Understanding Probability and Randomness in Computer Game Design

1. Probability and Risk CS 4730 – Computer Game Design Credit: Several slides from Walker White (Cornell)

2. Quick Recap • The game state is the current positioning/value of all entities in the game world • Actions a player takes is input into the current game state • An interaction is a function between game states as determined by the actions (of both player and world) in generating a new game state • The Game Loop 2

3. The Game Loop • In general: • Handling time • Gathering player input • Networking • Simulation • Collision detection and response • Object updates • Rendering • Other miscellaneous tasks 3

4. In MonoGame • Initialize() • Network, DB, etc. • LoadContent() • Graphics and other assets loaded • Update(GameTime) • Update the game state based on input • Draw(GameTime) • Refresh the screen 4

5. MonoGame • Design doc due Friday • Follow requirements for Written Word • Prototype due next Friday • We will expect at least one realized mechanic in the game • We do NOT expect great art, music, etc. • If you expect controller input later, we expect that now 5

6. It All Start With Numbers • Hope you remember your APMA classes… • Statisticsis the study of what HAS happened • Probability predicts what WILL happen • The interesting this is that people are reasonably good at statistics, but horrible at probability • “What’s the likelihood it will rain tomorrow?” 6

7. Randomness • Computers are actually horrible at being truly random • Which is both good and bad • Bad for security purposes • Good for networked games to have the same state without transmitting that state • Probability of any value is written as P(x) 7

8. Traditional Randomness • Cards and dice are still used as metaphors for randomness • (and non-metaphors…) • We still uses these items in both board and video games for randomness 8

9. Cards and Dice • Card notation: • 4S – 4 of Spades • AH – Ace of Hearts • Dice notation • xdn, where x is the number of dice to roll and n is the number of sides on each dice • 2d6 – roll 2 6-sided dice, values will be 2-12 • 3d8+4 – roll 3 8-sided dice and add 4 to the result, values will be 7-28 9

10. The Outcome Tree • We can calculate the probability of any random event by working out the outcome tree and counting the possibility • Monte Carlo simulations run the function for a large number of times and using that to determine percentages 10

11. Combining Probabilities • P(not A) = 1 – P(A) • P(A or B) = P(A) + P(B) • P(A and B) = P(A) * P(B) 11

12. Expected Value • The expected value is the average of all the possibilities of a random variable • Each value is weighted by its probability • E(x) = Sum(k * P(“x=k”)) over the possible values of k • E(1d2) = 1 * P(“x=1”) + 2 * P(“x=2”) = 1 * .5 + 2 * .5 = 1.5 12

13. Dice Expected Values • E(1d4) = 2.5 • E(1d6) = 3.5 • E(1d8) = 4.5 • E(1d10) = 5.5 • E(1d12) = 6.5 • E(1d20) = 10.5 13

14. Variance • What is the distribution of possible values about the expected value? • Var(x) = E((x – E(x))2) • Uniform (high variance) • Bimodal (low and high variance) • Gaussian (low and high variance) • Zero variance 14

15. Compound Expressions • E(x + y) = E(x) + E(y) • E(x + c) = E(x) + c • E(c * x) = c * E(x) • E(x * y) = E(x) * E(y) • Var(x + y) = Var(x) + Var(y) • Var(x + c) = Var(x) • Var(c * x) = c2 * Var(x) 15

16. Probabilities of Catan • Let’s look at the math of Catan to figure out how probabilities play into the game • Quick overview of the rules of Settlers of Catan • http://www.catan.com/service/game-rules 16

17. Settlers of Catan 17

18. Probabilities of Catan • It’s actually pretty easy to know what’s the “best” option • Just add up the dots! • Probability and randomness plays a HUGE role in Catan working “correctly.” • What about games in which probability and randomness is the entire game? 18

19. Chutes and Ladders 19

20. Chutes and Ladders • The game is ALL RANDOM. • But a video game that is ALL SKILL can eventually get boring! • You’ve learned every pattern • You’ve seen every level and enemy • Nothing varies! • We need to consider games that have some aspects of both! 20

21. Why do people gamble? • Let’s face it – gambling in Vegas is a losing proposition • Over time, everyone loses money • But in the (very) short term, it’s definitely possible to win • And besides – risk and uncertainty can be a lot of fun! 21

22. Psychology of Randomness • Player’s like longshots! • How many times have you gone for the “super move” to win the game? • Even if it’s a low probability, players will optimize for it! • Player’s suffer from too much Monte Carlo • “Oh, I’ve gotten bad results for so long… a good card/good roll has to come up soon!” • Probability does not care what the last roll was, but players will think the game is “unfair” otherwise! 22

23. Psychology of Randomness • But think about it another way – I bet you remember those big payoff moments • And THAT’S what gets you coming back to a game! 23

24. Other Forms of Risk • Imperfect information can add to the challenge/risk in a game without as much randomness • Perhaps you don’t know everything about the game state • Either AI or another player • Perhaps don’t know about the game might change 24

25. Fog of War - Partial 25

26. Fog of War - Total 26

27. Information Types • Info known to all players • Info known by one player • Info known by the game only • Some game state information, like next card in the deck • Info unknown • Next random number to be generated 27

28. Difference Between Video and Board • Table top games rely on randomness to work • Information that the game only knows can be hard to manage • D&D does this through a Dungeon Master • However, while computers aren’t as good at randomness, they are fantastic at managing information • Implementation and adherence to rules also varies greatly 28