On matters of world conquest.......

Factual Questions
1

This question has bugged me for many years:
While playing the popular word conquest game RISK, What has a statistical advantage: attacking or defending?

For those of you not familiar with the game, the way one attacks or defends is as follows:

The outcome of a battle is determined by the role of the dice.
The person attacking roles one less dice than the number of armies attacking, with a maximum of three dice.
The defendant roles a maximum of two dice and only one dice if there is only one defending army.
Considering that the attacker is rolling three dice and the defender two, the best two dice of the attacker are matched up against the two defending dice, the best of each matched up against the second best of each. The draw goes to the defending side.

for example: If the attacker roles 3-5-6 and the defender roles 3-5, the defender looses both dice because the attacker’s 6 beats the defender’s 5 and his 3 looses to the attacker’s 5. In this scenario, the defender would loose two armies. If the defender would has rolled a 5-6, the attacker looses two armies

The attacker has the advantage of choosing the best two out of three dice while the defender has the advantage of wining the ties, the question is: Does either side have a statistical advantage in a prolonged battle?

2

I once did a computer simulation and decided that it was very close, I think with a slight bias towards the attacker, but can’t find my programme to double check. But to a first approximation about the same

3

This thread looks at the questin. The article linked to there concludes the odds favour the attacker.

It kind of makes sense, as otherwise the game would be even more likely to end in prolonged stalemates.

4

quickly rewrote it. Out of 10,000 throws 10450 wins to the attacker, 9550 to the defender. About a 2% bias to the attacker. Not very significant.

5

talk about spooky darlings. I got the latest issue of new scientist and there was an article in it talking about the issue. It is not in the online version, but see http://www.sciencenews.org/20030712/mathtrek.asp for a discussion.
see also http://db.cs.helsinki.fi/t/ipuustin/webrisk/webrisk.jsp for a risk probability calculator

6

In the long run, the attacker has about 5-6 advantage, according to one online strategy guide that I read. That probability calculator has them approximately evenly matched at 85 vs. 99, which almost agrees. More like 6-7.

The best strategy for a defender is to leave 2 armies on a territory. This gives you the best odds per army. Territories with 1 army each have the worst odds per army. According to this guide, if you want to wipe another player out, you can expect to need the following number of armies:

(the number of armies they have × 5/6)

  • (the number of territories they have)
  • (the number of territories with only one army × 1/9)
  • (the number of territories with only two armies × 1/18)