Single Bet = -EV, Multiple Bets = +EV

[RocketManJames]

So, this might be really obvious, but I thought it was kind of neat. Anyway, here is a case where if you made a single bet you'd be -EV, but if you continued to make the same bet under the same structure, you'd be +EV.

You are a casino and you pay a 40% tax rate on your profit. Ignore everything else (operating costs, profits from non-gambline, etc).

Say you had a game that was 60/40 in your favor.

If you (the casino) were only allowed to play this game once, you'd be -EV.

EV of 1 play of $100: (0.6 * 100) * 0.6 + (0.4 * -100) = -$4
EV of 2 plays of $100: (0.36 * 200) * 0.6 + (0.16 * -200) = +$11.20

From here on out, the EV obviously remains positive. Again, maybe totally uninteresting, but I thought it was a neat observation.

-RMJ

[LetYouDown]

I am 99% sure there is a flaw in the logic here, but my brain is completely fried. The first case looks correct, but I don't see the EV shifting here. In the second case, you have the EV of winning twice and the EV of losing twice, but if you win one & lose one, you lose as well.

[RocketManJames]

[ QUOTE ]
I am 99% sure there is a flaw in the logic here, but my brain is completely fried. The first case looks correct, but I don't see the EV shifting here. In the second case, you have the EV of winning twice and the EV of losing twice, but if you win one & lose one, you lose as well.

[/ QUOTE ]

When you win one and lose one, you will show a profit of $0, and so you pay no taxes. Maybe there is a flaw in my logic somewhere, but if there is, then how do casinos make money with corporate tax rates as high as they are?

[mosdef]

[ QUOTE ]
I am 99% sure there is a flaw in the logic here, but my brain is completely fried. The first case looks correct, but I don't see the EV shifting here. In the second case, you have the EV of winning twice and the EV of losing twice, but if you win one & lose one, you lose as well.

[/ QUOTE ]

The "flaw", if you want to call it that, is that the scenario posed suggests that the house doesn't take their taxes out every time you win, rather only the cumulative winnings. If the tax is taken out step-by-step for each game played, then it's a negative game all along.

[LetYouDown]

That's what I was getting at. My wording is awful sometimes.

[RocketManJames]

[ QUOTE ]
The "flaw", if you want to call it that, is that the scenario posed suggests that the house doesn't take their taxes out every time you win, rather only the cumulative winnings. If the tax is taken out step-by-step for each game played, then it's a negative game all along.

[ QUOTE ]
That's what I was getting at. My wording is awful sometimes.

[/ QUOTE ]

[/ QUOTE ]

Yes, that's the mathematical reason why it works out given the scenario I posed. I just found it interesting that if casinos were only allowed to offer the wager exactly once per tax year, it'd be a losing proposition. But, if allowed any more than that, they'd make money.

-RMJ

I think the flaw is you are taxing the winnings not the profit.

[AaronBrown]

This is an important effect in business planning. One of the major reasons for a company diversifying is to save on taxes by guaranteeing all losses can be used. While losses can be carried forward to future tax years (and back in some cases) there are limitations to this and you lose the present value of the cash flows.

Similarly, individuals generally structure portfolios to have a high probability of making money every year. You can do that by diversifying, or by making sure you generate enough income from safe investments to offset any losses from risky ones.

People get screwed by this a lot in backing deals. If you agree to something like covering 100% of someone's daily losses and taking 50% of her daily winnings, you lose money if she goes +$1 one day and -$1 the next, which is why you should always make backing deals long-term.

[AaronBrown]

Hedge funds use "high water mark" to avoid this problem. If the player makes money on day 1, you get half and you can either call off the deal or continue. But if the player loses money day 1, you cover the loss, but now you get a higher percentage of winnings until the loss is made up from the excess.