In most countries, there are existing rules of the road and driving etiquette. For the most part, drivers follow the rules and maintain some semblance of order. This includes the notion of having "the right of way" in certain driving situation when interacting with other drivers on the road. Now, the concept of having "the right of way" is a simple one. For example, at an intersection where there are cars trying to cross, generally the right of way belongs to the vehicle that arrived at the intersection first. Once that car traverses the intersection the right of way then transfers to the car that arrived at the intersection after the first car. Simple, fair and effective.
Sounds crazy, right? Well, that's how other people will drive here and this is how you must drive here as well to survive. And despite the perceived chaos and insanity here there actually IS some semblance of order or a "way" that people drive here. Take this exact instance as a good example of this "order" as well. After you "make your right of way" in the above example, the other drivers that have been forced to stop will typically wait and let the cars that may be behind you to go for a bit as well as a kind of "courtesy" instead of bullying back in once you have gone. See, driver's in Manila aren't that cutthroat.
So like I said...do away with the notion of "right of way". Here you have to grab the bull by the horns and make it your right of way. Just remember that other drivers will be trying to do this as well. As long as you keep some balance between your aggressiveness and defensiveness like I mentioned in my previous post, you should be perfectly fine.