No problem Paul it's a quick and easy one and I know it's a topic close to many people's hearts!

paulharrow wrote:

I'm now guessing my skepticism comes from the fact that I never got to have so many rolls in such a short time playing risk at the dinner table, so now that I'm online, I should expect to see more of these strange events.

I think this is probably the biggest part of it for most of us.

Hugh wrote:

Details can be provided upon request :)

 Consider this a request!

asm wrote:

Hugh wrote:

Details can be provided upon request :)

 Consider this a request!

For 3v1, my claim is that 49.0% of attacker losses involve a tie.  We can analyze the situation where a particular value is the highest attacker die to count both total losses and losses involving a tie.  I will represent a single roll by listing the attacker's 3 die first followed by the defender's die.

For example, suppose the attacker's highest die roll is a 5.   To avoid overcounting, we separate the rolls into the following forms:  555z, 55xz (x < 5), 5x5z (x < 5), x55z (x < 5), 5xyz (x,y < 5), x5yz (x,y < 5), and xy5z (x,y < 5).  For ties, the defender's roll (z) must be 5, so we base the calculation on the configurations 555z, 55x5, 5x55, etc.  This gives 1 + 4 + 4 + 4 + 16 + 16 + 16 = 61 total ties when 5 is the highest die.  There are 122 total losses when 5 is the highest die (z can be 5 or 6).  To change the calculation based on the highest die being 4, we note that x and y will take on 3 possible values, so the same configuration types give 1 + 3 + 3 + 3 + 9 + 9 + 9 = 37 total ties when 4 is the highest die.  There are 3 possibilities for the defender die to produce a loss, so there are 37*3 = 111 total losses when 4 is the highest die.

Organizing the information into a table, we have:

Highest die:    |    1   |   2   |   3   |   4   |   5   |   6   |   Sum

# of losses           6       35    76     111   122    91      441

# of ties               1        7     19      37     61     91      216

For the conclusion, 216/441 ~ 49.0%

Note that what causes the prevalence of ties is that a tie is forced if the attacker loses with 6.  6 is the most prevalent highest die roll, and even when 5 is the highest die roll, 50% of such losses involve a tie.

The analysis for 3v2 die is similar, though accounting for the various configurations is a bit more complex.  For example, if 6 is the highest die and 3 is the 2nd highest attacker die, the defender must roll 63 or 36 to tie, so we count configurations like 63x63, 6x363, x6363, 63x36, etc, again being careful not to overcount (63363 belongs to both 63x63 and to 6x363).  Instead of brute forcing the tables, it is better to organize the patterns involved using summation notation and summation formulas.  I will post that as well upon request, though I hope this gives a good flavor of how to approach the calculations :)

With 3v2, again what drives the probability up is that the distribution of highest and second highest die are skewed toward the higher values where ties are more likely (there are fewer possibilities for straight defeat).

I hope what I just wrote is at least mildly sensible!

Hugh

By the way, did anyone ever take tom up on his offer for the dice rolling data to look for anomalies?

Pretty sure that was for you.