: RISK : Dice-throwing solution

: Generally, the best strategy is to always throw 2 dice, if you can, as a defender. Let's look at the way the dice can fall and the possible consequences of the throws:

: First we consider the odds of the defender losing K armies when the defender throws a single die. We may consider this problem in one of three ways. The first is to treat the dice values as random variables; the second is to count the relevant dice value combinations (note that this approach and the random variables approach are equivalent under the frequentist interpretation of probability); while the third is to run a numer simulation of the dice throwing and calculate the observed values. Let's look at all three.

: Let us start with the probability that the defender will lose some number of units by rolling a single die, based on how many dice the attacker roles:

: Recognizing that under the given RISK rules, the probability that a single defensive die beats a single offensive die is given by

: We can generate the first table of probabilities using the above formula. Counting the possible combinations of dice values yields the second table, while a numeric simulation yields the third.

: It is slightly more complicated to repeat this process for the case where the defender rolls 2 dice against the attacker.

: Applying a little combinatorics yields us the following equation for the A_1 case:

: It becomes tedious to derive the equations for the other cases, but diligence produces the following tables (Unfortunately, it's too complicated to describe the probabilistic derivations as they involve a long string of unions and intersections; thus they are omitted in favor of the combinatoric calculation):

: Comparing the numbers from the last table for both 1 and 2 dice, one can clearly see it is slightly more advantageous (mean improvement is 4.6%) to cast 2 dice instead of 1 when given the opportunity; if the attacker is foolish enough to use only 1 die, it is hugely to the defender's advantage to cast 2 dice (74% chance of winning vs. 58%). Somewhat surprisingly, it is only slightly more unlikely that the defender will lose at least one army if the defender rolls 1 die as opposed to 2 dice when the attacker is using 3. Thus, if it is clear that defeat is immenent, then one can further annoy the attacker by prolonging the battle in time by rolling a single defendering die.

I've made the numeric simulation code I wrote in Matlab publicly available:
RISK_dice.m

: © 2003, Aaron Clauset

: This and other topics of RISK have been explored by others on the web. Specifically, see the RISK FAQ for a very detailed discussion of probabilities, the expected number of armies needed to take a territory, etc.