Coming in 3.20 (exact date tbc but we're aiming for August this year), there will be some updates to how Balanced Blitz works in bigger battles. We are hoping this addresses the common complaints from the community that we have seen surrounding dice rolls. This has been a long time coming ever since we introduced the feature way back in 2019!

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Skip to section (3) if you don’t need the background context and just want to see an overview of the changes.

As a quick reminder, Balanced Blitz is a game setting that aims to provide more consistent probabilistic outcomes when utilizing the Blitz option in battles. Blitz instantly determines the result of a battle and avoids having to watch hundreds or thousands of dice rolls. Balancing is achieved by trimming out the extremely unlikely probabilities, making it more suitable for competitive play when compared to the more statistically accurate True Random setting.

Balanced Blitz turns losing battles like 10 vs 5 from a statistical possibility into an outright impossibility, and also limits how many troops you could lose in such a scenario. This is the setting you want when you’re looking to reduce the major factor of luck, and to focus more on strategic skill. Of course, True Random is still an option, but you will be at the full chaotic whims of RNG.

While the Balanced Blitz algorithm produced the results we were looking for, to do so, it required an immense amount of computational resources to output the brute-forced calculations that the balancing applied to. For the vast majority of battles which were small in size it was fine, however, it quickly scaled out of control as the troop count grew.

Unable to find a more efficient algorithm at the time, we employed the use of simple estimation techniques that could output approximations in a fraction of the time. This enabled us to extrapolate the results of smaller battles with the same troop ratio (attackers : defenders) to compute a rough estimation of large battles.

To implement this, we needed to determine a troop count threshold where the estimation algorithm would take over. For the sake of fairness and consistency in a competitive, cross-platform multiplayer game, we were limited by what the hardware in the oldest supported mobile phones could allow. Unfortunately this was determined to be at a disappointingly low 70 troops, and even fewer when you take into account special rules such as Capitals and Zombies.

It was the combination of both; a low estimation threshold and the quality of estimation techniques that lead to inaccurate results in larger battles. These inaccuracies became very evident in battles exceeding 100 troops or as little as 50 when attacking a Capital, and it only got worse as the battles became bigger.

An often experienced scenario that highlights these shortcomings is when you have a large stack of troops attacking a territory with only one troop on it. With Balanced Blitz enabled, you would expect a 10 vs 1 and 100 vs 1 to be effectively the same thing when it comes to calculating possible losses to the attacker side. However, due to the estimation method coming into play for the larger battle, it becomes possible to lose more troops than you would have, if you attacked with fewer troops.

Savvy players would use the Attack Limit slider as a workaround to this problem

The known ways to solve this would be to (a) find a more efficient Balanced Blitz algorithm that doesn’t require so much brute-force computational data, (b) drastically improve the performance of the current algorithm, and (c) improve the accuracy of the estimation method using more advanced techniques. Read ahead for more details, but we ended up going with (b) and (c) for the upcoming update.

Before diving into the details, here’s a quick rundown of the changes:

1. More accurate calculation and addressed attacker-side bias of troop losses in bigger battles*
   
2. More consistency when determining troop losses in lopsided battles**
   
3. Win chance will be displayed in all battles up to and including 1,000 vs 1,000 in size
   
4. Battles larger than 50,000 vs 50,000 will fall-back to using True Random logic

No changes to how battles play out if they are smaller in size or if you are using the True Random setting.

* Greater than 70 troops in standard battles or greater than 35 troops in battles involving dice augments (Capitals, Zombies, etc.)

** For example: the attacker will only ever lose up to two troops regardless of if the battle is 10 vs 1 or 100 vs 1

All battles referenced in this post exclude the one attack troop that remains behind. So when we say: 10 vs 10, we really mean a territory with 11 troops attacking another territory with 10 troops because one troop stays behind and doesn’t partake in the battle.

The first example shown is a relatively large 200 vs 200 standard battle with no special dice rule variations. The horizontal axis plots possible outcomes that each simulation could result in, where all the way to the left has the attacker winning with all their troops remaining - and all the way to the right has the defender winning with all their troops remaining. The vertical axis is just the frequency at which the one million simulations landed on a particular outcome for each method.

By comparing the bell-curve peaks, we can clearly see how the Old Balanced Blitz[(OBB)[/b] deviates from True Random (TR) in an undesirable way, where the attacker is favoured more than they should be. New Balanced Blitz (NBB) corrects this and is aligned perfectly with no bias in either direction.

Another point worth noting is that you can see the possible range of outcomes is smaller in NBB compared with OBB. While both guarantee the attacker will win; NBB ensures that the attacker doesn’t lose too few, or too many troops in the process. This is an effect that is greatly exacerbated in even larger battles as seen below.

While not very many Risk matches will feature battles of this size, it is still important to provide results that align with statistical expectations (especially so if a battle of this size decides the match). OBB performs appallingly in this simulation and is so attacker-favoured that it produces outcomes outside of the TR range.

NBB stays well within the bounds of what is considered a reasonable outcome while guaranteeing the win for the attacker. Interestingly, there were over 1,000 simulations that resulted in a win for the defender using TR; this is exactly the kind of statistical anomaly that is not possible with Balanced Blitz. Just need to remember that anything can happen with TR and it will show an even wider range of outcomes if we weren’t limiting it to a measly one million simulations.

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The examples used so far have been with similar sized forces, so let’s take a look at more lopsided battles and closely examine resulting troop losses. We’ll start with a 10 vs 1 where we don’t expect to see any difference between OBB and NBB because it is well below the 70 troop estimation threshold.

Because the battle is so small we can plot out all outcomes on this reverse logarithmic-scaled bar chart. (smaller bar length = more likely outcome) It’s interesting how linear the TR outcomes appear, and it shows that you really don’t need to be super unlucky to lose a 10 vs 1.

As expected, we see NBB is within the margin of error of OBB, and the only thing worth noting for the two of them is that losing more than two troops is impossible. Let’s see what happens when we increase the attacking troop count…

Quite evidently, OBB cannot be considered “balanced” at all, as it follows almost exactly with TR. What’s happening here is that there is a special-case handling for large battles (over 70 troops) with extreme troop ratios (100:1), where it fails to provide an outcome consistent with the smaller 10 vs 1. NBB on the other hand, is able to correctly maintain the same result.

The exact same behaviour can be seen when you reverse the ratio. OBB has potential to give more than expected losses on the defender side for a 1 vs 100 battle and again, NBB corrects this.

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Now, let’s take a look at how this affects Capitals! A general rule of thumb if you want a good chance at succeeding, is to not even think about attacking a capital until you have at least twice the number of troops. In this example we have a little less than double, but we get a clear view of how much a coin toss the outcome would be.

Despite OBB and NBB having a near identical overall win chance (65% compared to TR’s 59%), it is very clear that their outcome distributions differ enormously. The outcome range of OBB is comparable to even TR, however a big difference between the two is that TR shows this trending towards zero at the extremes whereas with OBB you are not that unlikely to land on a near worst/best case scenario compared to the median.

If you’re wondering why there is a dip in probability at the win-lose crossover, this is due to how rounds work in Risk battles, making it incredibly unlikely to end the battle with only one or two troops remaining on either side. This effect is reflected properly in NBB, whereas it has become unnaturally pronounced in OBB.

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That’s all the space for graphs that we can share today. Just know that we’ve got hundreds more spanning all sorts of sizes, troop ratios and dice augments and we just wanted to show the more interesting ones. If you only care about how this affects your game, you can safely stop reading here - stick around if you want to understand a bit more of the technical internal workings.

5. How is this possible now?

As previously mentioned in section (3), there are two primary factors that make New Balanced Blitz improved over the Old Balanced Blitz, and without which they would otherwise be identical.

1. Increased Estimation Threshold
• 70 ➔ 2,000 (in Standard Battles)
  
• 35 ➔ 2,000 (with Capitals and Zombies)
2. Better Estimation Techniques
• Old:Linear regression with a small sample size and edge-case handling for extreme troop ratios.
  
• New: Asymptotic regression with a large sample size, min/max bounding and no edge-case handling.

Both are related to the estimation of large battles and have nothing to do with the balancing algorithm itself.

Increased Estimation Threshold

Estimation threshold refers to the point where the troop count of a battle is so large that it is deemed too computationally expensive to calculate the probability of every outcome on older mobile devices. This was overcome by simply making it faster…

For any programmers out there you might recognize that the compute complexity didn’t exactly change as it still scales quadratically with troop count. However, the rate at which it scales is far, far slower - to the point where we could justify increasing the estimation threshold by several orders of magnitude.

CPU time doesn’t tell the full story of performance, as memory allocation played a severe limiting factor in the old version. That too saw similar levels of improvement and hopefully results in less memory pressure in those big Risk battle games.

Better Estimation Techniques

We won’t be able to go too in-depth with the new estimation techniques because there are quite a lot of steps to it, but one aspect that is shared and directly comparable with the old method is the need to derive a formula used to determine the overall win chance based on the troop ratio. (Attackers : Defenders) The old method achieved this by using linear regression on a sample set of battles below 1000 troops in size. This has been upgraded to using asymptotic regression on a sample set up to 2000 troops.

Above we have an example of a troop ratio that is favourable (9:10) and another that is unfavourable (6:7) to the old method. The reason that the linear method performs reasonably well in the 9:10 troop ratio is because the actual odds moves towards the limit faster. A 6:7 troop ratio is only very slightly in favour of the defender and so it takes a long time for the actual win percentage to get near 0%. Having far fewer samples also resulted in it having a very inaccurate calculated slope causing it to deviate quickly after a little extrapolation.

Regardless, the new method outperforms the old in both cases with lower average deviation from the actual win percentage, simply because an asymptotic curve can better fit the real form that trends to but never reached either 100% or 0%. The “squiggliness” of the line is simply due to the tested troop step size resulting in a ratio that doesn’t quite match the target 9:10 and 6:7 and so it is deriving multiple slightly different formulas per step.

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While the fact that this still requires an estimation method at all, means this could be improved in the future - at least we can say that for 99.999% of Risk battles that consist of fewer than 2,000 troops, the situation is vastly improved. And even beyond that size, it will be significantly more accurate.

A surprising amount of work went into making this a reality despite the fundamentals of it being just some dice rolling. Worth shouting out the several explorations into alternative algorithms and sheer amount of data to process which required a whole suite of custom tooling to help analyze.

So that’s about it for covering the basics of the new Balanced Blitz! If you were once burned by one of the issues in the old Balanced Blitz, we hope these changes restore your faith in the dice once again.

Over and out,

Aiken & Claire
Developers
Team RISK.