Risk Legacy Odds – Basic Combat Math with a Twist, Part 2

Risk Legacy

This week I’m going to look at some of the more advanced scenarios in Risk Legacy odds as a follow-up to the post here.  We’ll look at 3v1 and 3v2 combat.

3 Attackers vs. 1 Defender
This is a pretty common case when an overwhelming attacker is trying to clean out thinly-defended territories.  It’s obviously quite in favor of the attacker, but let’s see how many the attacker can expect to lose.  Due to the complex nature of these probabilities I’m going to switch to more computer calculations versus by hand.  For many of these, I am simply using code to enumerate the rolls and then count the rolls of interest.

Attacker rolls 3d6:
P(6 is highest) = 91/216
P(5 is highest) = 61/216
P(4 is highest) = 37/216
P(3 is highest) = 19/216
P(2 is highest) = 7/216
P(1 is highest) = 1/216

1 is highest: Attacker automatically loses one army. (1/216)(6/6) = 6/1296 Attacker loses one army, 0 Defender loses one army.
2 is highest: Attacker wins if the defender rolls a 1.  (7/216)(5/6) vs (7/216)(1/6) = 35/1296 Attacker loses one army, 7/1296 Defender loses one army.
3 is highest: Attacker wins if the defender rolls a 1-2.  (19/216)(4/6) vs (19/216)(2/6) = 76/1296 Attacker loses one army, 38/1296 Defender loses one army.
4 is highest: Attacker wins if the defender rolls a 1-3.  (37/216)(3/6) vs (37/216)(3/6) = 111/1296 Attacker loses one army, 111/1296 Defender loses one army.
5 is highest: Attacker wins if the defender rolls a 1-4.  (61/216)(2/6) vs (61/216)(4/6) = 122/1296 Attacker loses one army, 244/1296 Defender loses one army.
6 is highest: Attacker wins if the defender rolls a 1-5.  (91/216)(1/6) vs (91/216)(5/6) = 91/1296 Attacker loses one army, 455/1296 Defender loses one army.

For a grand total of 441/1296 Attacker loses one army, 855/1296 Defender loses one army.  Or more easily read as 34.03% Defender wins and 65.97% Attacker wins.  Of course, when the Defender wins, the Attacker only loses one army, not all 3 involved in the attack.

3 Attackers vs. 1 Defender w/ Bunker
Same as above, but the Defender now has a bunker.

Attacker rolls 3d6:
P(6 is highest) = 91/216
P(5 is highest) = 61/216
P(4 is highest) = 37/216
P(3 is highest) = 19/216
P(2 is highest) = 7/216
P(1 is highest) = 1/216

1 is highest: Attacker automatically loses one army. (1/216)(6/6) = 6/1296 Attacker loses one army, 0 Defender loses one army.
2 is highest: Attacker automatically loses one army. (7/216)(6/6) = 42/1296 Attacker loses one army, 0 Defender loses one army.
3 is highest: Attacker wins if the defender rolls a 1.  (19/216)(5/6) vs (19/216)(1/6) = 95/1296 Attacker loses one army, 19/1296 Defender loses one army.
4 is highest: Attacker wins if the defender rolls a 1-2.  (37/216)(4/6) vs (37/216)(2/6) = 148/1296 Attacker loses one army, 74/1296 Defender loses one army.
5 is highest: Attacker wins if the defender rolls a 1-3.  (61/216)(3/6) vs (61/216)(3/6) = 183/1296 Attacker loses one army, 183/1296 Defender loses one army.
6 is highest: Attacker wins if the defender rolls a 1-4.  (91/216)(2/6) vs (91/216)(4/6) = 182/1296 Attacker loses one army, 364/1296 Defender loses one army.

For a grand total of 656/1296 Attacker loses one army, 640/1296 Defender loses one army.  Or more easily read as 50.62% Defender wins and 49.38% Attacker wins.  Of course, when the Defender wins, the Attacker only loses one army, not all 3 involved in the attack.

3 Attackers vs. 2 Defenders
The classic attack case.

Attacker rolls 3d6:
P(6 is highest) = 91/216
P(5 is highest) = 61/216
P(4 is highest) = 37/216
P(3 is highest) = 19/216
P(2 is highest) = 7/216
P(1 is highest) = 1/216

1 is highest: Attacker automatically loses two armies.  36/7776 Attacker loses two armies, 0 each lose one army, 0 Defender loses two armies.

2 is highest: Attacker’s second highest roll is 1(3/216) or 2(4/216).

Result… If attacker’s second highest roll is…
1 2
Attacker loses two 35/36 25/36
Each lose one 1/36 10/36
Defender loses two 0/36 1/36

205 /7776 Attacker loses two armies, 43/7776 each loses one army, 4/7776 Defender loses two armies.

3 is highest: Attacker’s second highest roll is 1(3/216) or 2(9/216) or 3(7/216).

Result… If attacker’s second highest roll is…
1 2 3
Attacker loses two 32/36 24/36 16/36
Each lose one 4/36 9/36 16/36
Defender loses two 0/36 3/36 4/36

424/7776 Attacker loses two armies, 205/7776 each loses one army, 55/7776 Defender loses two armies.

4 is highest:Attacker’s second highest roll is 1(3/216) or 2(9/216) or 3(15/216) or 4(10/216).

Result… If attacker’s second highest roll is…
1 2 3 4
Attacker loses two 27/36 21/36 15/36 9/36
Each lose one 9/36 10/36 13/36 18/36
Defender loses two 0/36 5/36 8/36 9/36

585/7776 Attacker loses two armies, 492/7776 each loses one army, 255/7776 Defender loses two armies.

5 is highest:   Attacker’s second highest roll is 1(3/216) or 2(9/216) or 3(15/216) or 4 (21/216) or 5(13/216).

Result… If attacker’s second highest roll is…
1 2 3 4 5
Attacker loses two 20/36 16/36 12/36 8/36 6/36
Each lose one 16/36 13/36 12/36 13/36 14/36
Defender loses two 0/36 7/36 12/36 15/36 16/36

630/7776 Attacker loses two armies, 800/7776   each loses one army, 766/7776 Defender loses two armies.

6 is highest:   Attacker’s second highest roll is 1(3/216) or 2(9/216) or 3(15/216) or 4(21/216) or 5(27/216) or 6(16/216).

Result… If attacker’s second highest roll is…
1 2 3 4 5 6
Attacker loses two 11/36 9/36 7/36 5/36 3/36 1/36
Each lose one 25/36 18/36 13/36 10/36 9/36 10/36
Defender loses two 0/36 9/36 16/36 21/36 24/36 25/36

421/7776 Attacker loses two armies, 1045/7776  each loses one army, 1810/7776 Defender loses two armies.

For a grand total of the 2275/7776 Attacker loses two armies, 2611/7776 each loses one army, 2890/7776   Defender loses two armies.  Or more easily read as 29.84% Attacker loses two armies, 32.99% both lose one and 37.17% Defender loses two armies.

3 Attackers vs. 2 Defenders w/ Bunker
The toughest possible defense I’ve seen so far.  Let’s see how bad it is.

Attacker rolls 3d6:
P(6 is highest) = 91/216
P(5 is highest) = 61/216
P(4 is highest) = 37/216
P(3 is highest) = 19/216
P(2 is highest) = 7/216
P(1 is highest) = 1/216

1 is highest: Attacker automatically loses two armies.  36/7776 Attacker loses two armies, 0 each lose one army, 0 Defender loses two armies.

2 is highest: Attacker’s second highest roll is 1(3/216) or 2(4/216).

Result… If attacker’s second highest roll is…
1 2
Attacker loses two 36/36 25/36
Each lose one 0/36 11/36
Defender loses two 0/36 0/36

208 /7776 Attacker loses two armies, 44/7776 each loses one army, 0/7776 Defender loses two armies.

3 is highest: Attacker’s second highest roll is 1(3/216) or 2(9/216) or 3(7/216).

Result… If attacker’s second highest roll is…
1 2 3
Attacker loses two 35/36 25/36 16/36
Each lose one 1/36 10/36 19/36
Defender loses two 0/36 1/36 1/36

442/7776 Attacker loses two armies, 226/7776 each loses one army, 16/7776 Defender loses two armies.

4 is highest:Attacker’s second highest roll is 1(3/216) or 2(9/216) or 3(15/216) or 4(10/216).

Result… If attacker’s second highest roll is…
1 2 3 4
Attacker loses two 32/36 24/36 16/36 9/36
Each lose one 4/36 9/36 16/36 23/36
Defender loses two 0/36 3/36 4/36 4/36

642/7776 Attacker loses two armies, 563/7776 each loses one army, 127/7776 Defender loses two armies.

5 is highest:   Attacker’s second highest roll is 1(3/216) or 2(9/216) or 3(15/216) or 4 (21/216) or 5(13/216).

Result… If attacker’s second highest roll is…
1 2 3 4 5
Attacker loses two 27/36 21/36 15/36 9/36 4/36
Each lose one 9/36 10/36 13/36 18/36 23/36
Defender loses two 0/36 5/36 8/36 9/36 9/36

736/7776 Attacker loses two armies, 989/7776   each loses one army, 471/7776 Defender loses two armies.

6 is highest:   Attacker’s second highest roll is 1(3/216) or 2(9/216) or 3(15/216) or 4(21/216) or 5(27/216) or 6(16/216).

Result… If attacker’s second highest roll is…
1 2 3 4 5 6
Attacker loses two 20/36 16/36 12/36 8/36 4/36 1/36
Each lose one 16/36 13/36 12/36 13/36 16/36 19/36
Defender loses two 0/36 7/36 12/36 15/36 16/36 16/36

676/7776 Attacker loses two armies, 1354/7776  each loses one army, 1246/7776 Defender loses two armies.

For a grand total of the 2740/7776 Attacker loses two armies, 3176/7776 each loses one army, 1860/7776   Defender loses two armies.  Or more easily read as 35.24% Attacker loses two armies, 40.84% both lose one and 23.28% Defender loses two armies.  Ouch.  So a bunker makes quite a noticeable difference.

Next week I’ll look at the effects of the bunker, as well as expected losses / battle.

One thought on “Risk Legacy Odds – Basic Combat Math with a Twist, Part 2

  1. Pingback: Risk Legacy Odds - Basic Combat Math with a Twist, Part 3 - Gaming By The Numbers

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