Economic outcomes are not symmetric
Since economic systems are not contained like this, we know that in real life systems Person A and Person B will likely engage again in some form through the market, and it is possible that the person who gave up the $1 initially will see some percent of that value come back to them (and then maybe back to other person, and so on). Because Person A and Person B are different agents, it is not guaranteed that both the initial moves described above will result in symmetric outcomes over the long term.