"Stats question (help!)" Topic

In a combat system, you add up the number of units in combat and roll that many dice. Defender does the same. Dice are then assigned to each unit as the player wishes. Player with the lower rated leader goes first.

Die rolls are compared for each unit, modifiers accounted for, and results applied. Rolling high is good. Simple example: we each have two units so each roll 2 dice. I have the better leader. You roll a 10 and a 15 and assign the lower die to the unit on the right. I roll 11 and 16. I can assign the 16 to beat your 15 and the 11 to beat your 10. But results are dictated by differential so I prefer to win by 6 (my 16 vs your 10) and lose by 4 (my 11 vs your 15), rather than win twice by 1.

Now in this case it seems to me that rolling more dice is very beneficial because I have more chances to beat your dice.

In this case, other things being equal, having more dice is an advantage, right?

It's an advantage, but being able to choose allocation might be more of an advantage, depending on how many dice are rolled. Choosing where you win is kind of a big deal, but choosing where you lose might be more so.

Also, what happens if the leaders are of equal value? Because of your assignment rules, they have to be different, otherwise the tie-breaker is the critical event.

In case of tied leadership flip a coin (still TBD).

Yes the leadership is a big advantage but I was wondering about the number of dice.

RISK
dice probabilites
about half way down

link

assuming that "leftover" dice are discarded, the advantage lies in both more chances for high scores, but even more so in the ability to discard low scores. This advantage likely increases rapidly with more extra dice. the odds can be calculated for any finite number of dice vs. another, and I suppose if I worked at it I could derive a formuala for the general case… but not now! :-)

Dice are not discarded

Choosing allocation second is the advantage. The number of dice you roll makes no difference, it just gives you more options, your chances of high and low results are the same as your opponents. It's the fact that you get to react to his play which is the advantage. It's a bit like playing poker face up, you choose which hands to win an which to lose.
The main issue I see is the game slowing down. First, the defender has to choose whether to average out his results to bolster weak units and ignore stronger ones, then the attacker has to do the same. Lots of mental arithmetic going on. Don't really know without seeing the game and how many dice are involved.

I think the system would actually be very clean and fast. You assign one die to one unit. You know what the total will be but not what your opponent rolled.

Okay here's what gets me. My chances of high and low results don't change with the number of dice? I'm not more likely to get a 20 with 5 dice than 4? Or to roll a 1?

Yes, your chance of any specific number increases with the number of dice used per roll, purely because you are producing more results. You have a 1 in 20 (5%) chance with any single dice roll (i.e. with one dice), so if you use 5 dice, you should see a 20 one in every 4 rolls of all the dice as you have rolled 20 dice in total. If you use 4 dice, it would be 1 in every 5 rolls, 10 dice would be every second roll etc. Note that the odds don't increase if you haven't rolled a particular number before. As you are not combining dice (like 2D6) no number is more or less likely. The chance of rolling 3x20 with 5 dice is the same as rolling 3 x10 or 3 x1.

If dice are not discarded, what happens when one side has more? Say in a combat between two units on one side and three on the other ?
Also, is this done for each multi-unit combat separately or for all combats at once?

I think it would be slow and fiddly and prone to over analysis. What is the decision about how to allocate dice supposed to represent? It doesn't seem to relate to any decision a general would actually make.

How many dice would there be typically ?

You have a 1 in 20 (5%) chance with any single dice roll (i.e. with one dice), so if you use 5 dice, you should see a 20 one in every 4 rolls of all the dice as you have rolled 20 dice in total.

Not quite, but not far off. The probability of getting one or more 20s when you roll 5 dice is 22.6%.

10 dice would be every second roll etc

On ten dice the probabilty of at least one 20 is 40%.

Button Men by Cheapass Games has a mechanic that works the way you describe. There is a decent amount of analysis of the game online, as well.

In play, the allocation process does become a thoughtfest. In Button Men, that is the fun part.

The advantage of rolling more dice has a decreasing effect the more additional dice you roll. Think of it like this. If you roll 10,000 dice, you are still not guaranteed to roll a 20. The probability of rolling any number, or higher than any given number approaches, but never reaches 100%.

Going second becomes a better and better advantage the greater the difference between two players (no matter which side). For any given set of rolls, there is an optimal allocation for both players. If you add one die to one side, the optimal allocation changes and the potential advantage of going second gets better for both sides.

I think it would be slow and fiddly and prone to over analysis. What is the decision about how to allocate dice supposed to represent? It doesn't seem to relate to any decision a general would actually make.

Not my game, but in reading the rules it actually sounds pretty quick. Most of the time you will roll about 4-7 dice, then assign one to each unit. The idea is that where you put the better rolls represents where the higher formation puts emphasis. Given that individual units are part of a higher formation that has real game functions, which unit gets which die is actually less important than you might think most of the time.

I think it would be slow and fiddly and prone to over analysis.

In the Button Men game, it really doesn't. The first few times, you take a while to get the hang of how it works. After that, you learn what is what and how things work out. I don't think that's unreasonable; you have never done this before, you have to figure it out.

With 6 dice each, there are 720 allocations. So there are 720 possible responses to your decision (if you go first). But this gets geometrically reduced when you roll the same number (it doesn't matter which two goes on this unit).

The decision space also collapses for high and low rolls. If my opponent's lowest roll is a 4, then all 3, 2, and 1, rolls are (functionally) the same number. If my opponent's highest roll is a 16 … well, you get it.

So, with six dice you are likely to have two or three "easy calls", i.e. sure winners or loses.

Then you are left with real strategic choices. If I put this here, will my opponent sacrifice her high die to beat it, letting the rest of my hits succeed? Do I double a sure looser with a sure winner, or distribute them?

Before ever doing it, it seems like a large, complex decision space (just over half a million outcomes from 6 dice). But really, most players pick it up pretty quick and it really resolves down to a couple strategic decisions rather than dozens of tactical ones.

Using this mechanic in a mini game (with maneuvering, terrain, etc.) adds a level of obscuring to your strategy. If you don't know how the dice will be allocated, you don't know which parts of what formation are going to be more effective (by their stats, which are (or could be) known to everyone at the start of many other games).

Ultimately, you don't have to make any decision. You can get mete out dice. This is a great psyop to play on certain types of opponent.

The decision space also collapses for high and low rolls. If my opponent's lowest roll is a 4, then all 3, 2, and 1, rolls are (functionally) the same number. If my opponent's highest roll is a 16 … well, you get it.

That doesn't apply to the mechanic described by the OP though, as the difference does matter in the system he describes, not just who wins.
With 7 dice each, there are over 5000 possible allocations each (if all dice come up different). Even if a lot of the choices are obvious, there is still a lot of potential for analysis paralysis. Especially for the second person to place, if he wants to consider every possible match-up and the resulting outcome.

The idea is that where you put the better rolls represents where the higher formation puts emphasis

But I don't think the general should have this level of control. I can understand a mechanic where you get command points to dish out to units and these give you a combat bonus, representing the command influence on that combat, but I don't think you should be able to decide absolutely how well all of your units fight relative to each other on each turn.

Actually, Button Men is a bad example. You're not allocating all your dice at once, but taking one of your opponent's dice on your turn, and re-rolling the attacking dice. And, despite the number of possible combinations, it reduces down to a very few general ones in actual play.

Having just play-tested a similar system, I can tell you that the ability to assign dice in this situation is HUGE. Quality will be much more important than quantity.

In the end, I don't think it's the statistics that will be your problem — you need to see how the mechanic will play out, and you need to include people who want to abuse your rules.

That doesn't apply to the mechanic described by the OP though, as the difference does matter in the system he describes, not just who wins.

Good catch. I got too much into the parallel rather than the OP.

With 7 dice each, there are over 5000 possible allocations each (if all dice come up different). Even if a lot of the choices are obvious, there is still a lot of potential for analysis paralysis. Especially for the second person to place, if he wants to consider every possible match-up and the resulting outcome.

My point was that the size of the state space doesn't really represent the decision process. No one is going to sit down and write out all 5K allocations of their dice, 5K of their opponent's dice and then the full factorial cross-product.

There are really only a few decisions to be made, and a natural order for making them. Where do I put my biggest number? Then the second? And so on.

Actually, Button Men is a bad example. You're not allocating all your dice at once, but taking one of your opponent's dice on your turn, and re-rolling the attacking dice. And, despite the number of possible combinations, it reduces down to a very few general ones in actual play.

The decision is fundamentally the same, even though the mechanics are different. When you choose which die to apply, you need to think about the ones you are not using and your opponent's move next. While the sequencing is different, the decision implications are fairly similar.

Button Men was just an accessible example (free rules, analysis on the Internet).

Quality will be much more important than quantity.

Completely agree. Mostly because

it reduces down to a very few general ones in actual play.

in both cases.

The statistics question is does going second create the opportunity for that player to win an inordinate number of times. I.E., is the controlling determinant of combat outcomes the decisions the players make, or just the initiative?

In the range of six or so dice, a margin of three or so dice flips the advantage to initiative vice good decisions.