Game Ideas Unlimited: Cumulative
May 9, 2003 in Articles

Every time you roll the dice, you take a chance.
This may seem like the most obvious thing in the world; yet most of the time most of us don’t really recognize what it means. You see, each time you roll the dice, you take that chance again; and that means the odds increase.
It was called back to my attention recently as someone was trying to develop a mechanic for a game set in a Star Wars-like world which would tempt players toward the equivalent of the dark side of the force; but I recognized the problem long ago, and included some support for it in Multiverser. I’ll start with the way I first saw the problem.
Let’s suppose you’ve got your typical adventuring party, and they come into a room in which someone or something is hidden. Each member of the party gets a check to see whether he notices, right? That seems fair. After all, the question on the boards is whether the party is going to see whatever it is, usually before it’s too late.
But the dynamic of this is unexpected. Let’s say, for convenience, that each person in the room has a twenty percent chance to notice this object. That means that if there is only one person present, he’s going to fail to see it eighty percent of the time. But if you add a second person to this, that drops to a sixty-four percent chance of failure. Each additional person in the group increases the probability that someone will notice this hidden object. With a party of a dozen, the chance of failure is something below seven percent–about one chance in fifteen of not noticing the object. Get the size of the group up to twenty and you’re looking at almost one in a hundred of having everyone in the group miss it.
Yet intuitively we know that this doesn’t make sense. After all, assuming these people aren’t all searching for whatever this is, the odds of the third person noticing something that the first two missed are not as good as the odds of the first person noticing it in the first place. That is, if it’s hidden well enough that several people haven’t seen it, it’s probably well hidden. To put it in perspective, let us suppose that it is the player characters who are hidden. An enemy is passing their hiding place, and it seems appropriate for the enemy to get a chance to notice them. Now, if the enemy is two guys, and they each get a check, the chance of them missing the characters is pretty good; but if we’re talking about a hundred guys, and we’re going to roll thirty checks, there’s no chance of the characters escaping notice. Yet if we really were hidden there, once the first three or four guys passed us without noticing, wouldn’t we start to relax a bit, start to think that we were hidden well enough that they weren’t going to see us? In this case, the cumulative effect of so many checks creates an outcome which doesn’t fit our expectations. Two guys would actually be more likely to notice us than a hundred, because a hundred guys are going to be more focused on each other, and aren’t going to turn at the sound of a twig cracking among the trees by the road.
Multiverser’s solution is to allow a single die roll for such a check, which will be compared to every opponent’s chance of success. If any character in the group would have noticed given that roll, someone notices. This saves the players from the impact of cumulative odds in such a situation.
Yet once you recognize the power of cumulative odds, you can use them to your advantage. This idea of being tempted over to the dark side is an excellent example. The trick to the dark side is that it is seductive. You should be afraid on the first roll that you might fail; but each time you take the chance, you should be less afraid. To draw the player into this, a mechanic could be designed in which the chance of success on this roll increases each time the roll is made. Of course, there would have to be benefits to using the power of the dark side, or there would be no temptation; we’ll assume that if a player chooses to tap the dark side, he gets immediate bonuses on whatever he’s doing, but then has to roll to resist the dark side. The first time he rolls, he must roll seventy or less to succeed; he has a serious thirty percent chance to fail. Assuming he succeeds, he has escaped the lure of the dark side this time, and next time he draws on its power he will have to roll seventy-one or less, only a twenty-nine percent chance to fail. With each successful roll, he seems to increase his ability to resist the dark side; at this rate, if he successfully resists the dark side thirty times, he will be immune, and can never fail. That thirtieth roll will present only a one percent chance of failure–a chance anyone would take.
What is hidden in the fact that with each roll the odds of success improve is a cumulative chance of failure. The odds of successfully making all thirty of those rolls–of resisting the dark side through thirty individual uses of its power–is slightly greater than one half of one percent. Fewer than one in one hundred characters will survive twenty such rolls, and only about one in twenty will manage to resist the dark side through ten. The system is alluring, because all the player really sees is the odds of making this roll; he doesn’t see that this roll is one of a series he’s been quite lucky to have survived so far.
There is a sense in which that perception is correct; his chance of failure on this roll is lower than it has ever been before. Yet one of the features that drives Multiverser’s chance to botch is that eventually you will botch. As long as your chance of success doesn’t exceed ninety, you’ve got at least one chance in a hundred of rolling a botch. The more you do, the more rolls you make, and the more likely it is that the botch is going to occur. The fact that one character in two hundred might resist the dark side is an incentive to try; the fact that each roll offers a better chance of success than the last is an incentive to try again. Failure looms not in the odds of this roll, but in the accumulated odds of all of them.
If you’re designing a game, bear in mind this aspect of accumulated odds, and remember to use it where you need it and avoid it where it creates problems. If you’re running a game, be cognizant of when the characters should already be covered for something, such that another roll would unfairly penalize them.
If you’re only playing, remember that whenever the dice are passed to you, you are taking a chance that adds up over time.
Next week, something different.
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M. Joseph Young is co-author of Multiverser and Vice President for Development at Valdron Inc. His many contributions to online literature are indexed for convenience, and he looks forward to discussing these things by e-mail or on our Gaming Outpost forums.