Problem $1(20$ pts$)$

Your buddy proposes a game to you. They will write a number between $1$ and $10$ on a piece of paper. They will then tell you what number they wrote down, without showing you the piece of paper. Your buddy may be telling you the truth, or they may be lying, and you have to guess whether or not they've told you the truth.

If they're caught in a lie, then they must pay you $$10.$

If they are falsely accused of lying, you must pay them $$5.$

If you correctly guess that they've told you the truth, then they must pay you $$1.$

If you falsely assumed they told the truth, you must pay them $$5.$

If you cheat and they were telling the truth, you must pay them $$6.$

If you cheat and they were lying, they must pay you $$15.$