A mathematical model of the cRPG metagame

[Rhekimos]

Actually this is taken into account, this line :

fitness[myClass] = fitness[myClass] * (1-pow(pop[myClass],2))

means that the closer the current pop of the class is to 0, the higher it's fitness will remain, simulating the fact that the last players of a class will not move so easily.

I'm sorry, but that's not taking the human sitting in front of the computer into account. That's adding a parameter to your function that makes it look less flawed, while making the exact same assumption about the average player. Science wise, a cosmetic change.

Now there's a few nutjobs who are acting against their best interest in the population. Most of the population is still only after kills, and will switch over to anything that killed them. That's not how you play, is it?

Also, you are not forced to interpret the fitness matrix as a measure of battlefield effectivity of a class against a certain class. It can represent how funny playing a class is.

I won't have fun as a polearmer if I end up against a 2h player? That's what your matrix would say, isn't? Can you model different actors with different payouts? Not everyone likes the same things after all, and some have a principle of their own against going ranged for example. Can you factor culture into it?
I think it would be a little naive to claim that how a player acts can be represented with a single number for each class.

Also I applaud your efforts, but I feel I must point out that this doesn't prove much. Our interactions with the game and each other are much more complex than this, and most importantly not everyone plays only to win. Models are necessarily simpler than reality, but if they are based on an incorrect assumption, no amount of tweaking will help it make correct predictions.

[Vibe]

Spends 1 hour writing a quality renown-whoring post, gets +5
Vibe posts an image of a carrot in the same thread, gets +8

learn the way of renown you must, young padawan

[Armpit_Sweat]

I found some math too:

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EDIT:

Spends 1 hour writing a quality renown-whoring post, gets +5

Vibe posts an image of a carrot in the same thread, gets +8

+1 for you both

[Moncho]

This is a model. A quite thought through model, but as with any model, it cannot take into account every single element of the metagame. It still does a good job imo, from the numbers I saw in agweber post. And of course it can be improved, but even in the current state, it gives an overall idea.

These seem reasonable with the biggest amounts being 2h/pole, then ranged, shield, but with cav I think there is something pretty wrong, due to the fact that cav is not the best on 1v1 with aware people, but it shines in backstabbing and group fighting. You could represent this by adding a random factor to cav (representing the awareness of the target), or just take the average and increase its effectivity against other classes by a bit. I would suggest increasing cav v cav to 3, see what this does, as after all cav fights other cav at the beginning of the round often.

Unfortunately, I cannot run the script now to check what my changes do, might do it later though

[Kafein]

I'm sorry, but that's not taking the human sitting in front of the computer into account. That's adding a parameter to your function that makes it look less flawed, while making the exact same assumption about the average player. Science wise, a cosmetic change.

Now there's a few nutjobs who are acting against their best interest in the population. Most of the population is still only after kills, and will switch over to anything that killed them. That's not how you play, is it?

Actually, this tweak makes low pop classes inherently "attractive". This means if this class is getting more players, it will grow faster, but it will also lose fewer players in the opposite situation. Players do not switch classes linearly. The very hardcore single-class players will only leave if their class becomes unplayable. The most volatile, min-maxing players will switch as soon as they see an opportunity to do "better", and this represented more or less accurately. If a class is very popular, only a few players of that class are the super hardcore type, and it is therefore easy for that class to lose players. If a class is very unpopular, only the most determined will stay, and those guys are hard to convince to switch. If you discretise the population and think of each class as a stack of players, with the most volatile on top, it is easy to understand.

I won't have fun as a polearmer if I end up against a 2h player? That's what your matrix would say, isn't? Can you model different actors with different payouts? Not everyone likes the same things after all, and some have a principle of their own against going ranged for example. Can you factor culture into it?
I think it would be a little naive to claim that how a player acts can be represented with a single number for each class.

The model does not represent the behavior of an isolated player, but rather that of the whole population. Things like taste can be averaged and included in the payoff matrix.

Also I applaud your efforts, but I feel I must point out that this doesn't prove much. Our interactions with the game and each other are much more complex than this, and most importantly not everyone plays only to win. Models are necessarily simpler than reality, but if they are based on an incorrect assumption, no amount of tweaking will help it make correct predictions.

It gives an idea of how the metagame can evolve, and how things that seem unrelated (shields becoming stronger against 2h and the amount of mounted ranged for example) are connected. We lack the base data to establish correct parameters for the model, but doing simulations with values that seem "close enough" already displays interesting non-linear behavior. It also questions how do we think about the metagame, maybe helping people to formulate their ideas and suggestions better.


Valid remarks nonetheless.

[Polobow]

Where's WaltF4 when you need him?

[CrazyCracka420]

I was told there would be no maths

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[Lt_Anders]

The numbers have pretty much stabilized. Ranged Cav are no longer around. Cav's barely holding on.

Time :  164.2
Ranged  0.24
Shield  0.20
2h      0.27
Pole    0.27
Cav     0.03
HR      8.07676550724e-09

(rounded the numbers)

Interesting. 2h=pole followed just behind by range.

Wut do we have now...Shiedless heavy servers, with a lot of range. Sheilders are there, but not as much as 2h, pole, range.

Also, you should probably break up ranged a little more, Something like archers and then xbow/throwing. Becasue xbow is bad dedicated, but a lot carry so it should be somehow weighted separately.

Also, BUFF HA, they clearly suck and this model shows it too!

@Crazy:
WUT?

[Rikthor]

I feel this is an appropriate response to this thread

[Leshma]

I realize I should do this with matlab and post graphs but heh I did this for fun and I tought it would be nice to post it, if you are into that kind of things, enjoy

Do it with matlab and also on paper (using real math).

[bilwit]

Do it with matlab

Yeah, not sure why he would just post source code and not even have any comments explaining his reasoning on top of not showing any results or analysis either. Why do it in python and not matlab anyway where you can readily pull graphs and shit out of it if the purpose was truly to show it to the community?

No flowchart, no results, not even a detailed description outside of the vague OP. The OP is basically, "here is a wall of source code I came up with, compile and analyse it yourself if you can even though I can easily make the assumption that most people here aren't coders or mathematicians." If I was a peer reviewer and this was a presentation I would throw 0's at you.

[PhigNewtenz]

I think it's an interesting concept. I do modeling like this for a living, and I'm going to give the approach some more thought. First blush: it's a good model. I'm not sure if the 'Payoff' matrix is set up correctly, but without real data that's difficult to know.

For people who don't speak code, here're the results of Kafein's simluation in graphical form (only plotted to time = 50, because nothing interesting happens after that):

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For those who speak MATLAB:

(click to show/hide)

Code: [Select]

N = 500; %Number of iterations to run

s = .1; %Time step per iteration

tRec = s:s:(N*s); %Time vecot for plotting

classes = {'Ranged','Shield','2h','Pole','Cav','HR'}; 

P= [2 1 4 4 4 3; ... %Payoff matrix; needs analyzing
    5 3 2 2 1 1; ... %    Elements are effectiveness of class[row] against
    1 4 3 4 3 1; ... %    class[column]
    1 5 3 3 5 1; ...
    4 5 2 1 1 2; ...
    2 1 4 3 4 2];

%Initialize Variables
popRec = zeros(N,length(classes));
pop = ones(size(classes))/length(classes);

for i = 1:N
    %Evaluate fitness of each class against current population
    f = pop*P'.*(1-pop.^2);
    t = sum(f.*pop);
   
    %Update population
    f = f - t;
    pop = pop.*(1+f*s);
   
    %Record values for plotting
    popRec(i,:) = pop;
end

%Plotting
figure
area(tRec,popRec)
axis([-inf inf 0 1])
xlabel('Time Index')
ylabel('Portion of Population')
legend(fliplr(classes),'Location','EastOutside')
title('Population Evolution')
It's not pretty or very well commented, but it is equivalent to Kafein's code. I just translated it quickly, as I didn't have much time. I'll revisit it later.

I understand that the fitness = fitness * (1-pop^2) is trying to model the resistance of 'the last, hardcore players of a certain class to change'. The relationships are definitely non-linear, and this shows some of that. Can any other math people comment on whether they think this is the right way to do it?

[PhigNewtenz]

So I made one minor change to the 'Payoff' matrix and ran it again. I changed all the diagonal elements to '3', because I think every class is break even against itself. The results were amusing, but also convinced me that this model (or my assumption that classes are equal against each other) needs some tweaking.

Are you ready for an eerie (but clearly not realistic) prediction of the future of cRPG?

(click to show/hide)

40% ranged, 35% cav, 25% ranged cav; This should be fun...

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If people have their own cases they want to see plots for (a case being some combination of initial conditions and 'Payoff' matrix), post them here. I'll run them if I'm on, or maybe some else can grab the code and run with it.

[genric]

This is cool. I like it.
Not enough pure positive comments so this is mine to you

[Kafein]

Do it with matlab and also on paper (using real math).

I don't think anybody would like to compute each step on paper. A bullet in the head is faster. And afaik there's no easy way to directly derive results at an arbitrary time without doing all the steps with models such as this. But as PhigNewtenz showed, in matlab it's quite awesome (btw I don't have matlab at home, otherwise I would have used it in the first place :s)

Yeah, not sure why he would just post source code and not even have any comments explaining his reasoning on top of not showing any results or analysis either. Why do it in python and not matlab anyway where you can readily pull graphs and shit out of it if the purpose was truly to show it to the community?

No flowchart, no results, not even a detailed description outside of the vague OP. The OP is basically, "here is a wall of source code I came up with, compile and analyse it yourself if you can even though I can easily make the assumption that most people here aren't coders or mathematicians." If I was a peer reviewer and this was a presentation I would throw 0's at you.

I don't want to start an argument (even though this sentence is not a very good idea to acheive that goal), but when I started the thread I wasn't trying to do a presentation. If I wanted to release serious stuff only, I would have spent a lot of time, which I didn't. I only thought it would be interesting to share, and it seems to me I wasn't that far from the truth. Other than that you are right.

I think it's an interesting concept. I do modeling like this for a living, and I'm going to give the approach some more thought. First blush: it's a good model. I'm not sure if the 'Payoff' matrix is set up correctly, but without real data that's difficult to know.

For people who don't speak code, here're the results of Kafein's simluation in graphical form (only plotted to time = 50, because nothing interesting happens after that):

visitors can't see pics , please register or login



For those who speak MATLAB:

(click to show/hide)

Code: [Select]

N = 500; %Number of iterations to run

s = .1; %Time step per iteration

tRec = s:s:(N*s); %Time vecot for plotting

classes = {'Ranged','Shield','2h','Pole','Cav','HR'}; 

P= [2 1 4 4 4 3; ... %Payoff matrix; needs analyzing
    5 3 2 2 1 1; ... %    Elements are effectiveness of class[row] against
    1 4 3 4 3 1; ... %    class[column]
    1 5 3 3 5 1; ...
    4 5 2 1 1 2; ...
    2 1 4 3 4 2];

%Initialize Variables
popRec = zeros(N,length(classes));
pop = ones(size(classes))/length(classes);

for i = 1:N
    %Evaluate fitness of each class against current population
    f = pop*P'.*(1-pop.^2);
    t = sum(f.*pop);
   
    %Update population
    f = f - t;
    pop = pop.*(1+f*s);
   
    %Record values for plotting
    popRec(i,:) = pop;
end

%Plotting
figure
area(tRec,popRec)
axis([-inf inf 0 1])
xlabel('Time Index')
ylabel('Portion of Population')
legend(fliplr(classes),'Location','EastOutside')
title('Population Evolution')
It's not pretty or very well commented, but it is equivalent to Kafein's code. I just translated it quickly, as I didn't have much time. I'll revisit it later.

I understand that the fitness = fitness * (1-pop^2) is trying to model the resistance of 'the last, hardcore players of a certain class to change'. The relationships are definitely non-linear, and this shows some of that. Can any other math people comment on whether they think this is the right way to do it?

Awesome

About that line, I tried a few alternative formulas, but that one seemed to give the closest results to what I wanted (lower pop -> better fitness). A greater exponant leads to reducing the influence of this tweak, but 2 seemed "okay" (see that, I'm being unprofessional again ).

So I made one minor change to the 'Payoff' matrix and ran it again. I changed all the diagonal elements to '3', because I think every class is break even against itself. The results were amusing, but also convinced me that this model (or my assumption that classes are equal against each other) needs some tweaking.

Are you ready for an eerie (but clearly not realistic) prediction of the future of cRPG?

(click to show/hide)

40% ranged, 35% cav, 25% ranged cav; This should be fun...

visitors can't see pics , please register or login

If people have their own cases they want to see plots for (a case being some combination of initial conditions and 'Payoff' matrix), post them here. I'll run them if I'm on, or maybe some else can grab the code and run with it.

I had set different values to reflect that I think some classes tend to have a better time fighting internally. Fighting other 2h when you are 2h yourself is not really a problem. But for cav or shielders it can be a tedious process. This also reflects the fact that some classes have fewer counters than others. Mounted ranged can only be fully countered by ranged and other mounted ranged. If an enormous part of the playerbase was mounted ranged, the fitness of mounted ranged would stay high because classes like 2h would be "obsolete". Some classea are self-reinforcing and some others are self-regulatory, or at least that's how I see it.