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Reducing standard deviation in games involving single die rolls

My recently-published design Hastings 1066, which well-known game reviewer Marco Arnaudo calls a “lunchtime wargame”, and which I call a successor to old-time microgames, reflects the amount of chance that occurs in a real battle: a lot. As with any historical battle game, simulating the chaos and chances of war is more or less the opposite of what gamers want as they explore generalship.

(Commercial wargames are not representative of war, more of generalship). Gamers want to control the game, they want to feel that they succeed or fail by their own efforts; but war isn’t like that at all.

Some people won’t mind this, while others might hope there was less chance. Here I present a method of making sure that each player gets about the same number of good, bad, and middling die rolls during the play of the game. And it also might be faster than rolling dice.

To do this you need at least two decks of ordinary playing cards. Extract all the Ace through Six cards in a deck (24), shuffle them thoroughly, and draw from that deck when you need a D6 roll. Each player has his own draw deck, and each deck has an identical selection of cards. When you’ve exhausted your deck shuffle it and start over.

I think it’s more practical in some ways if you have two decks of cards per player, because it will be harder for players to memorize how many times a particular number has been drawn, and in the course of a seven turn game you’ll need more than 48 die rolls.

There’s still the chance that one player’s sixes will all be up front or all at the end of his deck, and one player might reshuffle well before the other; but in the long run this may be more satisfying than a lot of dice rolls. It’s up to you.

I make this suggestion because in one of my playtests I played someone who was not a wargamer and who was not a deep thinker, playing for the first time, but I couldn’t roll for shit and after a valiant fight I lost. It’s like the famous poker champion Doyle Brunson saying that if you consistently don’t get decent cards there’s not much you can do (when he went out of the World Series of Poker on the first day). Imagine how happy he might be if you could somehow be sure that the cards he was getting were about the same average value and frequency as the cards other players were getting, in the long run.

(Of course, dice rolling “evens out” in the long run; there is no such thing as a “bad roller”. What we’re doing here is trying to even it out in the shorter run.)

By the way, this method has flaws for rolling 2d6 or more. I wouldn’t use it for that.

(The game is available on Worthington Publishing’s website at $35. I haven’t looked for it on the usual online sellers.)

Comments

Correct me of I am wrong

I think you meant, increasing the sd.
You want to widen the curve with your method, right?

Which is a wonderful idea, no doubt. I always look for new methods as well.

Another good thing about

Another good thing about using cards is you have opportunities to give special abilities and advantages like:

Peek at the top card in your deck/ your opponent's deck
Shuffle any used 5s and 6s back into your deck
Insert weird nasties into opponent's deck

X3M wrote:I think you meant,

X3M wrote:
I think you meant, increasing the sd.
You want to widen the curve with your method, right?

Which is a wonderful idea, no doubt. I always look for new methods as well.


Well, it's no longer a random distribution of the sort you use to compute SD, so I'm not sure the term even really applies, but I'm not a statistician either.

What it does is prevent you from having a run of "good luck" or "bad luck" for too long.. so long as high rolls are always good and low rolls are always bad (or v.v.). Because if sometimes you want high and sometimes you want low, this doesn't ameliorate anything...

Edit: Btw, if you want to be famous, you could use this method to have the first-ever wargame based on rolling D7's :)

Jay103 wrote:Edit: Btw, if

Jay103 wrote:
Edit: Btw, if you want to be famous, you could use this method to have the first-ever wargame based on rolling D7's :)

Already did that years ago
;p

The reshuffling was cumbersome though.

X3M wrote:Jay103 wrote:Edit:

X3M wrote:
Jay103 wrote:
Edit: Btw, if you want to be famous, you could use this method to have the first-ever wargame based on rolling D7's :)

Already did that years ago
;p

The reshuffling was cumbersome though.


So you're famous!

;)

3 more thoughts

Just to expand on what Tom Edwards wrote with a couple ideas I had while reading today:

1) Decks could be improved by certain actions or conditions being met. "Improvement" would be simply discarding lower valued cards, or adding in higher valued cards. Thematically, this might reflect experienced troops fighting more effectively after seeing combat. Similarly, you could reverse the process to reflect lower morale after defeats or other negative events. The analog for this in dice would be +1/-1 modifiers, but using cards feels like it would have a subtler effect

2) I've seen something like this done to reduce the effect of "bad rolls" in Settlers of Catan's 2d6 system. There is a deck of results 2-12 that correspond to their frequency on a bell curve. However, there may be times where you want a flatter bell curve than 2d6 normally gives. By using cards, you can fine-tune the frequency of any particular result by simply adding and removing results until satisfied.

3) A preset stack of results opens the possibility of having pure luck reduced over the course of multiple games. A disinterested third party can record the order of Player A's deck of results, and then make sure that Player B gets that same deck order at some point during a later round of gaming. Likewise for Player B with Player A's deck order. Each player, over the course of multiple rounds, would at some point have to suffer through the pain of multiple low results in a row.

Variance and variability

This method doesn't reduce the standard deviation in the long run, but it can stabilize it in the short run (especially when a game is designed to use the deck a fixed integer number of times).

Standard deviation is the square root of the variance, which has a precise definition in statistics. You can calculate the expected variance (and standard deviation) for a distribution, which is what you'd see over the long term if the dice/cards performed properly. You can also calculate the "sample variance" of actual outcomes and compare this to the expected (a.k.a. asymptotic) variance, a mathematical exercise best left as a torture device for students taking required courses.

A deck of cards is much less likely to be unbalanced than physical dice, but even the latter is unlikely to be a problem unless someone is cheating. If the game includes custom dice, the chance of a player showing up with loaded dice falls to about zero. They still might try to stack a deck.

What a small deck of cards removes is medium-term variability, the perception of hot streaks and cold streaks. That in itself might hold value for players.

But I don't think this is fully taking advantage of the card medium. A card face is a LOT bigger than a die face. The game could include N different runs ("suits") of 1-6, each with a different condition on it (Sun Glare: range penalties doubled; Gap in Cover: Barrier penalties halved; etc.). This simulates a second die roll for the condition.

To really leverage the card medium, the conditions need not be independent of the result number: maybe the poor conditions pile up on the high numbers and the good conditions pile up on the low numbers. This will tend to push outcomes toward the middle, but more subtly than going from 1d6 to (2d6)/2.

Work-Around for Planned Imbalance

FrankM wrote:
But I don't think this is fully taking advantage of the card medium. A card face is a LOT bigger than a die face.
The game does utilize an "event deck," which would likely fit the description of what you're mentioning here.

What I think Dr. Lew is proposing here, however, is merely a work-around for the perception of the hot- and cold-streaks (perceived or legitimate) you also mention in your comment. I think it's pretty clear that a bit more chaos than usual was the intent of the design, not necessarily a level playing field. I suspect that's going to be the point og the whole Breaking the Line game series.

Those who would prefer a more evened-out contest would benefit from the work-around, I suppose.

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