Nature of Dice Luck and Streaks

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The Nature of Dice, Luck, and Streaks

How the Dice Works

At Major Command Risk game, in each assault, the attacker rolls up to three dice and the defender rolls up to two dice. Each die rolls random number 1-6. The highest two attacker rolls are compared to the defender’s rolls; if you roll a higher number than your opponent, you destroy one defending troop, but if you roll less than or equal to the defender, then the defender destroys one of your troops. Although the defender wins each tie, statistically the attacker has a slight advantage because of rolling three dice instead of two.

Mathematically, The Major Command Risk rolls are determined by a variation of the Mersenne twister, specifically, MT19937.

Wikipedia's article on Mersenne Twister

Controversy over dice

Any complaints that the dice are out of whack, unfair, too streaky, or disadvantage are really just constructs we make up in our head due to our pattern seeking brains unable to comprehend randomness.

There are a few major issues.

1. People compare Major Command Risk or online dice to real life. While when play Risk with your buddies, losing 6 times against a 1 will seem like a spectacular defeat. However playing Major Command Risk is a larger sample size. Multiple games allow for many more rolls. You have played hundreds of games in a year. That's certainly more than you could handle of table top Risk. So we are unaware of how much more we are actually rolling the dice.

2. The dice cannot be a disadvantage or unlucky. If you lose 6 v 1, then your opponent has won a 1 v 6. For every hot streak you have, your opponent(s) are on a cold streak, they just are not aware of it because its not synchronous.

3. People remember big losses more than big wins. Big wins feel deserved, losses feel unfair. This leads to thinking that over time, the dice are frought with lots of big losses, thus being unfair.

4. If you lose the game, it's not because of the dice. Faulty dice or bad luck is an arrow in all of our quivers that we must shoot from time to time. Resting the balance of the battle on a single arrow is a good way ensure defeat.

How to test if you hold some irrational but totally human assumptions about the dice? Have you ever been a situation where you are attacking 6v1 or 7v2, and you get some bad rolls and end up 2v1 or 3v2. Do you feel like your luck should turn since you have had some bad dice? Do you attack with a 2v1 or 3v2 despite it not being a critical move? If you have, then you are projecting your own human weakness onto an inert mathematical concept. It's never a good idea to attack 3v2 unless it's mission critical.

Common Misconceptions about Randomness

You’ve amassed an unstoppable juggernaut. Your 15 troops go up against 2 defending armies and…

Ice, ice, ice, ice. You’ve just lost 8 troops, and now you’re 7 vs. 2. Do you continue your assault or retreat like a coward? “Well,” you think, “I just lost a bunch, so mathematically I’m more likely to win if I keep attacking”.

But alas, those 2 troops seem invincible, and your once formidable army is decimated. “What’s wrong with Major Command?! This isn’t possible! This isn’t random!” Does this sound familiar? You, comrade, have just been bitten by the so-called “Law of Averages”.

Each roll of the dice is a random event, but even more important is that each roll is an independent random event. In a random process like the die roll, any string of results is not any more likely to be evened out by a string of opposite results in the future. Each dice roll is statistically independent, which means that the occurrence of one result makes it neither more nor less probable that another result will occur next time. For example, rolling a 6 (the best number you can roll) never ever influences the next roll. The first roll is independent of the second roll, which is independent of the third, into infinity and beyond.

Speaking of infinity, the only time that the Law of Averages would work is if you had troop numbers that approached infinity – for example, a gajillion - then you could expect the random results to average out over time. But when dealing with troop numbers you see at Major Command, the results are always random and unpredictable. Although we've seen some drawn out escalate games here on MC, we still haven't had any troop numbers approach a gajillion yet. You can use the dice probabilities below to guesstimate how many troops you need to win a battle or eliminate an opponent, but you can never be entirely sure when it comes to the dice.

Dice Probabilities

When rolling the dice in the classic board game Risk, or when clicking on the Attack or Blitz button in MajorCommand, the defending player always wins when there is a tie. This gives the defender the advantage when the two players roll the same number of dice. However, the attacker's ability to use more dice during an attack offsets this advantage.

See Wikipedia's article on Dice Probabilities for more information.

The table below states the probabilities of all possible outcomes of one attacker dice roll and one defender dice roll.  Green  indicates an advantage to the attacker and  red italic  an advantage to the defender.

Outcome probabilities of dice rolls (various numbers of dice)
Defender Attacker
one die two dice three dice
Defender loses one 42% 58% 66%
Attacker loses one 58% 42% 34%
Defender loses one 25% N/A N/A
Attacker loses one 75% N/A N/A
Defender loses two N/A 23% 37%
Attacker loses two N/A 45% 29%
Each loses one N/A 32% 34%

The attacker has a slight advantage when rolling:

  • 4 or more Troops vs 2 or more Troops (Three dice against Two dice)
  • 4 or more Troops vs 1 Troop (Three dice against One die)
  • 3 or more Troops vs 1 Troop (Two dice against One die)

Otherwise, the defender has an advantage when the attacker rolls:

  • 3 Troops vs 2 Troops (Two dice against Two dice)
  • 2 Troops vs 2 Troops (One dice against Two dice)
  • 2 Troops vs 1 Troops (One dice against One dice)

When large armies attack, a player will tend to gain a greater advantage over his opponent by attacking rather than defending.

The following table shows the probabilities that the attacker wins a whole battle between two countries or territories (a sequence of dice rolls).  Green  indicates an advantage to the attacker (i.e. that the probability to win is larger than 50%), and  red italic  an advantage to the defender.

NOTE: The number of attacking armies does not include the minimum one army that must be left behind in the territory (e.g. if the attacking territory has 10 armies total, it has maximum 9 attacking armies).

Probabilities of attacker winning a whole battle
Number of attacking armies
1 2 3 4 5 6 7 8 9 10
Number of
defending armies
1 42% 75% 92% 97% 99% >99% >99% >99% >99% >99%
2 11% 36% 66% 79% 89% 93% 97% 98% 99% 99%
3 3% 21% 47% 64% 77% 86% 91% 95% 97% 98%
4 1% 9% 31% 48% 64% 74% 83% 89% 93% 95%
5 <1% 5% 21% 36% 51% 64% 74% 82% 87% 92%
6 <1% 2% 13% 25% 40% 52% 64% 73% 81% 86%
7 <1% 1% 8% 18% 30% 42% 54% 64% 73% 80%
8 <1% <1% 5% 12% 22% 33% 45% 55% 65% 72%
9 <1% <1% 3% 9% 16% 26% 36% 46% 56% 65%
10 <1% <1% 2% 6% 12% 19% 29% 38% 48% 57%

Another common situation when playing Risk online is when an attacker wants to take over a whole region of countries or territories during the same round, by a series of battles. After each successful battle, the attacker leaves one army in that territory and continues with the remaining attacking armies into next region. The following table shows the average number of countries or territories that an attacker can take over, as well as the 90 percentile of confidence, starting with a certain number of attacking armies in the first battle. A fixed number of armies is assumed to defend each country.

One defending army
in each country
Two defending armies
in each country
Number of attacking armies in the first battle: Number of regions that can be defeated: Number of regions that can be defeated with 90% confidence: Number of regions that can be defeated: Number of regions that can be defeated with 90% confidence:
1 0.42 0 0.11 0
2 1 0 0.39 0
3 1.7 1 0.82 0
4 2.3 1 1.2 0
5 3 2 1.6 0
6 3.6 2 2 1
7 4.3 3 2.4 1
8 5 3 2.8 1
9 5.6 4 3.2 2
10 6.3 4 3.6 2
11 6.9 5 3.9 2
12 7.6 5 4.3 2
13 8.3 6 4.7 3
14 8.9 7 5.1 3
15 9.6 7 5.5 3
16 10.2 8 5.9 4
17 10.9 8 6.3 4
18 11.5 9 6.7 4
19 12.2 10 7.1 5
20 12.9 10 7.5 5

So, for example: If you want to conquer 6 territories in a row, with 1 troop defending each, and you want to do it with a 90% confidence of success, then you should start with 13 attacking armies (14 troops in total).

Streaks and Luck

Because each dice roll is a random event, sometimes you will have a hot streak and roll amazing dice over and over again, decimating your opponent. On the other hand, you will probably also have cold streaks, ruining your immediate plans for world domination. Because of the way the human brain works, you are much more likely to remember the cold streaks than the hot ones. This is analogous to being stuck in traffic - you notice when cars in the other lane are passing you, but you barely notice when your lane passing other cars. Likewise, in Major Command Risk you remember when you get stomped, but tend to forget successfully stomping others. The randomness of the dice will cause streaks to happen. Because we see the streak in the context of a battle or assault, we tend to connect all those independent results in our mind. To the dice, however, every roll is a completely separate event. There will be streaks, but you will never be able to predict when a streak will happen, how long the streak will last, or when the streak will end.

I just lost 10 attacking 1!?!?!?

Sometimes when playing Risk online a bad run of luck can seem totally absurd or unfair. You have 10 to defeat 1, and you lose 9 straight. In that situation, you have a 99.987% chance of victory. It feels so certain, but yet you pull that .00013% chance and come up with the loss. How is that possible? It's simple probability. Let's say you have been playing Risk for 2 months, and have played 20 games. In each game, you roll the dice about 100 times in total. We are talking 2,000 results at this point. For something that has a .00013% chance of occurring to be likely, you need to make about 7,692 attempts. If you have made 2,000 attempts then there is a 38% chance the event has occurred. The event being 10 losing to a 1. So, in the end, its just a matter of time and rolls before those crazy rolls show up. The one in a million outcome is likely to have happened after a million rolls... :-)

The good news is, if you just experienced an incredible streak of luck or misfortune, then head straight over to the Major Command Risk Hall of Records. You may be immortalized for your pain or gain!