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.

(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?

(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?

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?**