Improving the Odds in RISK

RISK is a classic board game of global conquest. First published in 1959, this war game remains a popular pastime–and continues to attract mathematical attention. Recent analyses reveal that the chances of winning a battle are considerably more favorable for the attacker than was originally suspected.

“The logical recommendation is . . . for the attacker to be more aggressive,” says statistician Jason A. Osborne of North Carolina State University in Raleigh, who presented his findings in the April Mathematics Magazine.

In RISK, a map of the world is divided into 42 territories. Each player has a certain number of tokens representing armies. The number of tokens a player has at the start of a turn depends on the number of territories occupied by that player’s armies. A player’s objective is to defeat an opponent’s armies to gain territory and eventually dominate the world.

At each turn, a player with armies in a given territory may attack any neighboring territory held by an opponent. However, the attacker must leave at least one army behind in the attacking territory. Successive rolls of dice decide the outcome of the battle over the invaded territory.

An attacker with three or more attacking armies rolls three dice, and one with two armies rolls two dice. A defender with two or more armies rolls two dice, and one with one army rolls one die.

At each turn, after the dice are rolled, each side’s dice are placed in descending order, and the two sets of dice are paired off. The attacker loses one army for each die that is less than or equal to the corresponding defender’s die. The defender loses one army for each die that is less than the corresponding attacker’s die. After the armies lost at that turn are removed, the dice are rolled again. This continues until one side loses all its armies or decides to withdraw.

Basic strategy guides typically focus on such issues as which continents to control (Australia is a good starting point), when to attack (attack with a large group), and how many dice to use (always roll as many as possible when defending).

Mathematicians have the analytical tools to address two important questions about individual battles: If you attack a territory with your armies, what is the probability that you will capture this territory? If you engage in a battle, how many armies should you expect to lose depending on the number of armies your opponent has on that territory?

Baris Tan of Ko University in Istanbul, Turkey, tackled these questions in the December 1997 Mathematics Magazine. He applied a technique involving so-called Markov chains to calculate the required probabilities over the course of a long game with many battles.

Based on his analysis, Tan concluded that, when both attacker and defender have the same number of armies, the probability that the attacker wins is less than 50 percent. When there are twice as many attackers as defenders, the winning probability exceeds 80 percent. Moreover, the expected loss by the attacker is slightly lower than the number of defending armies. For example, if an attacker has 20 armies and a defender has 10 armies, the attacker would win the war with a probability of 98 percent and lose about 9 armies doing so.

Keeping in mind the need to defend a newly conquered territory, Tan formulated the following rule of thumb: Based on how many defending armies you want to leave on a territory that you want to conquer, attack if you have twice as many armies on a neighboring territory and also if the number of armies your opponent has is at most half of the number of your armies.

However, Tan assumed that rolls of more than one die are independent events. Given that the dice are placed in descending order, then paired off, that assumption doesn’t actually hold. Osborne noticed this problem in Tan’s analysis, and he recalculated the probabilities of victory and the expected losses in battle.

TABLE: Probability that the attacker wins

No. of attacking armies (blue); number of defending armies (red)

1 2 3 4 5 6 7 8 9 10
1 .417 .106 .027 .007 .002 .000 .000 .000 .000 .000
2 .754 .363 .206 .091 .049 .021 .011 .005 .003 .001
3 .916 .656 .470 .315 .206 .134 .084 .054 .033 .021
4 .972 .785 .642 .477 .359 .253 .181 .123 .086 .057
5 .990 .890 .769 .638 .506 .397 .297 .224 .162 .118
6 .997 .934 .857 .745 .638 .521 .423 .329 .258 .193
7 .999 .967 .910 .834 .736 .640 .536 .446 .357 .287
8 1.000 .980 .947 .888 .818 .730 .643 .547 .464 .380
9 1.000 .990 .967 .930 .873 .808 .726 .646 .558 .480
10 1.000 .994 .981 .954 .916 .861 .800 .724 .650 .568

The new results point to a bigger advantage for an attacker over the long run than suggested by Tan’s analysis. For instance, even when the number of attacking and defending armies is equal, the probability that the attacker ends up winning the territory is actually greater than 50 percent, provided that both sides have at least five armies each. The attacker also suffers fewer losses on average than the defender.

Sharon Blatt, an undergraduate student at Elon University in North Carolina, also improved upon Tan’s analysis to obtain results similar to those produced by Osborne. Her model appears in an article published in the online Undergraduate Math Journal. Scott Bartell of the University of California, Davis has created an online calculator for obtaining battle odds for various numbers of attackers and defenders.

Of course, such results are useful only over a long sequence of many battles. Depending on the turn of the dice, you can still lose no matter how overwhelming your forces may be in any given instance. And small variants in the rules–such as whether you can withdraw from an attack if you so choose or the number of dice a defender is allowed to roll–can strongly affect the expected results.

So, know the rules, then go for it!

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