From: "Mircea Pauca"
Subject: Statistics for Impulse wargames (Storm over Arnhem)
Statistics for AH's Impulse Wargames and Analysis
especially applying to Storm over Arnhem
by Mircea-Valer Pauca (mpauca@fx.ro)
I received recently a copy of Storm over Arnhem (folio).
The fast-playing, chess-like Impulse mechanism quickly
attracted me (anyone willing to PBEM me ?)
Although strategic options are limited in this game, there is
much room for good tactical decisions, so I wrote this piece of
number-crunching and analysis. I believe this applies to
other Impulse games, too, as they are very similar.
Those disliking numbers and tables may jump to the
'conclusions and tactical tips' section at the end.
** Fire impulses **
The rules say:
AV = max.AF + bonuses + 2d6 (attack value)
DV = min.DF + bonuses + 2d6 (defense value)
If AV > DV then the defender takes
CP = AV - DV casualty points.
I rewrote it this way:
AA = (max.AF+bonuses) - (min.DF+bonuses) attacker's advantage
R = 2d6-2d6 the random part
Actually in SoA the 2d6 are simulated by chits with the same
distribution, pulled without replacement.
D = AA+R the difference
with CP=D if D>0 or CP=0 otherwise.
The following table includes:
(1) AA is the attacker's advantage, as above
(2) P(D=0) is the probability of getting exactly a D=0 difference:
P(D=0) = P(R= -AA). It is a very good aproximation of a
'normal' distribution, which is symmetric and bell-shaped.
(3) P(CP>0) is the probability of getting at least some damage,
obtained by summing the P(D=0) in columns above.
It also is the added effect of one more attack factor.
(4) E(CP) is 'expected casualty points' (the average)
E(CP)=1*P(CP=1)+2*P(CP=2)+...
(5) Var(CP) is the variance or dispersion, measured in 'points squared'.
I used a nice arithmetic artifice for this one, anyone interested can ask me.
As a nice property, if the draws are independent, these can be added
to give the variance of a sum of random variables (the CP's of more
attacks). See the example below.
With the chits pulled without replacement, the variance is
slightly lower as each pull restrains the possible results of future chits.
Anyone knows how to calculate this exactly in a simple way?
(6) StDev is the square root of Var(CP)
It's the standard deviation (also called 'root mean square deviation').
measured in CP's.
AA P(D=0) P(CP>0) E(CP) Var(CP) StDev
-10 0.1% 0.0% 0.00 0.00 0.00
-9 0.3% 0.1% 0.00 0.00 0.03
-8 0.8% 0.4% 0.00 0.01 0.08
-7 1.5% 1.2% 0.02 0.03 0.16
-6 2.7% 2.7% 0.04 0.08 0.29
-5 4.3% 5.4% 0.10 0.22 0.47
-4 6.2% 9.7% 0.19 0.48 0.69
-3 8.0% 15.9% 0.35 0.94 0.97
-2 9.6% 23.9% 0.59 1.66 1.29
-1 10.8% 33.6% 0.93 2.67 1.63
0 11.3% 44.4% 1.37 3.95 1.99
1 10.8% 55.6% 1.93 5.42 2.33
2 9.6% 66.4% 2.59 6.93 2.63
3 8.0% 76.1% 3.35 8.36 2.89
4 6.2% 84.1% 4.19 9.55 3.09
5 4.3% 90.3% 5.10 10.46 3.23
6 2.7% 94.6% 6.04 11.06 3.33
7 1.5% 97.3% 7.02 11.41 3.38
8 0.8% 98.8% 8.00 11.59 3.40
9 0.3% 99.6% 9.00 11.65 3.41
10 0.1% 99.9% 10.00 11.67 3.42
11 0 100.0% 11.00 11.67 3.42
A Fire example:
Attackers (Germans): 1HQ (can call 3 arty fires), 6x355 inf,
one 628 assault gun, one 677 Tiger, one 3-3-10 recce car
Defenders (British): 6x475 paras. They hold fire.
Arty at 9-7=+2 E=2.59 Var=6.93
Arty at 8-7=+1 E=1.93 Var=5.42
Arty at 8-7=+1 E=1.93 Var=5.42
Firegroup of 628+3310+3xinf+HQ:
6 lead attack+5 more units+1 integrity bonus=12 AV
12-8=+4 E=7.02 Var=11.41
Firegroup of 677+3(inf)+1=10 AV
10-8=+2 E=2.59 Var=6.93
Total: E=13.23 expected casualty points,
or some 4 units eliminated and 1 retreated
Var=34.25 or a standard deviation of 5.85 CP's.
These results assume that there are enough targets to
absorb all the result. If the targets are few, casualties are
limited at their complete elimination, less one. This will distort
the average result downwards.
(ex: 2 infantry units can absorb max. 5CP = both eliminated)
** Close Combat (Folio version with chits) **
A DR to kill Prob Extra effect
<=0 10+ 6/36=16.7%
1 9+ 10/36=27.8% +4/36
2 8+ 15/36=41.7% +5/36
3 7+ 21/36=58.3% +6/36
4 6+ 26/36=71.2% +5/36
5 5+ 30/36=83.3% +4/36
6 4+ 33/36=91.7% +3/36
** Close combat (Boxed version with 1d6) **
<=0 dr=6 1/6
+1 5-6 2/6
...
+5 1-6 6/6=100% certain
Every extra factor means 1/6 more eliminations.
** Urban fire setting **
Units DR to succeed Prob
2 2-3 3/36=8.33%
3 2-5 10/36=27.8%
4 2-7 21/36=58.3%
5 2-9 30/36=83.3%
6 2-11 35/36=97.2%
*** Conclusions and Tactical Tips ***
1) The usual attacker's tactic against entrenched targets
is to form _large_ firegroups from one leading, good attack
unit and many followers who increase the firepower,
also giving the +1 Formation integrity bonus.
2) Vulnerable targets (like Committed infantry) are best
attacked by individual units.
If each unit's attack is at -1 or better, shoot separately
with it. If not, put it in other (perhaps better) units' group.
For instance: 4-7-5 paras against adjacent (def 4) scouts.
Three -1 attacks (doing 3*0.93=2.79 average CP's) are better
than one +2 attack doing 2.59 CP's.
3) 'In defense with infantry, fire discipline is of the essence'
(John Keegan, _Face of Battle_ quoting a WW2 German general)
Infantry is 3 or 2 factors more vulnerable when 'committed' so
why give this bonus to the enemy for every attack ?
Endure enemy fire in the tougher Uncommitted state
(camouflage?) and don't return fire until (almost) every enemy
which could do damage has fired. So it pays to start an attack
only with a sufficiently large advantage, otherwise threaten only.
The defender could even Pass to threaten the attacker to
continue attacking him Uncommitted or else leave units
unused. Beware of nasty psychological games ! the attacker
should keep a fresh reserve (perhaps on a flank, or unused arty)
to exploit the more vulnerable defender after the return fire.
The exact solution can be given by an 'event tree' analysis
and depends on the willingness of both sides to take more
losses in order to inflict more on the enemy.
4) Close Combat (CC) in SoA[folio] does relatively little losses and is
ineffective as an attackers' tool - especially for the Germans.
An German inf unit (355, 466) in CC does at most 6/36=0.167
'eliminated' units on average.
A British para (475) does 10/36=0.278 against an inf (475).
If the same unit is added to a fire group, they do around
0.76-0.99 extra CP's which means 0.253-0.3 'eliminated'.
CC is effective for the British against isolated armoured cars
who dared and advanced too much.
5) The game rules encourages you to be a pyromaniac :-)
like the real Germans were in Arnhem. In urban areas with
forces from both sides, a German fire-setting attempt with
5 or 6 units works best to make the Brits Committed and
vulnerable. (one engineer counts as 3 units)
Any comments, additions are welcome. If anyone wishes
to adapt this analysis for other similar games, please contact me.
Best regards,
Mircea-Valer Pauca (mpauca@fx.ro)