Bill is a real estate agent, and Susan has engaged Bill to help her buy a house. Susan tells Bill that Susan can afford to pay a maximum of $$600,000$ for the house purchase, but that she would like the best deal possible. Bill shows Susan a house for sale that Susan likes very much $($nice neighborhood; good schools; close to work; etc.$)$ with an asking price of $$550,000.$ Susan $($with Bill$$s help$)$ makes an offer to purchase the house for $$500,000.$ The seller$$s agent receives Susan$$s offer to purchase the house for $$500,000$ and calls Bill $($Susan$$s real estate agent$)$ to discuss Susan$$s offer with Bill. During that discussion, the seller$$s agent casually inquires of Bill what the maximum amount Susan can pay is$.$ Bill replies $$Susan told me that the maximum she can afford to pay is $$600,000.($Note that Bill$$s real estate commission is calculated as a percentage of the real estate transaction price $($typically $3\%),$ and that Bill will make more money in commission with a higher priced transaction. If the house sells for $$500,000,$ Bill$$s commission for acting as Susan$$s agent will be $$15,000(3\%$ of $$500,000)$; if the house sells for $$600,000,$ Bill$$s commission will be $$18,000(3\%$ of $$600,000).)$ The seller then rejects Susan$$s offer of $$500,000$ and makes a counter$-$offer to Susan of $$599,950.$ Susan, who really wants the house, then reluctantly accepts the counter$-$offer and pays $$599,950($just $$50$ under the maximum she can afford$),$ wishing very much that she had been able to purchase the house more cheaply.

Has Bill done wrong on these facts? Explain.