After buying Risk: Legacy (by Hasbro) a few weeks ago, I’ve finally gotten a chance to play it.  The base game itself is very simple and I was a bit disappointed at first, but the large number of changes that happen after every game (or even during the game) drastically change things.  I’m impressed enough and, more importantly, excited to play more.  Of course, now I want to see how the math works.

Acronyms and Definitions:
Bunker – Defender adds 1 to their highest die roll.
dX – Die with X amount of sides.

Now, the basic Risk dice rules haven’t changed.  Attacker gets 3 dice (max) and the defender gets 2 dice (max), but the defender wins ties.  Since it’s competitive dice rolling, I’m going to do this a little differently. Let’s get into some Risk Legacy odds.

 1 Attacker vs. 1 Defender
Attack rolls 1d6:
1:  Attacker automatically loses.  0 Attacker wins, .1667 Attacker loses.
2: Attacker wins if the defender rolls a 1.  (1/6)(1/6) vs (1/6)(5/6) = .0278 Attacker wins, .1389 Attacker loses.
3: Attacker wins if the defender rolls a 1-2.  (1/6)(2/6) vs (1/6)(4/6) = .0556 Attacker wins, .1111 Attacker loses.
4: Attacker wins if the defender rolls a 1-3.  (1/6)(3/6) vs (1/6)(3/6) = .0833 Attacker wins, .0833 Attacker loses.
5: Attacker wins if the defender rolls a 1-4.  (1/6)(4/6) vs (1/6)(2/6) = .1111 Attacker wins, .0556 Attacker loses.
6: Attacker wins if the defender rolls a 1-5.  (1/6)(5/6) vs (1/6)(1/6) = .1389 Attacker wins, .0278 Attacker loses.

For a total of .4167 Attacker wins, .5834 Attacker loses.  Meaning that on your 1 v 1 basic attack, you have roughly a 41% chance of winning.

 1 Attacker vs. 1 Defender w/ Bunker
Attack rolls 1d6:
1:  Attacker automatically loses.  0 Attacker wins, .1667 Attacker loses.
2: Attacker automatically loses.  0 Attacker wins, .1667 Attacker loses.
3: Attacker wins if the defender rolls a 1.  (1/6)(1/6) vs (1/6)(5/6) = .0278 Attacker wins, .1389 Attacker loses.
4: Attacker wins if the defender rolls a 1-2.  (1/6)(2/6) vs (1/6)(4/6) = .0556 Attacker wins, .1111 Attacker loses.
5: Attacker wins if the defender rolls a 1-3.  (1/6)(3/6) vs (1/6)(3/6) = .0833 Attacker wins, .0833 Attacker loses.
6: Attacker wins if the defender rolls a 1-4.  (1/6)(4/6) vs (1/6)(2/6) = .1111 Attacker wins, .0556 Attacker loses.

For a total of .2778 Attacker wins, .7223 Attacker loses.  Meaning that with your 1 v 1 bunker assault, you have roughly a 28% chance of winning.

2 Attacker vs. 1 Defender
Now, for this case, only the highest die matters for the attacker, thus, let’s calculate the probability for each value to be the attack roll’s highest, and then calculate the probability that value wins.

Attacker rolls 2d6:
P(6 is highest): (1/6)(6/6)+(1/6)(6/6)-(1/6)(1/6) = 11/36
P(5 is highest): (1/6)(5/6)+(1/6)(5/6)-(1/6)(1/6) = 9/36
P(4 is highest): (1/6)(4/6)+(1/6)(4/6)-(1/6)(1/6) = 7/36
P(3 is highest): (1/6)(3/6)+(1/6)(3/6)-(1/6)(1/6) = 5/36
P(2 is highest): (1/6)(2/6)+(1/6)(2/6)-(1/6)(1/6) = 3/36
P(1 is highest): (1/6)(1/6)+(1/6)(1/6)-(1/6)(1/6) = 1/36

1 :  Attacker automatically loses.  0 Attacker wins, .0277 Attacker loses.
2: Attacker wins if the defender rolls a 1.  .0833 * (1/6) vs .0833 * (5/6) = .0138 Attacker wins, .0694 Attacker loses.
3: Attacker wins if the defender rolls a 1-2.  .1389 * (2/6) vs .1389 * (4/6) = .0463 Attacker wins, .0926 Attacker loses.
4:  Attacker wins if the defender rolls a 1-3.  .1944 * (3/6) vs .1944 * (3/6) = .0972 Attacker wins, .0972 Attacker loses.
5:  Attacker wins if the defender rolls a 1-4.  .2500 * (4/6) vs .2500 * (2/6) = .1667 Attacker wins, .0833 Attacker loses.
6:  Attacker wins if the defender rolls a 1-5.  .3056 * (5/6) vs .3056 * (1/6) = . 2547 Attacker wins, . 0509 Attacker loses.

For a total of .5785 Attacker wins, .4211 Attacker loses.  Meaning that with your 2 v 1 basic attack, you have roughly a 58% chance of winning.  Note that in the lose case, the attacker only loses one army.  If you want to attack again, you then shift to the 1v1 basic case, giving you a combined chance of winning at ~75%.

2 Attacker vs. 1 Defender w/ Bunker
Now, for this case, only the highest die matters for the attacker, thus, let’s calculate the probability for each value to be the attack roll’s highest, and then calculate the probability that value wins.
Attacker rolls 2d6:
P(6 is highest): (1/6)(6/6)+(1/6)(6/6)-(1/6)(1/6) = 11/36
P(5 is highest): (1/6)(5/6)+(1/6)(5/6)-(1/6)(1/6) = 9/36
P(4 is highest): (1/6)(4/6)+(1/6)(4/6)-(1/6)(1/6) = 7/36
P(3 is highest): (1/6)(3/6)+(1/6)(3/6)-(1/6)(1/6) = 5/36
P(2 is highest): (1/6)(2/6)+(1/6)(2/6)-(1/6)(1/6) = 3/36
P(1 is highest): (1/6)(1/6)+(1/6)(1/6)-(1/6)(1/6) = 1/36

1 : Attacker automatically loses.  0 Attacker wins, .0277 Attacker loses.
2: Attacker automatically loses.  0 Attacker wins, .0833 Attacker loses.
3: Attacker wins if the defender rolls a 1.  .1389 * (1/6) vs .1389 * (5/6) = .0232 Attacker wins, .1158 Attacker loses.
4: Attacker wins if the defender rolls a 1-2.  .1944 * (2/6) vs .1944 * (4/6) = .0648 Attacker wins, .1296 Attacker loses.
5: Attacker wins if the defender rolls a 1-3.  .2500 * (3/6) vs .2500 * (3/6) = .1250 Attacker wins, .1250 Attacker loses.
6: Attacker wins if the defender rolls a 1-4.  .3056 * (4/6) vs .3056 * (2/6) = . 2037 Attacker wins, . 1019 Attacker loses.

For a total of .4167 Attacker wins, .5833 Attacker loses.  Meaning that with your 2 v 1 bunker assault, you have roughly a 42% chance of winning.   Note that in the lose case, the attacker only loses one army.

2 Attackers vs. 2 Defenders
Here’s where things start getting complicated.  Each player is rolling two dice and comparing the highest of each roll, and then compare the lowest of each roll.  Thus there are now 3 possible outcomes – attacker loses two, defender loses two, or both lose one (where before either side could only lose one.)  Because of this, I’m going to break up the problem.  First, I’m going to look at the probabilities associated with the attacker’s higher die roll, and then the probabilities associated with the attacker’s lower die roll.

Attacker rolls 2d6:
P(6 is highest): (1/6)(6/6)+(1/6)(6/6)-(1/6)(1/6) = 11/36
P(5 is highest): (1/6)(5/6)+(1/6)(5/6)-(1/6)(1/6) = 9/36
P(4 is highest): (1/6)(4/6)+(1/6)(4/6)-(1/6)(1/6) = 7/36
P(3 is highest): (1/6)(3/6)+(1/6)(3/6)-(1/6)(1/6) = 5/36
P(2 is highest): (1/6)(2/6)+(1/6)(2/6)-(1/6)(1/6) = 3/36
P(1 is highest): (1/6)(1/6)+(1/6)(1/6)-(1/6)(1/6) = 1/36

1 is highest: Attacker automatically loses two armies.  36/1296 Attacker loses two armies, 0 each lose one army, 0 Defender loses two armies.

2 is highest: Attacker’s lowest roll is 1(2/36) or 2(1/36).

Result…	If attacker’s lowest 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

95 /1296 Attacker loses two armies, 12/1296 each loses one army, 1/1296 Defender loses two armies.

3 is highest: Attacker’s lowest roll is 1(2/36) or 2(2/36) or 3(1/36).

Result…	If attacker’s lowest 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

128/1296 Attacker loses two armies, 42/1296 each loses one army, 10/1296 Defender loses two armies.

4 is highest:Attacker’s lowest roll is 1(2/36) or 2(2/36) or 3(2/36) or 4(1/36).

Result…	If attacker’s lowest 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

135/1296 Attacker loses two armies, 82/1296 each loses one army, 35/1296 Defender loses two armies.

5 is highest:   Attacker’s lowest roll is 1(2/36) or 2(2/36) or 3(2/36) or 4 (2/36) or 5(1/36).

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

116/1296 Attacker loses two armies, 124/1296 each loses one army, 84/1296 Defender loses two armies.

6 is highest:   Attacker’s lowest roll is 1(2/36) or 2(2/36) or 3(2/36) or 4(2/36) or 5(2/36) or 6(1/36).

Result…	If attacker’s lowest 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

71/1296 Attacker loses two armies, 160/1296 each loses one army, 165/1296 Defender loses two armies.

For a grand total of the 581/1296 Attacker loses two armies, 420/1296 each loses one army, 295/1296 Defender loses two armies.  Or more easily read as 44.83% Defender wins, 32.41% both lose one and 22.76% Attacker wins.

2 Attackers vs. 2 Defenders w/ Bunker
Same situation as before, but now the defender has a bunker.  Remember, the Bunker adds +1 to the Defender’s highest roll.  Again, I’m going to start by looking at the probabilities associated with the attacker’s higher die roll, and then the probabilities associated with the attacker’s lower die roll.

Attacker rolls 2d6:
P(6 is highest): (1/6)(6/6)+(1/6)(6/6)-(1/6)(1/6) = 11/36
P(5 is highest): (1/6)(5/6)+(1/6)(5/6)-(1/6)(1/6) = 9/36
P(4 is highest): (1/6)(4/6)+(1/6)(4/6)-(1/6)(1/6) = 7/36
P(3 is highest): (1/6)(3/6)+(1/6)(3/6)-(1/6)(1/6) = 5/36
P(2 is highest): (1/6)(2/6)+(1/6)(2/6)-(1/6)(1/6) = 3/36
P(1 is highest): (1/6)(1/6)+(1/6)(1/6)-(1/6)(1/6) = 1/36

1 is highest: Attacker automatically loses two armies.  36/1296 Attacker loses two armies, 0 each lose one army, 0 Defender loses two armies.

2 is highest:   Attacker’s lowest roll is 1(2/36) or 2(1/36).

Result…	If attacker’s lowest 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

97 /1296 Attacker loses two armies, 11/1296 each loses one army, 0/1296 Defender loses two armies.

3 is highest:   Attacker’s lowest roll is 1(2/36) or 2(2/36) or 3(1/36).

Result…	If attacker’s lowest 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

136/1296 Attacker loses two armies, 41/1296 each loses one army, 3/1296 Defender loses two armies.

4 is highest:   Attacker’s lowest roll is 1(2/36) or 2(2/36) or 3(2/36) or 4(1/36).

Result…	If attacker’s lowest 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

153/1296 Attacker loses two armies, 81/1296 each loses one army, 18/1296 Defender loses two armies.

5 is highest:   Attacker’s lowest roll is 1(2/36) or 2(2/36) or 3(2/36) or 4 (2/36) or 5(1/36).

Result…	If attacker’s lowest 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

148/1296 Attacker loses two armies, 123/1296 each loses one army, 53/1296 Defender loses two armies.

6 is highest:   Attacker’s lowest roll is 1(2/36) or 2(2/36) or 3(2/36) or 4(2/36) or 5(2/36) or 6(1/36).

Result…	If attacker’s lowest 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

121/1296 Attacker loses two armies, 159/1296 each loses one army, 116/1296 Defender loses two armies.

For a grand total of the 691/1296 Attacker loses two armies, 415/1296 each loses one army, 190/1296 Defender loses two armies.  Or more easily read as 53.32% Defender wins, 32.02% both lose one and 14.67% Attacker wins.

Next week I’ll look at the more typical 3v1 and 3v2 scenarios, and will include analysis about the bunker itself next week or in a follow-up article.