A Game Theory Mathematical Analysis of Alliance War 2.0

As a statistician and analyst I intend to offer a suggestion or two regarding Alliance war.

"Game Theory is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers."

In Game Theory all complex decisions can be reduced to a series of Choices within a given Environment in order to achieve desired Outcomes.

I won't bore you with descriptive analogies.

As this applies to Alliance War:
Desired Outcomes = defeat opposing alliance.
Environment = Alliance War map with 9 necessary routes + a given Point Rubric
Choices = Constrained by the environment such that an Alliance has exactly 1 strategic option.

The third item (Constrained Choices) is the visible complaint that you have with Alliance War 2.0, and is seemingly why everyone is upset.

The second item (Environment) is the root cause of the of the Constrained Choices problem.

////////////////////////////////////////////////////

The Alliance War map allows the for the placement of 50 defenders across a map with 92 tiles, 54 of which are capable of containing a defender.
The Alliance War map can be traversed entirely by an alliance moving on as few as 9 paths.

Due to the relative high-value of Boss kills (20,000 each) and Exploration (300 per tile), there are no viable strategies that do not include full exploration. Therefore, those values (totaling 150,000)can be canceled and removed from the victory calculation.

Victory Calculation

= placement * (X1 - Y1) + diversity * (X2 -Y2) + kills (Y1 - X1) + rating * (X3 - Y3)

Where,
placement = 50
diversity = 125
kills = 50
uniquechamps = 106 (at the time of writing)*
rating = 0.02

X1 = Alliance A # defenders
X2 = Alliance A # unique defenders
X3 = Alliance A total defender rating (avg defender rating * X1)
Y1 = Alliance B # defenders
Y2 = Alliance B # unique defenders
Y3 = Alliance B # defender ratings (avg defender rating * X1)

* side-note this calculation is written with respect to the scoring system as presented within the Forums, not as presented within the game itself. I.E. this calculation is consistent with Alliance-Wide diversity.

With the above Victory Calculation in mind, there are now (2) potential strategic options, and only (1) viable option.

Option 1) Strong Defense, attempt to prevent Opposition from 100% Exploration
Option 2) 100% Exploration, Maximize Diversity, Maximize Defense Rating.

The Alliance War Map can be traversed and fully explored by 9 people on 9 routes. An Average Alliance with 10 members will have 1 member out of 10 reserved entirely as backup. So the Map environment does not prohibit 100% exploration.

There is no penalty for an attacker Knock Out so a given determined attacker could revive his entire team for 40% hp 15 times. More than enough to make it through any given path, albeit expensively.

The Alliance War node buffs are regarded as being weak and insufficient from prohibiting a determined assault.

Best possible outcome = Opposition is unable to take 1 or more boss tiles.
Most likely outcome = Opposition will 100% explore map.

Scenario 1 = Max Diversity, High Quality

Let's assume that two imaginary Alliances are able to field 150 defenders totaling 750,000 defense rating (avg defender = 5000). Both are able to provide 106/150 unique champions, both are able to 100% all three BGs.

The score for both teams would be 193,250

Scenario 2 = Minor Diversity, High Defense Quality

Let's assume that Alliance A executes strategic Option 2, while Alliance B forgoes diversity for maximum defender quality, fielding 3 identical battlegroups of 50 unique champions. Both alliance can field champs avg defender rating = 5000

Alliance A = 193,250
Alliance B = 186,250

Alliance B loses by a 7,000 spread, all Diversity points.

Scenario 3 = Minimum Diversity, Highest Defense Quality

Let's now assume that Alliance A executes strategic Option 2, while Alliance B miraculously is able to field the 30 teams of the exact same 5 strongest quality defenders. (probably full mystic teams). Both alliance can field champs avg defender rating = 5000

Alliance A = 193,250
Alliance B = 180,625

Alliance B loses by a 12,625 spread, all Diversity points.

Scenario 4 = Maximum Diversity, No Duplicates

Ok, let's say Alliance B thinks there is some advantage to ONLY fielding the 106 unique defenders. Both alliance can field champs avg defender rating = 5000

Alliance A = 191,050 (can only kill 106 defenders, but larger defense rating)
Alliance B = 186,650 (can kill 150 defenders, but smaller defense rating)

Alliance B loses by 4,400 points, all Defense Rating & Placement overcomes Kills.

When you chart those 4 strategies:
LHwKx2h.png

Only 2 Strategies actually have a viable outcome.

Max Diversity, High Quality can outmatch 3 of 4 potential opponent strategies, and tie against a matching strategy.
Max Diversity, No dupes has an opportunity to win against Min Diversity, High Quality.

GAME THEORY HAPPENING NOW

So, if you are to consider your own potential strategies, Min Diversity, High Quality has no chance of winning against any of the four opponent strategies.

When that happens in Game you discard that as a viable strategy, and you discard that strategy as a viable strategy for your opponent since they will arrive at the same conclusion. This effectively reduces the choice table.

IhAA3TO.png

Now you have 3 potential strategies rather than 4, as does your opponent.

Since we do not know in advance what the opponent will decide, we will assume that each potential strategy is equally likely for the opponent.

Max Diversity, High Quality has 1 chance to tie, and 2 chances to win, a 66% probability of success and a 33% probability to tie.
Minor Diversity, High Quality has 1 chance to lose, 1 chance to tie, and 1 chance to win, a 33% probability of success, and a 33% probability to fail.
Max Diversity, No Dupes has 2 chances to lose and 1 chance to win, a 33% probability of success, and 66% probability to fail.

ANOTHER REDUCTION

Since the Max Diversity, No Dupes strategy has a better chance to reward the opposition than me, I will remove that from my choice of viable strategies. Since I remove that choice for myself, I will assume that my opponent has come to the same conclusion:

dfczJ16.png

And now we have arrived at the root of the problem.

Under the environment and conditions of Alliance War 2.0, there is only (1) viable strategy any rational Alliance can pursue.

If the only way you could play chess was Torre Attack Wagner Gambit, then you would probably not want to play chess.

//////////////////////////////////////////////////////////////////

So what can be done?

Change the Environment such that more than 1 strategem is viable:
Potential options include
  • More PATHS than 10, such that no Alliance can possibly 100% exploration.
  • Introduce COSTS to the equation. Defense kills was a sufficient Cost in AW 1.0, but is not necessarily the only cost that could be introduced.
  • Restructure the points Rubric such that the rewards for Exploration (90,000) and Boss kills (60,000) are not so disproportionate to Diversity (13,250) & Rating (< 2000)
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Comments

  • VoluntarisVoluntaris Posts: 1,133 ★★★
    edited September 2017
    I love you @JJW

    [thank you so very much for clearly illuminating the crystal clear problem with AW 2.0]
  • We're not worthy...

    Amazing job explaining all of this. Thank you for your time and dedication.
  • DD2DD2 Posts: 309 ★★★
    Unfortunately this great thread will die off in this section barely seen by anyone.
  • This is an hours worth of podcast right here :dizzy:

    But seriously, we should talk about this.
  • THX135THX135 Posts: 83
    You should post in Discussion section if you want more people to see this
  • Dexman1349Dexman1349 Posts: 3,060 ★★★★★
    Love the write up.

    Free bump to keep this at the top of the list.
  • HeroBoltsyHeroBoltsy Posts: 763 ★★★
    Wow.
    Just wow.
  • HoidCosmereHoidCosmere Posts: 550 ★★
    @phillgreen I never like to see snowflakes so early in the fall.
  • BozwelBozwel Posts: 8
    Lots of write-up, but frankly it's largely wasted. There's obviously one superior strategy to war - max diversity and highest PI possible. The scoring is so simple, there's no game theory analysis needed. 100% clear, maximize your diversity in the highest PI way possible... and hope it's higher than your opponents' score. Minimal skill required, your ability to influence the result is highly restricted in the short-term (and long-term, it's all about getting new champs... yawn), and there's very little satisfying about a win other than the reward. The system is overly simplistic and, just about no matter how you look at it, broken. It's not fun, it's not based on skill, and it's horrible game design. This isn't a F*@# Kabam post either... it's just a plea to restore war to its former glory.
  • JaybeeRightJaybeeRight Posts: 23
    edited September 2017
    If you read it, it wasn't wasted.

    The merit is in the attempt, and this is a fantastic (and very informative) attempt. I agree on all counts with OP regarding the resolution to AW issues.
  • VoluntarisVoluntaris Posts: 1,133 ★★★
    edited September 2017
    Bozwel wrote: »
    Lots of write-up, but frankly it's largely wasted. There's obviously one superior strategy to war - max diversity and highest PI possible. The scoring is so simple, there's no game theory analysis needed. 100% clear, maximize your diversity in the highest PI way possible... and hope it's higher than your opponents' score. Minimal skill required, your ability to influence the result is highly restricted in the short-term (and long-term, it's all about getting new champs... yawn), and there's very little satisfying about a win other than the reward. The system is overly simplistic and, just about no matter how you look at it, broken. It's not fun, it's not based on skill, and it's horrible game design. This isn't a F*@# Kabam post either... it's just a plea to restore war to its former glory.

    pretty much what the OP game theory post said ... just in a more constructive fashion and through the lens of game theory.... and most certainly NOT wasted by any means.
  • BozwelBozwel Posts: 8
    > pretty much what the OP game theory post said ... just in a more constructive fashion and through the lens of game theory.... and most certainly NOT wasted by any means.

    It's wasted if I can tell you that answer without going through all that analysis. There's no point comparing obviously inferior options to an obviously superior option. The war scoring formula is so simple that it's incredibly easy to determine the superior way to win. Game theory analysis might be interesting where there are multiple viable ways to win, with a given strategy being superior to some other strategies but not others. But that's not the case with the overly simplistic scoring Kabam implemented for the current iterations of AW. There are only a few variables - assuming 100% completion, you will win if you maximize diversity and PI. Period. There is no counter strategy, there is no need for further analysis.

    I'm not trying to **** on the OP - I'm just saying this is a very easy problem to solve and there's no need for pages of analysis. When you have two variables that aren't in competition with one another, maximizing them both is obviously the best strategy.
  • Bozwel wrote: »
    > pretty much what the OP game theory post said ... just in a more constructive fashion and through the lens of game theory.... and most certainly NOT wasted by any means.

    It's wasted if I can tell you that answer without going through all that analysis. There's no point comparing obviously inferior options to an obviously superior option. The war scoring formula is so simple that it's incredibly easy to determine the superior way to win. Game theory analysis might be interesting where there are multiple viable ways to win, with a given strategy being superior to some other strategies but not others. But that's not the case with the overly simplistic scoring Kabam implemented for the current iterations of AW. There are only a few variables - assuming 100% completion, you will win if you maximize diversity and PI. Period. There is no counter strategy, there is no need for further analysis.

    I'm not trying to **** on the OP - I'm just saying this is a very easy problem to solve and there's no need for pages of analysis. When you have two variables that aren't in competition with one another, maximizing them both is obviously the best strategy.

    The different between detecting that there is a problem and understanding why that problem exists in the first place - those are two very different things.

    Many people here have moaned about AW2.0, but few have taken the time to break down the problem and identify the cause.

    AW2.0 is not problematic because of the introduction of Diversity, nor is it the point score system per-se.
    The problem with AW2.0 is that the design is such that there are no strategic options for any of us, so we will all by necessity choose the exact same tactic.

    I would not even go so far as to say the problem is 'easy' to solve.

    You could shift some points around, you could restore defender kills, you could redraw the map a little - all pretty easy things to do. The problem is that if Kabam does not fully consider the consequences of the little changes that they make then I, or someone like me, will quickly write an optimization formula and we will be right back where we are.
  • @JJW just hit people with your brain.

    You know! I shouldn't have thought you needed to be told, but anyone says anything - Hit. Them. With. Your. Brain.

    But thanks, now 2 awesome reads!
  • BozwelBozwel Posts: 8
    JJW wrote: »
    Bozwel wrote: »
    > pretty much what the OP game theory post said ... just in a more constructive fashion and through the lens of game theory.... and most certainly NOT wasted by any means.

    It's wasted if I can tell you that answer without going through all that analysis. There's no point comparing obviously inferior options to an obviously superior option. The war scoring formula is so simple that it's incredibly easy to determine the superior way to win. Game theory analysis might be interesting where there are multiple viable ways to win, with a given strategy being superior to some other strategies but not others. But that's not the case with the overly simplistic scoring Kabam implemented for the current iterations of AW. There are only a few variables - assuming 100% completion, you will win if you maximize diversity and PI. Period. There is no counter strategy, there is no need for further analysis.

    I'm not trying to **** on the OP - I'm just saying this is a very easy problem to solve and there's no need for pages of analysis. When you have two variables that aren't in competition with one another, maximizing them both is obviously the best strategy.

    The different between detecting that there is a problem and understanding why that problem exists in the first place - those are two very different things.

    Many people here have moaned about AW2.0, but few have taken the time to break down the problem and identify the cause.

    AW2.0 is not problematic because of the introduction of Diversity, nor is it the point score system per-se.
    The problem with AW2.0 is that the design is such that there are no strategic options for any of us, so we will all by necessity choose the exact same tactic.

    I would not even go so far as to say the problem is 'easy' to solve.

    You could shift some points around, you could restore defender kills, you could redraw the map a little - all pretty easy things to do. The problem is that if Kabam does not fully consider the consequences of the little changes that they make then I, or someone like me, will quickly write an optimization formula and we will be right back where we are.

    I agree. The solution isn't easy, as it's a game design question and it's a question of what's fun, what do they want to encourage/discourage, what's fair for new and old players alike, what requires skill, and so on. I absolutely agree that's not an easy problem to solve.

    What I think is simple is analyzing Kabam's current solution to that problem, which comes down to two variables assuming max exploration: PI and diversity, with diversity being significantly more important. Obviously if there are two variables that matter and you maximize both of those better than your opponent does, you win. But in any event, I absolutely agree Kabam didn't consider the consequences of their changes, as evidenced by them already having to tweak their new formula once and the tweaked formula still being (in my humble opinion) terrible from a game design perspective. Please don't take this as my crapping on your effort - I just think this sort of analysis is better suited for a rock-paper-scissors context, which would be way more complex than what Kabam managed to implement for the new AW.

    A more interesting analysis to me would be if they added back in defender kills. Now there's a real question as to are you better off placing max diversity but with weaker defenders, or less diversity and hoping you can stop your opponent from exploring 100% via difficult defenders. But that assumes Kabam makes further changes to the AW system (which I hope they do).
  • You had me until X Y
  • [quote="JJW;d-22173"

    "Game Theory is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers."


    [/quote]

    And this is where your theory went wrong.
  • vaderdaman wrote: »
    [quote="JJW;d-22173"

    "Game Theory is the study of mathematical models of conflict and cooperation between intelligent rational decision-makers."


    And this is where your theory went wrong.[/quote]

    In this context, YOU are the intelligent rational decision-makers.
  • I was just saying to a mate the other day when aw was "fixed", Kabam desperately needs to hire a mathematician...
  • GroundedWisdomGroundedWisdom Posts: 26,307 ★★★★★
    edited September 2017
    JJW wrote: »
    As a statistician and analyst I intend to offer a suggestion or two regarding Alliance war.


    As a current SVP I can attest the following:

    Kabam Management: make us more money and cost less, and hurry our end of year numbers needs to kick ass

    Game Developers: But but but fixes and bugs and hackers and stuff and...

    Content Developers: (working on Marvel IP spec screenplays) No one reads those blurbs anyway

    Kabam Management: hows about 6 star heroes? Ok half of you are being laid off, we need to prop up Transformers.

    Kabam Mike: (logs in as GroundedWisdom) Wonk wonk wonk

    I am not Miike. They don't hire shills or come in undercover. I have way too many opinions I like to be able to express freely to be a Mod.
  • GroundedWisdomGroundedWisdom Posts: 26,307 ★★★★★
    JJW wrote: »
    Bozwel wrote: »
    > pretty much what the OP game theory post said ... just in a more constructive fashion and through the lens of game theory.... and most certainly NOT wasted by any means.

    It's wasted if I can tell you that answer without going through all that analysis. There's no point comparing obviously inferior options to an obviously superior option. The war scoring formula is so simple that it's incredibly easy to determine the superior way to win. Game theory analysis might be interesting where there are multiple viable ways to win, with a given strategy being superior to some other strategies but not others. But that's not the case with the overly simplistic scoring Kabam implemented for the current iterations of AW. There are only a few variables - assuming 100% completion, you will win if you maximize diversity and PI. Period. There is no counter strategy, there is no need for further analysis.

    I'm not trying to **** on the OP - I'm just saying this is a very easy problem to solve and there's no need for pages of analysis. When you have two variables that aren't in competition with one another, maximizing them both is obviously the best strategy.

    The different between detecting that there is a problem and understanding why that problem exists in the first place - those are two very different things.

    Many people here have moaned about AW2.0, but few have taken the time to break down the problem and identify the cause.

    AW2.0 is not problematic because of the introduction of Diversity, nor is it the point score system per-se.
    The problem with AW2.0 is that the design is such that there are no strategic options for any of us, so we will all by necessity choose the exact same tactic.

    I would not even go so far as to say the problem is 'easy' to solve.

    You could shift some points around, you could restore defender kills, you could redraw the map a little - all pretty easy things to do. The problem is that if Kabam does not fully consider the consequences of the little changes that they make then I, or someone like me, will quickly write an optimization formula and we will be right back where we are.

    I agree with the majority of this statement. I found your Post to be well thought-out and impartial. I think with any new system, there can be fine-tuning needed. I'm glad to see someone who is seeing past the Diversity issue and examining the system overall.
  • Dexman1349Dexman1349 Posts: 3,060 ★★★★★
    Honestly, the simplest of fixes comes not from changing the points, but rather the ease in which points are earned.

    All of the above revolves around 100% exploration, and many of us agree that 100% is too easily accomplished. This is war, not everyone should get the participation trophy of 100% exploration.

    Simple solution, buff the nodes so that 100% is less common. Force players to double up lines. Provide nodes where true skill is needed because it's genuinely challenging. This is where the previous AW worked. Strategy came from how each team approached the attack.

    If Kabam really wanted diversity, they would key each of the nodes to only a handful of specific champions based on their passive and signature abilities. Maybe even go so far as to provide champ-specific boosts (Node 13: Khan gets ____, Fix it gets ____, etc).

    The way most alliances are approaching things is precisely as noted above: unless that champ stands a reasonable chance at exploration-block, go unique. So wars will continue being the same champs in the boss nodes, but the rest of the map will be a repeat of fighting through the proving grounds...
  • Bump
  • Great post. I just wanna say that, with AW the way it is, a defender, is no longer a defender. All these cool abilities that you guys worked so hard on doesn’t mean anything now. A “good defender”, which die hard players loved to debate and discuss, what’s good for what node etc etc, is no longer meaningful as it once was,or even relevant now that AW is a complete joke. It doesn’t even deserve to be called “war” anymore. It should be called “Daily AQ mode” or alliance vs alliance quest.” All a good defender is now is high PI? Seriously? That’s all. What is fun about that to you? We have enough quests, why turn war into one? You have totally destroyed the only competitive and exciting element of your game that kept it interesting and worth collecting new champs. (Even though I get kabamed on 95% of my spins) I really hope you see the damage you have done to your game and fix it ASAP!
  • Yeah it's roster wars.
    Or someone called it a parade, I think that term applies. Just standing there watching the whole thing slowly go by, and every float has to be different.
  • Great post. I just wanna say that, with AW the way it is, a defender, is no longer a defender.

    BB, fundamentally I agree with you. If 100% Explore and Boss kill is the baseline requirement in AW, then it does not matter at all what champions are in defense. At all.

    So much so that I do not think it is worth your time to set a defense arrangement. It's just going to get sweeper, so save yourself a half hour and let the defenders sit where they are played.

    Someone I interacted with today asked if it was worthwhile playing a boss tile. Essentially, I don't think it matters right now, because all those points are a wash.

  • Ya I was just talking with a teammate about how I think Medusa could be a great defender. But then it dawned on me? Defenders? They don’t matter any more
  • And as far as setting them, yes, I have totally lost all motivation to set them. Why? It doesn’t matter ! Who cares? Nodes are so weak and with the airplane, it makes no difference anyways.Let’s hope kabam makes a move soon.
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