A Game Theory Mathematical Analysis of Alliance War 2.0
JJW
Member, Content Creators Posts: 134 Content Creator
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.
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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:
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.
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:
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
"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:
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.
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:
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
[thank you so very much for clearly illuminating the crystal clear problem with AW 2.0]
Amazing job explaining all of this. Thank you for your time and dedication.
But seriously, we should talk about this.
Free bump to keep this at the top of the list.
Just wow.
Thankyou OP for such a detailed and informative post.
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.
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.
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!
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).
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
"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.
And this is where your theory went wrong.[/quote]
In this context, YOU are the intelligent rational decision-makers.
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.
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.
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...
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.
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.