: RISK : Dicethrowing 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.
