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Ergodicity: Difference between revisions
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{{a| | {{a|bi|<youtube>https://youtu.be/f1vXAHGIpfc</youtube>}} | ||
Fans of {{author|Nassim Nicholas Taleb}} and {{author|Rory Sutherland}} will know all about it, but for mathematics clots like me, it’s new, and it’s taken two weeks to get my head around and I probably have it wrong. I am no mathematician and I’m in [[Off-piste|way over my skis]] here — but since it’s the only time that’s going to happen in season 20/21, I’m going to enjoy it. | Fans of {{author|Nassim Nicholas Taleb}} and {{author|Rory Sutherland}} will know all about it, but for mathematics clots like me, it’s new, and it’s taken two weeks to get my head around and I probably have it wrong. I am no mathematician and I’m in [[Off-piste|way over my skis]] here — but since it’s the only time that’s going to happen in season 20/21, I’m going to enjoy it. | ||
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But anyway let me try. In what follows, to make it easier, I am assuming coins cleave exactly to their probabilities at all times. It would be possible to get different results by improbable sequences, but the longer the sequence the more improbable variations become. | But anyway let me try. In what follows, to make it easier, I am assuming coins cleave exactly to their probabilities at all times. It would be possible to get different results by improbable sequences, but the longer the sequence the more improbable variations become. | ||
Ergodicity compares the payoffs of ''series'' probabilities — ''one'' person repeating a single action several times — with those of ''parallel'' probabilities — ''several'' people performing a single action once each. If those two payoffs are the same, the event is “[[ | Ergodicity compares the payoffs of ''series'' probabilities — ''one'' person repeating a single action several times — with those of ''parallel'' probabilities — ''several'' people performing a single action once each. If those two payoffs are the same, the event is “[[ergodic]]”. | ||
Ergodicity is less common than you would think, and even where you have it, it doesn’t take much to scare it off. | Ergodicity is less common than you would think, and even where you have it, it doesn’t take much to scare it off. | ||
For example, the probable outcomes from flipping a coin are | For example, the probable outcomes from flipping a coin are ergodic. It doesn’t matter if ten people flip a coin once, or one person flips it ten times, the odds are the same: 50:50. | ||
But if our coin flippers ''bet'' on their flips, their expected return is ''not'' | But if our coin flippers ''bet'' on their flips, their expected return is ''not'' ergodic. | ||
=== When flipping coins isn’t | === When flipping coins isn’t ergodic === | ||
Imagine a game where you stake £10 on a [[coin flip]], and “heads” wins 50% but “tails” loses 40%. | Imagine a game where you stake £10 on a [[coin flip]], and “heads” wins 50% but “tails” loses 40%. | ||