative Balance Analysis . 2 1.3 The Challenge of Competitive Games . 3 1.4 Restricted Play . 5 1.5 Contributions . 5 1.6 Overview . 9 Part I: Characterizing Game Balance . 10 Chapter 2: Game Balancing Background . 11 2.1 Three Perspectives on Game Balance . 11 2.2 Game Balance and Society . 20 Chapter 3: Related Work in Quantitative Balance Analysis . 31 3.1 Instrumented Gameplay . 31 3.2 Automated Analysis . 35 Chapter 4: Introducing Restricted Play . 40 4.1 The Problem with Observational Quantitative Balance Analysis . 40 4.2 The Alternative: Restricted Play . 41 4.3 Restricted Play Balance Measures . 43 4.4 Applying Restricted Play . 45 Part II: Measuring Game Balance . 46 Chapter 5: Evaluating Game Balance without Players . 47 5.1 The Potential of AI-Based Balance Evaluation . 48 i 5.2 Exploratory Study . 49 5.3 Monte-Carlo Tree Search and Heuristic-Free AI . 54 5.4 Benefits of MCTS for Balancing . 58 5.5 A Balancing System using Human-Calibrated MCTS . 62 5.6 Wrapping Up Automated Analysis . 65 Chapter 6: Analyzing Human Gameplay with Domain-Specific Causal Models . 67 6.1 Retroactive Restrictions . 68 6.2 Overview of Playstation All-Stars Battle Royale . 70 6.3 Data-Guided Design at SuperBot Entertainment . 72 6.4 Case Study Part 1: Measuring Character Strength . 73 6.5 Case Study Part 2: Building Causal Models . 78 6.6 Case Study Part 3: Extending Models for Prediction . 91 6.7 Wrapping Up Causal Models . 102 Chapter 7: Contextualizing Gameplay with Player Skill . 103 7.1 Background: Elo Ratings . 104 7.2 Plotting by Skill . 106 7.3 Filtering by Skill . 108 7.4 Sorting by Skill . 110 7.5 Alternate Formulations of Skill . 117 7.6 Wrapping Up Skill Analysis . 130 Chapter 8: A Formulation of Metagame Balance . 131 8.1 Examples of Metagame . 132 8.2 Reasoning about Metagame with Zero-Sum Game Theory . 133 8.3 Estimating Fairness in a Metagame . 139 8.4 Wrapping Up Metagame Analysis . 145 Chapter 9: Conclusion . 146 9.1 Contributions . 147 9.2 Future Work . 148 9.3 Final Words . 156 Bibliography . 161 ii LIST OF FIGURES Figure Number Page 5.1 A basic analysis using our tool. The results show the performance of varieties of restricted play against a restriction-exploiting player. Players who never play Red 1/1, for example, win 11.54% of the time. 52 5.2 Screenshot from Monsters Divided. G1=1 and B5=6 are played, with +3=2 and +1=3 power cards applied to them, respectively. G's rule is in play: closest to 1/2 wins. 5=6 + 1=3 beats 1=1 + 3=2, so P2 wins a B trophy. 53 5.3 Balance parameters across variations. The gaps between Greedy and Random convey the relative roles of long-term strategy and short-term tactics. The designer can easily discover trends such as the power gap between G1/2 and R1/1 due to a 1/2 power card. 55 6.1 Placement percentages of each character. Each column represents one of the twenty characters; the four colors indicate the percentage of games in which they achieve first, second, third, or fourth place. The characters are ordered by their combined probability of getting first or second place. Observe that some characters are better at others than securing at least second place, even when another character may be better at getting first place. (This holds beyond the margin of error.) . 76 6.2 The average `win rate' of each character, defined by Wc = E[1 − (Rc − 1)=3], where Rc is c's placement in a given match. This aggregate statistic captures the probability that a character places a higher than a random given opponent. 77 6.3 Scatter plot of all each of the 20 characters according to two statistics. The y-axis specifies a character's actual win rate. The x-axis specifies their mean score per game. The linear regression is somewhat accurate. 81 6.4 Extension of Figure 6.3, in which x-axis is now a modified score, adjusted by the average score difference this character's presence effects on each opponent. The best linear model is substantially more accurate than it is on raw scores. 83 6.5 Plot of the relationship between average opponent scores for a character and estimated average opponent scores, solely as a function of a character's kills and deaths. 85 iii 6.6 Extension of Figure 6.4, in which the x-axis is no longer shifted by actual opponent score differences, but by the estimated differences accounted for by this character's unique pattern of kills and deaths. It is potentially more interpretively useful as a `direct' statistic of fundamental gameplay. 86 6.7 Slightly improved extension of Figure 6.6, with the addition of one more feature into the estimation. A character's effect on opponents' AP generation is factored into the predicted opponent score, by estimating opponents extra kills on each other through supers paid for with this AP. 89 6.8 Comparison of linear model of win rate to Poisson model. Each pair of lines represents the prediction value of win rate as a function of kill rate, for some fixed value of the death rate. The five pairs of line represents death rates (from top to bottom) 1.5, 3.0, 4.5, 6.0, and 7.5 mean deaths per game. 96 6.9 Range of kill and death rates at which a player can achieve between a 45% and 55% win rate, for κ = 2; δ = 1. The x-axis represents the death rate, and the y-axis represents the kill rate. The space between the two lines represents the valid range of kill and death rates. 99 6.10 Range of kill and death rates at which a player can achieve between a 45% and 55% win rate, for κ = 5; δ = 1. The x-axis represents the death rate, and the y-axis represents the kill rate. The space between the two lines represents the valid range of kill and death rates. 100 6.11 Range of kill and death rates at which a player can achieve between a 45% and 55% win rate, for κ = 0:5; δ = 1. The x-axis represents the death rate, and the y-axis represents the kill rate. The space between the two lines represents the valid range of kill and death rates. 101 7.1 Average number of kills by each character. 107 7.2 Average kills by a character for players of each Elo. 107 7.3 Average kills per game by players of each Elo. Orange show all games, and blue shows only games in which all players are within 100 Elo of each other. 109 7.4 Average difference in kills between two players of varying Elo differences. Note that the graph is necessarily symmetrical. 109 7.5 An example of Nariko's Level 1 Super Attack. 111 7.6 An example of Kratos's Level 2 Super Attack. 112 7.7 An example of Big Daddy's Level 3 Super Attack.