Why a 25-Year-Old Coin Puzzle Still Divides Philosophers and AI Researchers

Imagine you volunteer for an experiment. On Sunday, a scientist flips a coin but doesn't tell you the result. You go to sleep. On Monday, you wake up. If the coin was tails, you'll be given a drug that erases your memory of Monday, so when you wake up on Tuesday, you won't know if it's Monday or Tuesday. If the coin was heads, you'll be told on Tuesday that it's Tuesday and that the coin was heads. So when you wake up and haven't been told anything, what's the chance the coin was heads? This is the Sleeping Beauty problem, and it's been debated for decades.

Some argue the answer is 1/2, because the coin is fair and you learn nothing new when you wake up. Others argue it's 1/3, because there are three equally likely scenarios: heads and Monday, tails and Monday, tails and Tuesday. The author sides with 1/3, and explains that this matters for betting: if you bet on heads each time you wake up, you'll break even only if you price it at 1/3, since you get two chances to bet if tails and one if heads. So your subjective probability should guide your bets.

But the puzzle goes deeper: can we even assign probabilities to things like 'it's Monday' or 'I'm in the US'? The author says yes—these are 'centered worlds,' which include your location in time and space. It's like knowing which player you are in a video game. Without that, you can't predict what you'll see next. So subjective probabilities about your own time and place are valid and useful.

The author connects this to bigger questions: decision theory, anthropics (how to reason when you don't know your place in the world), reinforcement learning, and even consciousness. The full formal story isn't written yet, but the sketch suggests that understanding imperfect recall—like in Sleeping Beauty—could help us build better AI that reasons under uncertainty. For now, it's a fascinating philosophical puzzle with practical implications for how we make decisions when we're unsure of our own situation.