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To escape the friend zone, you must first realize that all relationships involve negotiation - and you are attempting to "re-negotiate" the current exchange. Essentially, you want "more" from the other person. Most likely, you are already giving too much and what you really want is for them to balance the scales.
--> Be Less Interested - The relationship is already imbalanced because you value it more than the other person. Take a step back. Being "needy" is no way to negotiate. Desperate people end up with what others give them, not what they want. So, be less interested and ready to walk away if you don't get the relationship you want. Those who are more willing to walk away have the power to guide the relationship (called the "Least Interested Principle" - Waller & Hill, 1951).
--> Make Yourself Scarce - Spend some time away from your "friend" and do less for them. If they truly appreciate you, then your absence will make them miss you and want you more. This is the principle of "Scarcity" - where people value something more when it is rare or taken away from them (Cialdini, 2009). When you are no longer around as much or tending to their needs, they will most likely feel the loss. This will increase their desire for you and their willingness to meet your needs back. If it doesn't, then they are just "not that into you"...and don't value you. In that case, find another "friend".
--> Get Them To Invest - Ask your friend to do things for you. Contrary to popular belief, people like you more when THEY do favors for you, rather than when you do the favor for them (for more, see here). This is called the Ben Franklin Effect (Jecker & Landry, 1969). The more they invest in the relationship, the more you will mean to them. So, stop doing favors...and start asking for them. Get them to give you a ride, study with you, fix something, etc. Heck, even asking them to get you a soda from the fridge has an impact!