Background: I'm currently writing a strategy guide for magic as my pet project, and I'm trying to explain decks with multiple colors. Here's what I have so far:
Dual color decks are useful because they allow you to take advantage of the best of 2 different colors instead of just 1. This allows you to minimize the weaknesses a mono-colored deck might normally have. For example, red decks usually have trouble dealing with enchantments, blue decks tend to have pretty bad creatures, green decks tend to be fairly bad with removal spells, etc. When combined with another color, most of these disadvantages are drastically reduced as you can use cards from each color to compensate for the other's weaknesses.
Some decks also have more than 2 colors, although they are much less common. With 3 colors your deck will have plenty of options to reduce the shortcomings of any individual color as odds are there's a card in at least one of the other colors you're playing that will fill that gap nicely. The downside of a tri-color deck is that it's substantially more difficult to get the mana you need to play the cards in your hand. This means you'll need to devote more of your deck to cards that can generate mana of multiple colors, which detracts from your overall focus of getting cards out onto the field that actually win the game for you. Tricolor decks often include green because it is the best color for generating mana in its own and other colors, making it great support for the rest of the deck. These decks often rely on tricolor spells for finishers as they are often extremely powerful to make up for the difficulty of casting them, to the point where some can almost win the game on their own.
Decks with more than 3 colors are extremely rare as there is usually relatively little advantage a 4th color will add that couldn't be covered by the others, and it is VERY difficult to create a successful 4 color mana-base that will let you play all of the cards you need to in a timely fashion.
When I read this, it sounds like I'm saying "Tri-color decks are bad because they're so hard to get to work correctly and people only play them because of powerful gold cards." However, I'm pretty sure this isn't true and is stemming from my own lack of experience with tri-color decks.
Question: Why would you want to play a tri-color deck? What advantages does adding the third color often add that makes the deck better than if it only had 2? What should I add to this section of my strategy guide to make it a little more complete/useful?