Last week, one tied mayoral race in a Houston suburb came down to selecting names from of a hat:

Sean Skipworth and Jennifer Lawrence were vying to be the next mayor of Dickinson, but they each ended up with 1,010 votes after a runoff election last month and a recount earlier this week.
According to Texas law, a tie in a race for public office can be resolved by casting lots.
Skipworth became mayor after a ping pong ball with his name was pulled out of a hat during a ceremony Thursday that lasted about 10 minutes, the Galveston County Daily News reported…
Dickinson, located about 40 miles southeast of Houston, has more than 21,000 residents.
What a way to peacefully resolve an election.
Two things strike me in reading this. First, many Americans like having local government at a relatively small scale. This is a medium size suburb: it could be small enough to be considered a small town, large enough to feel bigger than a small town. Such elections can be decided by a small number of voters: the article suggests each candidate had 1,010 votes, meaning the front-runners had slightly more than 2,000 votes out of over 15,000 residents who might be available to vote (rough calculations from Census data).
Second, Americans often feel better about their local politics – from their community through their representatives in Congress – compared to national politics. Perhaps people want to think better about those from their places or the stakes at the local level are lower (though local disagreements can get heated). The mayor of Dickinson, Texas may not be able to do much in the grand scheme of things but local officials are often non-partisan and say they are about getting things done.
Because this happened at a very local level, there is likely little from this particular solution – casting lots – to apply to the national level. Yet, the spirit and means of local politics may provide regular reminders of what is possible and how politics can be conducted.