Sawtooth Problems

Red Button, Blue Button On April 24th, 2026, Tim Urban put forth the following poll on Twitter/X:Everyone in the world has to take a private vote by pressing a red or blue button. If more than 50% of people press the blue button, everyone survives. If less than 50% of people press the blue button, only people who pressed the red button survive. Which button would you press?I love this dilemma, and I'm exhausted by it. I’ve been thinking about it for two straight weeks, and have spent nearly all that time refining my thoughts by writing this piece. It's consumed me in a way that I've never before experienced with any math problem, and I need to get it out of my head.Discourse surrounding the Button Dilemma reminds me of polarizingly political topics. In much the same way that political discussions make people go funny in the head, answers to the Button Dilemma tend to elicit vitriol from people of both Red and Blue conviction. Everyone feels their answer is clear, and everyone is confounded by the lack of consensus.I think this dilemma is pointing to something very important and fundamental about coordination problems. The variety of answers it receives, the arguments, and the emotional fervor feel raw in a way that I've never seen so cleanly extricated from political policy arguments.There's something very strange happening inside the Button Dilemma's universe, and the more I stare at it, the more I find.I'd like to share some of my observations.[1]What Are We Even Arguing About?I see a common thread whereby the Button Dilemma is approached as a sort of lateral thinking puzzle. Is everyone awake and lucid? Are blind people forced to pick at random? How do we account for people who don’t have hands and can’t press buttons? Are children expected to choose a button? Well if children are involved, then I’m definitely choosing blue!I endeavor to omit these sorts of questions from the exercise we're about to walk through.If you find yourself wondering about questions like