Friday, 10 September 2010

Dice, Field of Glory & Hard Sums

I’ve had some feedback on the original article on dice, DBMM & FOG which in turn led to an even more interesting telephone discussion about whether rolling more dice in Field of Glory is better for you.




At first glance questioning this seems stupid.  “Of course it is” I hear you cry!  Well I’ve had a look at this and the answer is both yes and no.


Let’s first look at the “more is better” side of things.  You certainly roll more combat dice in Field of Glory compared to DBMM.  The temptation is to view each dice rolled as a potential hit and assume more dice simply means more hits. Sadly, it doesn’t work out this way.

More dice are advantageous but it’s not a simple relationship because they are rolled together. The chance of rolling any result on 1d6 is 16.7% and the chance of rolling the standard hit in Field of Glory of 4 or more is 50%.  This is the only case where the probability of a particular result is evenly spread over all the outcomes as this table shows:

Hits (4 or more)
Dice 0 1 2 3 4 5 6
1 50.0 50.0
2 25.0 50.0 25.0
4 6.3 25.0 37.5 25.0 6.3
6 1.6 9.4 23.4 31.3 23.4 9.4 1.6

What it also shows is that the probability of an extreme result declines sharply as more dice are rolled and that the most probable outcomes begin to bunch around the average score (which is half the number of dice).  So more dice are better because you have a higher probability of rolling an average, or near average, result but at the cost of a much reduced chance of rolling a very high number of hits, or thankfully, misses.

To put this in a wargames context; I have often rolled four dice and been disappointed to get “only 2 hits” yet this is the most probable outcome.  Likewise, for a battle group of 4 knights rolling 8d6 in combat the most probable outcome is 4 hits (27.4%) closely followed by 3 and 5 hits (21.9% each).  I wonder how many of you would roll the knight’s dice silently hoping for 6 or more hits oblivious to the fact that increased number of dice actual favours the average results at the expense of the extremes?

Next let’s look at uneven combats where the same factors are in play but in reverse.  The easiest way to assess a combat outcome is to count dice.  As shown above this approach can lead to an over estimate of the chance of a significant win. However, there’s another twist when the combat involves low numbers of dice such as 2s and 4s.

Consider a potential combat between a battle group of 4 cavalry and one of 4 light horse.  If the light horse were yours would you stand or would you evade?  What’s the percentage play as the American’s say and do you really have a choice?  Standing would result in a competitive roll of 4d6 and 2d6 for the cavalry and the light horse respectively; assuming no net points of advantage or troop quality differences.   Are you thinking “that’s twice the number of dice so I’m going to lose”?  If you are then you are right but take care.  The odds suggest that you are indeed most likely to lose but most importantly not heavily.

The most probable score for the cavalry is 2 hits; for the light horse it is 1.  This makes the most probable combat outcome a 1 hit win for the cavalry.  In Field of Glory this isn’t too bad as the cohesion test penalties for the losing light horse are relatively slight and the death roll chances (2:1) aren’t too bad either. So standing might not be so terrible and a viable option if the game demands it.

However, because we are rolling low numbers of dice for the light horse extreme results are more likely. There’s a 1 in 4 chance of the light horse scoring a maximum 2 hits compared to 1:17 for the cavalry scoring a maximum 4 hits.  This makes the situation more complex and reliance on the average result alone unwise. Overall the outcomes are:

Cavalry Hits vs. Light Horse
Result -2 -1 0 1 2 3 4
% 1.6 9.4 23.4 31.3 23.4 9.4 1.6

So the most probable outcome is, as before, the light horse losing by 1. However, because only small numbers of dice are involved the use of the average result doesn’t represent the range of options well at all. In fact the average hides a 23.4% chance of a draw and an 11.0% chance of the light horse either winning or losing heavily. In rounded odds, you can present the odds for the light horse as:
  • 1 in 3 chance of winning or drawing,
  • 1 in 3 chance of losing by 1,
  • 1 in 3 chance of losing by 2 or more.
In the light of this presentation, the choice of whether to evade or not is a close one.   Tactically, I suppose it depends on whether, at any particular point in a game, you need to be conservative or can afford be aggressive.

To me it’s only the loss by 2 or more that carries any real risk; there’s that extra penalty in the cohesion test.  The combat odds are a lot closer than the numbers (4 vs 2) suggest and it’s one thing to evade because the time isn’t right or to lure you opponent into a trap; it’s another to evade because you think the odds are heavily against you when they aren’t really.

In Field of Glory it’s important to estimate combat risks and outcomes correctly.  The examples discussed here demonstrate that the probabilities aren’t easily estimated just by counting dice.  Under or over estimation is likely depending on the number of dice involved.  So from now on I’m going to try the following rules of thumb and see how I get on:
  • Don’t just look at the number of dice involved: it is misleading.
  • For an average combat result: work out the difference in the number of dice on both sides and half it.
  • Rolling  lots of dice (4 or more)? Then you can begin to rely on the average score.  The more dice rolled, the more reliable the average is as a predictor.
  • Rolling some dice (fewer than 4)? Think about all the outcomes and don’t rely on the average score too much.
As before the figures in this article were calculated using Jason Flick’s excellent AnyDice website.  The code used was:

\Results distribution of N d6\
output [count {4..6} in 2d6] named "2 d6"
output [count {4..6} in 4d6] named "4 d6"
output [count {4..6} in 6d6] named "6 d6"
output [count {4..6} in 8d6] named "8 d6"


and

\Results for 4 vs 2 no rerolls\
output [count {4..6} in 4d6]-[count {4..6} in 2d6]


Finally, in researching this article I found a great interactive demonstration of the "Law of Large Numbers” that’s well worth a visit.  Have fun.

3 comments :

  1. The other thing to consider is the long term position of the light horse versus the cavalry. While the prospect isn't so grim for them based on one turn of combat, unless they have friends who are going to come to the rescue, they are likely to be ground down and lose badly in the longer term. So standing really only makes sense when you only need the LH to hold out for a turn or maybe two. That's where you do begin to build up enough die rolls that the averages become dominant. And the LH don't get an option to break off like they would against foot so if nothing else intervenes they will be stuck in for the long haul.

    Chris

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  2. Bates is correct unless they are repulsed, retire in good order and avoid being ground down but that is a different system.

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  3. Firstly, thanks for the comment.

    Secondly, I agree with the first point entirely. You're right to say the longer the combat goes on the more chances the light horse take of suffering a significant loss, or failing a cohesion test, so supporting them is important.

    However you can't combine the rolls from different combats so that "the averages become dominant" for two reasons; one maths and one FOG related.

    The FOG reason is that failed cohesion tests for either side will change the number of dice between each round of combat.

    The maths reason is that, even if things stay the same, each round of combat is an independent event and previous results have no impact on their outcome. The average changes but not the odds for each combat.

    Have a play with "Law of Large Numbers" demo to see this in action. Be sure to show the mean and roll totals; the latter continue to vary whilst the mean converges on the average.

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