About the definitions of "strength" and "luck":

That strength is not well approximated by the sum of the dice is exposed when considering specific cases. The strength is held in the top two dice. For instance, 6,5,3 and 6,4,4 sum to 14, but are not equal in power.

But I agree that plenty of rolls should mollify the inconsistencies.

Luck depends on the circumstances of the event. That a banana falling from the sky is lucky depends on whether I'm stranded on a desert island with no food.

I know we're looking for something approximate, but I think the best way to calculate luck would be

Look at the defense dice. Say the defender rolled a 5,5 Then the outcome falls into 3 events

Pr (att takes 2, given defender rolled a 5,5) = x

Pr (1 and 1, given defender rolled a 5,5) = y

Pr (att loses 2, given defender rolled a 5,5) = z

Depending on the outcome, Then x,y,z measure the luck of the defender, 1-x, 1-y, 1-z measure the attacker's luck.

Lets say in this scenario the attacker takes 2 (doesn't matter the specific roll, just that the attacker got the best outcome). The rolls that would do it (and the number of ways of getting that roll) are

6,6,6 1 way
6,6,5 3 ways
6,6,4 3 ways
6,6,3 3 ways
6,6,2 3 ways
6,6,1 3 ways

16 ways total, 216 possible triples, So the probability that the attack takes 2 given the defender rolled 5,5 is 16/216, this would be the defender luck (unlucky), and the attackers luck is 200/216, very lucky.
Add up the luck for each player, divide by number of engagements.

So luck as it is calculated here will always be between 0 and 1, and a luck of .5 is a coin flip.

Can someone run some analysis on this series of turns for me

465 asm attacked GrimAvenger Alaskanesia > Alaska (6,4,4)(6,4) 06/05/2010 - 21:06:35
466 asm attacked GrimAvenger Alaskanesia > Alaska (6,5,1)(6,6) 06/05/2010 - 21:06:36
467 asm attacked GrimAvenger Alaskanesia > Alaska (5,4,1)(5,4) 06/05/2010 - 21:06:37
468 asm attacked GrimAvenger Alaskanesia > Alaska (5,3,1)(6,4) 06/05/2010 - 21:06:38

asm wrote: Can someone run some analysis on this series of turns for me

465 asm attacked GrimAvenger Alaskanesia > Alaska (6,4,4)(6,4) 06/05/2010 - 21:06:35
466 asm attacked GrimAvenger Alaskanesia > Alaska (6,5,1)(6,6) 06/05/2010 - 21:06:36
467 asm attacked GrimAvenger Alaskanesia > Alaska (5,4,1)(5,4) 06/05/2010 - 21:06:37
468 asm attacked GrimAvenger Alaskanesia > Alaska (5,3,1)(6,4) 06/05/2010 - 21:06:38

100%

asm wrote: Can someone run some analysis on this series of turns for me

465 asm attacked GrimAvenger Alaskanesia > Alaska (6,4,4)(6,4) 06/05/2010 - 21:06:35
466 asm attacked GrimAvenger Alaskanesia > Alaska (6,5,1)(6,6) 06/05/2010 - 21:06:36
467 asm attacked GrimAvenger Alaskanesia > Alaska (5,4,1)(5,4) 06/05/2010 - 21:06:37
468 asm attacked GrimAvenger Alaskanesia > Alaska (5,3,1)(6,4) 06/05/2010 - 21:06:38

TILT

asm wrote: Can someone run some analysis on this series of turns for me

465 asm attacked GrimAvenger Alaskanesia > Alaska (6,4,4)(6,4) 06/05/2010 - 21:06:35
466 asm attacked GrimAvenger Alaskanesia > Alaska (6,5,1)(6,6) 06/05/2010 - 21:06:36
467 asm attacked GrimAvenger Alaskanesia > Alaska (5,4,1)(5,4) 06/05/2010 - 21:06:37
468 asm attacked GrimAvenger Alaskanesia > Alaska (5,3,1)(6,4) 06/05/2010 - 21:06:38

No Whammies.............STOP!

Mongrel wrote: About the definitions of "strength" and "luck":

That strength is not well approximated by the sum of the dice is exposed when considering specific cases. The strength is held in the top two dice. For instance, 6,5,3 and 6,4,4 sum to 14, but are not equal in power.

But I agree that plenty of rolls should mollify the inconsistencies.

Luck depends on the circumstances of the event. That a banana falling from the sky is lucky depends on whether I'm stranded on a desert island with no food.

I know we're looking for something approximate, but I think the best way to calculate luck would be

Look at the defense dice. Say the defender rolled a 5,5 Then the outcome falls into 3 events

Pr (att takes 2, given defender rolled a 5,5) = x

Pr (1 and 1, given defender rolled a 5,5) = y

Pr (att loses 2, given defender rolled a 5,5) = z

Depending on the outcome, Then x,y,z measure the luck of the defender, 1-x, 1-y, 1-z measure the attacker's luck.

Lets say in this scenario the attacker takes 2 (doesn't matter the specific roll, just that the attacker got the best outcome). The rolls that would do it (and the number of ways of getting that roll) are

6,6,6 1 way
6,6,5 3 ways
6,6,4 3 ways
6,6,3 3 ways
6,6,2 3 ways
6,6,1 3 ways

16 ways total, 216 possible triples, So the probability that the attack takes 2 given the defender rolled 5,5 is 16/216, this would be the defender luck (unlucky), and the attackers luck is 200/216, very lucky.
Add up the luck for each player, divide by number of engagements.

So luck as it is calculated here will always be between 0 and 1, and a luck of .5 is a coin flip.

Huh?

I don't understand what you trying to say here at all.  Why would you base luck off of what the defender rolls to start with?  Sure if the defender rolls a 5,5 and the attacker ends up winning 2 then the attacker can be considered lucky, but why are you treating one event as happening first?  The defender rolling 5,5 to start with contains a certain amount of luck, the defender could roll a 1,1 just as easily.

IRoll11s wrote:

Mongrel wrote: About the definitions of "strength" and "luck":

6,6,6 1 way
6,6,5 3 ways
6,6,4 3 ways
6,6,3 3 ways
6,6,2 3 ways
6,6,1 3 ways

16 ways total, 216 possible triples, So the probability that the attack takes 2 given the defender rolled 5,5 is 16/216, this would be the defender luck (unlucky), and the attackers luck is 200/216, very lucky.
Add up the luck for each player, divide by number of engagements.

So luck as it is calculated here will always be between 0 and 1, and a luck of .5 is a coin flip.

Huh?

I don't understand what you trying to say here at all.  Why would you base luck off of what the defender rolls to start with?  Sure if the defender rolls a 5,5 and the attacker ends up winning 2 then the attacker can be considered lucky, but why are you treating one event as happening first?  The defender rolling 5,5 to start with contains a certain amount of luck, the defender could roll a 1,1 just as easily.

I think 11's might be right here. Consider in the above situation that we look at it the other way.

Attacker rolls 6,6,x

There is one way for defender to take 2..  (6,6), and the odds of doing that are 1/36

Based on defender's dice: 16/216 = .074  (Mongrel's math)

Based on attacker's dice 1/36 = .028

In fact, I wonder that this exposes a fatal flaw in any luck calculations of this type where the pips are considered.

What's the difference (luck-wise) between 6,6,x vs 6,6 and 3,5,1 vs 6,6?  Absolutely nothing. The best you can do is calculate the odds that attacker will win 2, win 1 or win 0 with 6 sided in a 3 vs 2 army attack.  There are only three possible outcomes and it doesn't matter one bit what the spots are. I lifted these numbers from an on-line site:

Attacker: three dice; Defender: two dice:

	Attacker wins both: 2890 out of 7776 (37.17 %)
	Defender wins both: 2275 out of 7776 (29.26 %)
	Both win one: 2611 out of 7776 (33.58 %)

So let's say there's a battle where both win one. Clearly the defender was luckier than the attacker. I'm not sure how to proceed from here.  Perhaps we could find the difference for luck numbers of 7.91 and -7.91

If the attacker wins both then the luck numbers could be (100 - 37.17) - 50 = 12.83 and -12.83.

While this is all well and good, it doesn't answer my basic questions.

Did I roll 'em good, bad, or ok?

Did my opponent roll 'em good, bad, or ok?

Using my simple count and average the pips method, I can answer these questions.

No wonder I lost.  Geez, my armies gave it their all fighting at 110% of what is normally expected, but my opponent's armies were superhuman, fighting at 117%.

Tom, I doubt you'll go with my count and average the pips idea for a primary method, but including some kind of relative dice strength number in the stats might be fun and simple for some of the more statistically challenged among us.

Here's another possible way to calculate relative dice strength.

I found this url: http://www.anwu.org/games/dice_calc.html

I don't know how easy the coding to do this might be to find or emulate but clearly it's do-able. There could be many ways to present this data, but here's one that appeals to me.

Let’s say that over the course of a turn, a player rolls 10 seven sided dice for a total of 33 pips. According to the calculator, they should roll worse than this only 12% of the time.  Double this number for a nominal efficiency rating from 0 - 200%

We could then say, “Your troops are performing at 24% nominal efficiency.”

Now because we are dealing with percentages, it should be easy to weight different sided dice performance. Let’s say the same player threw 3 additional ten sided dice for a total of 18 pips, yielding a 57.5% number with those dice. Then it's just matter of weighting these two numbers and then dividing by the number of dice thrown:

 ((57.5% * 3 ) + (12% * 10))/13 = 22.5%

22.5 * 2 gives us a 45% nominal efficiency rating over the course of 13 throws.

What really appeals to me about this type of stat is that in fogged games, I can get an idea of the how well I (and/or my teammates) are doing without compromising knowledge of others' performance.  Obviously in non-fogged games, I could also see how well my opponents are rolling independent of my rolls.

Edit: I'm realizing that it makes more sense to use the <= number.

That would change the results as follows:

10 seven sided at 33 pips = 15.4%
3    ten    sided at 18 pips = 64.8%

for a 26.8% *2 =  53.6% nominal efficiency . ..where 0% is akin to falling on your swords, 100% is expected and 200% implies divine intervention.

The easiest and most useful statistics are based on the outcomes of the rolls alone. If you want to know what the probability of taking someone out is, you _could_ analyze it at the atomic level of the dice rolls. However, all that is required to know is that 37.17% of rolls result in defender losing 2, 33.58% result in both losing 1, and 29.26% result in the attacker losing 2.

For example, from that, you can calculate the number of armies the attacker expects to lose per roll: .3358(1) + .2926(2) = .921

If you use 3 dice versus 2 dice 20 times, the attacker expects to lose 18.42 armies (and expects to kill 21.58 defenders). The difference between the actual outcome and this number is what the other site used for its luckstats plugin.

The other site failed to account for non 6v6 situations, but the outcomes for nonstandard dice are calculable by modern computers. One solution is to have the computer loop over all possibilities. Mongrel devised a piecewise defined function (the number of sides of attacking/defending dice as input) to calculate these outcomes even more rapidly than brute force.

I say this because my impression is that M57's averaging sums and other ideas were put forth under the impression that expected outcome is hard to calculate and that it would be desirable to have a "quick and dirty" solution. I don't care to speculate how well correlated this number would be over the long haul because we already have the technology to do better than that.

That said, another statistical curiosity the other site provided was a breakdown of the percentages of 6's, 5's 4's, etc rolled as attacker or as defender. This is fun, but again, does not give _direct_ insight into something you can relate to the game. Expected outcomes may not tell you about context, but it is easy to interpret: I should have only lost 30 armies in the attacks I've made, but I lost 45, and that sucks.

Summarizing, I side strongly with 11s and Alpha's outcomes-based position because it is easy to calculate and interpret.

Postscript: To improve on the other site further, we should calculate the standard deviation of the expectation for the number of attacks involved. Comparison with the probability distribution of outcomes is also calculable, and easy to interpret if you put it in graphical form, but the standard deviation gives a quick dirty number to tell you just how unusual it is that your outcome lies in a certain range of values.

Hugh's post made me smile.

Hugh wrote: 

Summarizing, I side strongly with 11s and Alpha's outcomes-based position because it is easy to calculate and interpret.

Postscript: To improve on the other site further, we should calculate the standard deviation of the expectation for the number of attacks involved. Comparison with the probability distribution of outcomes is also calculable, and easy to interpret if you put it in graphical form, but the standard deviation gives a quick dirty number to tell you just how unusual it is that your outcome lies in a certain range of values.

This is all very encouraging.

I would love to see some of these stats expressed in terms of standard deviation, either as a number or graphically.  Those familiar with interpreting these numbers would instantly have a strong handle on how to interpret them, and those who are not would come to an intuitive understanding of them pretty quickly IMO.

RiskyBack wrote: Hugh's post made me smile.

It made me jealous.

I suck at quoting posts and keep screwing it up, but I really liked the veiled tear on ToS by saying "Modern Computers can do this".
Not sure if he meant it the way I took it but for the purpose of joy & happiness, I will subscribe to a Post-Modern Reader Response interpretation of the Posting.

I was going to comment but then saw Hugh's post which pretty much summed my thoughts. The odds of rolling 6's 5's or 2's has very little meaning to me. I want to know how many units I lost and how many units I can predict to lose.

I don't care if it was 6,5 vs 5,5, or 2,1 vs 1,1. Okay, maybe I do. But it's not as meaningful as knowing the predicted outcome of 44 armies vs 32 armies.

The Luck Graph plug-in on TOS was great and was updated to work with modifiers. This is very useful when testing new boards, especially 1v1 games.

Though I don't think I've said anything incorrect, the devil is in the details, and there's no need to explain any further. Expected losses is the way to go.

I like your logic Rolls, at the end of a game I want to be able to look at what percentage of rolls I won vs what percentage I expected to win. What would be even more useful is if I could graph that against which turn I made that roll on. I'll take my good rolls at the start of a game any day :).

Is this just sitting way back in the queue because there are more important things, or because we don't have a solution? or is it a consensus thing?

Consensus.