Maximizing Rubberband Scoring - Way Too Much Math

Nemek

Ok, so this thread is going to be long and rough, but the thought exercise was interesting enough to share. A whole lot of credit goes to dannyshan for figuring out the formula for rubberbanding (so uhh...not sure if you wanted it to be public on the forum, but you've made it pretty public in the FB group...)

So everybody knows that to maximize rubberbanding, you need to wait until the last possible moment to do your missions, but which missions do you do first? The high point ones? The low point ones? Does it depend on your current modifier? It's tough to tell. 99% of the time, it really doesn't matter. The gains are AT BEST minimal. But, it could mean the difference between winning and losing.

What's the rubberbanding formula?

It actually became fairly apparent what the rubberbanding formula was in The Hulk PvE event. We had always assumed that the rubberbanding multiplier was determined by your distance from the leader, but it was kind of a guess. As The Hulk went on (especially the last non sub-event missions), it became more obvious. With The Hulk, we always knew who the global leader was in the event since it wasn't bracketed at the main level. While I don't know exactly how dannyshan came to the formula, I'd guess he just assumed the multiplier was linearly determined from the difference of points between yourself and the leader and noticed that the max multiplier seemed to go away at around the 4000 point mark.

So, you can make a guess that the formula for the multiplier would look like:
(<top score> - <your score>)/400

Then with the added condition that the multiplier could not be below 1 or above 10.

So, if you are looking to do a mission with base points of 100, and you are 3500 points away from the leader, then your multiplier would be: (3500)/400 = 8.75. So the mission would be worth 875.

The formula was tested many times across different times and players to know that it was accurate in The Hulk.

Well, that's for The Hulk. We don't know who the global leader is for the other PvE events. How do you know it's still true?

You're right - from a single player's perspective, they could never be truly sure it's the same, there just aren't enough data points, since we don't know what the top scorer's point total is. It's, however, highly likely that whoever leads your bracket is pretty close to the leader (within a few hundred, unless the bracket is new or has pretty lazy people.) You could notice that the max modifier changes at 'around' 4000 away from your bracket leader, but it doesn't actually prove it's the same.

dannyshan apparently has multiple accounts all competing in the PvE events. So, he was able to capture two or three data points for at the exact same time, which means that the top score would be the same for all data points.

So, you could assume that the old formula is still in use, and then compare answers:

(X - <player 1 score>)/400 = <modifier1>
(X - <player 2 score>)/400 = <modifier2>

Since you know both player scores and both modifiers (if you knew the base amounts of the missions you were playing), you could show that the top player score is the same for both calculations.

So, you can conclude (or, probably...just make a super awesome guess), that the formula holds true.

Ok great, now that we know the formula, what could we actually do with it?
When the formula was presented in the FB group, that's basically the response dannyshan got. Nobody seemed to care, which made me slightly sad. But, it provides the information necessary to come to what is likely the 'optimal' way of scoring points in the PvE events. Really...it's probably completely unnecessary to do unless it's the last refresh before rewards go out.

Get on with it, what's the most optimal route?
Wait, but you haven't seen the math yet!

There are a couple of different cases you need to consider:

• I just started the event during the last hour, and I'm 50000 points away from the leader. (Or something so far away it doesn't matter.)
• I'm just barely over 4000 points from the leader, one or two missions will get me below that.
• I'm less than 4000 points away from the leader.

Let's say we have to choose between three different missions, each worth different points:
A : 100
B : 75
C : 50

Case 1: I just started the event during the last hour, and I'm 50000 points away from the leader.

This is obviously the easiest case. It doesn't matter which one you do first, since you'll always get a 10X multiplier.

10 (100 + 75 + 50) = 2250

Case 2: I'm just barely over 4000 points from the leader, one or two missions will get me below that.
and
Case 3: I'm less than 4000 points away from the leader.

(Because it's easier to show both at the same time.)

For this example, let's say you are 4600 points away from the leader.

Here are the six different combinations you could have done:

ABC
A: 10x multiplier * 100 = 1000 points, 4600 - 1000 = 3600 points from leader
B: (3600/400) = 9x multiplier * 75 = 675 points, 3600 - 675 = 2925 points from leader
C: (2925/400) = 7.3125 * 50 = 366 points, 2925 - 366 = 2559
2041 points gained.


ACB
A: 10x multiplier * 100 = 1000 points, 4600 - 1000 = 3600 points from leader
C: (3600/400) = 9x multiplier * 50= 450 points, 3600 - 450= 3150 points from leader
B: (3150/400) = 7.875 * 75 = 591 points, 3150 - 591 = 2559
2041 points gained.

(Obviously, that doesn't look impressive, you get to the same spot!.)

BAC
This exercise is left up to the reader.
BCA
This exercise is left up to the reader.

CAB
C: 10x multiplier * 50 = 500 points, 4600 - 500 = 4100 points from leader
A: 10x multiplier * 100= 1000 points, 4100 - 1000= 3100 points from leader
B: (3100/400) = 7.75 * 75 = 581 points, 3100 - 581 = 2519
2081 points gained.

CBA
C: 10x multiplier * 50 = 500 points, 4600 - 500 = 4100 points from leader
B: 10x multiplier * 75= 750 points, 4100 - 750= 3350 points from leader
A: (3350/400) = 8.375 * 100 = 838 points, 3350 - 838 = 2512
2088points gained.

So, you can see that CBA gives the highest point total of all of the combinations (trust me on BAC/BCA).

What can we conclude from this?

We can see that there is no difference between the points gained from ABC/ACB. Knowing the multiplier formula, we should have actually known that to be true, though, since it's just linear (if I'm bored, I'll write a proof). The conclusion from that is:

• After passing the 4000 point mark, it doesn't matter which one you do first.

In CAB/CBA, a different conclusion can be made. In those two scenarios, doing the first mission doesn't go past the 4000 point mark, so the second mission is the one that crosses the 4000 point threshold. The mission you choose for that is the last mission that is important.

• When crossing the 4000 point threshold, do it with the lowest point mission possible (well, the mission that you actually plan on doing)

You can basically extrapolate those two points to conclude that you should be trying to stay as far away from 4000 points for as long as possible (in terms of picking high point missions in a run, I don't mean that you should never cross into the 4000 point threshold.)

• Save your high point missions for after you hit 4000 points away from leader.

This sounds SO unintuitive. We've already concluded that it doesn't matter what happens when you're within 4000 points, so why are we 'wasting' our high point missions at this point? Well, it's true.

Ok, so that gives us these simple rules to follow:

• After passing the 4000 point mark, it doesn't matter which one you do first.
• When crossing the 4000 point threshold, do it with the lowest point mission possible (well, the mission that you actually plan on doing)
• Save your high point missions for after you hit the 4000 points threshold

EDIT for those who might actually dig this thing up again:

It's said later on in this thread, but after you cross over the 4000 point threshold (or whatever the threshold is for when you lose 10x multiplier), it is still best to save your high point missions for absolutely last. There are two reasons for this:

• While the multiplier is completely linear between 10x and 1x, it does floor out at 1x. So, when you are within 400 points (or, well, possibly the leader), 1x is actually HIGHER than what your multiplier would be if it were a true linear function. So, missions done at 1x are actually given more 'weight' than the ones before it.
• A lot of the conclusions above assume a constant point value for the leader. This is obviously not likely to be true. The leader's points are highest right before the end of the event, so you'll get a higher multiplier at that point.

Nemek

More questions!

But!!! That all assumes that the leader never changes point total, that's so totally unlikely...he/she will likely have done a mission in the same span of time that I've done a mission. How does that change things?
That's even better! If the leader is going up in points, that makes the multiplier even better for the later missions (which are your high point ones!)

How else can I apply this information?
One thing that people mention every time an event is ending is that they need to wait until the last possible moment to begin your run through your refreshes. Well I'm here to tell you that's only partially true (ok, it's still like 90% true.)

Now that we know it's better to do your low point missions before you reach the 4000 point threshold, and we also know that the multiplier has a maximum value (of 10), we can see that doing your low point missions early is not only non-detrimental, but actually better. This is, of course, as long as you stay further than 4000 points out (which sometimes happens after only a couple of missions). And then, you have even more time to do those high point missions in the last 30-45 minutes (or whatever amount of time you allocate for your final push.)

Does this ALWAYS work?
Well, only if the event is fairly standard. For instance, in this last event (Thick as Thieves), the absolute biggest trick to finishing in the top spot was that last sub-event. If you didn't join the sub-event (or at least didn't do many missions in it), you could have ground all of your main event missions to <50 points in the last hour, and then switch over to the sub-event, where the last mission was worth around 800 points. This was because the main event missions had no effect on the sub-event rubberband, but the same was not true the other way around.

Is this even worth doing?
In my opinion, not really. I don't think the difference in points is going to be terribly drastic, but who knows. I do think it's worth doing your low point missions early (before 4000) since it will give you more time going into the final stretch. But, if you were super super efficient and your low point missions don't refresh until you are about to make your final push, anyway...it doesn't really matter.

Unknown

Keep in mind that the next event featuring DD, they said they are reducing rubberbanding effects and focusing the rewards on progression

Unknown

Great job with #'s and stuff. That was prolly hard to figure out.

I'm like 4 classes and 12 beers into my day/night so all I'm getting is:
Think back to "21 Jump Street" and Channing Tatum saying "fk you science!!!" Right after explaining something or other about radioactive spiders.

Yup. That's where this took me. It took me straight to "fk you math!"

Unknown

Glorious! Please copy/paste to Tips and Tricks. It might be worth emphasizing even more the importance of that threshold, since discounting all the unknowns it's relatively easy to grasp intuitively, I think. And please do write that proof. I'm practically innumerate, but such things bring me comfort.

Another thing I think can be relevant is the head-games aspect, if you're shooting for high placement. Even if you're losing multiplier, you might want one or two high-value payouts reserved until very late, as long as you're confident you can blow them away in a minute or less. If you're in a tit-for-tat one point grindfest with someone over #1 and #2, jumping at least 200 and probably more (assuming you're not the overall leader) with three minutes left might be a touch underhanded, but it will make them give up. And they can do the same to you, after all.

Unknown

For any new players that get turned off by (for instance) the length of this post I think most would agree with me:

If you're only going to read posts by one forum member Nemek is far and away the best choice.

Unknown

Well done! Easy to grasp and apply. Excellent analysis makes it easy to forget how I was robbed in this PVE event.

Unknown

Oversoul wrote:

Well done! Easy to grasp and apply. Excellent analysis makes it easy to forget how I was robbed in this PVE event.

By explaining part of how we got robbed makes you forget getting robbed?

Unknown

The ordering only matters if the number of stacks is unlimited (like The Hunt). In Thick of Thieves, each sub bracket has around 350X3 = 1050 total base points. I've never seen any competitive sub bracket where a guy still had a stack of a 50 point mission left when he took the lead, which should be possible if it uses the above formula. I really doubt this is because nobody thought of 'do hardest mission last', and by the way I was doing hardest mission first in these brackets, and yet in these competitive brackets the difference is always on the order of 5 or so points, which is simply the difference between the speed one can grind down 1 point missions. Since I did not do any 1 point mission at all (too tiresome), and I'm pretty sure the guys I"m competing did (their points were definitely going up 1 at a time) I don't see how the order could've mattered unless everyone just happened to choose exactly the same order.

However, saving the hardest mission for last has an important impact of being able to catch the bracket leader off guard. This is especially true in the unlimited stack missions like The Hunt, where you can in theory very slowly chip away the bracket leader's lead by doing the lowest point missions and then suddenly do all your highest missions to get a large lead. That said, I noticed most of the time the bracket leader is simply plugging away at the highest point mission (at least back when boosts were available) and you'll actually see his lead getting bigger over time.

Unknown

Interesting. I'd been assuming that the formula was something like (<leader score> - <your score> + 400) / 400, to make sure that the rubber-banding factor was never less than 1. You're saying that in the Hulk event the factor was just proportional to the score difference but that the formula was only applied over (400, 4000). I guess my question is: you're sure that the rubber-banding factor (at least in the Hulk event, blah blah) really is just proportional to the score difference, with no offset? Because if there is an offset built in in order to handle the case where your score is within 400 of the leader, you get a different result. Was it really the case that you had a rubber-banding factor of 1x when within 400 points of the leader?

Phantron wrote:

The ordering only matters if the number of stacks is unlimited (like The Hunt). In Thick of Thieves, each sub bracket has around 350X3 = 1050 total base points. I've never seen any competitive sub bracket where a guy still had a stack of a 50 point mission left when he took the lead, which should be possible if it uses the above formula. I really doubt this is because nobody thought of 'do hardest mission last', and by the way I was doing hardest mission first in these brackets, and yet in these competitive brackets the difference is always on the order of 5 or so points, which is simply the difference between the speed one can grind down 1 point missions. Since I did not do any 1 point mission at all (too tiresome), and I'm pretty sure the guys I"m competing did (their points were definitely going up 1 at a time) I don't see how the order could've mattered unless everyone just happened to choose exactly the same order.

However, saving the hardest mission for last has an important impact of being able to catch the bracket leader off guard. This is especially true in the unlimited stack missions like The Hunt, where you can in theory very slowly chip away the bracket leader's lead by doing the lowest point missions and then suddenly do all your highest missions to get a large lead. That said, I noticed most of the time the bracket leader is simply plugging away at the highest point mission (at least back when boosts were available) and you'll actually see his lead getting bigger over time.

If this exact formula (with the 400 scale factor) was used in the sub-events, then rubber-banding in the sub-events would just not have been very extreme. A 100 base value mission is always going to be worth more than a 20 point base value mission if nobody ever gets more than 2000 points.

Nemek

Gotchaye wrote:

Interesting. I'd been assuming that the formula was something like (<leader score> - <your score> + 400) / 400, to make sure that the rubber-banding factor was never less than 1. You're saying that in the Hulk event the factor was just proportional to the score difference but that the formula was only applied over (400, 4000). I guess my question is: you're sure that the rubber-banding factor (at least in the Hulk event, blah blah) really is just proportional to the score difference, with no offset? Because if there is an offset built in in order to handle the case where your score is within 400 of the leader, you get a different result. Was it really the case that you had a rubber-banding factor of 1x when within 400 points of the leader?

I believe so. Otherwise, we would never know what the base values of the missions really were unless we were the top player (or close enough to the top player so that it's rounded out). It's often the case that when I make a huge run-up to close to the top, the base values would reveal themselves. (Technically, I don't know if I wasn't the top player, but the likelyhood was low.) If there were an offset, only the person in front will know the values.

But, I guess we would really know it at the start of a PvE event. If we start seeing rubberbanding before the leader got to 400 points, then we would know there was some sort of offset. And...I feel like we've probably seen that, so you could be right.

Unknown

Psykopathic wrote:

Oversoul wrote:

Well done! Easy to grasp and apply. Excellent analysis makes it easy to forget how I was robbed in this PVE event.

By explaining part of how we got robbed makes you forget getting robbed?

Yup. It's weird, isn't it? It's like a form of closure, almost. You learn, take your licks and quit the game... ahem, I mean move on to the next challenge!

Unknown

Oversoul wrote:

Psykopathic wrote:

Oversoul wrote:

Well done! Easy to grasp and apply. Excellent analysis makes it easy to forget how I was robbed in this PVE event.

By explaining part of how we got robbed makes you forget getting robbed?

Yup. It's weird, isn't it? It's like a form of closure, almost. You learn, take your licks and quit the game... ahem, I mean move on to the next challenge!

Hell yeah! We just gotta piss off a few unmentioned idiotards in the mean time. Hehe.

Unknown

I'm assuming this applies to the sub events as well? I found it a little frustrating when trying get those Patch covers I kept falling short getting what I needed...

Eddiemon

I hate you Nemek. You made me do the actual math to work out why this is so.

For those who care, at a 9x multiplier, you are 2600 points away from the first place person.

You have two missions valued at y and z points respectively.

If you do y first, you get 9y points. The multiplier for z is set to (3600 - 9y) / 400. 3600 is the number used because to have a multipler of 9 you must be 3600 points away from the leader.
So total points is 9y + ((3600 - 9y)/ 400)z
= 9y + (3600/400)z - (9y/400)z
=9y + 9z - 9yz/400

For z first:
9z + ((3600 - 9z)/ 400)y
= 9z + (3600/400)y - (9z/400)y
=9z + 9y - 9yz/400

which is the same as the first formula just with the variables in a different order.

Eddiemon

But wait, there's more

As long as you don't get down to a 1x multiplier, then your order doesn't matter. But once you get there strange magic happens. I used 3 sets of 200,100,50 and 25 point missions to get these figures. Base points is the mission base points and Scored Points is that with the multiplier applied.

Big points first

Points
Behind Multiple Base Scored Total
4000 10 200 2000 2000
2000 5 200 1000 3000
1000 2.5 200 500 3500
500 1.25 100 125 3625
375 1 100 100 3725
275 1 100 100 3825
175 1 50 50 3875
125 1 50 50 3925
75 1 50 50 3975
25 1 25 25 4000
0 1 25 25 4025
-25 1 25 25 4050

Small points first

Points
Behind Multiple Base Scored Total
4000 10 25 250 250
3750 9.38 25 234.38 484.38
3515.63 8.79 25 219.73 704.10
3295.90 8.24 50 411.99 1116.09
2883.91 7.21 50 360.49 1476.58
2523.42 6.31 50 315.43 1792.01
2207.99 5.52 100 552.00 2344.00
1656.00 4.14 100 414.00 2758.00
1242.00 3.10 100 310.50 3068.50
931.50 2.33 200 465.75 3534.25
465.75 1.16 200 232.87 3767.13
232.87 1 200 200.00 3967.13

The problem is the math formula I used before is correct up until you get the multiplier below 1. At that point the game replaces the multiplier with 1, breaking the formula. In the Big points first table, we had broken out of the formula by the fifth mission. This happens when our total score gets beyond 3600 points.

From the first table you can see it took 700 base points to break through the 3600 point barrier. (Which also means we picked up about 2950 bonus rubberband points).
From the second table we spent 925 base points to achieve this, which means we only picked up 2842 bonus rubberband points in that time.

If the multipliers had been allowed to descend below 1, then the second table would have caught up with the first. But because it didn't, the first comes out well ahead.

Nemek

Oh that's interesting. We can use the fact the multiplier has a minimum value to prop up those last few missions.

Unknown

All this mathematical stuff makes my head hurt... I just use some old fashioned blood and thunder.

In other words, I beat what I can, when I can

Kiamodo

I'm with ya Ladder. But you can't argue with Nemek's results. He blew past me for a top ten spot last night for DD. Work smarter not harder I guess.

Unknown

This is where the fun has been taken out of the game. I don't want to have to keep track of all this garbage. I enjoy playing. Match three is fairly mindless and it's how I like to spend some free time when kids aren't crawling on me. Why oh why should I have to do math and calculate perfect playing times in order to get the most out of this game. That's called work and I do it for a living.

I was ranked 1 - 10 in my bracket mostly hanging around 1-4 the entire event except the last day. I was pushed out to 25 and it was an extreme struggle to place top 10 in the end. I don't understand how that is even possible (I do, your math is good) it just shouldn't be this way. Sigh.

Kiamodo

I hear ya. I do math all day at work. I have no intrest in marvel math quest.