House Selling Problem

You want to sell your house. Each day, a potential buyer arrives and makes an offer. The offers $(X_n)_{n \geq 1}$ are i.i.d. and uniform on $[0, 1]$ (in millions of euros).

Each day, you must immediately decide:

• Accept the offer — the house is sold, the game stops.
• Reject — the buyer leaves and never comes back.

There is a waiting cost: each additional day costs you $c$ (in millions of euros).

Q1. What is the optimal strategy? What is the maximum expected gain $V^*$ and the expected waiting time?