Determining when the Brokers are guaranteed to lose in The Index

[CK-1996]

In the Index, once the Tenno team has fulfilled their Team Deposit requirement, they have two choices: stand around, waiting for the timer to run out, or intentionally kill themselves repeatedly to give the Brokers points to shave seconds off of the timer. What if, instead, there was a prompt that, given the following conditions:

• The Tenno have achieved their Team Deposit requirement
• The Brokers are guaranteed to lose

allowed the Tenno to decide whether to extract victorious or to keep going (for John Prodman etc.)? The first condition is easy enough to account for, but how can the Brokers' guaranteed loss be determined? I offer a solution in this post.

First, the optimal strategy that the Brokers can use to gain the most score within a given time is to hoard all of their points until the last second. That way, they do not need to worry about cutting 10 seconds off of the time they have for every point that they bank. In order to calculate the Optimal Score Increase, labelled O, we must consider the Maximum Index Point Gain Rate, labelled M, the Time Left, labelled t, the Number of Points in Play, labelled P, and the Bank Bonus, labelled B. O can then be calculated as:

O = Mt + P + B

P is simply the sum of all Index Points either being held by players (whether Tenno or Broker) and the ones that haven't been picked up. t is simply the time left on the counter.

M can be determined by considering that upon death, a player will drop one Index Point. They will then respawn after a given Respawn Time, labelled R, and be invulnerable after respawning for a given Invulnerability Time, labelled I. After that, they can be killed again. Assume that a player is always killed immediately after they become vulnerable. Then the maximum rate at which this player can yield Index Points, which I will call the Individual Index Point Rate and represent with M_i, is:

M_i = 1 / (R + I)

For variables relating to the Tenno, I will use subscript T. Given that there are n Tenno in the Tenno team, the Tenno Index Point Rate, labelled M_T, is:

M_T = n_T / (R_T + I_T)

The variables on the right-hand side are already determined as:

n_T = 4, R_T = 8, I_T = 10

Thus the value for M_T is:

M_T = 4 / (8 + 10) = 4/18 = 2/9

For variables relating to the Brokers, I will use subscript C (for Corpus). The equation for the Broker Index Point Rate, labelled M_C, is:

M_C = n_C / (R_C + I_C)

I do not know the values for the variables on the right-hand side, but DE should have the information to be able to determine them (i.e. how many Brokers can be in play at any one time, how long do they take to respawn, and how long are they invulnerable for after respawning?).

So, given that every player is dying as fast as they can, M is:

M = M_T + M_C = 2/9 + M_C = 2/9 + (n_C / (R_C + I_C))

B is dependent on how many points the Brokers can gather within the time left, i.e., it is dependent on Mt + P. B It has different values depending on how large Mt + P is, defined as:

0  if 0 ≤ Mt + P < 5
2  if 5 ≤ Mt + P < 10
4  if 10 ≤ Mt + P < 15
8  if 15 ≤ Mt + P < 20
10 if Mt + P ≥ 20
    
We now have all the variables needed to determine O:

Let the score of a team at any point in time be represented by S, where S_T is the score of the Tenno and S_C is the score of the Brokers. Define ΔS as:

ΔS = S_T - S_C

At a given time left t, if O < ΔS, then the Brokers cannot win.

This is my solution for deciding when the Brokers have lost and, thus, when the prompt should come up to give the Tenno the choice to either leave or stay.

As for the prompt itself, it might say something like:

You have achieved your Team Deposit goal and at this point in time, the Brokers are guaranteed to lose. Would you like to extract victorious or keep going?

or something, I'll let DE decide. Could be accompanied by Nef Anyo saying how this is a waste of time and scolding the brokers (more). Just ideas.

Anyway, I hope this explanation hasn't been too confusing. Feedback and/or corrections would be appreciated.

Edit: I posted the first version of this calculation to Reddit, to which I got a reply from u/nametype_generic offering corrections, hence the complete re-calculation on both platforms. I thank them for their correction.

Edited August 13, 2017 by CrasherK
Complete re-calculation after realising the Brokers could use a better strategy, formatting

[Nez-Kal]

I started reading this, agreeing with the points the OP brought up, scrolled down a bit......and suddenly MATH!!~

 

(Good idea though, in my opinion. X'D)

[True_Naeblis]

Pretty sure the current system is just for the sake of making the grind take longer. I would love to see it removed, as it's silly, so I'm right there with you.

[Wiergate]

I'm an Arts&Humanities guy, and I'd rather not know what 'formatted in latex' means, but yes: I agree that once certain conditions have been met we should be able to extract before the timer runs out. As you suggested, there are several creative ways to have that work with a simple narrative device.  

As things stand, a 'bad spawn' to me is one that takes a long while to kill me effectively once I've secured my victory, which can't exactly have been what DE intended.  

Edited August 15, 2017 by Wiergate

[CK-1996]

LATEX is a maths formatting code that generates nice-looking equations like the one above. Also, after posting this on Reddit, I have been getting feedback and have noticed a large error in my assumptions. Edit to ensue.

[CK-1996]

Re-calculation is complete.

[mrhapps]

@CrasherK

I too wish for the index to let you leave without having to wait for the timer.

Why would [if(tenno.points => 50){startavote();}] not work instead of your complicated math. (for medium index that is)

startvote() could include something where every 25-50 points would start another vote to leave (similar to how defense works)

Edited August 15, 2017 by mrhapps