A related concept in finance/trading is “drawdown”. A single trade can have a positive expected value. But over time, if you take a loss you have to get a bigger win to end up back where you started, because you have less capital to work with.
It does! You take insane amounts of risks (that you want to avoid), but the expected value stays positive, even if you repeat it a million times. You might have a 99.9999999999...% chance of going broke, but you'll also have a 1/2^1mil chance of making an absurd amount of money; on average (and not median), you will come out ahead.
Yes, the article shows that almost certainly, any individual's wealth will approach 0 from repeatedly taking this gamble. However, the comments I replied to say:
> But this is purely a result of the distribution of returns from a single toss.
It will only approach zero because you lose more than you gain.
If the loser got 0.6666c instead of 0.6c, and the winner got $1.50, then over time you'd break even, on average.
And yet the expected return would apparently be 1.08333. If think the conclusion is that 'expected return' is a fallacy, you just can't add proabability-outcomes in this way to get an 'expected outcome'.
> If the loser got 0.6666c instead of 0.6c, and the winner got $1.50, then over time you'd break even, on average.
For what definition of "average"? Yes, the most likely outcome would be to break even. But if we mean "expected value" when we say average, then on average, that repeated wager would be massively profitable for us (in terms of wealth). Maybe a better way to view it is that it would be massively unprofitable for the casino offering it (this is more straightforward since the casino's outcomes are more ergodic than the individual's).
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Maybe a more clarifying discussion would be: what if you could accept these wagers (let's say 0.6x and 1.5x) if you can use a bet amount of your choice each time? That is, you don't have to wager your entire life savings, but you could choose to make the bet with $1, with the outcomes being $0.60 and $1.50.
Then, this wager is clearly a free money machine, right? We can come to the conclusion that it is a free money machine by looking at its expected value. That's why I disagree that "you lose more than you gain" and I disagree that "this is purely a result of the distribution of returns from a single toss".
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I guess I should concede that I agree with your reasoning, if we make the reasonable assumption that utility is the logarithm of wealth. In that case, I agree that I would be indifferent to taking a 0.666 and 1.5 bet for my entire networth, and that I would not take a 0.6 and 1.5 bet for my entire networth. However, I still contend that we can analyze the value of these wagers using expected value -- we simply look at the expected value of what we actually care about (utility), not wealth.
I think the conclusion is that 'expected value' calculated this way is a bogus metric, even for a single toss (what is the 'average' of a single toss, it already makes no sense). You can't simply take the mean average of summed probability-outcome products, or at least, it does not mean 'expected value', it means the 'probable limit' of the total.
In the $60/$150 bet the 'expected value' is $105, but no-one is getting $105. What is that number? It indicates a 'probable limit' to the total money in the system, but says nothing about what any individual should expect, or about the actual total size of the system, which will obviously tend towards the aggregate individual outcomes. And that is clearly exposed when you start repeating the bet.
In fact 'expected value' is quite clearly 0.6 * 1.5 = 0.9, ie $90, or I think more generally: (outcome-1 * probability-1 * ... outcome-n * probability-n) / (probability-1 * ... probability-n)
It's maybe counter-intuitive that the 'limit' of the system should keep growing, while the 'actual' size of the system tends toward the 'real' aggregate expected value, but I think it's much less counter-intuitive than what I think is claimed here, that repeating a bet turns it from being a 'good' bet to a 'bad' bet.
