{"id":3760,"date":"2015-03-26T19:00:05","date_gmt":"2015-03-26T19:00:05","guid":{"rendered":"http:\/\/bucksurdu.com\/blog\/?p=3760"},"modified":"2015-03-26T19:01:49","modified_gmt":"2015-03-26T19:01:49","slug":"dice-progression-mechanic","status":"publish","type":"post","link":"https:\/\/bucksurdu.com\/blog\/?p=3760","title":{"rendered":"Musings on a Dice Progression Mechanic"},"content":{"rendered":"

For many years I have had interest in implementing an opposed-die roll dice progression mechanic in a game. \u00a0Many years ago Cory Ring and I wrote a small set of rules for the HMGS MidSouth Dispatch (newsletter) that featured such a mechanic. \u00a0The problem is that there isn’t enough variance between a d4 and a d12 and then there is the big gap between d12 and d20. \u00a0The gap can be filled with two dice, but then you don’t get the same uniform distribution of results that a single die achieves.<\/p>\n

\"24-sided<\/a>
24-sided dice from http:\/\/mathartfun.com\/shopsite_sc\/store\/html\/DiceShop.html<\/figcaption><\/figure>\n

Recently, I found a company (http:\/\/mathartfun.com\/shopsite_sc\/store\/html\/DiceShop.html)\u00a0 that sells d14, d16, d18, d22, and d24. \u00a0I wrote to them, and they were able to sell me 10 of each such that each type of die was a unique color. \u00a0Since these are uncommon shapes I wanted to be able to say, “roll the blue one and always mean the d14 — or whatever shape was blue. \u00a0They arrived recently, and I have begun to think about how to employ them.<\/p>\n

\"22-sided<\/a>
22-sided dice from from http:\/\/mathartfun.com\/shopsite_sc\/store\/html\/DiceShop.html<\/figcaption><\/figure>\n

The basic notion is that abilities would have a base die as a part point. \u00a0 Modifiers would then change the die rolled. \u00a0The attacker and defender would each roll a die, with the higher roll winning. \u00a0I have also thought it might be interesting if the difference in the rolls somehow indicated the level of success. \u00a0For instance if the attacker’s roll is three times the defender’s that might indicate some sort of critical hit.<\/p>\n

\"A<\/a>
A 16-sided die from http:\/\/mathartfun.com\/shopsite_sc\/store\/html\/DiceShop.html<\/figcaption><\/figure>\n

On a recent flight for work, I began to wonder about the probabilities of winning under these types of rules. \u00a0One of the reasons that this die progression approach appeals to me is that someone rolling a d4 COULD<\/em><\/strong> defeat someone rolling a d24. \u00a0But what is that probability? \u00a0So out came Excel. \u00a0The table below shows the chance of the attacker (rolling the dice along the left of the table) defeating a defender (rolling the dice across the top of the table).\u00a0<\/span><\/p>\n

\"\"<\/a><\/p>\n

So, if an attacker roll d4 and the defender rolled d24, the attacker would have just a 6% of winning. \u00a0Note that the attacker must roll higher<\/em> than the defender to get a hit, so ties go to the defender. \u00a0On the other hand, if the attacker rolled d24 and the defender rolled d4, the attacker would have a 905 chance of winning. \u00a0Again, ties go to the defender.<\/p>\n

Looking at this chart, I was pretty happy with the way the probabilities laid out. \u00a0Then I stated wondering why things weren’t summing to 100%. \u00a0For instance, why was P(Victory, d4 vs. d24) + P(Victory, d24 vs. d4) not equal to 1? \u00a0Then Duncan made a comment that helped me figure it out. \u00a0It’s those ties. \u00a0Since some rolls are losses for both\u00a0d4 vs. d24 and d24 vs. d4 those were the missing percentages.<\/p>\n

\"\"<\/a><\/p>\n

The table (above) shows the probabilities of ties that are always failures. \u00a0For a d4 vs. anything, there are 4 rolls that are always ties: 1:1, 2:2, 3:3, and 4:4. \u00a0For d4 vs. d4, this is 25% of the total rolls possible (16). \u00a0To check my math, I then inverted the first table…<\/p>\n

\"\"<\/a><\/p>\n

so the defender is down the left and the attacker is across the top. \u00a0Then I added all three tables together, yielding this:<\/p>\n

\"\"<\/a><\/p>\n

Except for one cell (it looks like two, but this table is symmetrical about its diagonal) at 99%, all the math adds up. \u00a0I rechecked all the math and didn’t find an error, so I’m chalking it up to round-off errors.<\/p>\n

\"18-sided<\/a>
18-sided dice<\/figcaption><\/figure>\n

So, if anyone has stayed with me this far, I think the math shows that from a probability standpoint, the die progression mechanic is viable.<\/p>\n

I am planning to implement this with something melee heavy so that weapons get a base attack die and skill and circumstances modify the die. \u00a0The defender’s armor gets a base defense die, with skill and circumstances modifying it. \u00a0I may try this in a couple of weeks with some Robin Hood figures.<\/p>\n