Hello Idle MinMaxers!

Common knowledge is that for market strategy one should focus on Alloy Values (AV), Mercenary Discount (MD) and Additional mercenaries (AM).

But in what proportions? Again, common knowledge is to "Have money -> Invest in AM; Not enough money -> Invest in AV/MD". But what are the numbers exactly?

To answer the this questions, I did some modelling, based on the data and knowledge, that I have about WZ Idle.

TL,DR: First max AV, then do MD

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Goal of the research:
- Maximize total purchased mercs by redistributing the AP spent between "Alloy Values" (AV), "Merc Discount" (MD), "Additional Mercs" (AM) advancements within a set AP limit.

Assumptions:
- All money from markets goes to merc purchases and all mercs are purchased with market money only. This implies that hospitals are upgraded with cache money. At a first glance this seems to be 'fair enough' assumption
- We have all AV (+30%) and MD (50%) techs factored in the calculation
- No other artifacts or CW bonuses are taken into account
- No level or tech routing costs are taken into account
- The step for AP limit increase is 50 AP

I will be examining scenarios for 3 levels of AV artifact:
- No AV - typical "New Player"
- Epic AV
- Legendary AV - typical "Player who has super-ascended"

And I will take a look at 3 levels:
- Old Town - level which, imho, starts the sequence of levels great for market-merc strategy
- Afro-Eurasia Gargantuan - usually it is China, when people start having issues with money. But China also screws people by having 1 market with no alloys, so I selected the next level after it.
- Europe Huge - the pinnacle of regular ascensions
For level research I used the merc camps and markets avaiable on those levels. Mercs are bought in order from cheapest to most expensive.

Charts:
So here you will see what are the best combinations of AV/MD/AM for every total AP spend limit which maximize the purchased mercs (displayed as a % of total armies required to beat a level).

- Level: Old Town:






- Level: Afro-Eurasia Gargantuan:






- Level: Europe Huge:






Key Insights:
- AV is generally superior to MD. A safe casual strat is to max AV first, then do MD.

- The theoretical limit for mercs/level is 37.5%, we can clearly see, that some levels have a "tougher" market economy, so that the limit may not be reached. One may alreay spot the trend:
-- For Old Town: you only need epic AV to be able purchase all mercs. Leg AV makes MD advancement obsolete.
-- For AEG: you need Leg AV to to purchase all mercs. Insane AV is expected to make MD advancement obsolete
-- etc...

- When the line start oscillating, that probably means that a few alternatives are so close to each other in terms efficiency, so that it all comes down to what exact advancement level you are able to purchase with you last AP. In practice, this shows a coridor of acceptable options.

- Increasing AM on its own helps to buy more mercs by shifting purchases to cheaper camps, but this exact effect is small. (You key reason to increase AM is easier routing and to dump other money sources for level completion)

- The early spike in AM > AV,MD for AP < 450 is caused by the significant unlock costs AV and MD.

- When spend AP incrementally you cannot properly balance Advancements in every case (you literally cant get from 100% AV + 50% MD to 90% AV + 70% MD), so just stick the the general trend.

Final notes:
- Technically it is possible to make a calculator, which would take more inputs and present you with the next most AP-efficient advancement. But it seems like to be an overshoot for the issue and still would be hard to account for level- and tech-routing. (unless we aim at a complete "Idle-Solver" which would obviously take all fun away).

Thanks for your attention.

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Usually here I post a fancy call-to-action recruitment banner for Prime. I will not do that now. :p
We are open to new applicants though.

Edited 2/25/2023 16:23:33

why can't we just play risk/warlight like the good old days

I like how you’ve set this up but the results unfortunately don’t resonate for me at all.

I find that I have excess money on every non-Hardened level except the final three (Trisk, US, Europe) from Insane Alloy Values alone. In other words, I consider the optimal AP allocation to pass through the +150% Mercs, +0% Alloy Values, -0% Merc Discount inflection point for nearly every level.

I’m not sure where the disconnect comes from but I hypothesize it’s the “tech routes are ignored” assumption. I don’t think this is a good assumption when you take into consideration that tech can give you +40% alloy values (read: a *lot* more than +40% cash) and -60% merc discount.

That said, if you mentally stack a “free” ~50% on the Insane Alloy Values graphs, then these do pass the sniff test for me so that’s reassuring.

I think it could make sense to repeat this with “reasonable” techs (merc discount and alloy values) baked in.

@graemes

That's probably because insane Alloy Value is indeed insane. That's like having a free built-in 100% Increased Alloy Value on top of a legendary Alloy Value.

As someone with legendary Alloy Value who always start running out of money in the Reconquest level, this post is interesting to me. I've always done Mercenary Discount first, but that's probably not the right thing to do. Maybe in my next run I shall try doing Increased Alloy Value first.

Will be nice if there's a graph for Reconquest. That's always the point when I start having money issues.

@Mathematician

Will be nice if there's a graph for Reconquest. That's always the point when I start having money issues.

That's what I get for Reconquest 1065 with Legendary AV:

I've always done Mercenary Discount first, but that's probably not the right thing to do. Maybe in my next run I shall try doing Increased Alloy Value first.

Yea, AV seems like way better. How I understand the fundamental reasoning:
- MD artifact effectively allows for linear scaling to effective mercs on top of other MD sources
- AV scales profits quadratically. Some good math expert might probably derive the exact formula, while I resort to iterative modelling and charting:


So when you already have an epic/leg/insane artifact, you better further push the total AV % with the AV advancement.

@graemes

I like how you’ve set this up but the results unfortunately don’t resonate for me at all.
....
I’m not sure where the disconnect comes from

The disconnect comes from that, I did not include the charts for Insane value artifact, because wanted to limit the amount of charts in post. There might be too much of the already. So I focused on more common cases of None/Epic/Leg AV artifact.

One may alreay spot the trend (will include it to my insights in the first post):
- For Old Town: you only need epic AV to be able purchase all mercs. Leg AV makes MD advancement obsolete.
- For AEG: you need Leg AV to to purchase all mercs. Insane AV might make MD advancement obsolete
- etc...

I’m not sure where the disconnect comes from but I hypothesize it’s the “tech routes are ignored” assumption. I don’t think this is a good assumption when you take into consideration that tech can give you +40% alloy values (read: a *lot* more than +40% cash) and -60% merc discount.

I think it could make sense to repeat this with “reasonable” techs (merc discount and alloy values) baked in.

Sorry, I believe, that there is some misunderstanding of the stated assumptions:
- The techs have been already factored in: AV at 30% and MD at 50%
- "No level or tech routing costs are taken into account" means that I did not include:
-- costs of materials to unlock the AV and MD techs (in my calculation we get them for free)
-- money "lost" due to that you have to get to all markets first, so you are forced to partly trade at markets with non-optimal rates and purchase mercenaries without full discount

Edited 2/25/2023 11:08:04

Got it. Completely understand and agree with what you’ve done. I had mistakenly thought the third set of charts was Insane and not Legendary.

This is good stuff! Insane Alloy Values is indeed insane and these charts are an indirect argument for upgrading ACB over AV since in almost all cases it’s implied a player would prefer “more than +150% mercs” over “more than +100% AV”.

The data I have paints a different picture. I did this manually for different markets (not that many) and they +/- have the same multiplier within the margin of rounding error so this should extrapolate to all markets since I'm measuring the relative increase. I will explain the format of the 1st sample date:

0% AV <--- total AV
Buy=Sell @ 618000 <---- number of the best alloy I bought to get Buy price = Sell price
Spent: 21 B <---- Input I want to buy 618000 and take the rounded number from the buy window
Sold: 21.935 B <--- Sold 0 of any alloy AND have 0 of that alloy before buying. Use statistics # of money from alloy sold for extra precision.
Profit: 0.935 B <---- Sold - Spent

50% AV
Buy=Sell @ 4490000
Spent: 193B
Sold: 239.0494
Profit: 46.0494

100% AV
Buy=Sell @ 8400000
Spent: 436
Sold: 596.2925
Profit: 160.2925

50% MD => Half price => 100% more money => Always a 2x multiplier to money (only applies to mercs)
000% to 050% AV => 49.25 multiplier to money <--- relative increase => NewProfit/OldProfit => 46.0494/0.935 = 49.25
050% to 100% AV => 3.48 multiplier to money
================================================================================
131% (Epic + Tech + 45%)
Buy=Sell @ 10770000
Spent: 617
Sold: 883.0347
Profit: 266.0347

141% (Epic + Tech + 55%)
Buy=Sell @ 11600000
Spent: 686
Sold: 992.2592
Profit: 306.2592

146% (Epic + Tech + 60%)
Buy=Sell @ 11800000
Spent: 704
Sold: 1030.3
Profit: 326.3

186% (Epic + Tech + 100%)
Buy=Sell @ 15040000
Spent: 1010
Sold: 1526.7
Profit: 516.7


131% to 186% AV => 990 AP spent => 1.94 multiplier to money
50% MD => 1002 AP spent => 2 multiplier to money

141% to 186% AV => 855 AP spent => 1.665 multiplier to money
45% MD => 892 AP spent => 1.818 multiplier to money

146% to 186% AV => 780 AP spent => 1.58 multiplier to money
40% MD => 792 AP spent => 1.666 multiplier to money

If you have an epic artifact you are most likely doing techs therefore stop at 131% AV (Arti + 30% Tech + 45% advancement) and max MD from advancement. You gain 100% more money for mercs with 1002 AP via MD versus 94% more money via maxing alloy values.

MD also works with money from other sources (caches, ore/items sold, etc) and that is usually sizeable. For example in Old Town there is enough money in caches that I could buy all mercs at 150% AM if I desired with only cache money. Due to this for similar multipliers I always give the advantage to MD over AV.

================================================================================
112% AV (Leg)
Buy=Sell @ 9200000
Spent: 494
Sold: 692.2672
Profit: 198.2672

142% AV (Leg + Tech)
Buy=Sell @ 11600000
Spent: 686
Sold: 996.3764
Profit: 310.3764

162% AV (Leg + 50%)
Buy=Sell @ 13100000
Spent: 820
Sold: 1218.2
Profit: 398.2

172% AV (Leg + 60%)
Buy=Sell @ 13950000
Spent: 901
Sold: 1346.8
Profit: 445.8

202% AV (Leg + Tech + 60%)
Buy=Sell @ 16200000
Spent: 1130
Sold: 1736.5
Profit: 606.5

112% to 172% AV => 2.25 multiplier to money
142% to 202% AV => 1.954 multiplier to money

For legendary artifacts usually getting the 2 techs is enough. You gain 95.4% more money spending 976 AP via AV while you get 100% more by spending 1002 AP on MD.

================================================================================
224% AV (Insane)
Buy=Sell @ 18000000
Spent: 1333 B
Sold: 2070 B
Profit: 737 B

254% AV (Insane + Tech)
Buy=Sell @ 20300000
Spent: 1610
Sold: 2550.6
Profit: 940.6

274% AV (Insane + 50%)
Buy=Sell @ 22000000
Spent: 1830
Sold: 2920.4
Profit: 1090.4

284% AV (Insane + 60%)
Buy=Sell @ 22600000
Spent: 1910
Sold: 3080.3
Profit: 1170.3

314% (Insane + Tech + 60%)
Buy=Sell @ 24900000
Spent: 2230
Sold: 3658.9
Profit: 1428.9

224% to 284% AV => 1.59 multiplier to money
254% to 314% AV => 1.52

For insane if you ever need money do MD. If you still need money then do AV. If you still need money change playstyle because I doubt money cache boost will save you at this point xD. Should just stack AM tbh.


tl;dr version

AV is additive and therefore suffers from relative increases what people call "the more you have the less you get out of it" thing. Getting more AV will get you more profit obviously but the relative amount you get before and after upgrade shrinks. MD is multiplicative and simple you always get what it promises aka 2x your money for 50% MD.

The formula for MD% to money multiplier is 1/(1- MD) so for 25% MD => 1/(1-.25) = 4/3 ~= 1.333. Multiplicative discount are of the type "the more you have the stronger they become" therefore MD% is sort of backloaded.

tl;dr #2
Epic artifact => 2 techs + arti + 45% AV from AP then do MD
Leg artifact => 2 techs then do MD

Anyway this is grasping at straws tier min/maxing since there are other factors more important imo.

Edit: Fix lots of typos

Edited 2/26/2023 07:44:48

Thanks for feedback.

While I have double ckecked and 100% sure that:
- the underlying math is correct (it both aligns with my ingame observations and Xeno's calculations)
- the charts are technically correct too

There is clearly something questionable with my interpretation of the data and the derived conclusions.
I might be examining the issue in closer details, and for now please disregard the initial post (idk, maybe I'd better delete it for now, so that no more confusion is spread)

Sorry for false alarm.

Edited 2/26/2023 15:25:01

What I noticed long ago is that it's a little tricky since AV is frontloaded gains while MD is super backloaded. I don't know your method of simulation or data you used but an algorithm that searches for optimal path will tend to pick AV since well the 1st few % are super strong if looked at them individually on a vacuum and it will get stuck investing on them due to the lackluster start of MD. To combat that I decided comparing in batches and used 60% AV which is typically ~1000 AP with 50% MD which equals 1005AP.


The best analogy I have is like playing a game with different characters that have 10 levels for simplicity sakes:

Toon AV starts super strong for the 1st 3-4 levels but for the rest of the levels it has lackluster gains in power. Toon MD starts really really slow as in the 1st 5-6 levels feels weaker than the 1st 2 levels of toon AV but in the last few levels it becomes a god of destruction. XD

In practice the slow start of MD never bothered me so it's more of adapt as you play kind of thing since I can't generalize to how everyone plays but if we are going to be extremely picky and optimize these 1000AP then ya if I'm at the AV breakpoints then I'm going MD >.>. Most people will eventually end up maxing them both anyway.

Edited 2/26/2023 15:35:43

Ok, I have figured out what went wrong.

What went wrong?
(This is for self-reflection mainly. You may skip this part and go straight to the part after the horizontal line.)

1. Misleading charts

Take a look at this chart (posted before):



It is technically correct. The initial math behind it is correct and the chart does show the combinations of AV/MD/AM advancements, which allow for maximum of effective mercs purchase.

Where it fails:

- It fails to show multiple combinations which allow to achieve the same output of effective mercs. It only selects one and does that not effectively.
- It doesnt show, whether you are merc-limited or money-limited. It is when you are merc-limited, then you actually have various combinations of money advancements, which achieve the same effect. The oscillating effect on some of the previous charts is the results of this too.
- It does not allow for the second optimization based on actual AP expenditure. It implies that you spend all AP possible at that limit, even if 1) that may an overkill 2) there are AP leftovers.
- In case of multiple combinations which lead to the same mercs, the optimization process is prone to prioritizing AV (just because, it was the first in the order of listed advancements)

2. Premature conclusions

The TL,DR was too quickly derived, without properly looking at many levels data and charts of modelling insane artifacts.

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Second Attempt

First of all let's look at the same level/arti, but with improved optimization and more data

What's on the charts:
- Besides optimization only for max mercs, I then search for cheaper AP-combinations, which allow for the same max mercs values + when there are multiple combinations for both optimizations, I show them all.
- I also charted:
-- the leftover mercs in camps (thanks to AM cheapening effect, when you dont have money to buy all mercs, you are incentivized to improve AV anyway, which leads to more mercs in camps)
-- the leftover free AP below AP limit on the secondary y-axis
- Now instead of increasing AP limit in fixed steps, I chart results for all unique AP costs of possible combinations. (omg, how long it takes to model all that now)



What we here, is that literally up to a point of total money/mercs (~140% AM) the AV adv is better, but then MD adv becoms superior.
Comparing two charts, on may notice that for example the AV line in the new chart roughly corresponds to the lower limit of AV oscillation in the old chart.

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Not going to bother you all with different charts, etc. (unless someone requests anything in particular)

Basically I get similar conclusions to what Xeno wrote:
- Up to a certain amount of money needed for mercs, AV is is superior, then MD. Whether a certain level fall before of after this threshhold depends on level's economy: how much money can be generated from markets and how much mercs cost. Usually this threshold may be experienced with Leg AV (see the chart on Reconquest, the threshold is literally there) but for some easy levels (Africa), you can meet the threshold even with Epic AV.

Some recommendations:
- With Epic AV or less
-- AV is a safe bet, then MD
- With Leg AV:
-- Depends on level, but you might want to go 50% AV adv -> 50% MD -> Sometimes some more AV
- With Insane AV:
-- Up to Netherlands you basically dont need neither AV, nor MD advancements
-- For later regular levels, go straight to MD, most of the times (except Trisk) it is enough

One other consideration for this thread: we’re focused on the late-level aspect of how many drafts are needed at the end of the level but MD (and not AV) also helps marginally with the early-level aspect of getting to the first market.

That’s makes for a decent tiebreaker since it seems like there isn’t a clear cut winner on the late-game side.