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Cumulative Probability

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Original Post

Which is better, a weapon that's easy to use and automatically hits but needs 5+ to kill a target, or one that needs several rolls, 2+ to equip it, 2+ to fire, 2+ not to jam, 2+ to hit, 2+ to penetrate armour and finally a 2+ roll to kill? How many steps would the second weapon need in order to change this? I'll give you a moment to think about it and stick the answer in the next paragraph, before anyone gets clever we're rolling a dice with six sides bearing the integers one to six (inclusive).

The answer is that the weapons are pretty similar in quality, but the second weapon is very slightly better (33.5% vs 33.3%), adding a seventh step to the second weapon makes it the worse weapon (27.9% vs 33.3%). I've got a hypothesis that most gamers will have gotten this wrong and tried to subtract steps, though we can't really test it since people are more inclined to comment when they get things right This is drawn from two related observations:

1) Most people are horrible at cumulative probability and underestimate how unlikely several likely events are in combination.
2) Most gamers know this and overcorrect for it.

The first point is fairly well observed. Take the slippery slope informal fallacy, in the modern usage this occurs where someone claims that one event will follow from another because every step in between is probable. You start at "If I eat this chocolate there's a 90% chance I'll want to eat another" take a few steps to get to "There's a 80% chance that I'll steal a chocolate from my housemates box if mine is empty" which is only another series of probable links to "If he's out of chocolate and I've previously taken one of his, there's a 95% chance he'll think nothing of taking one of mine" which after a few more steps gets us to "Now given that I've punched him it's 90% likely he's going to punch me back." this very quickly escalates to "There's a 80% chance that the fight is in the room with the katana" and eventually concludes "If I eat this chocolate I'm going to prison."

Of course the argument is invalid, while every step is independently likely the odds of them all happening are miniscule in conjunction. It doesn't prevent this from often being a persuasive form of argument. I've not seen formal evidence on the topic, but it feels like there's some heuristic which leads people to assume that each probable event will happen.

The second point, I think comes from the nature of games. I previously wrote about how people experience probability and suggested that game designers might take advantage of this in their games. Of course this has been happening for years on one level or another and cumulative probability is a reliable tool. Nobody wants options that are likely to fail, but a degree of failure makes a game more interesting. So sometimes designers make actions likely to succeed, but then make it possible to take so many actions that one will fail sooner or later.

For me this is epitomised by Blood Bowl, though I'm sure everyone has their own example. Failure on any given check is normally quite unlikely, a generic model with no skills performing a basic task will manage it on a 3+. While suboptimal conditions can make things harder, you also have a high quantity of specialists and some rerolls to make it easier, it balances out such that most rolls you make are likely to succeed. However the consequences for failure can be dramatic, for most rolls a failure means that every piece you hadn't moved yet misses their turn. There are also a lot of rolls to make, roll to pick up the ball, roll to dodge past that guy, roll to run an extra space, roll to throw the ball, roll to catch the ball. Despite the rolls being easy, cumulative probability catches up with people. New players think they lose every game to luck, since the critical moments in the game always look like a single event in which they needed to make an easy roll and failed. Once people get used to the game they start to see it in terms of minimising the number of rolls they need to make and become hesitant about plans that require lots of rolls even if each one is easy.

All of this leads to the conclusion that few people will treat a series of simple obstacles rationally, but that you can't rely on inaccuracies in this assessment being in a particular direction. Most people will underestimate cumulative probabilities, many will overestimate and some are able to assess the threat accurately. How to implement this effect in your game will depend a lot on your target audience and what experiences they are likely to have had, but I'd like to highlight the importance of making overestimating the threat a problem. It seems common for games to set up cumulative probability situations in which underestimating the chance of failure leads to disaster, but as players become more experienced (both in your game and in general) they will shy away from such mistakes. If you're looking to offer a challenge that's rewarding to overcome, it'll be more effective to make inaccuracies of threat assessments in both directions punishing.
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