I guess it may come as a surprise to you, but people don't just get green cards. You can try to get one either through job, investment, family ties or a DV lottery. Actually, as a US-born person, you are probably not qualified for a DV lottery.

This is absolutely not true. People regularly relinquish without having to pay. If you VOLUNTARILY commit an expatriating act WITH THE INTENT of relinquishing U.S. citizenship, you can lose it at no cost. The most common method is to voluntarily become naturalised in another country with the intent of losing your U.S. citizenship. The State Department has a questionnaire (form DS-4079) that has to be filled out which helps them determine your intent. Among the questions are whether you own any property in the United States, if you hold any professional licenses from the United States, whether you have used your U.S. Passport, voted in any U.S. elections, paid U.S. taxes and whether you have visited the U.S. since you committed your expatriating act, in this case became a naturalised citizen of another country. There are countless former citizens who have, and continue to, relinquish their U.S. citizenship without having to pay the fee to renounce. The key is once you have committed the potential relinquishing act, do not do anything that would make you look like a U.S. citizen (e.g. travel on your U.S. passport, vote in a U.S. election, pay U.S. taxes). If you have any of those things after you have committed the relinquishing act then you are out of luck and have to pay the fee to renounce.

The Issac Brock society in Canada is a great resource if you are considering renouncing or relinquishing. Among the resources are testimonials about the relinquishing and renunciation experiences from former Americans for various U.S. Embassies and Consulates around the world. Many of them are able to obtain their Certificate of Loss of Nationality (CLN) without having to pay the renunciation fee.