Take a simple case: toss a coin repeatedly, and whenever it comes up heads, add 50 percent to your wealth. When it comes up tails, subtract 40 percent of your wealth. If the sole criterion were expected value, you should play the game. In fact, you should borrow money and bet more than you have.
But you exist in just one of those worlds
“If a small investment increases my expected wealth, then a large investment increases it even more,” Peters said of this faulty reasoning. “Result: it looks as if I should leverage as much as I can, borrow 100 times the money I have, (or even better 1000 or 1,000,000 times), and invest it all. This mathematically naive perspective would fool me, and I will be bankrupt pretty soon.”
The problem is that the expected value is typically averaged over parallel worlds. But you exist in just one of those worlds.
Plot this coin-flipping game out over 1000 or 1 million iterations using the parallel-worlds approach, and gradually the random fluctuations smooth out, showing a clear overall upward trajectory. Sounds great, right? But when Peters and Gell-Mann took just one world’s average over time, their models showed the opposite conclusion. Instead of ending up with a clear upward trajectory, they saw a pronounced downward trajectory in the resulting plot.
That’s the disconnect. Crunch the numbers in aggregate (the parallel worlds, or ensemble method) and we all collectively appear to win. Do the same with the trajectory of a single world line using the time-centric method, and we lose as individuals.