orb suggestion: cost reduction

[vulderbeak]

(10-11-2015, 11:27 AM)Aaron Storm Wrote: In that example it is multiplicative. Doesn't matter if it's multiplied inside parenthesis or outside as long as it's never added. For example 3*3*3 is the same as 3*(3*3) but 3*(3+3) and 3+(3*3) are different. If it was included in the event rate additions then it would not be multiplicative.

This is incorrect.

When two bonuses of +x% and +y% stack mutliplicatively, the total bonus is given by
(1+x%)*(1+y%) = 1+x%+y%+xy%% (i.e. 1+x/100+y/100+xy/10,000),
for a total percentage bonus of (x+y+xy/100)%

It's called multiplicative because you multiply the ratios together to get the result.  You have a 2% increase stacking multiplicatively with a 3% increase?  The result is a ratio of (1.02)*(1.03).

When two bonuses stack additively you don't multiply the ratios, you add the differences.  A 2% increase stacking additively with a 3% increase yields 2%+3%=5% increase.

Or, more generally an x% and y% increase stacked additively yield an (x+y)% increase.

The situation with the experience and gold orbs is more complicated.  The stacking is neither multiplicative nor additive with any other factor.  Rather it is a percentage increase on the eRate Brouwer described.

[NOTE: Brouwer describes eRate=(1+orb%)*(1+global%)*(event*+support+account%+etc%).  Not to confuse things, but in what follows i have written
erate% =(1+global%)*(event%+support%+account%+etc%), without the (1+orb%).  If we can agree on terminology i can replace my erate% with a more suitable term.]

At first glance it may appear to be multiplicatively stacking with the erate, but that in fact would look like
(1+orb%)*(1+erate%), whereas what we actually have is (1+(1+orb%)*erate%).

To calculate the true effect of the xp orb we should examine the ratio of the experience with an orb to the experience with no orb:
[1+(1+orb%)*(erate%)] / (1+erate%) = (1+erate%+orb%*erate%)/(1+erate%) = 1+orb%*[erate%/(1+erate%)]

So we can see that the actual impact on our xp is not an increase of orb% (which is what we would expect from true multiplicative stacking, i.e. if it were independently mulitplicative with each other factor, the way party boost is), but instead has a coefficient of [erate%/(1+erate%)].

In other words, a 2% xp orb will not increase our xp received by 2%, but rather by 2*[erate/(1+erate)]%, or if you prefer, 2*[1 - 1/(1+erate)]%.  As erate becomes quite large, this value will approach 2%, but if we have very little erate (e.g. if we have low account upgrade for experience), the value of the experience orb will be greatly diminished.