[Discussion] Speed Scaling

[DarkstaR]

So, some chatter in the shatxbix got me thinking about the current system which increases speed in a linear fashion as you level up. While this system was nice when the average level wasn't over 80 and top levels were below 200, it is outdated and leaves lower levels (even level 100's) at a huge disadvantage.

Let's discuss ways this can hypothetically be modified to better suit the increasing average level of the game.

My idea is below:

old (actually, current) formula = 220 + (2 * (level-1))
new (actually, proposed) formula = 220 + (1.4 * (level-1)) + (1000 - level) / 15

Level 1: {new: 286, old:220}
Level 25: {new: 318, old:268}
Level 50: {new: 351, old:318}
Level 75: {new: 385, old:368}
Level 100: {new: 418, old:418}
Level 125: {new: 451, old:468}
Level 150: {new: 485, old:518}
Level 175: {new: 518, old:568}
Level 200: {new: 551, old:618}
Level 250: {new: 618, old:718}
Level 300: {new: 685, old:818}
Level 350: {new: 751, old:918}
Level 400: {new: 818, old:1018}
Level 450: {new: 885, old:1118}
Level 500: {new: 951, old:1218}
Level 550: {new: 1018, old:1318}
Level 600: {new: 1085, old:1418}


From this, we can draw a few conclusions:
1. With the proposed system, a level 1 is definitely faster, now about the speed of a current level 30.
2. With the proposed system, a level 50 is noticeably faster, but only about the speed of a current level 60.
3. With the proposed system, a level 100 is exactly the same speed.
4. With the proposed system, a level 175 is the speed of a current level 150.
5. With the proposed system, a level 250 is the speed of a current level 200.
6. With the proposed system, a level 400 is the speed of a current level 300.
7. With the proposed system, a level 550 is the speed of a current level 400.

[Farsa]

There isn't enough data so far, but, anyway, I looked into the information given by the update teaser and fitted these points(level/speed):

proposedPoints = [1, 220;
                  30, 1.25*currentFormula(30);
                  100, 1.13*currentFormula(100);
                  175, currentFormula(175);
                  200, 0.96*currentFormula(200);
                  300, 0.83*currentFormula(300)];

to this function:

speed(level)=a*log(b*level+c)/log(d)+e*level+f

using genetic algorithm optimization on MATLAB. These are the values I got for the coefficients [a..f]:

8.4011    0.9807   10.2916    1.1007    0.5820    7.3080

with a summed squared error of

44.1228

which is pretty decent considering the number of points used.

Comparison plot:


Curious random fact about this error value:

mean squared error = 44.1228/6=7.3538
sqrt(7.3538)=2.7118
2.7118 is pretty close to e

MATLAB code:

speed_scaling.m

close all;
clear;

uptolevel=2000;
n_opt_attempts=50;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

level=1:uptolevel;
currentFormula=220+(2*(level-1));
darkstarFormula=220 + (1.4 * (level-1)) + (1000 - level) / 15;
proposedPoints = [1, 220;
                  30, 1.25*currentFormula(30);
                  100, 1.13*currentFormula(100);
                  175, currentFormula(175);
                  200, 0.96*currentFormula(200);
                  300, 0.83*currentFormula(300)];
ppX=proposedPoints(:,1);
ppY=proposedPoints(:,2);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

options=gaoptimset();
options.CrossoverFraction=0.8;
options.Display='iter';
options.FitnessLimit=0;
options.Generations=1000;
options.PopInitRange=[0;10];
options.PopulationSize=1000;
options.UseParallel='always';
options.Vectorized='on';
options.TolFun=10^-15;


X=zeros(n_opt_attempts,6);
FVAL=zeros(n_opt_attempts,1);

matlabpool(4);
parfor attempts=1:n_opt_attempts
    display(['==============ATTEMPT N#' num2str(attempts) '==============']);
    [X(attempts,:),FVAL(attempts)]=ga(@(x)obj_fun(x,proposedPoints,uptolevel),6,[],[],[],[],[],[],[],options);
end
matlabpool close;

best=find(FVAL==min(FVAL));best=best(1);
x=X(best,:);
fval=FVAL(best);
fitPoints=x(1)*log(x(2)*level+x(3))/log(x(4))+x(5)*level+x(6);

fprintf('SQE result: %f\n',fval);
display('Optimization results:');
display(x);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

hold on;
grid on;
plot(level,currentFormula,'r');
plot(level,darkstarFormula,'b');
plot(ppX,ppY,'ok');
plot(level,fitPoints,'g');

set(gca,'XTick',linspace(0,uptolevel,20));
legend('Location','NorthWest','Current','DarkstaR','ProposedPts','FitPts');
ylabel('speed');
xlabel('level');

obj_fun.m

function sqe = obj_fun( x , proposedPoints,uptolevel)
sqe=zeros(size(x,1),1);
for ii=1:size(proposedPoints,1)
    sqe=sqe+(mathFun(x,proposedPoints(ii,1))-proposedPoints(ii,2)).^2;    
end
notinf=sqe~=Inf;


speeds=zeros(size(x,1),uptolevel);
for ii=1:uptolevel
    speeds(notinf,ii)=mathFun(x(notinf,:),ii);  
end

crescent=rowscrescent(speeds);
reals=rowsreal(speeds);

sqe(~(crescent & reals & notinf))=Inf;

end

function speed=mathFun(x,level)
%speed=a*log(b*level+c)/log(d)+e*level+f
speed=x(:,1).*log(abs(x(:,2)*level+x(:,3)))./log(abs(x(:,4)))+x(:,5)*level+x(:,6);
speed(speed~=real(speed))=Inf;
end

function r=rowscrescent(m)
    r= all(m(:,1:end-1)<m(:,2:end),2);
end
function r=rowsreal(m)
    r= all(m==real(m),2);
end

[Sketchy]

Instead of changing the speed point formula you could alternatively change the formula used to calculate the walking step delay. Based on the current test server CIP appears to be making their step delay formula follow a logarithmic pattern, they are also keeping the speed point formula linear with half the value & rate of the current formula (ie: 110 + level - 1).

[Blequi]

I think the current formula is good and doesn't need changes, but if they want some formula to do low level speed closer to high level speed, just setup a formula with a increasing injective function with concavity to down. Ex:

speed(level) = 220 + sqrt(level)
speed(level) = 220 + ln(level) (with base e)

I guess these two formulas are underneath the current formula. If possible, Farsa plot it please.

Maybe those formulas need math.floor or math.ceil if cipsoft doesn't want floating points.

@offtopic: Farsa, are you engineer? If no, where do you use MATLAB? I like programming in MATLAB, I never made some program on it, but its syntax looks like Lua.

[Farsa]

Those are pretty lower speed values indeed, way too low in comparison with what Tibia has nowadays:


@offtopic
*Almost* engineer lol, need to finish final project someday. I used MATLAB for research("iniciação científica") and my final project too, both in the field of machine learning.