Date: Tue, 26 Jul 94 10:36:05 CDT
From: johnson@ferret.cig.mot.com (Brad Johnson)
Subj: We The People: Empirical Probabilities
A while back, I posted a note describing my hypothesis that no method of
battle card play in We The People could be appreciably better than just
making random attacks. I based this guess on the fact that empirical
evidence showed that random attack selection already gave the aggressor (i.e.
initial attacker) an appreciable advantage. I decided to look into this a
bit further, and I've generated some data that might be interesting.
I created computer programs to empirically generate the odds of winning any
battle, given the number of cards in each hand, plus the two battle ratings.
I then studied 4 attack philosophies:
1) Random Selection - The current attacker always chooses a random card from
his hand with which to attack. This uses a standard random number
generator which gives good results.
2) Depth-First Selection - The current attacker always chooses an attack card
of the type of which he currently holds the highest percentage, and then
exhausts that type before proceeding to his next attack type. For
example, if a player holds 3 Frontal Attacks, 3 Right Flanks, and 2
Probes, he would play his attacks in the following order, unless
interrupted by a counterattack: 2 Probes (2/7 = 0.29), 3 Right Flanks
(3/14 = 0.21), and then 3 Frontal Attacks (3/15 = 0.20).
3) Breadth-First Selection - The crrrent attacker always chooses an attack
card by rotating through a list of attack types, starting with his
highest-percentage one and proceeding to his lowest-percentage one, then
starting over. For example, if a player holds 3 Frontal Attacks, 3 Right
Flanks, and 2 Probes, he would play his attacks in the following order,
unless interrupted by a counterattack: Probe, Right Flank, Frontal
Attack, Probe, Right Flank, Frontal Attack, Right Flank, Frontal Attack.
4) Fixed-Order Selection - The current attack always chooses an attack card
that is as high as possible on the following list: Double Envelopment,
Bombardment, Probe, Flank (Left or Right), Frontal Attack. In many cases,
this will be similar to playing a Depth-First Selection.
All algorithms assume that the defender will always attempt to counterattack,
unless the attacker is out of cards. (Since I'm only concerned with win/lose
at this point, it doesn't really matter -- It might if we were talking about
average armies lost in addition.)
The tables following this message show my results. In all tables, the
initial attacker (aggressor) is shown down the left side, and the initial
defender is shown across the top. The first set of tables keeps battle
ratings equal and constant, and varies number of battle cards. The second
set of tables keep battle card counts equal and constant, and varies battle
ratings. Each data point in all tables is the percentage chance that the
initial attacker (aggressor) will win the battle, taken as the actual number
of wins out of 10,000 trials.
I'm sure a statistician could argue about my data collection methods, but I'm
personally fairly confident in them. I think there are a lot of interesting
conclusions that can be drawn, but in general, I think this indicates the
following:
1) An aggressor with a numerical advantage in battle cards will win at least
60-90% of the time, even if playing his cards completely randomly.
2) The fixed-order selection method, favoring the rare attack types, is the
best of the ones I studied, but is still only a few percentage points
better than completely random card play.
3) Each battle card in the initial defender's hand subtracts more from the
aggressor's chance of winning than each battle card in the initial
attacker's hand adds. (I don't have enough data to estimate what the
formulae are.)
4) Each battle rating point of the initial defender's general subtracts more
from the aggressor's chance of winning than each battle rating point of
the initial attacker's general adds. (Again, I don't have enough data to
estimate the formulae.)
I intend to continue studying these figures, ideally coming up with some
formulae that could estimate the aggressor's winning percentage, given the 4
variables (battle card counts and battle ratings). I would then like to
extend this to estimate armies lost on each side. (As we know, battle card
play differs if you intend to inflict maximum damage. I'm ignoring that for
now, because in my experience, my primary concern is normally to just win the
battle.)
I'd love to hear any comments or suggestions anyone might have. Also, if
anyone has any favorite battle tactics that are significantly different from
the ones I used, please let me know and I'll code them up for analysis. I'm
pursuing this in order to evaluate the effects of various rules variants
ideas that I have -- My goal is to find a way to penalize random play and
reward good tactics without destroying the simplicity and flavor of the game.
(Someone once mentioned that the game designer has stated that his analysis
of Revolutionary battles led him to design in the advantage for aggressors.
I have no quarrel with that, but I would like to see more options for
tacticians during battle.)
Have fun,
Brad Johnson (johnson@cig.mot.com)
Copyright 1994
Varying battle card counts (both battle ratings = 2)
====================================================
Method A: Random Selection
Defender
3 4 5 6 7 8 9 10 11 12
----------------------------------------------------------------
3 84 76 68 60 54 46 41 35 32 27
A 4 87 80 73 65 59 53 48 42 38 32
t 5 88 82 76 69 63 57 52 47 41 37
t 6 90 85 78 72 66 62 56 51 45 41
a 7 91 86 80 74 69 65 60 53 49 44
c 8 92 88 83 77 72 67 62 57 52 48
k 9 94 90 84 79 75 69 64 60 55 52
e 10 94 90 86 81 76 72 68 63 58 54
r 11 95 92 87 83 79 74 70 66 61 57
12 96 93 89 86 81 76 72 68 64 60
Method B: Depth-First Selection
Defender
3 4 5 6 7 8 9 10 11 12
----------------------------------------------------------------
3 86 78 71 64 57 51 44 38 34 30
A 4 90 84 76 71 66 58 53 47 42 37
t 5 90 85 80 74 70 64 60 53 47 44
t 6 91 86 82 77 73 68 63 58 54 49
a 7 92 87 82 79 74 70 65 61 57 53
c 8 93 89 84 79 76 72 68 65 60 56
k 9 94 90 86 81 78 74 70 66 62 60
e 10 94 91 87 83 80 76 72 68 65 62
r 11 96 92 88 84 80 77 73 70 67 64
12 96 92 89 85 81 79 75 72 68 65
Method C: Breadth-First Selection
Defender
3 4 5 6 7 8 9 10 11 12
---------------------------------------------------------------
3 87 78 71 63 57 50 45 38 34 29
A 4 89 82 76 70 63 55 50 45 39 34
t 5 90 85 78 73 67 59 55 48 43 38
t 6 92 86 80 75 69 63 58 52 47 42
a 7 93 88 82 77 72 67 61 57 51 45
c 8 94 90 84 80 74 70 63 58 53 50
k 9 95 91 86 81 77 72 67 62 56 52
e 10 96 93 88 83 78 73 69 64 60 55
r 11 97 93 90 86 81 77 71 67 63 58
12 97 94 91 87 84 78 73 69 66 60
Method D: Fixed Order Selection (Double Envelopment, Bombardment,
Probe, Flank (Left or Right), Frontal Attack)
Defender
3 4 5 6 7 8 9 10 11 12
---------------------------------------------------------------
3 87 79 71 65 59 51 45 40 35 31
A 4 91 84 77 73 65 60 54 48 44 38
t 5 92 87 81 75 70 64 59 54 48 44
t 6 93 89 84 78 74 68 63 58 53 48
a 7 95 90 86 81 76 71 66 61 57 52
c 8 95 92 87 82 77 73 69 64 59 55
k 9 96 92 88 85 80 76 71 67 62 58
e 10 96 93 89 85 81 78 72 69 65 61
r 11 97 94 91 87 83 80 75 71 67 63
12 97 95 92 88 85 81 77 73 69 65
Varying battle ratings (both battle card counts = 5)
====================================================
Method A: Random Selection
Defender
0 1 2 3
--------------------------
0 99 84 72 61
A 1 99 85 73 64
t 2 99 86 76 66
t 3 99 87 78 70
Method B: Depth-First Selection
Defender
0 1 2 3
--------------------------
0 99 88 76 68
A 1 99 88 79 70
t 2 99 89 80 73
t 3 99 89 81 74
Method C: Breadth-First Selection
Defender
0 1 2 3
--------------------------
0 99 85 75 64
A 1 99 86 76 68
t 2 99 88 78 70
t 3 99 89 80 73
Method D: Fixed Order Selection (Double Envelopment, Bombardment,
Probe, Flank (Left or Right) Frontal Attack)
Defender
0 1 2 3
--------------------------
0 99 88 79 72
A 1 99 89 81 73
t 2 99 90 81 74
t 3 99 90 82 75