Tries to determine how many dice to roll as a defender in the game of Risk given an attacker's rolls. This takes advantage of the rule that the attacker must roll first and then the defender gets to choose how many dice to roll in response. If the defender chooses to roll only one die, then he will only lose one army if he loses.

Apparently this is a common misprint in the first Dutch edition of the rules; I have seen it once in an English anniversary edition.

Examples

1. Battle 5 attackers against 2 defenders many times for all available selection methods. Matrices describing whether the defender will roll one or two dice (dependent on the attackers' rolls) are contained in various should_roll.txt files.

octave:1> [win_pcts, avg_atk_left, avg_def_left, atk_left_std, def_left_std] = run_risk_a_lot(5, 2)
win_pcts =

   0.82000
   0.76000
   0.77000
   0.79000
   0.86000

avg_atk_left =

   4.1707
   3.6842
   3.9740
   4.0000
   4.3372

avg_def_left =

   1.5000
   1.7500
   1.7826
   1.6667
   1.2857

atk_left_std =

   1.06348
   1.39724
   1.33746
   1.22997
   0.64371

def_left_std =

   0.51450
   0.44233
   0.42174
   0.48305
   0.46881

2. Battle 20 attackers against 20 defenders many times for two dynamic selection methods: should_roll.txt, and should_roll_joost.txt.

octave:1> [win_pcts, avg_atk_left, avg_def_left, atk_left_std, def_left_std] = run_risk_a_lot(20, 20, 10000, [1; 1; 0; 0; 0]);

Algorithm

The simulation will attack until either the territory is conquered or the attacker only has one army left and therefore cannot attack any more.

A decision matrix looks like:

     |  1   2   3   4   5   6
  ---+-----------------------
   1 |  2   2   2   2   2   2
   2 |  2   1   2   2   2   2
   3 |  2   2   1   1   2   2
   4 |  2   2   1   1   1   1
   5 |  2   2   2   1   1   1
   6 |  2   2   2   1   1   1

where the entry at i,j indicates how many dice the defender should roll given that the attacker's top two rolls are i and j. (Notice its symmetry.)

My decision matrix above (should_roll.txt) was generated by looking at the probabilities of winning outright (i.e., losing no armies) by rolling either one or two dice as recorded in o-vs-d.txt.

Joost Rijneveld hypothesizes a decision matrix here:

     |  1   2   3   4   5   6
  ---+-----------------------
   1 |  2   2   2   2   2   2
   2 |  2   2   2   2   2   2
   3 |  2   2   2   2   2   2
   4 |  2   2   2   2   2   2
   5 |  2   2   2   2   1   1
   6 |  2   2   2   2   1   1

Ger Koole hypothesizes a decision matrix here:

     |  1   2   3   4   5   6
  ---+-----------------------
   1 |  2   2   2   2   2   2
   2 |  2   2   2   2   2   2
   3 |  2   2   2   2   2   2
   4 |  2   2   2   1   1   1
   5 |  2   2   2   1   1   1
   6 |  2   2   2   1   1   1

Results

Some interesting preliminary results over 10,000 iterations given my dynamic matrix, Joost's matrix, Ger's matrix, a defender who always throws two dice when they're available, and a defender who always throws one die:

Successful conquer rate (lower percentage corresponds to better defense):

Attack	Defense	Andrew	Joost	Ger	2	1
5	2	80.7	79.8	79.8	82.9	87.2
10	10	41.2	39.6	37.0	48.1	80.1
15	10	79.7	77.4	76.6	84.8	98.5
25	15	93.4	91.7	90.1	95.4	99.9
50	50	57.7	50.1	45.1	70.5	99.8

Average armies the attacker has left when the territory is successfully conquered (lower is better for the defender):

Attack	Defense	Andrew	Joost	Ger	2	1
5	2	4.1 (1.0)	4.0 (1.2)	4.1 (1.1)	4.0 (1.2)	4.3 (0.8)
10	10	4.9 (2.0)	4.9 (2.0)	4.8 (2.0)	5.2 (2.0)	5.8 (1.9)
15	10	7.7 (3.0)	7.5 (3.0)	7.4 (3.0)	8.0 (3.1)	10.0 (2.6)
25	15	12.3 (4.6)	11.9 (4.7)	11.6 (4.6)	13.2 (4.7)	17.3 (3.4)
50	50	10.8 (5.9)	10.4 (6.0)	10.0 (5.6)	12.7 (6.6)	24.3 (6.1)

Average armies the defender has left when the territory is fully defended (higher is better for the defender):

Attack	Defense	Andrew	Joost	Ger	2	1
5	2	1.6 (0.5)	1.7 (0.5)	1.6 (0.5)	1.7 (0.4)	1.3 (0.5)
10	10	4.4 (2.1)	4.6 (2.2)	4.6 (2.2)	4.4 (2.2)	3.0 (1.8)
15	10	3.3 (1.8)	3.4 (1.8)	3.4 (1.8)	3.5 (1.8)	2.2 (1.5)
25	15	3.4 (2.0)	3.6 (2.1)	3.5 (2.1)	3.2 (2.1)	3.2 (1.0)
50	50	8.6 (5.5)	9.4 (5.9)	9.7 (6.0)	7.9 (5.5)	3.3 (1.6)

Looks like Ger's matrix has the edge.