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Randomness without Replacement

Case Study: Massively Multiplayer RPG

For a simple example in a massively multiplayer role-playing game (MMORPG), there could average 1000 attacks per player per hour. To be crude, this could be coarsened to 100 strings of 10 attacks (Actually, only one of the ten possible combinations of ten consecutive failures conform to this demarcation). If-admittedly oversimplified-there were, for a reasonably small MMORPG, an average of 100 simultaneous players per day with uniform distribution of attacks, and independence of attacks, then there would be 100 strings / hour (24 hours / day) 30 days / month. The arithmetic equals 72,000 strings per month. Since this is very large and the probability is very small, the Poisson distribution approximates, with a rate of λ = n p. The expected number of frustrations is solved in Equation 7.

Equation 7. Expected number of frustrations per month, with replacement.

So there's going to be, on average for a low-traffic MMP, given these simplistic preconditions, 70 strings of frustration (i.e., 10 consecutive misses) per month using a mechanism analogous to a 1d20 die roll. Whereas, without replacement (a mechanism analogous to a 20-card deck), the expected number of frustrations is computed in Equation 8. For approximation, the number of trials is divided by two, because this is a sequence of 20, instead of a sequence of 10.

Equation 8. Expected number of frustrations per month, without replacement.

So, there are about 18 times fewer occurrences of frustration. A distribution of the frustration may be estimated by the Poisson distribution, as shown in Figure 1. Because only whole numbers can occur, the distribution has been stepped at the rounded (midpoint) value of a continuous Poisson distribution.

Figure 1. Distribution of frustration without replacement (blue) or with replacement (magenta).

From comparing the distributions, it is obvious that without replacement, there are almost certainly going to be fewer cases of frustration.

Case Study: Single-Player RPG

There's no theoretical or practical reason why such methods could not be applied to single-player and multiplayer RPG combat. The fact is, a game that sells a million copies, can expect similar figures to an MMP. It is a difference of penetration and time. The same Poisson distribution applies. Suppose the average player of a single-player RPG spends 10 hours with the game (since many more players quit sooner than those that play longer). If the rate of attacks remains constant (1000 attacks per hour), then only 7200 copies of the game need be sold to have the equivalent of a month of the modestly populated MMORPG above.

Every Mechanism Should be Designed as Simple as Possible, but Not Simpler

Einstein also said, "Everything should be made as simple as possible, but not simpler." Don't let mathematical precision dominate your analysis of a mechanism. While understanding the subtleties of a mechanism can make or break the design of a game, the relevance of the number crunching must be maintained.

In this example, we computed the difference, not in terms of all kinds of player frustration that may occur, but only with one single variation to a mechanism within the game. We simplified the situation in order to keep the math brief. We did not consider players with probabilities to hit at above or below 50%, nor did we strictly adhere to the definition of independence for a Poisson distribution in the die mechanism. Astute readers will have also noted that the deck mechanism does not randomize the first card drawn. Doing so would further complicate the calculation of probability. Therefore, our result is only an approximation for a special deck.

A more precise analysis was not necessary to prove the fitness of the deck mechanism. Bear in mind that a designer can correctly solve a problem but fail to solve the right problem. In this article, we honed in on one mechanism, ignoring the rest of the game and its effect on the player's satisfaction. Even if it were mathematically tractable, computing the distribution of loss of players due to this careful definition of one type of frustration is a tougher problem.

About the Author

David Kennerly directed five massively multiplayer games in the US and Korea. He localized Korea's first world, The Kingdom of the Winds, and designed the social system of Dark Ages: Online Roleplaying. Before joining Nexon in 1997, he designed The X-Files Trivia Game for 20th Century Fox, and troubleshot US Army networks in Korea.

David encourages creativity among developers and players. He helped organize MUD-Dev Conferences, and founded an online library of fan fiction. David has authored on game design for Charles River Media, ITT Tech, Westwood College, Gamasutra.com, and IGDA. To discuss this article with the author, please visit his website www.finegamedesign.com

Further Reading

Jay L. Devore, Probability and Statistics for Engineering and the Sciences. "Chapter 3: Discrete Random Variables and Probability Distributions." 6 ed. Brooks/Cole: USA, 2004.

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