Monday, 22 May 2017

Charging Elephants

There are a few apparently unrelated areas of Explore I've long wanted to address - grappling, animal attacks, and spell area attack saves - but I've come to understand how they're strongly connected through a rather circuitous path. Starting off considering climbing, you'll see how answering that comes round to the simpler issue of charging elephants, and how answering that has wider applications.

Climbing Power

Back in January I half finished a blog post on climbing - it isn't strength that's important but your power-to-weight ratio - but in Explore your size is derived from your strength and constitution. This lead to some odd consequences (your con gave you a negative bonus on climbing as high con implied being well built and in climbing being slimmer is better), so I determined that it should be the reverse: your height and build should influence your strength and constitution. This changes around how stats work during character creation, but not how the game works - whichever system I decide upon, when you need a climbing stat bonus you sum it up and write it on the character sheet and it doesn't alter.

This meant I started revising what the stats were. Instead of getting up to +3 from Strength, you get up to a +1 from each of Height, Build, and Muscle. Con similarly gets a bonus of up to +3 from Height, Build, and Toughness. (Note that this means that Str and Con have the same range as before and still cannot differ by more than 2). It is then only the Muscle that gives you a bonus on Climbing, and that's your power-to-weight bonus. The bonuses also tie into the sizing system. A height of 0 means you're between 5'6'' and 5'11''.  As per my revised size categories, height of +1 would put you in the 6' to 6'5'' range, and a height of -1 would put you in the 5' to 5'5'' range. If you also had +1 build that would push you up two weight categories from say 160lb to 192lb (assuming the character is not overweight!) or -1 build and -1 height would knock you down to 96lb.

Do you Dodge or Parry a Giant?

From this I wondered what other advantages / disadvantages you might get from being big / small. To work this through I considered large differences - with a human versus a giant is the giant slower to swing and thus it's harder to hit the human? Or is the sword a big attack which is tricky to dodge and impossible to parry? If you give bonuses how would this affect giant versus giant combat? Should it be the same as two humans, or would it be qualitatively different? I started to think about the different situations and what the desired outcome might be, and to temporarily stop considering mechanics.

When examining a big problem with lots of variables it's easier to break it down and consider one variable at a time, so I concentrated on dodging. Looking for extreme situations where dodging was clearly the only option, I considered an Elephant charging.

The Charging Elephant

When an Elephant charges you, you're not going to be able to outrun it.

I've already assumed that size doesn't affect your speed - so by default a huge animal like an Elephant would not get any bonuses or penalties due to its size - but you can get many bonuses due to other factors than size. In fact according to this speed of animals resource, an Elephant's top speed is 24.9mph . Usain Bolt's top speed is about 30mph, so Elephants are indeed very close in speed to a well-trained, naturally fast human, so aren't getting much in the way of bonuses. However, even if you are that fast, an Elephant will tire much less quickly than you - you'll soon tire and slow down and be caught and be squashed.

Alternatively you could try and face it down, like this man did (who doesn't even seem that fazed by the experience), but for today I'm going to assume you wait until the last moment and then try and dive out of the way. I observed that as the Elephant is 7' wide you have to move a fair amount to dodge it.

Alternatively consider a bee attacking you - it is hard to hit the bee because it is small, but it is hard to dodge because you are large.

Hence a charge attack is something along the lines of a bonus for speed and agility and skill plus a bonus for the size of the larger individual (attacker or defender), versus a dodge based upon speed and agility and skill.

Dodging Fireballs

This lead to a realisation that the Elephant misses because you have to move out of the way! So when grappling a giant, it will win easily in a contest of strength, but it has to catch you first - and avoiding it means moving around (not just staying stationary and rolling a dice and saying the giant missed).

If that's the case, then the reason why the fireball missed is because you jumped out of the blast zone. (I'm thinking 5' radius blast here incidentally, not D&D's mass effect fireballs from Chainmail and war-gaming). Similarly for a rock from a trebuchet, or thrown by a giant. Treating dodging an arrow in the same fashion would mean that you got a bonus for the larger of the arrow and the target, so you'd get a bonus for the size of the target, which is exactly what I already have.

In conclusion, I don't know whether I'll ever differentiate mechanically between "I dodge" and "I parry", but I can see combat effects becoming far more dynamic and size bonuses being applied more consistently. I can see different combat tactics being effective against different size adversaries. Giants are slow but deadly and are best taken on with missile attacks, smaller creatures have to be taken on hand-hand, but area effect attacks become more useful.

Wednesday, 1 March 2017

Complexity and Simplicity

Complex rules in games are in general seen as a bad thing. If it is complicated to apply a rule at the table, then this makes playing the game difficult. Not that it makes the game difficult to do well at, but it gets in the way of actually playing the game.

Simple rules are in general a good thing, as they enable you to get on with the business of playing the game.

But this is different to the complexity behind the rule.

Making Simple Complex

A rule can be a complex implementation of a simple idea.

Knight Hawks is a cool game - space ships flying around the table top shooting each other. In the advanced game there are ten different types of weapon, and five different types of shield, and to attack you cross reference these to see the percentage chance to hit. All weapons have a range, some weapons can only be fired in certain directions, some only go every other round, some get 5% knocked off per hex distance. All of these rules are based upon simple - even trivial - concepts, but in practice the implementation adds so much complexity to the game that combat makes my brain hurt.

Making Complex Simple

Rules can also be simple implementations of a complex idea.

In Explore I wanted skills to increase in cost, based originally upon the skill points system from Star Frontiers. But it quickly proves difficult to count up all the points in total you've spent. Star Frontiers has a table with two columns - one for the points for that level, one for the total so far.

In addition I wanted the cost to rise exponentially, so +1 rank always meant a fixed increase in the amount spent, which meant making the rules more complex - but...

I realised that if I chose the skill points carefully, they could both be exponential and remove the adding up. If the points were 1, 1, 2, 4, 8, 16 etc. then the total cost is just the next number in the sequence. If you're fifth rank you've spent 1+1+2+4+8 = 16 points. So you cross off the points as you spend them, and the next number is the total.

Mathematically that's quite a complex rule, but it results in a simpler system.

Is Explore Too Complex?

Explore attempts to (in some sense) "abstractly model reality"; the rules are chosen so that as much as possible the rules give you a reasonably consistent and non-arbitrary result; if the rules have to be overridden to avoid the fiction of the game becoming nonsensical then they've failed.

To achieve this has (as those of you reading recent posts in particular can attest to) taken a fair amount of maths and hard thinking.

However - in the end all this has boiled down to a couple of tables in the game that aren't even needed in play most of the time. You know a giant has +8 damage because double height is +8 height, so +8 size, so +8 damage. A -6 Strength penalty on throwing a spear is -6 damage and -6 range, which is half range. The only lookup were double height=+8, -6=half range, and both those are from tables looked up in character or monster creation.

Obviously the rules are more complex than many games, but they are also far simpler than many games based upon far simpler principles.

To me, the rules add to the game, without dragging it down. Any rules that dragged the game down have been ejected unceremoniously (and there were lots).

Complexity Tradeoff

Explore originally had have two scales - one for Height/Length (doubles every +2) and one for Weight/Strength/Range/Density (doubles every +4). Later I added a third for Speed (doubles every +8).

My latest change means I now have five scales, so that is more complex, but what is the trade off?

In play the increase in complexity is negligible. If you're looking up Range, you look up the range table as you did before. How it is calculated doesn't really matter.

On the other hand denser projectiles now get a damage bonus, so you get a bonus for a lead sling shot.

It also has many knock on effects which have to be considered:

Weak characters now lose far less range due to their penalties - a -6 penalty halves the range rather than reducing it to one eighth  - which is both more fun and more believable.

Previously in On Archery I had double the range gives you a -6 penalty on to hit. Now due to the recent changes double range is +6 range categories. Hence +1 range equals -1 to hit. A very pleasing coincidence!

In On Archery II: Longbows I calculated
for the arrow with the furthest range (and fastest initial velocity) in the experiment, kinetic energy is reduced by root 2 approx every 140.6m (461 feet), which in Explore means a -1 on damage per 448 feet (nearest length category).
So I said that at max range the kinetic energy was halved so an arrow got -2 damage, which didn't seem enough to me. Now it is -6.

So the knock-on effects all have a positive effect on the game, for a minimal cost.

Sunday, 19 February 2017

Lethality of Guns

My last post was about scaling projectile damage – how to relate changes in size, density, and velocity. As Explore is a fantasy RPG the focus was on low speed projectiles such as arrows and sling stones, but Leland commented about bullets, which I hadn’t considered (as they're not part of Explore), so after some further discussions via email I’ve been digging into the behaviour of bullets.

We discussed several bullets Leland had gathered data for, and in particular compared the slowest, a .32 S&W, and the fastest, a .17 Rem. Whether you measured KE, momentum, or KE/radius, the .17 Rem always came out as much better, which contradicted the opinion on the actual effectiveness of the .17 Rem. (Not any personal experience of mine I should add - I'm vegetarian and have never held a gun!)

Firstly I should clarify something about my formula – KE/radius. It’s a compromise between damage done (KE) and depth of penetration (KE/area). Without any KE there is no damage caused, but without any penetration it’s at best a shallow wound.

High Velocities

Does this formula make sense for high velocity bullets? Let’s consider how penetration is affected by high velocities. At high velocities bullets behave like the target body is a fluid, and we can easily model this like you calculate effects of wind resistance, but unfortunately we cannot calculate total penetration in this fashion since if we model penetration of a projectile into a fluid by a drag force proportional to the velocity squared, then the bullet will get slower and slower but never stop.
Looking for solutions to these equations online, the best resource is actually a GURPS page http://panoptesv.com/RPGs/Equipment/Weapons/Projectile_physics.php which combines equations for both low and high speed behaviour to calculate the depth of penetration:
Km/A * ln (1+v^2/C)
where m is bullet mass, A is cross sectional area, v is impact speed, and K and C are constants calculated from the target material (sectional density of the bullet has already been accounted for). The page I referenced suggests 100m/s for C.

When v is small (<100m/s) this is roughly Kmv^2/AC, i.e. proportional to KE/A. This means the penetration formula for Explore holds true for the range of speeds in that game.

When v is larger (>200m/s), although it is still proportional to m/A, it becomes proportional to ln(velocity), so increases in velocity have a reduced effect.



How does this work out when applied to the bullets Leland and I were discussing? We discussed bullets with speeds 200 – 1200 m/s. That sixfold increase in velocity with the simple method from Explore gives 36 times more penetration, so +32 damage. In the new system for high velocity projectiles it’s only 3 times more deadly, and gives only +10 damage. This means that now the .17 Rem comes out slightly worse than the .32 S&W, as expected.

(Note that since the ineffectiveness of the high velocity bullet was due to friction, I'd rule that the excess energy is used up as heat, so is lost to damage. That is, lethality is penetration * radius, not KE/radius).

Importance of Radius on Damage

It looks like my formula implies that increasing the radius of a projectile reduces its lethality, but actually the reverse is the case; we can rebase it to be density * length * radius * velocity squared. That is, increasing radius increases lethality (it increases in mass more than it increases in radius).

Not however that if you alter radius and length to keep the same mass, my formula implies that long thin bullets are more lethal than short stubby ones of the same mass.
However, there is an opinion that expanding bullets are more deadly, so it was suggested that radius should play a more important role.

Expanding bullets were introduced in 1897 (under the name Dum Dum bullets) as the then new high velocity low calibre bullets were proving ineffective. It was argued by Germany that they were inhumane and they got them banned, but this was based upon poor evidence. Expanding Bullets are actually used today by police forces as the expansion of the bullet slows it down and prevents it exiting the target and injuring bystanders. It seems reasonable to assume that the problem with the old bullets was simply that they went through the target (particularly an arm or leg) and straight out the other side. If you missed a vital organ then the shot was not as effective as it should be. That is, I think the purpose of the Expanding bullet is purely to avoid over-penetration. 

Thus an Expanding Bullet would be ineffective against opponents with tough armour / hide (as it would expand upon hitting the armour and fail to penetrate it) or large opponents, but it might be important in other situations to avoid over penetration and consequent loss of damage. So I'm happy with the role of radius in the formula, but there's a place for situational modifiers.

Deflection Of Bullets

A bullet can also fragment, be deflected, or tumble. All of these appear to have a similar effect to expansion of the bullet - reducing penetration thus giving a shallower but larger wound. The angle of deflection in a collision is determined by the angle of collision and the ratio of the masses of the two objects, so it would appear that less massive bullets are more easily deflected, so this could be a further cause of lower penetration amongst lighter bullets.

Further, for the same reasons, it appears that a lightweight bullet hitting off-centre, hence hitting the armour / hide at an angle, is more prone to ricochet. This would be another reason to avoid lightweight bullets.

Wind Resistance

As a side note it should be mentioned that a faster lightweight bullet with the same KE as another bullet is more accurate over short distance (due to the reduced drop) but over long distance it is affected more by drag, and not only slows down but can be blown off course.

Grazes

Leland also commented that he’d like any system with guns to include a chance of minor wounds or grazes even from a very powerful gun whereas in Explore you can have a sufficient bonus that any hit against a particular target is automatic death. Given the lethality of hits from guns, I agree that you’d have to include something like this, as it changes the dynamic of the game quite considerably.

So, in summary, you could probably have quite realistic rules for guns, and mostly they would simply be in the form of realistic bonuses or situational modifiers for the weapons (as they can be quite simply calculated), but it would take a bit of work. You might want to separate out penetration to be a separate bonus.

Sunday, 5 February 2017

How Lethal is a Projectile?

Rebasing the Scales

In How strong is an Elephant? I talked about how height, weight, and power scale in relation to each other. Explore is a game that uses Logarithmic Scales which double every +n categories. Due to the Elephant post I've revised the scale so that weight doubles every +3, height doubles every +8, and power doubles every +4 (instead of power scaling with weight):

Weight (lbs)
Power
Height (ft)
4
4
4
5
5
4.5
6
6
5
8
7
5.5
10
8
6
12
10
6.5
16
12
7
20
14
7.5
24
16
8
32
20
9
40
24
10
48
28
11
64
32
12
80
40
13
96
48
14
128
56
15
160
64
16

That is, when resizing creatures, moving one category up/down in weight matches to moving up/down one category in height and power, and giving +1/-1 in Strength, Constitution, Damage.

This means power is weight ^ (3/4), and height is weight ^ (3/8). This is as per the Elephant post, but 8/3 = 2.67 which is halfway between my 2.61 value and the 2.79 value in the paper I cited.

Power does not have units as it is applied in different situations. For example it would be the weight in pounds of missiles thrown.

Thus for example a Giant is double the height of a human, so +8 categories. Hence they are approx six times as heavy, and they are four times as powerful.

What affects Lethality

As I discussed in On Throwing, no matter the size of the humanoid, the starting assumption is that they can throw things the same distance - it's just the size of the projectile changes. Hence if a human throws a 6lb spear, a giant is size +8, so throws a 24lb spear with +8 damage due to the increase in mass.

A strong individual gets a bonus on throwing the same spear, because the spear hits the target at a higher speed, because they threw it faster (and hence further). We should choose range so that +1 max range = +1 damage.

Also the lethality of a projectile is obviously affected by its shape.

Hence damage from a projectile is based upon the velocity, mass, and shape of the projectile.

Kinetic Energy and Size

In Sizing Things Up I said that damage of projectiles (given the same shape) was based upon kinetic energy, 1/2 * m * v^2.

Kinetic Energy is clearly the key aspect of the damage potential of a missile, not momentum. If you fire a gun then you experience recoil and and forced backwards with the same momentum as that of the bullet, and this does not hurt you. Most of the kinetic energy is given to the bullet (as it is much smaller) and it is this which makes the bullet lethal.

The problem is that this implies a large wooden ball and a small lead ball of equal weight thrown at the same speed are equally dangerous, when clearly they are not.

(I tried to resolve this problem a couple of years ago but did not arrive at a satisfactory conclusion).

It is clear that the small ball is more lethal because the impact area is smaller. There are two obvious possibilities - divide the kinetic energy by the impact area, or by the square root of the impact area (scaling as the perimeter of the impact area, or the diameter of the cross section).

Dividing by the area is clearly wrong, as in this system being hit by several missiles at the same time would be identical to being hit by a single missile. (5* missiles means 5* KE but also 5*Impact area. Thus KE/Impact area remains the same).

Hence we should assume that we divide by the square root of the impact area, i.e. the damage is proportional to the KE divided by the diameter of the projectile.


Effect of projectile shape

I previously attempted to address this issue, but failed, and kept damage as being KE + a bonus for shape. The difficulty I had before was combining both blunt and sharp projectiles into the same system. Does it make sense to talk of the impact area of a sharp spear, or a round ball? Would they scale in the same fashion as each other?

I recently returned to this problem, and considered - if damage from projectiles is mostly due to penetration - how much energy is required to do this?

Now for blunt projectiles the energy required for this is proportional to the circumference of the hole, as this is how much skin has to be torn, which is proportional to the diameter of the projectile.

My recent observation is that arrow heads rip a hole, the length of which is the width of the arrow head, hence the energy required is also proportional to the diameter of the projectile.

Hence I am now happy that the diameter of the projectile is a reasonable value to divide by, regardless of the type of missile.  That is, a spear and a stone may have different damage bonuses, but it is reasonable to apply the same modifiers to scaling both of them. 

Hence the damage can be implemented as (KE / cube root of size) plus a fixed bonus for the shape.


Range

For a given shape of projectile:

Doubling the range: doubles the kinetic energy (same size), so doubles the deadliness.

Size +12 gives you +12 damage, and you have eight times the power, which means you throw a projectile eight times the mass the same range. That's eight times the kinetic energy, but the missile is twice as wide, so it is only four times the deadliness.

Hence +6 damage = double deadliness = double range or double density. Thus the scale for range and density both double every +6.

Note that for simplicity I'm now assuming that all weapons scale isometrically, so all dimensions of weapons double every +12. This means that giants have shorter swords with respect to their height, whilst halflings have longer swords relative to their height - this doesn't seem unreasonable.

Density

For a given shape of projectile, if you double the density and double the weight (+4), then you've doubled the kinetic energy but kept the same size. So that's identical to double the range (+6).

Hence density doubles every +2!

The New Combined Scale

Combing the results above we have extra columns in our table for Range & Density. For simplicity of figures I'm making Range 10* the numbers as it only applies to large quantities.


Density (lb/sq ft)
Weight (lbs)
Weapon Weight (lb)
Range (ft)
Height (ft)
4
4
4
40
4
6
5
5
45
4.5
8
6
6
50
5
12
8
7
55
5.5
16
10
8
60
6
24
12
10
70
6.5
32
16
12
80
7
48
20
14
90
7.5
64
24
16
100
8
96
32
20
110
9
128
40
24
120
10
192
48
28
140
11
256
64
32
160
12
384
80
40
180
13
512
96
48
200
14
768
128
56
220
15
1024
160
64
240
16

As you can see, a Giant is +8 size (twice as tall, approx six times as heavy), throws a spear +8 size (four times as heavy = 24lb), and gets +8 damage.

Consider a human with a magical throwing gauntlet which gives +8 range, so +8 damage. That's approx 2.5 times as far.

So the four times mass of the giant spear (4*KE) has the same damage potential as 2.5 times range (2.5*KE) because a large spear has a larger impact area (despite being just as sharp and pointy).

Apologies for the maths heavy analysis - but it all leads to a simple table in the rules. Next post I'll discuss complexity and simplicity in rules.

Saturday, 28 January 2017

How Strong Is An Elephant?

The Question

My game, Explore, is supposed to give (vaguely) realistic results to questions such as how varying the size of creatures affects their strength. I had assumed that the strength of a monster was proportional to its weight - thus a doubling in size doubled the weight it could lift. This then has a knock on effect through other rules of the game (if I want to retain consistency) so it is quite fundamental.

When working through my rules on climbing, where power to weight ratio is important (not raw strength) I noticed that the Olympic Weight Lifting records were odd.

Here are all the world records for each class in Snatch and Clean & Jerk (in kg):

Weight Class
Snatch
Clean & Jerk
56
139
171
62
154
183
69
166
198
77
177
214
85
187
220
94
188
232
105
200
246

You can clearly see that the weight lifted is not linear with the weight of the competitor - it only changes by half the rate expected. The lowest category has snatch being 2.5* the weight, so you'd expect the top class to have 263kg, 124kg more than the lowest class - which is twice the actual increase. With a log-log plot you can see that it's proportional to the weight class to the power 0.57 / 0.55. That is, the weight lifted is proportional to the square root of the weight class.

Nothing explains this result, so I decided to go back to first principles.

Return to Allometry

The fundamental scaling law for Explore is how weight scales with height/length. I previously wrote controversially about how the square-cube law does not appear to hold for animals and I noted that for the species I could find data about the average mass of adult animals in a species varied with the 2.62th power of the length of the animal. This is not a contradiction of Galileo's square-cube law, it is simply that larger animals tend to be thinner. The science behind this is called Allometry, but at the time I couldn't find any studies to back up this specific observation.

Well, last week I stumbled across a paper ALLOMETRIC SCALING OF BODY LENGTH: ELASTIC OR GEOMETRIC SIMILARITY IN MAMMALIAN DESIGN which measured length and weight for 1733 different species of mammals to see if the scaling exponent was 0.333 or 0.250. The scaling exponent is the reciprocal of the power, so my prediction was 1/2.62 = 0.382. The paper showed that it was not close to their two expected results, but unexpectedly much higher at 0.359, and 1/0.359 = 2.79. This is higher than my value, but this is for all mammals, whereas I'm looking for how small animals of a given type scale to large animals of the same type. That is I'm scaling from a pony to a horse, this study is about estimating the weight of an Elephant and a Giraffe from that of a hamster.

BMI (body mass index) bizarrely assumes that ideal weight is proportional to height squared, which self evidently wrong. This quotes from wikipedia says:
However, many taller people are not just "scaled up" short people but tend to have narrower frames in proportion to their height. Nick Korevaar (a mathematics lecturer from the University of Utah) suggests that instead of squaring the body height (as the BMI does) or cubing the body height (as the Ponderal index does), it would be more appropriate to use an exponent of between 2.3 and 2.7
So I'm happy with my 2.62 value, and I'm happy to use it both for estimations of the weight of a small or giant version of a species as well as estimations of the weight of individuals of a species of varying sizes.

Elephant Power

Returning to the question of strength I started to look for how strong different species of animals were. I wasn't going to find snatch & jerk records for animals, so I needed an alternative. The power of Horses is known - a draft horse has one horsepower - and I hoped to see how larger species compared, but for other species it proved remarkably difficult. I decided that as Elephants are used for their strength I might be able to find answers for them, but although I could find how much they could lift with their trunk, how much they could have piled on their backs, or how much they could drag - none of these were particularly helpful (or scientifically measured) or comparable to a horse. I started to wonder what I meant by Strength, and found the answer with the origins of horsepower.

James Watt standardised and popularised horsepower as a means for selling his Steam Engine. He agreed to take royalties of one third the savings in coal that people made by switching to his more efficient Steam Engine - but this could not apply for people who still used horses. He took the idea that a steam engine could do the work of several horses, and standardised it. He calculated the force that a draft horse could pull at a particular speed, and force*distance = energy. This gives you the power output of a horse, in watts, or joules/second.

This power output is what I wanted, and I noted a connection with my reading on Allometry.

Metabolic Rate

Basal Metabolic Rate of an organism is defined as the rate of energy burned by an organism at rest, i.e. it is also measured in watts. It seems reasonable to posit (given the complete absence of data) that on average the maximal power output (the strength) of a creature is a fixed multiple of their basal metabolic rate.

The allometry of metabolic rates is a well know result in Allometry:
In plotting an animal's basal metabolic rate (BMR) against the animal's own body mass, a logarithmic straight line is obtained, indicating a power-law dependence. Overall metabolic rate in animals is generally accepted to show negative allometry, scaling to mass to a power ≈ 0.75, known as Kleiber's law, 1932.
Hence I posit that the strength of a creature, its maximal power output, is proportional to its mass to the power 0.75.

(Note that many sources cite that strength scales with 2/3 the power of weight, which is inferred from the scaling of muscle cross-sectional area, which is a confusion of strength=power with the structural strength of materials. It is precisely this fallacy that lead people to insist for years that Kleiber's law was wrong and the value was 0.67, as their logic had told them it must be).


The Strength of an Elephant

Given a draft horse, 730kg, can pull with one horsepower - how much does a 5000kg Elephant pull with? My suggested scaling gives (5000/730)^0.75 = 4.2 horsepower. That is, although the Elephant is nearly seven times heavier it would only be four times stronger. This seems to accord with general consensus (that a horse is stronger, pound for pound) but the reduction is not very pronounced as the ratio of the weight of a draft horse to an elephant is not actually that big.

The Strength of an Ant

Consider instead the Asian Weaver ant, 5mg in weight, which was photographed carrying a 500mg weight - one hundred times its own weight! I'm 178 times taller than the ant is long, and 15 million times heaver than this ant. The ant compared to me, is like me compared to a 1000ft tall giant! Scaling for its size, the ant's power-to-weight ratio should be 60 times that of our 77kg weightlifter.

Is This Proof?
This is not proof of the formula, since to prove it we'd have to have good data, but we can compare this with the alternate theories - linear scaling, 0.67 scaling.

The linear rule would give ants as having no better power-weight ratio than a human, which is self-evidently wrong.

The 0.67 rule would give the power of an Elephant as only being 2.6 times as that of a horse, which is also obviously far too low.

Hence the rule is not only plausible, it also appears to give far more reasonable predictions than the alternates.

Return to the question

I've calculated how power varies with weight - 0.75, but at the beginning we saw clearly that the weight liftable varied with exponent 0.5. How to reconcile?

Well, now I know I'm talking about Power and Energy, which leads to the answer.

To lift a weight above one's head you have to lift the weight through a certain distance, that is you expend a certain amount of energy - m*g*h in fact.Hence taller weight lifters have a disadvantage - they have to do more work to lift the weight higher.
The average height of the weight lifter, and the height they lift it through, should be proportional to weightCat^(1/2.6).
This additional scaling (factoring in height as well as weight) predicts that the 56kg cat weight lifting records would be 146kg/176kg, and the 105kg cat records would be 214kg/259kg.
So not perfect, but pretty close.

What was the point again?

Firstly I'm interested in these answers for their own sake; trying to understand how the world works. Secondly I'd like values in the game not to be arbitrary if possible. But most interestingly applying the scaling rules in the game allows me to work out answers such as how much damage bonus a giant gets compared to how much damage penalty you get from air resistance at max range with an arrow. Once codified into the rules no-one needs understand how I came up with the values, but they're consistent and scalable.