Why Do Lockdown Weapons Always Work?

Fellow Pilots,
You are piloting your Kumiho, at Yamantau.  The battle begins, and it’s a race to the center beacon. Your race, and you are playing to win it.  Suddenly, your legs freeze; you are locked down!  Shamefully, a series of blue energy bolts from a Dragoon Fury destroys you in the five short seconds you wear the “lightning pants.”

If it appears that so-called “lock down” or “root” weapons work almost all the time, you might take the trouble to look up the official percentages.  Halo, the light shotgun, is described by Pixonic as having a 33% lock down probability per particle at level 9.  Shredder, the light energy cannon, has a 2.6% lock down probability per bullet, again at level 9.  These numbers seem reasonably low, so why do you get stuck in the mud every time someone shoots at you?

The answer lies, unfortunately, in the human brain’s inability to comprehend the awesome power of exponents.  We are good at story telling, and we can remember vast amounts of conversational/linguistic information.  Images, smells, and sounds imprint in our memories so well that we have trouble sleeping with them sometimes.  But repeatedly multiplying numbers turns our brains to mush.  As an example, a job paying a penny on the first day and doubling for a month would pay enough to retire ($5.38M on the last day alone).  Famously, if you could fold a paper on itself 50 times, its original .1mm thickness would be over 112 MILLION kilometers - the distance from Earth to the Sun!

As usual, I won’t bore you with a ton of math (the models are explained below, PLEASE help us out if you actually understand probability). A few simple examples with the Pixonic numbers will suffice. You already have the battlefield experience of either reliably locking the enemy down, getting locked down, or both.  The U is here to tell you why.

Going back to Halo, let’s assume a robot has two of them at level 9. The fire rate is twice per second, so they can get you eight times in two seconds (2 x 2 x 2).  Imagine a coin flip resulting in a “heads” causing lockdown.  This, of course, is a 50% chance.  What’s the difference if then eight coins are flipped with ANY “heads” resulting in a lockdown?  The answer is 2*2*2*2*2*2*2*2 (2 to the 8th power), or 256.  Thus, avoiding a lockdown from eight hits with a 50% probability weapon would be a VERY small 1 in 256: only a quarter of a percent.  Halo, with a 33% lockdown probability, results in almost all enemies being locked down after 8 hits (we believe the chances to be above 90% at close range).

Locking the whole team down ...

The Shredder has a lower lockdown chance, only 2.6% percent per plasma bullet at level 9.  But it fires much faster - ten rounds per second.  Even if half the rounds miss, ten still hit it two seconds.  We estimate that two level 9 Shredders have a 60% chance of locking opponents down after two seconds of firing.

Sometimes YOU wear the lightning pants ...

An important caveat is the weapon accuracy - while some items in War Robots are "all or nothing," like Magnums (full damage per hit, regardless of distance), all the lockdown weapons disperse over distance.  The shotgun particles spread (and do less damage/have lower probabilities), and the plasma bullets miss more at greater distances with the same negative effects.  

We did some experiments to test our theoretical math - Dr. Z and I lined up at 100 meter increments and took Halo shots at each other, measuring both lockdown probability and damage.  No matter the distance, after you "flip the lockdown coin" a couple times, you have a frozen opponent.  The recommendation from us is straightforward - get lockdown weapons using Royale tokens and the workshop, and don't worry about leveling them up unless you need them to do a lot of damage.  If you want to be a lockdown specialist, it's cheap or free to load up a Patton or Griffin with these rapid-fire toys and go to town.

For example, two level 1 Halos have a lockdown chance of 25%, a seemingly big difference from the 36% they sport at level 12.  But if you hit an enemy for two seconds (again, 2 weapons x 2 shots x 2 seconds), our model says you will lock them down 66% of the time.  The level 12 weapons only increase this effect to 77%.  Granted, they do much more damage, but you can use your Ember for that!

Another practical implication is for weapon mixers - because the shotgun weapons are all continuous reloaders with 500m range, you can throw one on virtually any existing build and not worry about trying to fire individually.  A single Corona on a Taran/Ember Inquisitor will result in many frozen targets for you to melt!  One Halo on a Russian Death Button Griffin means enemies who can't jump away from inbound Tulumbas.

In the video example, my Pursuer and Haechi are each equipped with a pair MK2 level 12 locking shotguns.  They are located on different planes to better hit robots in stealth or descend mode; you will note that I successfully lock each and every target.  We can expect a nerf to these weapons, I think, but if it comes in the form or lowered lockdown probabilities, it won't matter much.  The compounding effect of multiple hits is simply too big!

Hope you are having a great Day of the Dead,
Dr. Yat


MATHS
As a professional educator in a non-mathematics field, I'd LOVE to have these models torn apart and rebuilt by people who know their stuff!

For the Halo, we assumed half of the particles hit the target; the War Robots Wiki suggests this is the accuracy at 200m.  So we took the stated 33% lockdown rate (level 9) and cut it in half, for a 16.5% chance of locking, and a 83.5% chance of avoiding the lock.  Taking .835 to the fourth power equals .486, or a 48.6% chance of avoiding lockdown.  This is a model for one weapon/two seconds, or two weapons/one second.  Taking .835 to the eighth power equals .236, or a 23.6% chance of avoiding lockdown.  This is a model for one weapon/four seconds, or two weapons/two seconds.  Our custom game experiment yielded results consistent with our estimates.

For the Shredder, we assumed half of the plasma bullets hit the target; the War Robots Wiki suggests this is the accuracy at 300m.  So we took the stated 2.6% lockdown rate (level 9) and cut it in half, for a 1.3% chance of locking, and a 98.7% chance of avoiding the lock.  Taking .987 to the twentieth power equals .769, or a 76.9% chance of avoiding lockdown.  This is a model for one weapon/two seconds, or two weapons/one second.  Taking .987 to the fortieth power equals .592, or a 59.2% chance of avoiding lockdown.  This is a model for one weapon/four seconds, or two weapons/two seconds.  Because we don't have Shredders, we are not able to test this model and we think the accuracy of the weapon is somewhat better than our guess.  If we assume a 2.1% chance instead of 1.3%, the chance of avoiding lockdown after four seconds of one weapon would fall to about 42%.


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