While on vacation, I picked up the new version of Risk called Risk Reinvented (or maybe Risk Reinvention, it doesn’t actually say anything on the box except that it offers 3 ways to play), which has an optional set of rules to make the gameplay different. If you accomplish a certain objective (like controlling all of Asia at the end of your turn), you may earn the ability to attack or defend with a bonus die. So in the typical instance of 3 attacking dice vs. 2 defending dice, where the defenders win ties (this slightly favors the attackers), you might attack with 4 dice or defend with 3 dice even though only two troops are still at risk.
So I wanted to know what the odds are when you add dice to the mix. It was pretty obvious that the odds got a lot better when I earned the extra defense die. Then I got the extra attacking die and I was wondering how much of an effect it would have. I couldn’t find this on the internet (this was before I made the Risk Odds Calculator program).
So I built a spreadsheet that would generate every dice combination for six dice (3 attackers vs. 3 defending dice). This meant I had to have 6x6x6x6x6x6 or 46,656 rows. I had to check what the maximum number of rows that Excel can deal with and fortunately it was 65,536.
Next I generated all the combinations with each die in a different column and set up some Base 6 counting, then used the Max() function to pick out the highest value of the 3 attacking dice and compared that to the Max() of the defending dice. Then I used the Large() function to compare the second highest of each set of dice. Then I said that if the attacker won both, the total was 2, if they split the total was 0, and if the defense won both the total was -2. The average of all combinations was -0.491. That means the advantage is with the defense, whereas the number would be positive if the attacker had the advantage, and 0 if it was a 50-50 thing. I also counted up the percentages of each type of roll and used that to figure how many rolls it would take to knock out 100 defenders (100 divided by the fraction of splits plus two times the fraction where the defender loses two). With the number of rolls and the percentages I could then figure the number of attacking troops that would be lost (number of rolls times the fraction of splits plus two times the fraction where the attackers lose two). When the defender gets an extra die roll, 165 attackers will be lost in wiping out the defenders. That’s on average. Plus you would need one attacker to stay behind and 3 to advance into the contested country.
How does that compare with regular rolling? I did another spreadsheet and came up with an average of 0.158, which means the attacker has a slight advantage. Put another way, the attacker would need 85 troops to eliminate 100 defenders. The percentages of each outcome for this conventional battle agree with what is published at Wikipedia, so I think my methodology is good.
Now I wanted to check 4 attacking dice vs. 2 defending dice. The average for this scenario is 0.500 which more than doubles the advantage of 3 vs. 2 dice. Now you would lose 60 armies to defeat 100 defenders.
While researching some of this on the internet, there was some discussion about whether you had a bigger advantage defending with 1 troop and 1 bonus die against 3 attackers or if you should defend with 2 troops and a bonus die. So I ran numbers for 3 attackers against 2 defending dice, but this time you ignore the second highest roll for the defense, making the maximum the attackers or defenders can lose just 1. To compare that to defending with 2 troops, you just double the number. So the number I got for 3 vs. 1+1 was -0.057 which would double to -0.114, which is substantially less than -0.491 that you get when defending with 2 and a bonus roll. With two dice and a bonus die 112 attackers will be lost defeating 100 defenders, so this is much easier for the attackers than the 165 troops required to defeat 100 defenders who roll 3 dice at a time.
There is another bonus in the game where you can construct an airfield. If you have an airfield on a country, you can add 1 to your highest dice roll in both attacking and defending that country and any country the airfield country touches. So if the airfield is attacking, their factor goes up to 0.654 vs. 0.158 when attacking without an airfield. That’s a huge advantage as only 51 attackers will be lost defeating 100 defenders. On defense, the factor goes to -0.226 to the defense’s advantage from 0.158, again a pretty big swing, but not as good as getting a bonus die. With the airfield on defense, the attacker can expect to lose 126 armies to defeat 100 defenders.
There are further combinations. You could potentially have someone with a bonus die attacking an airfield or defending themselves from an airfield attack. Lastly, if you were really lucky you might have an airfield and a bonus die. So if you attacked with 4 dice and with the airfield bonus adding one to your highest, the factor is 1.232 (24 armies to defeat 100). But if you attack with 4 dice against an airfield, the odds are almost equal: 0.084, to the advantage of the attacker (92 attackers lost in taking out 100 defenders). There aren’t enough rows to see the odds of attacking with 4 dice and defending with 3 dice, but later I used a program I wrote to determine that this favors the defenders and the attacker would lose 115 troops defeating 100 defenders.
|Extra Die and Plus One||32||2 Defenders|
|Plus One||51||2 Defenders|
|3 Attack Dice||52||1 Defender|
|Extra Die||60||2 Defenders|
|2 Attack Dice||73||1 Defender|
|3 Attack Dice||85||2 Defenders|
|Extra Die||92||Plus One|
|3 Attack Dice||126||Plus One|
|1 Attack Die||140||1 Defender|
|2 Attack Dice||157||2 Defenders|
|3 Attack Dice||165||Extra Die|
|3 Attack Dice||251||Extra Die and Plus One|
|1 Attack Die||293||2 Defenders|