1,427 karma · joined April 13, 2014
scientist.wang
Wandb and React can get honorable mentions.
But the original content is talking about the past, not the modern era.
No part of this is true or even really makes sense.
It's been satisfying!
Quick correction:
The article uses 0.6 and 1.5. In this setting, the population wealth increases and the wealth of almost any individual decreases.
In the comment you replied to, they propose 0.66 and 1.5. In this setting, the population wealth doesn't stay the same, it increases even more than before.
(In order for population wealth to stay the same, the payoffs should be 0.5 and 1.5)
An intuition is that log(wealth) represents utility. An intuition for why log(wealth) represents utility is that wealth has decreasing marginal returns. The logarithm is a function that fits this. If you play around with it a bit you might see that it conforms to your intuitions: increasing your net worth from $100k to $200k increases your utility significantly, maybe about as much as increasing your net worth from $1m to $2m, and much more than increasing your net worth from $1m to $1.1m.
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.
> If you lose, it should then be divided by 1.5 (multiplied by 0.66..).
In what sense "should" this be true?