34

I'm looking for an way to estimate the percent chance of winning a specific Risk battle. Assume the maximum number of dice will be used.

It doesn't need to be a perfect calculation, just an estimate to give a general idea of whether to attack or not. It needs to be easy to memorize.

I'm sure there is a trade off between accuracy and easy of use, so I'm hoping there is more than one method proposed. I would be using this to estimate after each die roll so I can stop attacking when my winning percentage drops below a certain threshold.

Bonus free wild card if you can also describe a method of estimating armies left.

33

All of the detailed probability calculations and Markov analysis posted by Eric P. and ire_and_curses can be distilled into a simple set of Risk attack heuristics:

  • Large battles favor the attacker but only very slightly.
  • For small battles, attack if you have more armies, stop if you don't.

The rationale for these guidelines is outlined below.

A large battle is 3 or more attackers and 2 or more defenders. For large battles the army count has nothing to do with the outcome of any single battle, which means you only need to remember a single number: the net attrition rate, defined as the expected value of the difference between defender and attacker losses. This rate can be calculated from data in Table 2 of the Jason Osborne paper in Eric P.'s answer:

  • The expected value of the defenders lost is 1.08. The number of defenders lost by events pi_32x weighted by their probability.
  • The expected value of the attackers lost is 0.922.
  • This means the net attrition rate is 0.158 (i.e. over 10 rounds of combat, the defender will lose 1.58 more armies--on average). Over 20 rounds 3.16.

Leading us to our first heuristic:

  • Large battles favor the attacker but only very slightly. As the attacker, it takes on average 20 rounds to make up a three army deficit. Whether you should attack in any given situation is a strategic not a tactical decision, but tactically the attacking advantage is on average very slight. Remember as well that past events have nothing to do with future rolls.

For small battles, the full probability matrix is supplied above, but our second rule distills this knowledge:

  • For small battles, attack if you have more armies, stop if you don't. When outnumbered your probability of winning is no more than 0.417 and probably less. Otherwise your probability of winning is at least 0.656 but as high as 0.916.
  • 3
    I upvoted several answers, but I think this one actually does the best at giving a usable, memorable heuristic. – JSBձոգչ May 25 '11 at 12:53
  • NO, I this heuristic neither brief, nor memorable, nor correct. In 1v1 games (at least) the concept is to always attack if you have 3 attackers (4 or more troops on the attacking territory). There is no intrinsic difference between one large and many small battles. – TheChymera Jul 22 '14 at 0:06
  • 8
    @TheChymera -- I don't think what you said is inconsistent with this answer: Adam defined a "large battle" as 3 or more attackers and 2 or more defenders. – Hao Ye Oct 28 '14 at 17:01
  • 1
    Perhaps it's worth noting that "attackers" means the number of armies actually attacking; this would not include the 1+ armies that must remain on the attacker's original territory. If you have 3 armies and your opponent has 2, you have only 1 or 2 attackers, not 3. – Kyralessa Aug 23 '18 at 17:19
17

A good paper by Jason Osborne can be found here. (It's a correction to an earlier paper by Tan.) He uses Markov chain calculations to get the exact probabilities. You'll especially want to look at Table 3 on page 6, which has these probabilities rounded to three decimals for up to 10 armies per side. I've reproduced it below:

enter image description here

For quicker visual scanning, here's a colorized version using conditional formatting in Excel and percentages:

enter image description here

As to ease of use: just print it out and look up your odds after every roll!

  • 1
    I've often found players don't like to use such aids during play (thinking it cheating), this is why I prefer mental calculations. – Neal Tibrewala May 26 '11 at 3:26
  • Looks like you get a reasonably good approximation by taking 0.42 + 0.11 * A - 0.09 * D. So, start with 0.42, add 0.11 for every attacker, subtract 0.09 for every defender. (I tried to get a good fit for the results between 0.1 and 0.9; if you get outside of that range it's less important anyway.) – Erik P. May 26 '11 at 14:25
  • Your statement "look up your odds after every roll" is actually very deep. The chart assumes you'll fight until you've won or until you're down to one troop. If you stop once the odds are against you, the chart isn't accurate. If you make a chart to compensate, you can then stop when that NEW chart tells you the odds are against you, and so on, ad infinitum. I've done some inconclusive work along these lines: github.com/barrycarter/bcapps/blob/master/bc-calc-risk-odds.m – barrycarter May 27 '11 at 4:14
  • @barrycarter - interesting work. Essentially there is a third parameter at play here: the minimal probability of winning at which you're willing to continue. You set that at 0.5, I think (from a quick glance over your code). I think in different situations, you'd want to use different probabilities here: sometimes it's worth to risk a few armies for a long shot that would help you a lot. So we would need different tables for different values of this parameters... interesting, interesting. – Erik P. May 27 '11 at 18:33
  • You have a typo at 4 vs 4 - it should read more like 0.488 than 0.477. – Forget I was ever here Oct 28 '14 at 21:56
10

As stated in the Risk FAQ the expected losses per attack for standard Risk rules is about 6 to 7. This means the attacker is expected to lose 6 armies for every 7 defender armies destroyed. Since we're talking about expected values, this represents the mean (average), which is most akin to a 50th percentile or 50% chance that that is what will happen. (this is not techically correct in terms of statistics, but I'm trying to explain it in more layman's terms).

As for 'chances to win a battle' this is very difficult to produce a rule of thumb for, since the numbers vary wildly depending on # of armies in play. For example, an 'even match' of 100 to 100 is won with over 85% chance, but 10 to 10 is only about 50%-50%.

It would be easier if you specified a particular odds at which you want to attack or not, then a simple matrix of attacker/defender army counts is possible to create (perhaps with an easy formula), but when you start with N attackers and M defenders the best 'answer', statistically, is a probablilty distribution function which isn't easy to calculate.

Assuming you want to attack at, at least 50% expected value (as above), use the 6 to 7 rule. This will also tell you how many you are expected to have left, so, using this rule, if you have 20 armies (to attack with, so 21 in the country), and he has 21, the 6 to 7 rule would say that you're expected to lose 18, killing his 21 (6*3, 7*3), thus leaving 2 left over.

9

Although the full calculation to discover whether you will win a sequence of battles is difficult to make, it is easy to calculate the chances of winning any particular combination of attacker and defender dice. I reproduce here the table of expected losses described in this paper.

                       Defender Dice
                       1           2
Attacker Dice
1                  0.58/0.42   0.75/0.25
2                  0.42/0.58   1.22/0.78
3                  0.34/0.66   0.92/1.08

The first number in each cell is attacker losses, the second defender losses. So, if you attack with 1 die against a 2 dice defence, you have a 3/4 chance of losing the battle, which in this case will cost you 1 army. Remember also that if each of you have 2 or more dice, then exactly 2 armies will be lost in the battle. This is why in 2 dice vs 2, the attacker will lose on average more than one army.

So, you could easily memorise this table, and use it to decide at each step of a battle whether to proceed or not. Or you could just remember the broad rule of thumb described in the paper:

  1. When both attacker and defender have a large number of armies, the attacker will, on average, lose armies at a 15% slower rate than the defender.

  2. Towards the end, when either the attacker or the defender must shake fewer dice, the advantage swings more strongly toward the player with the most armies.

The details of the full calculation are explained in the paper. The author also provides an online Javascript implementation of the calculations that you might find interesting to play around with.

4

If you are online, the easiest way to determine whether you have a good chance of winning a battle is to use this calculator: http://armsrace.co/probabilities

It emphasizes a non-trivial conclusion: if you have the choice, always attack the big guys first in your sequence!

For instance, if you have 6 on a territory, and want to attack a 2 and a 1 (and you have the choice to start with either one), you have:

  • 73,60% chances of winning if you attack 2 then 1
  • 69,56% if you attack 1 then 2

That's a fairly large difference!

The reason why we're seeing this is basically because the more armies there are in front of you, the more armies you need yourself. If you start by attacking the 1 with 5 armies, you'll be at your best with 4 armies left to attack the remaining 2. And there is a chance that you have in fact only 3 or 2. Meaning that if you lose only one army (in attacking the 1 or the 2) you'll be in the disadvantageous situation where you have an equal number of dice against the 2.

On the other hand, if you start by attacking the 2, you have more cases where you get 3 dice against 2.

This is a very litteral answer, but I think it helps getting the idea why you should attack the most powerful territories first.

3

Here is an incredibly simple way to figure this out without graphs or calculators. Simply count one of your soldiers defeated for every other soldier you intend to defeat & one of your soldiers left behind for every territory your conquer. Example: If you wish to eliminate another player that has 12 soldiers spread over 5 connected territories & that player is right beside the territory that you will place your soldiers on, you will need 12+5=17 soldiers in addition to the one that must stay behind from the original attacking territory. If there are other soldiers or territories in the way of your intended goal then, after figuring out the least expensive route, simply count one of your soldiers defeated for every other soldier you intend to defeat & one of your soldiers left behind for every territory your conquer. This method slightly favours the attacker in big numbers & slightly favours the defenders in small numbers which brings me to my next point. From time to time you will have to split your forces in different directions in the path you have chosen to pursue. This event that I call a fork will force some of your soldiers to stop being able to attack usefully. For every dead end fork add 2 soldiers as they will be required to ensure you can always roll 3 dice right to the end of each fork.

2

Analyzing Dice Rolls in the Game of Risk

The Goal of this research is to answer this question: Who has the probable advantage in the game of Risk? Does the Attacker or the Defender have this advantage?

Initial Thinking:

With two dice rolled, what is the probability that at least one six will appear?

36 possible combinations when two dice are rolled

There are 36 combinations of dice numbers possible when you roll two dice in one instance. There are 11 instances where 6 appears once or more in this set of 36 possibilities. The answer is: 11/36=0.305.

What follows is a Permutation Set covering the combinations of dice rolls that are possible when 3 attack dice are used against 2 defense dice:

The Number Headings are used to number a logical structuring of the permutation set. The permutation set of numbers contains 6 defence instances (where the defence is assumed to have rolled at least 1 instance of a 1) times 6 respective defence instances (where the defence rolls a number from to 1 to 6) times 1 instance (where the Attacker only rolls ones) + (6-1) (this accounts for whats left (1 to 5) after the attacker rolls only ones) times 3 (number of Attacker dice used). Each of the number headings represent a number block from 1 to 16 based on the attackers total set of possible rolls.

Total Possible Rolls when the attacker uses 3 dice and the defence must use 2 dice.

S=6 x 6 x (1 + 15) = 6 x 6 x 16 = 576

If you study this rather large set of permutations, you should be able to understand the logical structure in this enumerating approach to answering who has the advantage in Risk, the attacker or the defender.

A means Attacker Dr means Draw D means Defence

Enumerating the Possible Dice Combinations in Risk

The defender has 2 dice and the attack 3 dice.

[D1, D2, A1, A2, A3]

*Sorry if the tables are hard to read, the original formatting could not be preserved for this post. I can reformat the tables using a color coding system that will make them easier to read. Let me know if you are interested and I will provide them as quickly as possible.

1

D wins 36 times resulting in A losing 2 units in each instance

[1,1,1,1,1] [1,2,1,1,1] [1,3,1,1,1] [1,4,1,1,1] [1,5,1,1,1] [1,6,1,1,1]

[2,1,1,1,1] [2,2,1,1,1] [2,3,1,1,1] [2,4,1,1,1] [2,5,1,1,1] [2,6,1,1,1]

[3,1,1,1,1] [3,2,1,1,1] [3,3,1,1,1] [3,4,1,1,1] [3,5,1,1,1] [3,6,1,1,1]

[4,1,1,1,1] [4,2,1,1,1] [4,3,1,1,1] [4,4,1,1,1] [4,5,1,1,1] [4,6,1,1,1]

[5,1,1,1,1] [5,2,1,1,1] [5,3,1,1,1] [5,4,1,1,1] [5,5,1,1,1] [5,6,1,1,1]

[6,1,1,1,1] [6,2,1,1,1] [6,3,1,1,1] [6,4,1,1,1] [6,5,1,1,1] [6,6,1,1,1]

2

D wins 35 times resulting in A losing 2 units in each instance. In 1 instance, D and A both lose 1 unit.

[1,1,2,1,1] Draw [1,2,2,1,1] [1,3,2,1,1] [1,4,2,1,1] [1,5,2,1,1] [1,6,2,1,1]

[2,1,2,1,1] [2,2,2,1,1] [2,3,2,1,1] [2,4,2,1,1] [2,5,2,1,1] [2,6,2,1,1]

[3,1,2,1,1] [3,2,2,1,1] [3,3,2,1,1] [3,4,2,1,1] [3,5,2,1,1] [3,6,2,1,1]

[4,1,2,1,1] [4,2,2,1,1] [4,3,2,1,1] [4,4,2,1,1] [4,5,2,1,1] [4,6,2,1,1]

[5,1,2,1,1] [5,2,2,1,1] [5,3,2,1,1] [5,4,2,1,1] [5,5,2,1,1] [5,6,2,1,1]

[6,1,2,1,1] [6,2,2,1,1] [6,3,2,1,1] [6,4,2,1,1] [6,5,2,1,1] [6,6,2,1,1]

3

A wins two units in 1 instance, 5 draws + 5 draws = 10 draws, D wins 25 times

[1,1,2,2,1] A wins twice [1,2,2,2,1] Draw x5 [1,3,2,2,1] [1,4,2,2,1] [1,5,2,2,1] [1,6,2,2,1]

[2,1,2,2,1] Draw [2,2,2,2,1] D wins x 5 [2,3,2,2,1] [2,4,2,2,1] [2,5,2,2,1] [2,6,2,2,1]

[3,1,2,2,1] Draw [3,2,2,2,1] [3,3,2,2,1] [3,4,2,2,1] [3,5,2,2,1] [3,6,2,2,1]

[4,1,2,2,1] Draw [4,2,2,2,1] [4,3,2,2,1] [4,4,2,2,1] [4,5,2,2,1] [4,6,2,2,1]

[5,1,2,2,1] Draw [5,2,2,2,1] [5,3,2,2,1] [5,4,2,2,1] [5,5,2,2,1] [5,6,2,2,1]

[6,1,2,2,1] Draw [6,2,2,2,1] [6,3,2,2,1] [6,4,2,2,1] [6,5,2,2,1] [6,6,2,2,1]

4

A wins two units in 1 instance, 5 draws + 5 draws = 10 draws, D wins 25 times

[1,1,2,2,2] A wins twice [1,2,2,2,2] Draw x 5 [1,3,2,2,2] [1,4,2,2,2] [1,5,2,2,2] [1,6,2,2,2]

[2,1,2,2,2] Draw [2,2,2,2,2] [2,3,2,2,2] [2,4,2,2,2] [2,5,2,2,2] [2,6,2,2,2]

[3,1,2,2,2] Draw [3,2,2,2,2] [3,3,2,2,2] [3,4,2,2,2] [3,5,2,2,2] [3,6,2,2,2]

[4,1,2,2,2] Draw [4,2,2,2,2] [4,3,2,2,2] [4,4,2,2,2] [4,5,2,2,2] [4,6,2,2,2]

[5,1,2,2,2] Draw [5,2,2,2,2] [5,3,2,2,2] [5,4,2,2,2] [5,5,2,2,2] [5,6,2,2,2]

[6,1,2,2,2] Draw [6,2,2,2,2] [6,3,2,2,2] [6,4,2,2,2] [6,5,2,2,2] [6,6,2,2,2]

5

A wins two units in 3 instances, 4 + 5 x 1 draws = 9 draws, D wins 24 times,

[1,1,3,2,2] A wins 2 units [1,2,3,2,2] A wins 2 units [1,3,3,2,2] Draw [1,4,3,2,2] Draw [1,5,3,2,2] Draw [1,6,3,2,2] Draw

[2,1,3,2,2] A wins 2 units [2,2,3,2,2] Draw [2,3,3,2,2] D wins 2 units [2,4,3,2,2] D wins 2 units [2,5,3,2,2] D wins 2 units [2,6,3,2,2] D wins 2 units

[3,1,3,2,2] Draw [3,2,3,2,2] D wins 2 units [3,3,3,2,2] D wins 2 units [3,4,3,2,2] D wins 2 units [3,5,3,2,2] D wins 2 units [3,6,3,2,2] D wins 2 units

[4,1,3,2,2] Draw [4,2,3,2,2] D wins 2 units [4,3,3,2,2] D wins 2 units [4,4,3,2,2] D wins 2 units [4,5,3,2,2] D wins 2 units [4,6,3,2,2] D wins 2 units

[5,1,3,2,2] Draw [5,2,3,2,2] D wins 2 units [5,3,3,2,2] D wins 2 units [5,4,3,2,2] D wins 2 units [5,5,3,2,2] D wins 2 units [5,6,3,2,2] D wins 2 units

[6,1,3,2,2] Draw [6,2,3,2,2] D wins 2 units [6,3,3,2,2] D wins 2 units [6,4,3,2,2] D wins 2 units [6,5,3,2,2] D wins 2 units [6,6,3,2,2] D wins 2 units

6

A wins two units in 4 instances, 4 x 2 + 4 x 2 draws = 16 draws, D wins 16 times,

[1,1,3,3,2] A wins 2 units [1,2,3,3,2] A wins 2 units [1,3,3,3,2] Draw [1,4,3,3,2] Draw [1,5,3,3,2] Draw [1,6,3,3,2] Draw

[2,1,3,3,2] A wins 2 units [2,2,3,3,2] A wins 2 units [2,3,3,3,2] Draw [2,4,3,3,2] Draw [2,5,3,3,2] Draw [2,6,3,3,2] Draw

[3,1,3,3,2] Draw [3,2,3,3,2] Draw [3,3,3,3,2] D wins 2 units [3,4,3,3,2] D wins 2 units [3,5,3,3,2] D wins 2 units [3,6,3,3,2] D wins 2 units

[4,1,3,3,2] Draw [4,2,3,3,2] Draw [4,3,3,3,2] D wins 2 units [4,4,3,3,2] D wins 2 units [4,5,3,3,2] D wins 2 units [4,6,3,3,2] D wins 2 units

[5,1,3,3,2] Draw [5,2,3,3,2] Draw [5,3,3,3,2] D wins 2 units [5,4,3,3,2] D wins 2 units [5,5,3,3,2] D wins 2 units [5,6,3,3,2] D wins 2 units

[6,1,3,3,2] Draw [6,2,3,3,2] Draw [6,3,3,3,2] D wins 2 units [6,4,3,3,2] D wins 2 units [6,5,3,3,2] D wins 2 units [6,6,3,3,2] D wins 2 units

7

A wins two units in 4 instances, 4 x 2 + 4 x 2 draws = 16 draws, D wins 16 times,

[1,1,3,3,3] A wins 2 units [1,2,3,3,3] A wins 2 units [1,3,3,3,3] Draw [1,4,3,3,3] Draw [1,5,3,3,3] Draw [1,6,3,3,3] Draw

[2,1,3,3,3] A wins 2 units [2,2,3,3,3] A wins 2 units [2,3,3,3,3] Draw [2,4,3,3,3] Draw [2,5,3,3,3] Draw [2,6,3,3,3] Draw

[3,1,3,3,3] Draw [3,2,3,3,3] Draw [3,3,3,3,3] D wins 2 units [3,4,3,3,3] D wins 2 units [3,5,3,3,3] D wins 2 units [3,6,3,3,3] D wins 2 units

[4,1,3,3,3] Draw [4,2,3,3,3] Draw [4,3,3,3,3] D wins 2 units [4,4,3,3,3] D wins 2 units [4,5,3,3,3] D wins 2 units [4,6,3,3,3] D wins 2 units

[5,1,3,3,3] Draw [5,2,3,3,3] Draw [5,3,3,3,3] D wins 2 units [5,4,3,3,3] D wins 2 units [5,5,3,3,3] D wins 2 units [5,6,3,3,3] D wins 2 units

[6,1,3,3,3] Draw [6,2,3,3,3] Draw [6,3,3,3,3] D wins 2 units [6,4,3,3,3] D wins 2 units [6,5,3,3,3] D wins 2 units [6,6,3,3,3] D wins 2 units

8

A wins two units in 8 instances, 3 x 2 + 1 + 3 x 2 draws = 13 draws, D wins 15 times,

[1,1,4,3,3] A wins 2 units [1,2,4,3,3] A wins 2 units [1,3,4,3,3] A wins 2 units [1,4,4,3,3] Draw [1,5,4,3,3] Draw [1,6,4,3,3] Draw

[2,1,4,3,3] A wins 2 units [2,2,4,3,3] A wins 2 units [2,3,4,3,3] A wins 2 units [2,4,4,3,3] Draw [2,5,4,3,3] Draw [2,6,4,3,3] Draw

[3,1,4,3,3] A wins 2 units [3,2,4,3,3] A wins 2 units [3,3,4,3,3] Draw [3,4,4,3,3] D wins 2 units [3,5,4,3,3] D wins 2 units [3,6,4,3,3] D wins 2 units

[4,1,4,3,3] Draw [4,2,4,3,3] Draw [4,3,4,3,3] D wins 2 units [4,4,4,3,3] D wins 2 units [4,5,4,3,3] D wins 2 units [4,6,4,3,3] D wins 2 units

[5,1,4,3,3] Draw [5,2,4,3,3] Draw [5,3,4,3,3] D wins 2 units [5,4,4,3,3] D wins 2 units [5,5,4,3,3] D wins 2 units [5,6,4,3,3] D wins 2 units

[6,1,4,3,3] Draw [6,2,4,3,3] Draw [6,3,4,3,3] D wins 2 units [6,4,4,3,3] D wins 2 units [6,5,4,3,3] D wins 2 units [6,6,4,3,3] D wins 2 units

9

A wins two units in 9 instances, 3 x 3 + 3 x 3 draws = 18 draws, D wins 9 times,

[1,1,4,4,3] A wins 2 units [1,2,4,4,3] A wins 2 units [1,3,4,4,3] A wins 2 units [1,4,4,4,3] Draw [1,5,4,4,3] Draw [1,6,4,4,3] Draw

[2,1,4,4,3] A wins 2 units [2,2,4,4,3] A wins 2 units [2,3,4,4,3] A wins 2 units [2,4,4,4,3] Draw [2,5,4,4,3] Draw [2,6,4,4,3] Draw

[3,1,4,4,3] A wins 2 units [3,2,4,4,3] A wins 2 units [3,3,4,4,3] A wins 2 units [3,4,4,4,3] Draw [3,5,4,4,3] Draw [3,6,4,4,3] Draw

[4,1,4,4,3] Draw [4,2,4,4,3] Draw [4,3,4,4,3] Draw [4,4,4,4,3] D wins 2 units [4,5,4,4,3] D wins 2 units [4,6,4,4,3] D wins 2 units

[5,1,4,4,3] Draw [5,2,4,4,3] Draw [5,3,4,4,3] Draw [5,4,4,4,3] D wins 2 units [5,5,4,4,3] D wins 2 units [5,6,4,4,3] D wins 2 units

[6,1,4,4,3] Draw [6,2,4,4,3] Draw [6,3,4,4,3] Draw [6,4,4,4,3] D wins 2 units [6,5,4,4,3] D wins 2 units [6,6,4,4,3] D wins 2 units

10

A wins two units in 9 instances, 3 x 3 + 3 x 3 draws = 18 draws, D wins 9 times,

[1,1,4,4,4] A wins 2 units [1,2,4,4,4] A wins 2 units [1,3,4,4,4] A wins 2 units [1,4,4,4,4] Draw [1,5,4,4,4] Draw [1,6,4,4,4] Draw

[2,1,4,4,4] A wins 2 units [2,2,4,4,4] A wins 2 units [2,3,4,4,4] A wins 2 units [2,4,4,4,4] Draw [2,5,4,4,4] Draw [2,6,4,4,4] Draw

[3,1,4,4,4] A wins 2 units [3,2,4,4,4] A wins 2 units [3,3,4,4,4] A wins 2 units [3,4,4,4,4] Draw [3,5,4,4,4] Draw [3,6,4,4,4] Draw

[4,1,4,4,4] Draw [4,2,4,4,4] Draw [4,3,4,4,4] Draw [4,4,4,4,4] D wins 2 units [4,5,4,4,4] D wins 2 units [4,6,4,4,4] D wins 2 units

[5,1,4,4,4] Draw [5,2,4,4,4] Draw [5,3,4,4,4] Draw [5,4,4,4,4] D wins 2 units [5,5,4,4,4] D wins 2 units [5,6,4,4,4] D wins 2 units

[6,1,4,4,4] Draw [6,2,4,4,4] Draw [6,3,4,4,4] Draw [6,4,4,4,4] D wins 2 units [6,5,4,4,4] D wins 2 units [6,6,4,4,4] D wins 2 units

11

A wins two units in 15 instances, 3 x 2 + 1 + 2 x 3 draws = 13 draws, D wins 8 times,

[1,1,5,4,4] A wins 2 units [1,2,5,4,4] A wins 2 units [1,3,5,4,4] A wins 2 units [1,4,5,4,4] A wins 2 units [1,5,5,4,4] Draw [1,6,5,4,4] Draw

[2,1,5,4,4] A wins 2 units [2,2,5,4,4] A wins 2 units [2,3,5,4,4] A wins 2 units [2,4,5,4,4] A wins 2 units [2,5,5,4,4] Draw [2,6,5,4,4] Draw

[3,1,5,4,4] A wins 2 units [3,2,5,4,4] A wins 2 units [3,3,5,4,4] A wins 2 units [3,4,5,4,4] A wins 2 units [3,5,5,4,4] Draw [3,6,5,4,4] Draw

[4,1,5,4,4] A wins 2 units [4,2,5,4,4] A wins 2 units [4,3,5,4,4] A wins 2 units [4,4,5,4,4] Draw [4,5,5,4,4] D wins 2 units [4,6,5,4,4] D wins 2 units

[5,1,5,4,4] Draw [5,2,5,4,4] Draw [5,3,5,4,4] Draw [5,4,5,4,4] D wins 2 units [5,5,5,4,4] D wins 2 units [5,6,5,4,4] D wins 2 units

[6,1,5,4,4] Draw [6,2,5,4,4] Draw [6,3,5,4,4] Draw [6,4,5,4,4] D wins 2 units [6,5,5,4,4] D wins 2 units [6,6,5,4,4] D wins 2 units

12

A wins two units in 16 instances, 4 x 2 + 2 x 4 draws = 16 draws, D wins 4 times,

[1,1,5,5,4] A wins 2 units [1,2,5,5,4] A wins 2 units [1,3,5,5,4] A wins 2 units [1,4,5,5,4] A wins 2 units [1,5,5,5,4] Draw [1,6,5,5,4] Draw

[2,1,5,5,4] A wins 2 units [2,2,5,5,4] A wins 2 units [2,3,5,5,4] A wins 2 units [2,4,5,5,4] A wins 2 units [2,5,5,5,4] Draw [2,6,5,5,4] Draw

[3,1,5,5,4] A wins 2 units [3,2,5,5,4] A wins 2 units [3,3,5,5,4] A wins 2 units [3,4,5,5,4] A wins 2 units [3,5,5,5,4] Draw [3,6,5,5,4] Draw

[4,1,5,5,4] A wins 2 units [4,2,5,5,4] A wins 2 units [4,3,5,5,4] A wins 2 units [4,4,5,5,4] A wins 2 units [4,5,5,5,4] Draw [4,6,5,5,4] Draw

[5,1,5,5,4] Draw [5,2,5,5,4] Draw [5,3,5,5,4] Draw [5,4,5,5,4] Draw [5,5,5,5,4] D wins 2 units [5,6,5,5,4] D wins 2 units

[6,1,5,5,4] Draw [6,2,5,5,4] Draw [6,3,5,5,4] Draw [6,4,5,5,4] Draw [6,5,5,5,4] D wins 2 units [6,6,5,5,4] D wins 2 units

13

A wins two units in 16 instances, 4 x 2 + 2 x 4 draws = 16 draws, D wins 4 times,

[1,1,5,5,5] A wins 2 units [1,2,5,5,5] A wins 2 units [1,3,5,5,5] A wins 2 units [1,4,5,5,5] A wins 2 units [1,5,5,5,5] Draw [1,6,5,5,5] Draw

[2,1,5,5,5] A wins 2 units [2,2,5,5,5] A wins 2 units [2,3,5,5,5] A wins 2 units [2,4,5,5,5] A wins 2 units [2,5,5,5,5] Draw [2,6,5,5,5] Draw

[3,1,5,5,5] A wins 2 units [3,2,5,5,5] A wins 2 units [3,3,5,5,5] A wins 2 units [3,4,5,5,5] A wins 2 units [3,5,5,5,5] Draw [3,6,5,5,5] Draw

[4,1,5,5,5] A wins 2 units [4,2,5,5,5] A wins 2 units [4,3,5,5,5] A wins 2 units [4,4,5,5,5] A wins 2 units [4,5,5,5,5] Draw [4,6,5,5,5] Draw

[5,1,5,5,5] Draw [5,2,5,5,5] Draw [5,3,5,5,5] Draw [5,4,5,5,5] Draw [5,5,5,5,5] D wins 2 units [5,6,5,5,5] D wins 2 units

[6,1,5,5,5] Draw [6,2,5,5,5] Draw [6,3,5,5,5] Draw [6,4,5,5,5] Draw [6,5,5,5,5] D wins 2 units [6,6,5,5,5] D wins 2 units

14

A wins two units in 24 instances, 4 draws + 2 draws + 4 draws = 10 draws, D wins 2 times,

[1,1,6,5,5] A wins 2 units [1,2,6,5,5] A wins 2 units [1,3,6,5,5] A wins 2 units [1,4,6,5,5] A wins 2 units [1,5,6,5,5] A wins 2 units [1,6,6,5,5] Draw

[2,1,6,5,5] A wins 2 units [2,2,6,5,5] A wins 2 units [2,3,6,5,5] A wins 2 units [2,4,6,5,5] A wins 2 units [2,5,6,5,5] A wins 2 units [2,6,6,5,5] Draw

[3,1,6,5,5] A wins 2 units [3,2,6,5,5] A wins 2 units [3,3,6,5,5] A wins 2 units [3,4,6,5,5] A wins 2 units [3,5,6,5,5] A wins 2 units [3,6,6,5,5] Draw

[4,1,6,5,5] A wins 2 units [4,2,6,5,5] A wins 2 units [4,3,6,5,5] A wins 2 units [4,4,6,5,5] A wins 2 units [4,5,6,5,5] A wins 2 units [4,6,6,5,5] Draw

[5,1,6,5,5] A wins 2 units [5,2,6,5,5] A wins 2 units [5,3,6,5,5] A wins 2 units [5,4,6,5,5] A wins 2 units [5,5,6,5,5] Draw [5,6,6,5,5] Draw

[6,1,6,5,5] Draw [6,2,6,5,5] Draw [6,3,6,5,5] Draw [6,4,6,5,5] Draw [6,5,6,5,5] D wins 2 units [6,6,6,5,5] D wins 2 units

15

A wins two units in 25 instances, 5 x 1 + 5 draws = 10 draws, D wins 1 times,

[1,1,6,6,5] A wins 2 units [1,2,6,6,5] A wins 2 units [1,3,6,6,5] A wins 2 units [1,4,6,6,5] A wins 2 units [1,5,6,6,5] A wins 2 units [1,6,6,6,5] Draw

[2,1,6,6,5] A wins 2 units [2,2,6,6,5] A wins 2 units [2,3,6,6,5] A wins 2 units [2,4,6,6,5] A wins 2 units [2,5,6,6,5] A wins 2 units [2,6,6,6,5] Draw

[3,1,6,6,5] A wins 2 units [3,2,6,6,5] A wins 2 units [3,3,6,6,5] A wins 2 units [3,4,6,6,5] A wins 2 units [3,5,6,6,5] A wins 2 units [3,6,6,6,5] Draw

[4,1,6,6,5] A wins 2 units [4,2,6,6,5] A wins 2 units [4,3,6,6,5] A wins 2 units [4,4,6,6,5] A wins 2 units [4,5,6,6,5] A wins 2 units [4,6,6,6,5] Draw

[5,1,6,6,5] A wins 2 units [5,2,6,6,5] A wins 2 units [5,3,6,6,5] A wins 2 units [5,4,6,6,5] A wins 2 units [5,5,6,6,5] A wins 2 units [5,6,6,6,5] Draw

[6,1,6,6,5] Draw [6,2,6,6,5] Draw [6,3,6,6,5] Draw [6,4,6,6,5] Draw [6,5,6,6,5] Draw [6,6,6,6,5] D wins 2 units

16

A wins two units in 25 instances, 5 x 1 + 5 draws = 10 draws, D wins 1 times,

[1,1,6,6,6] A wins 2 units [1,2,6,6,6] A wins 2 units [1,3,6,6,6] A wins 2 units [1,4,6,6,6] A wins 2 units [1,5,6,6,6] A wins 2 units [1,6,6,6,6] Draw

[2,1,6,6,6] A wins 2 units [2,2,6,6,6] A wins 2 units [2,3,6,6,6] A wins 2 units [2,4,6,6,6] A wins 2 units [2,5,6,6,6] A wins 2 units [2,6,6,6,6] Draw

[3,1,6,6,6] A wins 2 units [3,2,6,6,6] A wins 2 units [3,3,6,6,6] A wins 2 units [3,4,6,6,6] A wins 2 units [3,5,6,6,6] A wins 2 units [3,6,6,6,6] Draw

[4,1,6,6,6] A wins 2 units [4,2,6,6,6] A wins 2 units [4,3,6,6,6] A wins 2 units [4,4,6,6,6] A wins 2 units [4,5,6,6,6] A wins 2 units [4,6,6,6,6] Draw

[5,1,6,6,6] A wins 2 units [5,2,6,6,6] A wins 2 units [5,3,6,6,6] A wins 2 units [5,4,6,6,6] A wins 2 units [5,5,6,6,6] A wins 2 units [5,6,6,6,6] Draw

[6,1,6,6,6] Draw [6,2,6,6,6] Draw [6,3,6,6,6] Draw [6,4,6,6,6] Draw [6,5,6,6,6] Draw [6,6,6,6,6] D wins 2 units

Summary of the Permutation Set

1

D wins 36 times

2

A wins two units in 1 instance, 1 draw, D wins 35 times,

3

A wins two units in 1 instance, 5 draws + 5 draws = 10 draws, D wins 25 times,

4

A wins two units in 1 instance, 5 draws + 5 draws = 10 draws, D wins 25 times,

5

A wins two units in 3 instances, 4 + 5 x 1 draws = 9 draws, D wins 24 times,

6

A wins two units in 4 instances, 4 x 2 + 4 x 2 draws = 16 draws, D wins 16 times,

7

A wins two units in 4 instances, 4 x 2 + 4 x 2 draws = 16 draws, D wins 16 times,

8

A wins two units in 8 instances, 3 x 2 + 1 + 3 x 2 draws = 13 draw, D wins 15 times,

9

A wins two units in 9 instances, 3 x 3 + 3 x 3 draws = 18 draws, D wins 9 times,

10

A wins two units in 9 instances, 3 x 3 + 3 x 3 draws = 18 draws, D wins 9 times,

11

A wins two units in 15 instances, 3 x 2 + 1 draw+ 3 x 2 draws = 13 draws, D wins 8 times,

12

A wins two units in 16 instances, 4 x 2 draws + 2 x 4 draws = 16 draws, D wins 4 times,

13

A wins two units in 16 instances, 4 x 2 draws + 2 x 4 draws = 16 draws, D wins 4 times,

14

A wins two units in 24 instances, 4 draws + 2 draws + 4 draws = 10 draws, D wins 2 times,

15

A wins two units in 25 instances, 5 x 1 draws + 5 draws = 10 draws, D wins 1 times,

16

A wins two units in 25 instances, 5 x 1 draws + 5 draws = 10 draws, D wins 1 times,

Conclusion

576 total number of combinations as stated earlier

Let's now total the results we found in the permutation set

A wins = 1+1+1+3+4+4+8+9+9+15+16+16+24+25+25 = 161

Draw = 1+10+10+9+16+16+13+18+18+13+16+16+10+10+10 = 222

D Wins= 36+35+25+25+24+16+16+15+9+9+8+4+4+2+1= 193

Number of Sucessful Attacks/ Total Combinations = 161/576 =0.279 = 28% of the time Attacker will win 2 units in a 3 dice on 2 scenario.

Draws/Total Combinations = 222/576 = 0.385 = 39% of the time both sides will lose 1 unit in a 3 dice on 2 scenario.

Number of Successful Defenses/Total Combinations = 193/576 = 0.335 = 34% of the time the defense takes 2 units in a 3 dice on 2 scenario.

So to answer the primary question, the Defender is 6% more likely to win 2 units rather than that for the attacker.

Here are some more analyses, they should be fairly self-explanatory:

A+Dr/T=(Attacks + Draws) / Total Combinations = (161 + 222) / 576 = 0.665

D+Dr/T=(Defences + Draw) / Total Combinations = (193 + 222) / 576 = 0.72

Advantage in Favor of the Defense = (D+Dr/T)/(A+Dr/T) = 0.72/0.665 = 1.083

D/A = 193/161= 1.198

A/Dr = 161/222 = 0.725

D/Dr = 193/222 = 0.869

A Rule for Calculating the Number of Attacking Units Required

The following explanation covers the nature of the formula I provide as a rule to use when attacking with large forces in Risk. So, 5/4 relates to D/A = 1.19 (approx. = 6/5). I chose 5/4 because it is slightly larger than 6/5. This was chosen, because it should counteract any bad luck the user of this method may experience in most cases. The exponential function provides another additional buffer to ensure a win on the attacker's side. This buffer minimizes to e when DF is equal to infinity. This component of the formula accounts for the advantage that the attacker gains when both him and the defender wage battle with forces that are very large. The 1 added to the end of the equation accounts for a single unit being left behind on the territory from which the attacker originated.

AF means number of Attacker's Forces

DF means number of Defender's Forces

AF=(5/4)(DF-2)+e^(1+(1/DF))+1 [The number of Attacking units required]

Table showing the results for the above formula:

Table that shows the number of attack units required to defeat a given number of defense units

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