Tug-o-War Territory Capture Model

[Malorn]

This is a proposed model, not the current design.

For the current design model, please see this thread:
http://www.planetside-universe.com/s...ad.php?t=40416

To address some concerns expressed in the current model (expressed in the thread above), I have thought through some alternative ideas for a Tug-o-war type system to wrestle control of territories gradually as opposed to initiating a capture and then having a control race.


The Idea

The core idea behind the tug-o-war model is to move a control indicator point from one faction to another by having a higher net capture strength than the opposing faction(s). One way to think of it is a metal object is between two magnets, with each magnet pulling the object. The magnet with the stronger pull will move the object towards itself until the object reaches the magnet. In the case of Planetside2 there are 3 magnets (one representing each empire), each vying for control of the metal control object. The strength of the pull of each magnet depends on how much friendly territory borders the contested territory, and how many control points each faction has.

The goal is to create the same behavior as Higby and the other PS2 designers have intended, but doing so with a tug-o-war system because it is more natural and avoids some of the quirkier problems of a race-style system (the current design).

One of the challenges in a tug-o-war design is the presence of 3 factions. Many games have a tug-of-war system in the past, Old school WoW had a tug-o-war system in Eye of the Storm (I think this has since changed). In WoW's example, the number of people indicated the strength of the capture. Then the direction the control moves is simply the difference of the two populations within the capture area. This is trivial with two factions because it's a point along a line, a 1-dimensional space. Three factions makes it a two-dimensional space, specifically a triangle. I will address this later on, but first we must work through how territory adjacency affects the strength of the capture and how capture times are calculated with a tug-o-war system. I'll start with two empire scenarios and then describe how to handle a 3-way contest.



The Tug-O-War Model

Strength of capture is proportional to friendly adjacent territory, just as the current design for capture. The main difference is that instead of adjacent territory granting starting tickets it will directly impact the rate of ticket accrual.

In order to make things simple, I have a basic "Tickets per-second" calculation, which when combined with maximum tickets gives us a time to capture. From what Higby has indicated their intended design was, I came up with the following rough constraints.

It should take 30 minutes to capture if there is zero adjacent territory.
It should take 15 minutes to capture if the adjacent territory is exactly half (50%).
It should take 4 minutes to capture if the adjacent territory is entirely friendly.

(Note on the last one - since this is a tug-of-war system, the actual re-secure time in a back-hack situation could be significantly shorter than 4 minutes, but it would be 4 minutes if the enemy succeeded in capturing and they were re-capturing. This might seem a bit off from Higby's expectations but I think it will work out well in-practice).

Since I am looking for a target 15 minutes and looking at tickets-per second, instead of using 1000 tickets, I shortened it to 900. 900 tickets is a nice number because a rate of 1 ticket/second is exactly 15 minutes to generate 900. I made this my target rate for exactly a .5 territory adjacency ratio.

If 1 ticket / second is a 15 minute capture time, then .5 tickets / second is a 30 minute capture time, which gets me the backhack-scenario rate. For the last one I played around a bit with a few formulas until I found numbers that looked right.

T = Territory Adjacency ratio (# of friendly edges / # total edges)

For a facility, the total edges is 18.

Now to for the magic formula that fits our constraints.

R = (T+.5)^(2(T+.5))

I arrived at this formula by much fiddling with the numbers until I found a growth rate that fixed expectations. Below is a table showing some territory adjacency values and the resulting maximum capture rate.

Formula Results Table

0 = .5 (30 min to cap)
.1 = .5417
.2 = .6069
.3 = .6998 (21.4 min to cap)
.4 = .8272
.5 = 1 (15 min to cap)
.6 = 1.233
.7 = 1.548 (9.7 min to cap)
.8 = 1.978
.9 = 2.565
1 = 3.375 (4.5 minutes to cap)

The number on the left is the % adjacent edges that are friendly, the number on the right is the resulting maximum tickets generated per-second that is possible (like if all control points were owned). Notice that .5 = 1, as designed earlier, and 0 = .5, also as designed. The top constraint of 3.375 worked out well to have a cap time of 4.5 minutes, which is just about right. You can see how other rates grow mostly linearly until about 70% territory ownage when it grows rapidly to fit the model where adjacent territory really matters. This encourages battle at the front and makes non-front territories more difficult to capture.

If a territory has multiple control points, then the rate per control point is determined by dividing the rate by the number of control points.

Now for some scenarios & examples to illustrate how this works.


Example 1: Near-even adjacency matchup (55-45)

Suppose the NC own 55% around the edges in a facility, and that facility has 5 control points. The VS own the other 45% and are attacking. This is a fairly close matchup territory-wise, with a small advantage to the NC.

The rate generated per control point for the NC is 1.1079 / 7 = .1582

The rate generated per control point for the VS is .9071 / 7 = .1296

If the VS were to control all 3 of the 5 points, they would generate .3888 tickets every second, but the NC would generate .3164 from the other two. The net rate for the VS is .0724, which would take the VS about 3 hours. They are winning, but not significantly enough. For the VS to really start moving the capture along they need at least 4 of the 5.

And this is the expected result - VS have control over the facility but not a strong grip. They would need a stronger grip to move that capture along faster.

If the VS controlled all 5 points they would capture the facility in about 16.5 minutes.


Example 2: Large Adjacency advantage (70-30)

Suppose the TR own a facility and 70% of the edges around that facility, and that facility has 7 control points. The NC own the other 30% of the edges.

The rate generated per control point for the TR is 1.548 / 7 = .2211 tickets per second, per control point.

The rate generated per control point for the NC is .6998 / 7 = .1000

In this example, the TR (due to adjacent territory advantage) have a little over 2:1 capture rate advantage over the NC. This means the NC will need to control more of the facility to compensate.

If the NC own 5 of the 7 capture points, they will gain tickets at a rate of .5 per second. However, the TR will be pulling against that rate directly, with a rate of .4422 because they own the other two capture points. That means the NC are slowly gaining control of the territory, but at a slow rate of .0578 tickets / second. At this rate, the NC will capture the facility in about 4.32 hours. The rate is so slow they are effectively being held off by the TR's foothold and would have to take another capture point to secure the territory in a timely manner.

If they did capture one more control point, they would be gaining control of the facility at a rate of .3789, which is way faster and will capture the facility in about 40 minutes. If they exercised complete dominance of the facility and had all 5 control points it would be captured at a rate of .6998 and be captured in 21.4 minutes (which is the maximum listed above).

The major obstacle for the NC in this instance was the territory advantage which the TR enjoyed.


Example 3: Behind-lines Capture (100-0)

Suppose the TR own a facility and 100% of the edges around that facility, and that facility has 7 control points. The VS attempt to captures this facility with a daring gal drop way behind TR-VS lines.

The rate generated per control point for the TR is 3.375 / 7 = .4821 tickets per second, per control point.

The rate generated per control point for the VS is .5 / 7 = .0714

Clearly the odds are against the VS pulling this off, and even if they capture all 7 control nodes they are looking at a long 30-minute cap. Further complicating things for them is the high cap strength of the TR. If TR do show up they will only need a few control points to thwart the VS effort.

If the VS own 6/7 points, they are effectively stopped, as their capture rate is .4285, and the TR capture rate for their one point is .4821 - a gain for the TR. The only way the VS will succeed is by holding all the points for the duration of the 30 minute cap. It is OK for them to lose control of a node for a short time, but if they lose more than one any progress made will be quickly undone in a very short time.

This does not prohibit a back-hack in completely hostile territory but it certainly makes it difficult and will likely require a lot of manpower to secure the territory. Even once it is secured it only takes 4.5 minutes for the TR to reclaim that territory, so they must continue to defend it.


Modeling Three Empires

As mentioned above a significant challenge is modeling 3 empires. The above design will work to model three empires, but the modeling becomes a little more complicated (which is why I suspect the devs may have avoided this implementation). Conceptually, a 3-way Tug-O-War is a 2-dimensional space, with the control "object" having different forces acting upon it to move it towards one empire or another. This can be represented as an equilateral triangle, with each empire in a corner of the triangle. Calculating movement of the object is a matter of assigning a vector pointing to an empire's corner. The magnitude of the vector is the capture strength of the empire.

Here's a simple example of the capture object being "tugged" in two dimensional space. The vector notation shows that in this example the object has a strong VS component and so the object is moving towards the VS.



Like any vector, in a 3-way situation the empires are all pulling on the object and the one pulling the hardest makes it move towards them. In this case the "pulling the hardest" is the effective capture rate.

In this example each "Side" of the equilateral triangle is 900 units long, which is our "ticket" size expressed earlier. So it really isn't tickets/s, its a vector magnitude, measured in distance / s, o units / s traveled.

Normally this might be fairly difficult to model, but I expect with the physics libraries available to the PS2 team they can actually model this like they would a simple physics system, with an object having vectors pointing towards each of the empire and the vector components are the result of the capture magnitude.

Also since we know the current position of the capture object and it's movement vector, we can also calculate an ETA for capture for easy representation to players. This ETA would change if control points or adjacent territories changed hands.

It's also worth noting that in this 2-dimensional representation, the "Neutral" state of a facility not owned by anyone could simply be represented as an object that starts in the center of the triangle. The rest of the capture mechanic functions like the empires. This means neutral bases would be captured about 50% faster than non-neutral territories if they are uncontested.

Most contests will likely not need the 2-dimensional model as it will be 2 empires contesting over one piece of territory. In those instances the control object will move along one of the edges of the triangle rather than through the middle of it.


UI Representation

So now that I've described the mechanic - how do we express it to players in a meaningful way?

Well there are several key things we want to convey to users:

1) Capture strength of each empire
This could be represented simply as a bar showing the adjacent territory ratio. For example, a bar 50% full would indicate that the empire has 50% adjacent territory. Since capture strength is a direct function of adjacent territory this should give a good at-a-glance indication of how strong a claim each empire could have to a territory.

2) Control point ownership status
Ideally the control points should be labeled so we know which points need to be captured/defended, but at the least we should see how many control points in the current territory a given empire has.

3) Control object status
What is the position of the control object? Is it close to being captured? Far from capture? This is a quick visual indicator showing at-a-glance how close an empire is from capturing the territory.

4) Dominant Empire
Quite simply which empire has the strongest claim to this territory right now.

5) Time-To-Capture
Based on the current situation, how long before the dominant empire will capture the territory?

I have a mock-up of the control UI that indicates all of these key piece of information below.



The triangle area itself doesn't necessarily need to be that large. It could be shrunk down a bit. The VS/TR/NC labels may be overkill but there' here for illustrative purposes. These could easily be represented as small empire icons.

As mentioned above this mockup shows the control status for each control point at the bottom. Top-center shows the dominant empire & capture ETA. The dominant empire is represented by the color of the timer. An icon next to the timer might also be warranted.

Here's a more minimalist mockup where the size of the colored corner is the representation of capture strength. I think this one is easier to read and it clearly tells me the critical information - Which empire has the strongest claim? How long before they cap? Which control points do they have?





Augments

This model can be augmented fairly easily with modifiers to capture strength.

If we wish to give an empire a handicap or a bonus to capture rates they can be applied flatly to the capture strength formula.

For example, perhaps an empire has an excessively low population and we wish to give them a 20% capture time bonus. It's a simple multiplier to the capture strength formula. That empire will now have an easier time capturing territories.

Same goes for handicaps. If we want to make it difficult to own a lot of territory we could assign a capture time penalty based on the number of hexes an empire controls. In that case the capture time could be increased by reducing capture strength, which would also make it easier for other empires to capture that empire's territory. This is one good way to help the Rich Get Richer problem by making larger empires harder to maintain and slowing their ability to capture more territory.

Another possible augment is to introduce the concept of neutrality. If control moves too far away from an empire that owns the territory it might go neutral and move to a contested state where neither empire controls it. This might have impact on other territory control struggles also but could make "neutralizing" a territory a viable tactic to deprive it from assisting neighboring territories for control purposes.



Capture/Contest/Resecure Experience

Since this model can result in contests for territories that may span into hours for a large battle at a tightly contested territory rewarding players only for a successful capture is insufficient.

One idea to combat this is periodic experience awards for those in the capture vicinity as long as they remain capturing the area. The greater the contest (i.e. the more enemy present, the more control nodes the enemy has), the greater the periodic experience awards. In this way players are being rewarded for participation AS they participate. Actual capture of course should yield a bonus proportional to the amount of time spent in the struggle, which could be substantial.

Things that might yield more experience
- Being near a control point when it is flipped to your team
- Gaining experience near a friendly control point in a contested territory might have a defense-bonus
- Gaining experience near a hostile control point might have an offensive bonus applied

Also fully capturing or fully re-securing a base should yield xp proportional to the time it was contested while the player was around it. Sort of like the time-slices from PS1.

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I've really done it this time, reached post character limit, lol, TBC in post # 2.