Use the Multiply transparency option.
Just FYI, be careful of your terminology. "Additive color" refers to color derived by combining (adding) multiple different frequencies of light (in other words, to things that glow.) Subtractive color refers to color derived by removing (subtracting) multiple different frequences from white light that falls on them (in other words, things that absorb light and reflect only what's left).
So in your example, wherein overlaid colors become darker, that's a description of subtractive color, not additive color.
The difference between additive and subtractive color is why your TV set glows green and red whenever it needs to display yellow (additive); yet you can't make yellow by mixing crayons--you have to have a yellow crayon to begin with.
I guess some others don't understand the terminology difference between additive, subtractive, and multiply transparency modes.
In an additive mode, the color values literally add together. So if I have 3 layers, one of #FF0000 color, one of #00FF00 color, and one of #0000FF color, and they are all in additive mode on top of eachother, they should turn into white or #FFFFFF. This is like a RGB spectrum color wheel, and I'm assuming it's the effect you are looking for. This type of transparency is available commonly in many imaging platforms including Lightwave 3D.
Multiply transparency does what the name suggests: multiplies the color values, so this is not at all what you want to do.
I'm assuming that additive color is what you're looking for. Unfortunately in Illustrator and PS this is not available. What is available though is Subtractive color which is the opposite. So if you want to get the additive effect in AI or PS, you have to put a Black (#000000) underlay under everything, and then set them all to Subtractive color. This will subtract them all - and then subtract them from black, which will invert the result and turn them into Additive. Another option is to use Subtractive, and then Invert the result.
First, this thread is three years old.
This is like a RGB spectrum color wheel, and I'm assuming it's the effect you are looking for...Multiply transparency does what the name suggests: multiplies the color values, so this is not at all what you want to do.
The thread originator showed a screenshot of what he was wanting to do. And in Illustrator, using Multiply Transparency Mode is the way to do it.
Additive colour means starting off with darkness and adding light.
So make a black area and add a red light (red ellipse).
Now copy the ellipse, make it green and set its trasnsparency mode to Screen. Put it on top of the red and you've got yellow.
Screen means TV or movie screen and the term has been wrongly translated on some foreign versions of Illie. Screen in this instance has nothing to do with halftones.
As JET rightly points out what you are showing is subtractive mixing, best reached with Multiply transparency mode on a white background. The more colours you add this way, the more light you subtract, hence the term.
I have to agree with JETalmage as he is interpreting what the OP misunderstood as additive.
And Ihave to agree with you not everyone understands additive but in this case it is the OP and not JETalmage.