# Is there a method that gets beneficial diminishing returns when adding more dice, yet stays random?

Short Story: I want to find a dice system where multiple dice are rolled and compared at once, but rolling more dice is only slightly more effective than rolling fewer dice, while still being fairly chaotic no matter how many dice are rolled.

Long Story:

I'm designing a game that involves assigning dice to various cards for bonus stats to those cards. Cards fight, victor wins, etc.

The formula right now involves starting at 5d6, getting a die each time you are hit, up to 10d6.

Problem is, it's hard to balance a card around an expectation of having both one die (a 3.5 average) and 10 dice (35 average). Set an average expectation of "10 points", and now the person with 5 dice is only effective with 1 card, while the person with 10 dice is effective with 3.

We ended up with an issue where it mostly became about dice-trading, where the victor was almost always the one with more dice, as well as the dice mattering more than the card itself. The cards ended up feeling like "placeholders" for stacking dice on rather than strategic choices to invest in.

What we really want is to have the system reward assigning dice to multiple things on the board, so that the game becomes more chaotic and stressful the later into the game it gets, not more predictable or even necessarily much easier for the player with more dice.

We've looked at rolling all the dice you would for a card, keeping your highest roll. The good thing about it was that it was relatively balanced, but it was far too predictable.

We've looked at exploding dice + keeping high, where you keep the highest value + any dice that match it. It was chaotic, but it basically ended up being an excessive "crit" system that we didn't like and wasn't easy to balance.

In case it helps, dice are colored differently for each player.

Does anyone know of a dice resolution system that can do this?

(EDIT) This is the RPG.SE question that turns the resolution method that solved this question (Roll high, add +1 for each match of that die) into an Anydice formula that can be reviewed.

• One option you could try is having the dice go from 0 to 5. This would make it so that the minimum with 5 dice and 6 dice are both 0. You could alternatively try going from -1 to 4 (or -2 to 3) to add a push your luck element to having more dice.. Commented Oct 9, 2020 at 1:06
• A website like anydice.com will help you calculate complex probabilities including functions for exploding dice. Commented Oct 9, 2020 at 7:15
• What do you mean when you say that the highest value method was "too predictable"? Commented Oct 9, 2020 at 7:49
• @ArcanistLupus Rolling 1d6 has a 33% chance of getting a 5 or higher. Rolling 2d6k1 (2d6, keep 1) has a 55% chance of getting a 5 or higher. Rolling 3d6k1 has a 70% chance of getting a 5 or higher. The more dice you add, the more predictable they get, and this still holds true even if you're only keeping a single die. Commented Oct 9, 2020 at 15:15
• Look into "Pocket Tactics" by Ill Gotten Games. It uses a system of dice rolling where more dice isn't nearly as helpful as simply adding dice amounts together. To summarize it, you roll x amount of dice (x=1-3) and the difficulty/opposing force rolls x amount of dice. You compare the highest results. If the defense is higher, the roll is a failure. If the offence is higher, the roll is a success. If they are tied, move onto the next highest results, comparing them in the same way. Commented Nov 9, 2020 at 19:07

1. How can I design a function that computes a result on a dice roll that gives asymptotically decreasing benefit to adding more dice?

2. How can I have a system where adding more dice doesn't lead to a predictable result?

The first question is simple. Take the highest X dice of a roll. Alternatively, set some threshold that you need to roll above, such as roll at least one six, or roll a total of at least 15. Under any of these schemes, as you roll more dice, you are more likely to have dice with better results, but there is a fundamental limit on how high you can go. In the case of highest X dice, your limit is X times the highest value on one die, and in the case of targets, the limit is a 100% chance of success.

By way of a simple example, let's say you use D6's, and you need to roll at least one 6 to succeed. Your chance of getting a 6 based on how many dice you roll looks like this:

But while this satisfies your goal of the first question, it is directly contradictory to the goal of your second question. As you add more dice, the result gets more predictable. This is a general principle in probability; the more independent random events you aggregate, the lower the collective variance.

Given this, I believe that the answer to your second question is that "it's impossible without a re-framing of the question". Adding more dice will decrease the variance of the result, and therefore increase the predictability.

The way to re-frame the question is to not have all of the dice be the same. Here's a simple example, to illustrate the concept, though you likely will want to do something more complex.

You have two kinds of dice, white and red. White dice contribute to success, red represent risk. Each item, skill, etc can add some amount of dice, in one color or both. Most will add small amounts of white dice (1 or 2), but some better items may add larger amounts of white dice (3 - 5) but some red dice as well (1 - 2). Some skills could even do something non-roll related (like double the effect of a spell) but only add red dice. When you roll the dice, you take the highest three white dice... unless any of the red dice came up as a "1", in which case you treat the whole role as a 0. Through a system like this, you have to make trade-offs between getting increasingly marginal benefit from more white dice vs the risk of catastrophic failure from an additional red die. Maybe even have everyone roll 1 red die at baseline to prevent people from getting too powerful from stacking exclusively white die bonuses.

That's the type of idea you're probably looking for. Obviously you have to make it your own and fit it to your game. You can tweak the number of top dice you take. You could change the failure condition based on the circumstance. You could use different sized dice (D8s, D12s, etc.). You could add a third kind of die. The list goes on.

There's a lot of ways to weaken the effects of the dice.

1. Only the best X dice count - this means the more dice you have the more likely you will get a better overall result, but you are still stuck in the same x to 6x range of values - kind of like how D&D uses rolled ability scores (roll 4 keep 3) or how texas holdem works with hands (best 5 out of 7 cards).
2. Count each die that beats a target - Shadowrun rolls a lot of dice, as does the game Zombicide in some situations (I've rolled 163d6 for one attack in that game) but in both games there is a target for each dice to be a hit, counting as 1, and anything under that is a miss, counting as 0. Maximum number of hits is set by total number of dice, and you can have the cards set the target number, zombicide does this, some weapons hit on 4+ some on 5+. (Optional exploding 6s, where 6s are hits and you reroll to try for more)
3. Smaller dice - the 1 to 6 d6 are standard, but not the only dice out there. a lot of tabletop RPGs use D4 as well, which change how much each die can add to the overall result. You can enhance this by having the base dice everyone gets be larger, then adding smaller dice as more dice get added, say everyone gets 5d10, then add 2d8, then 2d6 and any more are d4 - guaranteeing diminishing returns by diminishing dice size.
4. Custom dice - In addition to dice not all being the same size, not every die goes from 1 to X for X sides. Betrayal at House on the Hill uses 6 sided dice with 0, 1 and 2 each on two faces, leading to an roll between 0 and 2x for x dice, with an average result equal to x, the number of dice rolled.
5. Have a penalty for additional dice, say -2x for x dice, so a roll would give you between -1 and 4 for each d6, still a positive average, but much lower than the default. This also creates the possibility of dice reducing effectiveness over all, if a player rolls all 1s and 2s.
• Only your first suggestion actually achieves the desired result. Counting each die that beats a target, using smaller dice, or subtracting an amount for each die rolled (which is effectively your third and fourth suggestion) all result in linear increases in the expected value based on number of dice rolled. While these suggestions reduce the linear gain from each die, non of them actually create diminishing returns. All you've done is reduced the first derivative, but you still have a second derivative of 0. Diminishing returns requires a negative second derivative
– Zags
Commented Nov 6, 2020 at 14:52
• Your second suggestion could create diminishing returns if paired with a "target number of successes", but it's the target number for the dice overall, not the target per die, that creates the diminishing returns.
– Zags
Commented Nov 6, 2020 at 14:53
• @Zags The asker talks about wanting to have rolling "more dice is only slightly more effective than rolling fewer dice" outside the diminishing returns in the main question (which by the way I will say option 3 does, by reducing dice size and thus the maximum and average results) meet the effect requested without being pedantic on the wording in the title. Commented Nov 7, 2020 at 1:45

How about "number of dice that show a 6", as is used in games like Neanderthal by Phil Eklund?

• With 5 Dice, you can expect to roll at least one 6 about 60% of the time
• With 10 Dice, you can expect to roll at least one 6 about 84% of the time

So you are more effective with more dice, but it's diminishing returns.

You can tweak the probabilities by requiring more 6s, or allowing 5s or 6s, etc.

• "Number of dice that show a 6" is statistically increases linearly with the number of dice rolled. Each die adds 1/6 to the expected total. It sounds like you're talking about rolling a target number of 6s. If that's the case, you should edit your answer for clarity
– Zags
Commented Nov 6, 2020 at 14:48

You don't have to actually use all of the dice that you roll. For example you could be looking at the results of the best 3 dice of the 5 that you rolled.

Example for needing a roll of 12 on 3 dice:

• Roll 5 dice get 1, 2, 3, 6, 6
• Chose the 3, 6, 6 for a total of 15
• Meet the goal

Example for Rolling 10 dice

• Roll 10 dice get 1, 1, 2, 2, 2, 3, 5, 5, 6, 6
• Chose the 5, 6, 6 for 17
• Meet the goal

There are some quick things you can do to balance it out such as having a roll of 1 remove another dice of the highest value

Example for needing a roll of 12 on 3 dice:

• Roll 5 dice get 1, 2, 3, 6, 6
• Remove the 1 and a 6
• Chose the 2, 3, 6 for a total of 11
• Don't meet the goal

Example for Rolling 10 dice

• Roll 10 dice get 1, 1, 2, 2, 2, 3, 5, 5, 6, 6
• Remove the 1, 1, 6, 6
• Chose the 3, 5, 5 for 13
• Meet the goal

Another limitation on it is you can limit how often extra dice can be rolled but keep a minimum amount around. Players can regain dice naturally over time or need to require them from some mechanic in the game.

This would allow you to roll more dice to get an advantage but not get a straight benefit from summing the total off all the extra dice and can even make it a little more risky to roll more if you get back luck.

For every x number of dice, one of them e.g. a different coloured one, counts negative to the total. It's still random and won't have a huge impact if x is set sensibly.

You might also consider having good and bad results from a roll. For instance, Blades in the dark uses d6, but a 1 is a failure, 4-5 a success, and 6 a crit. But if you have more fails than successes (critical or otherwise) it's a critical failure. Otherwise every fail cancels a success. More dice is generally good, but it's possible to fail worse as well as succeed better. Add in bonus/malus dice (which remove the worst or best rolls respectively) or DM/Player Rerolls and more dice isn't always good. Especially if a reroll affects ALL rolls of a particular number.

I think the key would be finding some interaction that gives bonuses.

At tier 1, you just get 5d6 - this has a low of 5, a high of 30, and an EV of 17.5

At tier 2, you get 5d6 but if you get a triple, you get an extra 1. This has a low of 9, a high of 36, and an EV of 17.7746. This isn't a huge improvement, but it is an improvement nevertheless.

At higher tiers you could give bonuses for pairs, mini-straights, and straights. And obviously the more dice you add, the more powerful these bonuses become. For example, upon reaching 7d5 you are guaranteed a pair (although, you were almost virtually guaranteed a pair before...)

Ask yourself what can you do with the dice you have already? Could you make one of them a different color that acts as a bonus die what special interactions? If you pair your bonus die you get +3?

Have a set number of dice per roll: say, 10 or so. Rather than cards giving you more dice, they give you more rerolls. Playing 5 rerolls worth of dice lets you reroll five die.

If you want to make it more tactical (that is, more microdecisions required), start with 5 dice per roll. Cards may either give you rerolls or additional dice (with additional dice generally more rare/valuable than rerolls), or some combination of the two.

Here's a quick simulator I whipped up. Just press run. https://replit.com/@hengstam/dicererollexample