Problem 1260. RISK board game battle simulation

Created by Jeremy

Solve Later

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Given two positive integer inputs, a (attacker army units) and d (defender army units) return the probablity of victory (from 0.000 to 1.000) to +- 0.02 accuracy. The rules are given below for those unfamiliar with the game.In the board game RISK battles are determined by the conflict of armies, namely the attacking army and the defending army. The results is determined as follows: the attacker rolls 3 six-sided die and the defender rolls 2 die. The highest two numbers of each player are compared respectively, and the higher number wins (this means the opposing army loses one unit). In the case of a tie the defender wins. For example:Attacker has 10 units
Defender has 10 unitsAttacker rolls [6 3 2]
Defender rolls [4 3]The first comparison is attacker - 6, defender - 4. Since the attacker is higher, the defender loses one unit. Hence Attacker has 10 units, Defender now has 9 units.The first comparison is attacker - 3, defender - 3. Since the defender is higher, the attacker loses one unit. Hence Attacker has 9 units, Defender now has 9 units.This is continued until either the attacker has only one unit left, in which case the defender wins the battle; or the defender has no units left, in which case the attacker wins the battle.This is one further rule: the number of die any player may roll cannot be more than the their units in case of the defender, or their units + 1 in case of the attacker.Example:
Attacker has 3 units,
Defender has 1 units.Attacker rolls 2 die (3 - 1),
Defender rolls 1 die.

Given two positive integer inputs, a (attacker army units) and d (defender army units) return the probablity of victory (from 0.000 to 1.000) to +- 0.02 accuracy. The rules are given below for those unfamiliar with the game.

In the board game RISK battles are determined by the conflict of armies, namely the attacking army and the defending army. The results is determined as follows: the attacker rolls 3 six-sided die and the defender rolls 2 die. The highest two numbers of each player are compared respectively, and the higher number wins (this means the opposing army loses one unit). In the case of a tie the defender wins. For example:

Attacker has 10 units
Defender has 10 units

Attacker rolls [6 3 2]
Defender rolls [4 3]

The first comparison is attacker - 6, defender - 4. Since the attacker is higher, the defender loses one unit. Hence Attacker has 10 units, Defender now has 9 units.

The first comparison is attacker - 3, defender - 3. Since the defender is higher, the attacker loses one unit. Hence Attacker has 9 units, Defender now has 9 units.

This is continued until either the attacker has only one unit left, in which case the defender wins the battle; or the defender has no units left, in which case the attacker wins the battle.

This is one further rule: the number of die any player may roll cannot be more than the their units in case of the defender, or their units + 1 in case of the attacker.

Example:
Attacker has 3 units,
Defender has 1 units.

Attacker rolls 2 die (3 - 1),
Defender rolls 1 die.

Solve

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Last Solution submitted on Apr 24, 2023

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James on 11 Feb 2013

Can you double check your values, Jeremy? I've run your sims against a couple of on-line Risk calculators, and they're coming up with different values for some of your test cases. For example, the one at http://recreationalmath.com/Risk/ using 10 attackers and 10 defenders comes up with 0.47993525647768026, which doesn't meet your error criteria.

Jeremy on 12 Feb 2013

Thanks, James, you were right - I generated the values myself with a Matlab script. Unfortunately I did not do enough simulations - 10000 doesn't always get within 1% accuracy. Have updated them all now and rescored all solutions too.

Jeremy on 12 Feb 2013

Your solution is now the leading solution btw :)

James on 12 Feb 2013

I didn't submit a solution yet...mainly because I couldn't match up your test values and wanted to make sure that the problem wasn't on my end. (Most of the time, it is! :-)

James on 20 Feb 2013

I've got a working submission now, but you'll need to check the high end of the size chart for it. As usual, Richard and Alfnie are far more clever than I.

Richard Zapor on 24 Feb 2013

James, you give me too much credit. The next ot best solution was a very minor tweak to Jeremy's random solver. My larger solutions, 250, find the exact solution recursively.

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Problem 1260. RISK board game battle simulation

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Problem 1260. RISK board game battle simulation

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