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Comment by confidantlake | Hacker News Reader

[Edit] As has pointed out I have misread the parent comment and agree with it. Leaving my original response below.

For number 2 it depends on strategy. The strategy you outlined has you risk $5 to win $1000. Of course in a fair game you will go broke most of the time. For every time you win you should lose 200 times to come out even. But you can easily take the reverse strategy. Bet $5 if you win, leave. If you lose double your bet, if you go down $1000 dollars leave. In this case you are almost guaranteed to win. But the times you do lose you lose enough to make up for all of your wins.

EV is EV, it doesn't change based on who has the larger bankroll. The casino doesn't magically get an edge because they have more money. They get the edge because the game odds are in their favor.

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I don't think you're talking about the same thing as the GP. If you bet $5 and lose, you can't double your bet, you have no money (which is exactly the GP's point).

It's a random walk, and it's much more likely to take you to -$5 way before it takes you to +$1000.

Yes exactly, completely agree with you. I am seeing this does not matter. It is not why a casino makes money.

Well... that's bordering on oversimplifying it.

The key distinction is whether you count the EV of a single wager, or the expected growth of compound returns.

As you say, the single-wager EV is independent of bankroll provided you can afford it in the first place. This is also what most people mean when they say EV, and indeed the common mathematical definition of it.

However, skilled risk takers know that the arithmetic EV isn't what matters more generally. What matters in the long run is geometric EV, or expected growth of compound returns.

And for geometric EV, bankroll absolutely matters. The larger your bankroll, the better your geometric EV. I suspect this is what the parent comment referred to.

(However, when the wager shrinks in comparison to your total wealth, the arithmetic EV approaches the geometric EV by Taylor series approximation. For "everyday affairs" (as I think Bernoulli put it) you can think of them as equal.)

I would have to disagree that I am oversimplifying it. We are talking about a casino vs a player. Most players betting $5 at a time are not losing because of bankroll considerations. They are losing because the game they are playing is negative ev for each individual bet.

A casino would much rather play against a player with $1,000,000,000 than a player with $5. Sure the player with $5 is going to go broke nearly 100%. And the player with more money will never go broke betting $5 at a time, they will die from old age way before that happens.

The fact that a player with only $5 has almost a 100% chance of ruin is absolutely not why a casino is making money. A casino is always going to want a player with a lower risk of ruin than a higher one. When a player goes broke they can no longer make money from them.

Do people really think a casino would prefer someone going broke 100% of the time with $5 then a player going broke 0% of the time with $1,000,000,000? That they are making money because of players losing their bankrolls and unable to bet?

Aha, I see what you're saying. You're right, but I still think you're reading something into the top-level comment that isn't there.

The way I understand it, they're not saying "casinos make money because their counterpart is going broke" (and you're right that they don't.) They're saying "in the casino vs. player situation, the player will almost always run out of bankroll before the casino does, and therefore the casino does not risk ruin, even in highly volatile games."

I reread the parent and the article and see what you are saying. I agree that I misread the top level comment, and they were correct. (I only looked at the first roulette example where bankroll did not even exist). Thanks for pointing this out.

From my (limited) experience watching folks at the roulette table, I think there are two kinds of players.

There's the players making small bets, they win a bit, lose a bit, and at the end of the day they may go home with a bit more or a bit less money than they came with. The casino wins on average because of the 0.

Then there are the gamblers. They come to the table, make a few large bets, and within a few minutes they lost their monthly salary (or whatever was the limit on the ATM). They lost because the player will continue playing until they can't.

I'm pretty sure that you could make a completely fair roulette, and the house would still win big every day. I'm not sure why there aren't any fair roulettes, though.

Speculation: because it would draw attention to where the real profit comes from?

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