The Ideal Battle Stats - A Mathematical Approach

So - I have seen so many questions out there asking "What is the best battle stat?" or some variant thereof. I figured - since Chedburn has told us the effect on battle stat ratios (see post here), there must be a way to determine this mathematically. Before you go on - understand that this is more of a thought exercise than a recommendation of what stat build you should choose. There are many factors such as temporary weapons, stat boosters, weapon accuracy and damage, overall strength, life, and much more.

This exercise effectively looks at two players with the same weapon with 50/50 accuracy and damage, and no armor, using no boosters, with the same effective stat totals, to determine who, on average, would do more damage under that specific scenario.

STOP HERE IF YOU ARE NOT A NERD LIKE ME AND SKIP TO THE END FOR THE RESULTS

The first step of course is taking his data, and putting it in excel format, as follows:

Speed	Dexterity	Speed:Dex Ratio	Hit Chance
1	10	0.1	11
5	10	0.5	33
10	10	1	50
15	10	1.5	60
20	10	2	67
30	10	3	74
40	10	4	78
50	10	5	81
60	10	6	84
70	10	7	85
80	10	8	87
90	10	9	88
100	10	10	89
110	10	11	90
120	10	12	90
130	10	13	91
140	10	14	92
150	10	15	92

Defense	Strength	Def:Str Ratio	Damage Reduction
1	10	0.1	16
5	10	0.5	40
10	10	1	50
15	10	1.5	57
20	10	2	63
30	10	3	71
40	10	4	75
50	10	5	80
60	10	6	83
70	10	7	86
80	10	8	89
90	10	9	91
100	10	10	93
110	10	11	95
120	10	12	96
130	10	13	98
140	10	14	100
150	10	15	100

Great! Now we can use an online regression tool like this one here and play around with it until we find something that is a good enough fit for purposes. Using this, I devise two formulas that closely approximately the relationship of Speed/Dex ratios to hit chance, as well as Defense/Strength ratios to damage reduction. These formulas are as follows:

Chance to Hit = 96.7400371 * e^(-0.871886852 / (SPDDEXRATIO + 0.3066613321))

Damage Reduction = 50.05976375 * DEFSTRRATIO^(0.3537365235 * DEFSTRRATIO^-0.1187762685)

I then extrapolated these out beyond the extremities of Ched's data and found that the Chance to hit formula holds up very very well, while the Damage Reduction formula holds up very well at very low Def:Str ratios but predicts >100 damage reduction at very high Def:Str ratios. That's ok - I will just limit it to 100 with an IF function.

I can plot these against Ched's Data to show that they predict fairly well:

[image: snag.gy]

So as you see - the formulas are pretty darn close. I'll call that good enough.

So now - how to use them in a way that helps to figure out which ratio of battle stats is the best? Well - for this, I did four iterations.

In each iteration, the attacks are against a defender with perfectly balanced stats (I used 25,000 in each for a total of 100,000, but the number doesn't actually matter, just the ratio). For the attacker, however:

Iteration #1: Vary speed between 1,000 and 99,000, then the remaining three stats are balanced to equal 100,000 total

Iteration #2: Vary dexterity between 1,000 and 99,000, then the remaining three stats are balanced to equal 100,000 total

and then the same for Defense and Strength.

In this way, we can take a look at an even fight from a stats total perspective, and get a look at way varying levels of imbalance in all the difference stats.

For each of these cases, in each iteration, I used the formula to come up with the hit chance for the attack and defender, and the damage reduction for both the attacker and defender. I then use these in the following way, to come up with "damage" values that I can use to compare:

Attacker Damage Done = (AttackerHitChance / 100) * (100 - DefenderDamageReduction)

and then the reverse for the defender. Using these, you can then come up with 4 sets of plots that compare the unbalanced attackers against the balanced defender

SPEED COMPARISON

From the data, the maximum advantage that the unbalanced attacker had against the balanced defender (both with 100k total stats) was 100.66% relative damage, occurring at 28% speed.

You can also see that you really should have at least 15% of your stats in speed, otherwise you're taking a big hit.



[image: snag.gy]

[image: snag.gy]

DEXTERITY COMPARISON

From the data, the maximum advantage that the unbalanced attacker had against the balanced defender (both with 100k total stats) was 101.34% relative damage, occurring at 32% dexterity.

You can also see clearly that being a dexterity whore is a bad idea... increasing dexterity much above this point does not buy you much - your opponents damage goes down to a point and then actually INCREASES at high levels of imbalance because when they do hit you, they hit you for much more.

[image: snag.gy]

[image: snag.gy]

DEFENSE COMPARISON

From the data, the maximum advantage that the unbalanced attacker had against the balanced defender (both with 100k total stats) was 101.42% relative damage, occurring at 18% defense.

Again, you can see clearly that whoring defense (which it might get you stalemates against unsuspecting individuals) will make you lose horribly against anyone with even stats. Increasing it to high levels actually INCREASES the damage your opponent will do, since they can hit you more often.

[image: snag.gy]

[image: snag.gy]STRENGTH COMPARISON

From the data, the maximum advantage that the unbalanced attacker had against the balanced defender (both with 100k total stats) was 100.74% relative damage, occurring at 22% strength.

As with the other four stats... whoring is clearly not a good strategy.

[image: snag.gy]

[image: snag.gy]

SO - WHAT IS THE BEST RATIO?

Using these data sets, the best ratio appears to be the following:

32% Dexterity

28% Speed

22% Strength

18% Defense

I expected after I did the calculations for each statistic that I would get a relative idea of which stats were more valuable and would have to play with the data, but as it turns out, the maximum points in each calculation perfectly added to 100%. So... when I run the simulation with these statistic values against someone with perfectly balances stats, I calculate an advantage of... Drum Roll Please.... 3.62%.

I know... not much of an advantage, but still an interesting result.

But Vladar, you are saying - this must be wrong because I like the way my stats are, or I lost to a ::FILL IN STAT HERE:: whore, or blah blah blah. I have news for you - THIS PROBABLY IS NOT THE BEST REAL WORLD STAT BUILD. Many things happen in real fights that I cannot account for, I am simply looking a theoretically perfect, balanced case to see what the results are.

So, a couple notes and disclaimers:

1) This is negating the effect of weapons (or assumes a 50/50 weapon is used) and negates the effect of armor. It also assumes that the person you are attacking has perfectly balanced stats, has the same effective stat total as you, and negates plenty of other things as well such as crit rate and other modifiers.

2) This data is going to be the AVERAGE result over many attacks. There is a HUGE variation in randomness with each individual attack. You can obviously get lucky or unlucky in any given attack, so results will vary considerably on a single fight basis.

3) This does not take into account game mechanics such as temporary boosters, smoke grenades, multiple players in an attack, etc. Obviously those can have a HUGE effect on the outcome.

4) As several people have commented - life obviously plays a huge roll in attack success. This is looking at the expected damage ratios over a large number of fights, not anticipated outcome of the battle.

5) As Alexstrasza pointed out, your base damage increases the higher your strength gets. This also does not account for other than the extreme case of someone with almost no strength, the ratios aren't so out of whack that I think this probably makes a big difference in the result.

5) This could of course be totally wrong! But I think the math behind it is as sound as it could be, so without doing some seriously large scale difficult testing, I think this is as close as we're going to get to an answer.

Just as a little bit of extra fun... I made a google spreadsheet for you guys to play around with and break! Enter the % of each stat (make sure they total to 100%!) and the see what the result is against someone with the same effective overall stats, but perfectly balanced.

Play around with it HERE

. . . And my stat distribution looks nothing like this . . .

My ratio is similar to that but with defense lacking at 8% and strength/speed balanced as it allows for using Frontline and Elite's for training. At this ratio I usually have the problem of taking more damage than the other person in close fights so it helps to be in a clothing (buffs armor and dex) or in a faction with good suppression during warring.

Good data, info in general. R+

Very interesting read. The only issue is the unknowing of the additional damage provided by strength to factor into the defense calcs, but otherwise I'm with you on dex & speed being better to weight. I opt for a higher speed than dex build due to the amount of dex whores that perpetuate Torn, but if I were to start training again, I'd definitely go for a similar build to what you suggested here.

Definitely a great guide, but you totally discounted life. A higher life player would need less defense/dex because they can absorb more damage.

well, clearly this man loves Econemetrics ( YOU MY BOY BLUE!!))



give him a couple more Variables to add to his equation.

That's a moot point in his calcs, as he's just doing a flat comparison of two equal players and varying the battle stat percentages. Their level, life, and other variables are considered equal.

To quote chedburn "balance is king". After I balanced my stats I've even been beating higher stat total people that are 95-100 and with careful temporary selections I can beat most people.

Edit: it was a good read though rated +

As Alexstrasza said, I am assuming everything else is equal. Equal battle stat totals, equal armor, equal weapons, etc. Obviously there are many many factors that come into play. I was just trying to answer the question of "is balanced really the best" in an objective, math based way.

Balanced is good, no doubt about it. Slightly unbalanced in the favor of dex/speed seems to be better, however.

As for temporary weapons - yes, of course - as I said I needed to discount those for the purposes of my assessment.

Thanks for the comments :)

Absolutely.To rephrase my statement, because it sounded rather harsh: I am suggesting a further variable to include should you update this guide in the future. I'd personally love to see a spreadsheet where you put in you own battle stats, your opponent's battle stats, life, and other factors, and it spits out the chance that you win. This would be a more practical application of the calculator.

I was talking to someone about that last night. The problem is for it to be meaningful you would need for both players:

Base Battle Stats

Battle Stat Modifiers

Average Armor Value

Weapon Damage/Accuracy/Crit Rate

Mods / Weapon Modifiers

Hit Points

Temporary Boosters / Weapons

Plus to top it all off, there are hidden factors that no one is REALLY sure of.

The only way to do it would be to again assume all else is equal and just look at base battle stats vs. base battle stats. Anything else would be too complicated and inaccurate I think.

Good read, very interesting stuff right here.

From my own experience I'd say I don't really need that much speed (so I leave it equal with str & def), as a very high accuracy weapon will compensate for that. But maybe I'll have to think about slightly changing my build and try this! :P

Very interesting.

As said, a disclaimer or acknowledgement of life would benefit its accuaracy.

Its not something you actually can account for.

Having done these calculations myself, I need to reiterate what Alexstrasza said: you're missing data with respect to max damage from strength, which throws your results off a bit. https://www.torn.com/forums.php#!p=treads&f=3&t=16007832&b=0&a=0

If the defender has only 10 defense, then whether the attacker has 1k or 1b strength, no damage reduction will occur, but the damage done will be very different. Your calculations assume the 100% damage value is the same regardless, which means your results are skewed to some degree in favor of whoever has the lower strength value.

Additionally, it's not correct to assume that equal weapons and armor simply offset each other and don't affect the validity of the calculations. I believe 50.00 damage/accuracy on weapons is the same as having no final effect on damage or hit rate, so it would make more sense to assume neither player is wearing armor, using mods, or getting any boosts, plus they are both hitting with 50/50 weapons.

I do agree with the general conclusions that dex and speed are quite important. The problem is that almost everybody's weapon has over 70 accuracy and so many people use accuracy mods and flash/smoke temps. There are a lot more questions to be answered before it's clear what kind of build is ideal.

I've claimed Dex whoring to be a valid strategy since the last change to attacking, just based on my own experience. But these graphs and data analysis are awesome. Thanks for the hard work, good job!

Aren't we missing the elephant in the room? Battle stat totals are virtually never even between two combatants. Focusing two or three stats will leave you with a higher total stat number. Enough to compensate for having a low value in the fourth stat. This tells us in a mathematical way what we already knew - to be the best you, don't go perfectly even in stats.

With the way drugs work; xanax use leads to higher total stats than the SSL gym and LSD. In my experience LSD has quite a high OD rate as well. If a player isn't going to use the SSL they're shoehorned into favoring one stat 25% higher than the rest to make the most of the gyms.

TLDR: As we know Balance is the lower total stat choice for fools.

EDIT. This neglects late game factors. Some day every player hopes to be able to use Stat Enhancers. Consider

Scenario 1: A player with 1B in all four stats selects Defense. They use 500 Stat Enhancers on defense only. 1B Speed, 1B Strength 1B Defense, 1B Dexterity grew to 1B Speed, 1B Strength, 152,119,732,792 Defense, 1B Dexterity. Around 155 B new stat total

Scenario 2: A player with 1B in all four stats selects Strength and Speed. They use 250 stat enhancers on each. 1B Speed, 1B Strength, 1B Defense, 1B Dexterity grew to 12,333,682,856.00 Speed, 12,333,682,856.00 Strength, 1B Defense, 1B Dexterity. Around 26.7 B new stat total

Scenario 3: A player with 1B in all four stats uses 125 stat enhancers on each stat. 1B Speed, 1B Strength, 1B Defense, 1B Dexterity grew to 3,511,934,346.77 Speed 3,511,934,346.77 Strength 3,511,934,346.77 Defense 3,511,934,346.77 Dexterity. Around 14 B new stat total

Which of these is most attractive to you I suppose depends on play style. Which is the most successful? Certainly not the balanced approach given time

Thanks for the feedback!

To your points:

1) Damage growth from higher strength totals - I agree that this throws off the numbers slightly but I don't think by that much. There is a huge difference in 1K strength vs. 1B strength vs. 1T strength, but not such a huge difference between say, 220m and 250m strength. Since the comparison I'm doing is looking at people with the same stat total, their overall difference in strength is never so enormous. If you ever found the data you were asking for in the other thread, I'd love to see it - I totally misunderstood what you were talking about the first time around!

2) Weapons and Armor - Great point! After having a good nerdy discussion with of you faction mates yesterday, I have been convinced that accuracy actually affects your speed before looking at the ratio, which means it does indeed need to be considered, since this exercise is looking at stat ratios! For this reason I will revise my thread to say "assuming they are both using 50/50 weapons and is wearing no armor". Thanks for that one.

3) Accuracy Mods / Temps - Yes, I totally agree - I am by no means trying to say that this build will actually work the best in an actual battle - I am saying theoretically, given my set of conditions, this seems to be the ideal build. Of course there are many other factors and it's impossible to consider them all. Every build has strengths and weaknesses. I'll try to reword my intro to make this clear.

Thanks for your response. To your point - at lower levels, focusing on one or two stats to the exclusion of others does indeed allow you to reach much higher stat levels much faster. However, once the stat cap is reached, whether your strength is at 1 billion or 50 million, you gain at the same rate, so this advantage goes away. This thread was not intended to be a discussion on training considerations, however.

Also - while I know that people are rarely perfectly evenly matched, this was a thought exercise to see if you could model a perfectly "fair" fight to see what unbalanced build would be ideal. I fully acknowledge that my numbers do not consider each and every possible variant.

Finally - regarding stat enhancers - again, this was not intended to be a recommendation of what build someone should choose, it was looking mathematically at the data we have available to look at an idealized fight.