# Risk

PBE6
Posers and Puzzles 29 Apr '08 16:47
1. PBE6
Bananarama
29 Apr '08 16:472 edits
In the game of RISK, players try to capture territory from other players by rolling dice. The attacker may roll between 1 and 3 dice, and the defender may roll either 1 or 2 dice, depending on the following rules:

ATTACKER

1. If the attacker has 2 armies, they can only roll 1 die.
2. If the attacker has 3 armies, they may roll either 1 or 2 dice.
3. If the attacher has 4 or more armies, they may roll 1, 2 or 3 dice.

DEFENDER

1. If the defender has 1 army, they can only roll 1 die.
2. If the defender has 2 or more armies, they may roll 1 or 2 dice.

Scoring is determined by comparing the attacker's highest roll to the defender's highest roll, and the attacker's second highest roll to the defender's second highest roll. Whoever rolls higher wins, with ties going to the defender. For example:

ATTACKER - 6, 3, 1
DEFENDER - 5, 3

RESULT - 6/5 attacker wins, 3/3 tie and the defender wins. Both players lose 1 army.

Q1: What is the expected number of armies lost on either side if:

(a) the attacker rolls 1 die and the defender rolls 1 die?
(b) the attacker rolls 1 die and the defender rolls 2 dice?
(c) the attacker rolls 2 dice and the defender rolls 1 die?
(d) the attacker rolls 2 dice and the defender rolls 2 dice?
(e) the attacker rolls 3 dice and the defender rolls 1 die?
(f) the attacker rolls 3 dice and the defender rolls 2 dice?

Q2: Rank the above strategies from most favourable for the attacker to least favourable for the attacker.

Q3: Assuming both players play their optimal number of dice each turn, what is the probability that:

(a) an attacker with 4 armies will conquer a defender with 2 armies?
(b) an attacker with N armies will conquer a defender with M armies?

Q4 - BONUS QUESTION! What is the expected number of armies left on either side if the attacker has N armies and the defender has M armies to start with?
2. 29 Apr '08 21:511 edit
Optimal strategy is to roll as many dice as possible as attacker or defender.

You're potentially risking more loss, but you're also improving your odds, particularly if you're adding a dice not matched by the opponent.

Will work out the numbers here in a bit.

Predicting the loss with a particular set of numbers gets tricky whenever one side or the other gets to the point they have to reduce the dice rolled, as then odds shift a bit, but perhaps the simplified version will provide numbers close enough to make good decisions by.

I will assume for this exercise, "attackers" are those armies the offensive side could move into the territory if it conquers it, and that the 1 extra "attacking" army is, in fact, homeland defense and not involved with the hostile takeover of the poor, innocent, peace-loving neighbors whom you happen to be attacking.

(In other words, the attacking country has to keep at least one army behind to keep the peace, and hence you cannot roll for that army, hence the one less..).
3. 29 Apr '08 23:422 edits
My analysis, excluding the "attacking" armies who, in reality, have to stay home to keep the peasants in line (and therefore not available to attack the enemy army, or to enslave the other country's peasants after the invasion succeeds).

1 Attacker vs 1 Defender
Attacker kills 1 Defender - 5/12.
Defender kills one Attacker - 7/12.

Avg Deaths - 0.583 Attacker, 0.417 Defender

1 Attacker vs 2+ Defenders
Attacker kills 1 Defender - 55/216.
Defender kills 1 Attacker - 161/216.

Avg Deaths - 0.745 Attacker, 0.255 Defender

2 Attackers vs 1 Defender
Attacker kills 1 Defender - 125/216.
Defender kills 1 Attacker - 91/216.

Avg Deaths - 0.421 Attacker, 0.579 Defender

3+ Attackers vs 1 Defender
Attacker kills 1 Defender - 95/144.
Defender kills 1 Attacker - 49/144.

Avg Deaths - 0.340 Attacker, 0.680 Defender

Now here it gets a little trickier, as we have two armies at stake..

2 Attackers vs 2+ Defenders
Attacker kills 2 Defenders - 295/1296.
One of each killed - 420/1296.
Defender kills 2 Attackers - 581/1296.

Avg Deaths - 1.221 Attacker, 0.779 Defender

3+ Attackers vs 2+ Defenders
Attacker kills 2 Defenders - 482/1296.
One of each killed - 435/1296.
Defender kills 2 Attackers - 379/1296.

Avg Deaths - 0.921 Attacker, 1.079 Defender

From highest kill ratio to lowest..

1.939: 3 or more attacking 1.
1.374: 2 attacking 1.
1.172: 3 or more attacking 2 or more.
0.714: 1 attacking 1.
0.638: 2 attacking 2 or more.
0.342: 1 attacking 2 or more.
4. PBE6
Bananarama
30 Apr '08 17:41
Originally posted by geepamoogle
My analysis, excluding the "attacking" armies who, in reality, have to stay home to keep the peasants in line (and therefore not available to attack the enemy army, or to enslave the other country's peasants after the invasion succeeds).

1 Attacker vs 1 Defender
Attacker kills 1 Defender - 5/12.
Defender kills one Attacker - 7/12.

Avg Deaths ...[text shortened]... or more.
0.714: 1 attacking 1.
0.638: 2 attacking 2 or more.
0.342: 1 attacking 2 or more.
Nice job!
5. 30 Apr '08 17:53
I'm not sure, but don't you forget now that the defender has preknowledge, i.e. if the attacker throws three times a six, it's not so smart to defend with two armies. Something to think about further...
6. forkedknight
Defend the Universe
30 Apr '08 21:071 edit
Originally posted by heinzkat
I'm not sure, but don't you forget now that the defender has preknowledge, i.e. if the attacker throws three times a six, it's not so smart to defend with two armies. Something to think about further...
The attacker and defender must roll at the same time by the rules I play. I'm pretty sure it's that way in the rule book as well.

*Edit* Verified...I checked the rule book.
As a side note, it also says "Hint: the more dice you roll, the better your odds of winning"
7. 30 Apr '08 21:41
I assume there exists a multitude of versions. The one I have always played so far, is quite old (and a Dutch version of Risk too), in which the attacker rolls first, after which the defender can decide whether he defends with two or one dice.
So I suppose other/newer versions (with updated rules) have been around for a long time, so never mind indeed. When I play it, it's always a matter of luck anyway. I can't remember ever having actually won.
(this doesn't make sense, but I can't make more sense of it, so I'll post it anyway)
8. forkedknight
Defend the Universe
30 Apr '08 22:171 edit
Originally posted by heinzkat
I assume there exists a multitude of versions. The one I have always played so far, is quite old (and a Dutch version of Risk too), in which the attacker rolls first, after which the defender can decide whether he defends with two or one dice.
So I suppose other/newer versions (with updated rules) have been around for a long time, so never mind indeed. Wh won.
(this doesn't make sense, but I can't make more sense of it, so I'll post it anyway)
I checked the 1959 and 1999 rulebooks from the Hasbro.com website. No doubt there are other versions and makers of similar games, since its such a classic.
9. 30 Apr '08 22:541 edit
Simultaneous rolls best mimic real battles, where you don't have the luxury as defender to see how effective the enemy attack is before defending against it, regardless of the dice setup.

--------------------------------------
| A attackers versus D defenders |
--------------------------------------

As for expected losses, with large battles, it would depend on whether 1.172 * Attackers is greater or less than the number of defenders.

If the attackers are favored, they can expect to have A - D/1.172 troops to move into the new territory.

If the defenders are favored, they can expect to have D - 1.172*A survivors for the next go round.

If the calculations show it to be very close, whoever wins will tend to have relatively few armies left, although whoever loses the ability to throw full dice first is likely to lose without doing much more damage.

Of course, all this is only approximate odds, as there is a shift in effectiveness as one side gets very close to defeat.
10. 01 May '08 07:561 edit
for Q3 should we play under the assumption that after turn #1, the defending team attacks in return? or is it multiple instances of the attackers pummeling the defenders? i.e. 4v2 may become 4v1, in which case it's 3 dice to 1 defense die? or is it 4v2 becomes 1v4 which is 1 attack to 2 defense? not sure if that's at all clear, it's late here ðŸ™‚

*edit* i am of course not saying this is what would happen, only describing a particular instance in which the defense survives the initial onslaught... what happens on the next turn? or does this qualify as a "A does not conquer D" case?
11. 01 May '08 11:49
In Risk, it's an invasion. The attackers/invaders remain the invaders while the defenders get the tie bonus for knowing the terrain better, and having the invaders come to them.

There is no role reversal unless the battle is finished the and the defenders decide to attack the attacking side back on their turn.
12. 01 May '08 12:51
Originally posted by geepamoogle
In Risk, it's an invasion. The attackers/invaders remain the invaders while the defenders get the tie bonus for knowing the terrain better, and having the invaders come to them.

There is no role reversal unless the battle is finished the and the defenders decide to attack the attacking side back on their turn.
And the preknowledge?
13. 02 May '08 01:001 edit
Originally posted by Thomaster
And the preknowledge?
Pre-knowledge would further slant the battle towards defender, based on the roll of the second highest dice.

Pre-knowledge isn't the standard way of doing it as far as I know, but we can look at how it might affect things.

(All the one defender numbers would remain the same, as it is assumed that 2 is impossible for those cases. Cases with one only one Attacker dice would be similarly unaffected since rolling twice would improve defender odds without increasing the potential loss at all.

However, the kill ratios for 2+ Attackers, 2+ Defenders would go down, as defenders would be capable of intelligently cutting losses by intentionally going with one die, based on knowing the roll of the attacker's dice.)
14. 02 May '08 01:38
And now for some numbers for a sample battle.

I would give exact numbers here, but the denominator is 7 seven digits long, so I won't.

Here is the expected breakdown for 3 invaders attacking a 2 defender country, assuming no pre-knowledge and optimal rolling. (It is assumed with the original set of questions that the fourth attacker has to remain behind, and cannot participate.)

37.19% - 3 Attackers move in.
19.42% - 2 Attackers move in.
8.99% - A lone Attacker moves in.
12.59% - A lone defender survives.
21.80% - Both defenders survive.

Reorganizing these numbers in a more meaningful way, this gives the following information on expected results.

Attacker wins 65.61% of the time with an average of 2.430 men left.
Defender wins the remaining 34.39% of the time with 1.634 men left.

The most important and interesting thing to note with these numbers isn't the heavy attacker advantage, but rather that the results tend to be lop-sided in either direction.

In other words, if one side wins big with the first toss of the dice, they will tend to have minimal losses. An initial "tie" roll (one man loss for both sides) favors the attackers.

The odds for results for the first roll are close to even for all 3 possibilities, but do favor the attacker by a small margin.