Meeting in the middle

A seller wanted fifty tokens for a hat. The buyer offered thirty.

The seller asked $50$ tokens; the buyer offered $30$.

Asked $A=50$, offered $O=30$.

Each of them had a secret limit: the seller would not sell below thirty-six, and the buyer would not pay above forty-five. They met in the middle, at forty tokens.

The seller's limit was $s=36$, the buyer's $b=45$. The middle is $\frac{50+30}{2}=40$, and $36\le 40\le 45$.

Limits $s=36$, $b=45$; price $p=\frac{A+O}{2}=40$, so $p\in [s,b]$.

The buyer handed over a coin worth fifty and got ten tokens in change.

The buyer paid with a coin worth $50$ and got $50-40=10$ in change.

Change $50-p=10$.

Both gained. The buyer paid five less than his limit and the seller got four more than hers. A bargain works when the seller's limit is not above the buyer's limit; any price from thirty-six to forty-five would have done.

The buyer gained $45-40=5$, the seller $40-36=4$. A deal exists when $s\le b$.

Gains $b-p=5$ and $p-s=4$; a deal needs $s\le b$.