Dice Odds in the Board Game Risk

An article by Plainsboro.COM owner Kennedy Lemke
March, 1999

This web page has to do with the board game Risk*, specifically, it deals with the odds of various dice combinations winning "battles". Follow this link to skip the introduction and rules of the game and go directly to the information about dice odds.

Introduction

Like lots of folks my age and younger (born in '61), I played the board game Risk growing up as a kid in the midwest. I haven't played much as an adult, maybe a couple times at friends' houses.

However, I do own a few video game systems including a Sony Playstation®. And one night not long ago I was shopping for some software and ran across the PSX version of the board game Risk. The PSX version is apparently pretty similar to the PC/Windows version, so the information in this page applies to both versions.

It's a pretty darn good game (and addicting, at least for a single human player versus multiple computer players). They have the "classic Risk" mode, a.k.a. "domination", which is like the board game except that there are 5 different and fun maps available instead of just one, plus they have an "Ultimate Risk" mode (which I haven't played yet) that is more intricate.

You can play the classic game with anywhere from 3 to 6 players, which can be any mix of human or computer players with the caveat that there must be at least one human player. I've played with other human friends, but I've definitely spent much more time playing against other computer opponents. You can set a difficulty called the "AI level" to easy, medium, or hard, but I do not really notice a difference between these difficulty levels.

In the next few paragraphs, I'm going to describe some of the rules of the game for those who don't know how to play. If you wish, you may skip right to the "why I think the game is biased" section.

Setup and Object of the Game

The game Risk is played on a board (or virtual board) that looks like a map. In the classic game, the map is a map of the entire world, and I'll use that classic map as an example during this description. The map is divided up into countries, or parts of countries. Each separate country (or part of a country) is called a territory. The classic map has 42 different territories. Each territory borders on one or more other territories, either by directly touching a neighboring territory, or by a line on the board that makes a link between two territories. Each territory is also a member of a group of territories near each other that form a continent; special awards are given to players who own all the territories that form a continent.

At the beginning of the game, each player is assigned ownership of a certain number of territories (total number of territories divided by the number of players). Each player has a certain number of game pieces also known as armies that are placed in each of the territories they own.

The game progresses by players taking turns "attacking" ("battling") other players' territories (see below). When a player wins an attack against an opposing territory, that player takes possession of that territory by moving armies into the territory. Each territory must contain at least one army. At the beginning of the game, most players will have between 1 and 5 armies in each territory they own (for example).

Battles

At the beginning of a player's turn, they are assigned some new armies, according to how many territories they own, whether the own any entire continents, and on "cards" they have acquired (see below). The player can choose to place the new armies in any territory they own, or split the armies between multiple territories they own.

After the player has placed his new armies that turn, they then can choose to do battle with an opponent's territory. In order to attack an opposing territory, a player must have at least 2 armies in the attacking territory, and the opposing territory must be connected to (i.e., a direct neighbor of) the attacking territory.

The attacker and the defender then roll a single die or multiple dice to do battle. The attacking player rolls either one, two, or three dice, depending on the number of armies they have in the attacking territory (one die with two armies, two dice with three armies, and up to three dice with four or more armies). A defender can roll either one die if they have only a single army in the defending territory, or a maximum of two dice with two or more armies.

After the dice are rolled (the dice should be different colors), the attacker's dice are compared to the defender's dice. The highest-value attacker's die is compared to the highest-value defender's die, and the second-highest attacker's die is compared to the second-highest defender's die (of course, only if both attacker and defender have more than one die). If the attacker's dice are higher in value than the defender's dice, the attacker wins the battle, and conversely if the defender's dice are greater than or equal to the value of the attacker's

For each win, one of the opposing armies are removed from the board.

Examples:

  • Attacker and defender both use one die:
    • Attacker rolls a 6, defender rolls a 5, attacker wins the battle and one of the defender's territories is removed from the board.
    • Attacker rolls a 5, defender also rolls a 5, defender wins the battle and one of the attacker's territories is removed from the board.
  • Attacker uses three dice, defender uses one die:
    • Attacker: 1-1-1, defender: 1: attacker loses one army
    • Attacker: 6-6-6, defender: 6: attacker loses one army
    • Attacker: 6-2-1, defender: 5: defender loses one army
  • Attacker uses three dice, defender uses two dice:
    • Attacker: 6-5-2, defender: 6-5: attacker loses two armies
    • Attacker: 1-1-1, defender: 1-1: attacker loses two armies
    • Attacker: 6-6-6, defender: 6-6: attacker loses two armies
    • Attacker: 2-2-1, defender: 1-1: defender loses two armies
    • Attacker: 2-2-1, defender: 2-1: attacker and defender each lose one army
    • Attacker: 5-4-3, defender: 5-3: attacker and defender each lose one army

When all of the defender's armies are removed from the board, the attacker moves one or more armies from the attacking territory into the now-empty defender's territory. When an attacker has only one army left in a territory, he can no longer mount an attack (because he would not have a spare army to move into an empty territory if he were to win the battle).

A player is out of the game when their last army is removed from the board. The game is won when there is one player remaining.

Cards

If a player is successful in defeating all the armies in at least one territory (and thus taking ownership of that territory) during their turn, they are awarded a card at the end of their turn. The deck consists of one card per territory, plus a couple "jokers" in the deck. Each card displays two symbols: a country symbol matching one of the countries on the board, and a character (either a cannon, a horse, or a soldier).

As a player accumulates cards, cards can be mixed and matched together to form "sets". Valid sets can be exchanged for certain numbers of armies at the beginning of a player's turn. A set is a group of three cards, and consists of three matching characters (like three cannons), or one of each character, or a combination that includes a joker. If a player trades in a set and also owns one or more of the countries displayed on the cards in the set he trades in, he receives an additional 2 armies that are placed on that territory only.

You can trade in a set of three cards at the beginning of your turn if you have exactly three cards that make a valid set, but you MUST trade in a set of cards when the total number of your cards reaches 5. This prevents players from hoarding sets and trading in multiple sets at the same time.

When a player takes another player's last army from the board, that player receives all of the defeated players' cards and can (or must) trade in valid sets of cards at that time.

There are different rules for how many armies a player receives when trading in a set of three cards. In the classic board game, sets are considered "increasing", meaning that as the game progresses, each set traded in is exchanged for more and more armies. For the computerized version, I prefer the "fixed set" option, for which different numbers of armies are received depending on the configuration of the cards (up to a maximum of twelve armies).

Luck

Luck plays a factor in this game in several ways:

  • Luck of the throw (dice): first, and most obviously, the roll of the dice on each turn affects the outcome of the game. If a player consistently rolls high numbers, they will do better in the long run than if they roll low numbers.

  • Luck of the draw (turn order): the order in which the players act can affect the outcome of the game. This is more obvious with a larger number of players, but suffice it to say that if a player can establish their strategy early, before other players have acted, they may have a better chance of winning the game.

  • Luck of the draw (territory ownership): At the beginning of the game, players can agree to pick territories themselves in turn, or to be assigned territories randomly (by dealing out all the cards). Random assignment is my preference, but of course some territories and continents are intrisically more strategic in the course of the game and thus you can gain an advantage by being assigned these strategic territories.

  • Luck of the draw (number of armies placed): At the beginning of the game, players can also agree to place armies in their assigned territories by hand or randomly. The number of armies placed in each territory at the beginning of the game can also affect the outcome of the game so luck is a factor here as well.

  • Luck of the draw (cards): Drawing cards is (in theory) random, just like throwing dice. However, the cards you draw affect how many armies you can get in exchange; therefore there is an element of luck involved in drawing cards.

Strategy

Numerous opinions exist concerning which of several different strategies is a better strategy for winning. I'm going to mention a few possible strategies here and discuss the positives and negatives of each strategy:

  • Attack early and often strategy: One possible strategy, especially early in a game and on a game map with lots of territories, is to attack always or almost always when possible, in order to own as many territories as possible, as quickly as possible. This strategy is good because the more territories you own, the more armies you receive at the beginning of your turn. And (therefore) if your opponents own less territories, they will receive fewer armies at the beginning of their turn. Also, the more territories you attack and conquer, the less that are left that could possibly attack you back! A negative of this strategy is that you will be spreading your armies and territories thinly. That is, the more territories you own, the less armies you will have for defense in each of those territories.

  • Conservative approach strategy: Another possibility is to approach the game more conservatively. Instead of attacking wherever possible, a player may wish to attack only one or a few territories during their turn, preferring to build a cache of armies in particular strategic territories as the game progresses. The advantages to this strategy are: with more armies per territory, the territory is more difficult to conquer. And, if you build up armies in very strategic territories (for example on the borders of continents), you can therefore protect these areas better.

  • Mixed strategy: A third possibility (and one that I personally prefer) is a mix of the above two strategies. A player may wish to build up caches of armies in a few particularly strategic territories, while attacking as much as possible in other territories that the player deems as less strategic. Diversifying your strategy in this fashion can work particularly well in the early stages of a game.

  • Get continents at all costs quickly strategy: Different continents on a particular map each have their own characteristics. Smaller continents contain fewer territories and are therefore generally easier to obtain and defend. Of course, such continents are highly desirable and therefore several players may be attempting to obtain the same continent you are pursuing at the same time.

  • Attacking opponents' continents strategy: If your opponent owns an entire continent, they will benefit by receiving additional armies at the beginning of their turn. They will lose this benefit if any one of their opponents owns even a single one of the territories in that continent. Therefore, if possible, try to attack at least a single territory in continents that are entirely owned by an opponent. One variation of this strategy that the computer players use often (and I have found to be successful) is not only to just attack a territory in such a continent, but if possible, leave a large number of armies in that territory to make it very difficult for your opponent to regain the continent with a counter-attack.

  • General strategies and concepts: There are some general strategies to keep in mind as the game progresses. These include:

    • Try to get your opponents to attack each other instead of attacking you.

    • Keep some armies "in reserve". If an opponent breaks through one of your critical strategic territories (for example, you are defending a continent), you will want to have some armies in a nearby territory to mount a counter-attack when it is your turn. You will be unable to do this if you keep only one army in each non-critical territory.

    • If you are using the "attack early and often" strategy, you might wish to obtain territories that are less susceptible to re-conquering, such as those that have few neighbors, or that do not directly border other continents.

    • Attack when you have a greater advantage, and be less inclined to attack when you are an underdog. For example, if you can use three dice to attack against a single defender, you are about a 2-1 favorite to win the battle. But if you are using a single die to attack against two defending dice, your odds fall to a dismal 1 chance in 4 to win the first battle. See more on dice odds below.

OK. Now, on to the information this article is really about, dealing with dice odds, and demonstrating that the PC/Playstation version of Risk does NOT use random dice!


Dice Battle Odds (or: Why I think the computerized game is biased)

While playing Risk, have you ever wondered what your chance of winning a battle is, given the various dice combinations (and assuming fair dice)? Have you ever wondered, all things being equal, whether or not it is statistically correct for you to attack a particular country?

Well, in the course of re-learning this game as an adult on the Playstation, I started noticing that defenders seemed to roll 6's pretty often, much more often than the odds dictate.

Noticing this caused me to do two things: first, I wrote to the Hasbro company via email to ask them why they didn't write the game to produce random dice results. Second, I wrote some code to figure out the precise odds of winning or losing battles (given fair dice). Here are the results of both of these actions:

What I asked and what Hasbro said

In my email to Hasbro, I basically asked simply why they didn't make the dice random (they're not, at least in a game with a single human player versus multiple computer players).

I have not actually sat down with a VCR or a pad and pen while playing the PSX Risk game to keep track of each roll of the dice, but anyone who has played this game much at all will notice frequent anomalies. The best example of this, which I mentioned in my email to Hasbro, is an attacker attacking a territory with a single army. Frequently, the defender will "happen'' to roll several 6's in a row, and then will "happen" to roll a 1. I've seen this way too often for this to be a random occurrence. (By the way, it doesn't really matter if the defender is the computer or human player, whoever the defender is has a chance of rolling several 6's in a row).

You might be thinking that even if the dice are not random, as long as they are equally not random to each player that it will not affect the game. I disagree. The point is, the dice should have been programmed to be random, and they were not.

The response I received basically simply stated that the dice are random.

Follow this link to see the complete text of the email I sent Hasbro, then follow this link to see Hasbro's response. I know the full name of the customer support person who responded to my email, but I've x'ed it out here to protect his privacy.

I suspect that the person from Hasbro who responded to my email was a customer support person, and likely not one of the engineers who was actually involved in writing the program. So I don't hold a grudge against him in particular, but I sure wish that the programmers of this game would see this article, my email inquiry, and respond as to why the dice are indeed loaded.

Update: December, 2000: someone sent me an email recently noting that it is possible that the folks who programmed the PSX version of Risk may have indeed used a standard random number generator in the game, therefore hoping and assuming that fair dice would result. But it apparently may be possible that RNG code running on certain hardware could produce somewhat non-random results.

So it could be the case both that I'm right in that the actual gameplay does not produce fair dice but that Hasbro is right in that the game may indeed have RNG code in it.


Code and results

Finally, here are the results of the program I wrote to calculate the odds of various dice combinations in attacking and defending territories in Risk. I'm including the full output of the program on this page since it's short, but if you're interested you can also view the code.

Update: December 2000: someone sent me an email recently suggesting that the output of my program below does not give the actual precise odds of various dice rolls. However, I believe this person probably did not examine my code, and was likely assuming that I was just doing some very large number of random dice rolls and reporting the result.

In fact, the program I wrote does NOT do random dice rolls and calculate the result. Instead, the program finds the TRUE odds by doing a brute-force calculation. In other words, it calculates the results of all possible outcomes of the dice (there are a finite number of dice combinations and outcomes), then reports the precise odds of either the attacker or defender winning or losing.

Odds of winning various dice combinations in Risk

Attacker: one die; Defender: one die:

	Attacker wins 15 out of 36 (41.67 %)
	Defender wins 21 out of 36 (58.33 %)


Attacker: two dice; Defender: one die:

	Attacker wins 125 out of 216 (57.87 %)
	Defender wins 91 out of 216 (42.13 %)


Attacker: three dice; Defender: one die:

	Attacker wins 855 out of 1296 (65.97 %)
	Defender wins 441 out of 1296 (34.03 %)


Attacker: one die; Defender: two dice:

	Attacker wins 55 out of 216 (25.46 %)
	Defender wins 161 out of 216 (74.54 %)


Attacker: two dice; Defender: two dice:

	Attacker wins both: 295 out of 1296 (22.76 %)
	Defender wins both: 581 out of 1296 (44.83 %)
	Both win one: 420 out of 1296 (32.41 %)


Attacker: three dice; Defender: two dice:

	Attacker wins both: 2890 out of 7776 (37.17 %)
	Defender wins both: 2275 out of 7776 (29.26 %)
	Both win one: 2611 out of 7776 (33.58 %)

Sample interpretation of the last data above (three vs. two). If an attacker starts with 1000 armies and a defender starts with 1000 armies and a 3 vs. 2 attack is ensued, the results should be (given fair dice): after 100 rolls, each side will have lost 1 army about 34 times. The defender will have lost 2 armies about 37 times, and the attacker will have lost 2 armies 29 times. Therefore, after 100 rolls, the attacker should have 908 armies left, and the defender should have 892 armies left.

Conclusion: heads up with three dice versus 2 dice, the attacker has an advantage in the long run. Similar interpretations can be made for the remainder of the data, which can be summarized as follows:

  • Attacker 1 versus defender 1: defender has the advantage, winning about 4 out of 7 battles
  • Attacker 2 versus defender 1: attacker has the advantage, winning about 4 out of 7 battles
  • Attacker 3 versus defender 1: attacker has the advantage, winning about 2 out of 3 battles
  • Attacker 1 versus defender 2: defender has the advantage, winning about 3 out of 4 battles
  • Attacker 2 versus defender 2: defender has the advantage, winning about 3 out of 5 battles
  • Attacker 3 versus defender 2: attacker has the advantage, but the advantage is much more narrow than any of the battles described above. The attacker's advantage is such that he will win about 7 out of 13 battles on average.

*Note: the name Risk is apparently not a registered trademark of the Hasbro® company. At least, I checked their web page and it was not listed as a registered trademark there.