Note that it is often disadvantageous to fight to the last man. The "single-roll" odds (see the -s flag below) show how an attacker is likely to fare on just one roll of, say, 3 dice vs. a defender's 2 (answer: attacker expects to lose 2387/2592 (~= .921) armies, defender expects to lose 2797/2592 (~= 1.079) armies, meaning in the limit an attacker and a defender are tied when the attacker has 2387/2797 (~= .853) times as many armies as the defender.

Usage:
# To simulate a battle to the death between forces of two given sizes:
$ ./risk

# To see "steady-state" single-roll outcome probabilities:
$ ./risk -s

Examples:

# Probable single-roll outcomes:
$ ./risk -s

1 attacker die vs. 1 defender die:
Attacker loses 1, defender loses 0: 21 / 36 = 0.583
Attacker loses 0, defender loses 1: 15 / 36 = 0.417

1 attacker die vs. 2 defender die:
Attacker loses 1, defender loses 0: 161 / 216 = 0.745
Attacker loses 0, defender loses 1: 55 / 216 = 0.255

2 attacker die vs. 1 defender die:
Attacker loses 1, defender loses 0: 91 / 216 = 0.421
Attacker loses 0, defender loses 1: 125 / 216 = 0.579

2 attacker die vs. 2 defender die:
Attacker loses 2, defender loses 0: 581 / 1296 = 0.448
Attacker loses 1, defender loses 1: 420 / 1296 = 0.324
Attacker loses 0, defender loses 2: 295 / 1296 = 0.228

3 attacker die vs. 2 defender die:
Attacker loses 2, defender loses 0: 2275 / 7776 = 0.293
Attacker loses 1, defender loses 1: 2611 / 7776 = 0.336
Attacker loses 0, defender loses 2: 2890 / 7776 = 0.372

3 attacker die vs. 1 defender die:
Attacker loses 1, defender loses 0: 441 / 1296 = 0.340
Attacker loses 0, defender loses 1: 855 / 1296 = 0.660

# Sample outcome of a rather large battle
$ ./risk 20 10

When 20 armies attack 10:

Probability of winning:

Attacker: 0.965
Defender: 0.035

Expected losses:

Attacker: 8.267
Defender: 9.900

Probabilities of various outcomes:

Attacker wins by 20: 0.007 | xxxx
Attacker wins by 19: 0.021 | xxxxxxxxxxxxx
Attacker wins by 18: 0.039 | xxxxxxxxxxxxxxxxxxxxxxxx
Attacker wins by 17: 0.059 | xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Attacker wins by 16: 0.077 | xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Attacker wins by 15: 0.087 | xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Attacker wins by 14: 0.096 | xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Attacker wins by 13: 0.092 | xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Attacker wins by 12: 0.092 | xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Attacker wins by 11: 0.079 | xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Attacker wins by 10: 0.074 | xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Attacker wins by 9: 0.059 | xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Attacker wins by 8: 0.052 | xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
Attacker wins by 7: 0.039 | xxxxxxxxxxxxxxxxxxxxxxxx
Attacker wins by 6: 0.033 | xxxxxxxxxxxxxxxxxxxx
Attacker wins by 5: 0.024 | xxxxxxxxxxxxxx
Attacker wins by 4: 0.020 | xxxxxxxxxxxx
Attacker wins by 3: 0.010 | xxxxxx
Attacker wins by 2: 0.005 | xx
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Defender wins by 1: 0.007 | xxxx
Defender wins by 2: 0.011 | xxxxxx
Defender wins by 3: 0.008 | xxxx
Defender wins by 4: 0.005 | xx
Defender wins by 5: 0.003 | x
Defender wins by 6: 0.001 |
Defender wins by 7: 0.001 |
Defender wins by 8: 0.000 |
Defender wins by 9: 0.000 |
Defender wins by 10: 0.000 |