A Hypothetical

Suppose you had the following scenario.

You have a bankroll, B.

If you do nothing, then you will lose some portion of B. Say 1/3rd. Call, it, oh, I dunno, a "tax on previous winnings."

If you bet and win (adding your winnings to B) then you will still lose 1/3rd of B (including your winnings).

If you bet all of B and lose it all then -- well, you all know how that goes.

So:

Should you do nothing, in which case you wind up with 2/3rds of B?

Or should you bet B (however you wish) figuring that you already have a built in edge of 1/3rd B?

Yes, I know: if you take B and bet it in a lot of small low-HE bets, you' could actually wind up betting a lot more than B, et cetera.

But what if you, I dunno, just made bets equal to exactly B and ended it right there so you never bet more than B, total?

Would you really have a 1/3rd edge minus whatever edge the house has for your chosen bets?

When you say, "If you bet and win, then you will still lose 1/3 of B (including your winnings)", does that mean that you will lose 1/3 of the original B plus 1/3 of the winnings?

If this is true, then the problem can be restated as:
You have an initial bankroll of 2/3 B.
When you win, it pays at 2/3 of the stated amount (e.g. if it is a 3-1 bet, it pays 2-1 in effect because of the 1/3 taken away).
Whether or not you bet is based on the house edge, taking the reduced odds into account. Since no bet is going to have a return of 150%, which is what you would need to offset the "tax", I would do nothing.

Quote: ThatDonGuy

When you say, "If you bet and win, then you will still lose 1/3 of B (including your winnings)", does that mean that you will lose 1/3 of the original B plus 1/3 of the winnings?

Correct. Sorry for any ambiguity. At the end of the game we pay 1/3rd of everything we walk away with as "tax."

Quote: ThatDonGuy

If this is true, then the problem can be restated as:
You have an initial bankroll of 2/3 B.
When you win, it pays at 2/3 of the stated amount (e.g. if it is a 3-1 bet, it pays 2-1 in effect because of the 1/3 taken away).
Whether or not you bet is based on the house edge, taking the reduced odds into account. Since no bet is going to have a return of 150%, which is what you would need to offset the "tax", I would do nothing.

Okay, fine! So far so good.

Now change things slightly to make it into two cases.

The first case behaves just like I've described: you owe that 1/3rd tax from your bankroll; you also owe 1/3rd tax on anything with which you walk away.

In the second case you don't owe the tax on your bankroll to start (just at the end, for your winnings only).

Now for the really important point: this really happens due to paying taxes on winnings. I happen to have won a lot this year (just lucky, I guess), and will pay about 1/3rd of it to the IRS if I don't lose it all back (which I hope I do not!). So let's contrast the case where I play with a bankroll of winnings-only compared to a bankroll of non-winnings. For the non-winnings bankroll we don't owe that 1/3rd amount up-front presuming we don't lose it all. We only pay tax on what we've won. Put another way: the IRS takes the 1/3rd vig on wins only.

Let's say we have B=$45K. Further, let's say we bet the pass line once (to keep the calculations simple with no loss of generality) so we either lose all of B or double it.

For the bankroll of winnings:

Lose: $45K entails a real loss of $30K.
Win: $45K turns into $90K minus 1/3 tax = $60K.

(If you want to figure it your way, then:
Lose: $30K entails a real loss of $30K.
Win: $30K turns into $60K.)

For the bankroll of non-winnings:

Lose: $45K entails a real loss of $45K.
Win: $45K turns into $90K minus 1/3 tax = $75K.

In the first case we risk $30K to win $60K after taxes (100% profit). In the second case we risk $45K to win $75K after taxes (67% profit).

That first case seems a lot better, doesn't it?

So what have I got wrong here?

Quote: Concinnity

For the bankroll of winnings:

Lose: $45K entails a real loss of $30K.
Win: $45K turns into $90K minus 1/3 tax = $60K.

(If you want to figure it your way, then:
Lose: $30K entails a real loss of $30K.
Win: $30K turns into $60K.)

That first case seems a lot better, doesn't it?

So what have I got wrong here?

What you have wrong is the fact that it's still better not to bet at all.

(30000 * .5071) = -15213

Your effective profit on a win is 30K, so: (30000 * .4929) = +14787

-15213 + 14787 = -426

Thus, your expectation is 30000 - 426 = $29,574, better to just pay your taxes and get the 30K back.

The negative expectation of the game is still there. Ultimately, you can either lock up the 45K (30K profit) or you can risk your, "Real loss," of 30K for a, "Real gain," of 30K with less than a 50% probability of success.

My consulting fees on this matter are whatever will put you into a lower tax bracket, but I think you have to pay by 12/31/14. (Kidding)

Quote: Mission146

What you have wrong is the fact that it's still better not to bet at all.

No, I think I understand that part. :)

I haven't done a good job explaining this. Let me try again.

Suppose I have a bankroll of $45K. Right now I have $15K of it entailed for taxes because I won it at a craps table this year (well, not all at once, or on the same table). Of course, if I lose the $15K (well, the entire $45K) I don't have to pay that $15K. I have a sort of silent partner now in Uncle Sam.

Now we have 2 cases (presuming I decide to bet) and, again, to make it simple let's presume I just bet it on the pass line, and happen to win, so we can look at a positive outcome here.

In the first case, I have that tax-entailed $45K. That means if I bet $45K then from a risk analysis standpoint I only risk $30K (think about it: if I lose $45K on the pass line then I have only lost $30K in profits, and Uncle Sam loses $15K in taxes which, I presume, Mr. Wynn gives to them eventually). Okay. So we could look at it this way: I bet $45K only risking $30K. I win (yay!) and I wind up with a total of $90K. Of which I must pay a total of $30K in taxes now, for a net of $60K (winnings of $30K). For a risk (to me, not the government) of $30K.

In the second case, I decide to risk the same amount ($30K) but not tax-entailed; out of my own consulting fee profits (I actually do work as a consultant). I risk/bet $30K and I win (yay!) and I wind up with a total of $60K. Of which I must pay a total of $10K in taxes (1/3rd of the $30K winnings), for a net of $50K.

I net $60K on the tax-entailed money that I have sitting in the bank at the moment (well, not exactly $45K -- a number I chose to make the arithmetic easy -- but close to it). Compared to netting $50K if I bet with the usual type of money.

A difference of $10K in winnings (for this example). If I win, of course. Presuming I win.

Now, again, where have I gone wrong? I could do the expected value computation of course, but I don't want to bore you all. But it seems like I have a 1/3rd advantage betting with the house money (at least, before December 31st -- I really need to visit Las Vegas before then I think, regardless of how the math works out on this).

You have to pay taxes on that extra 15k you bet when you win.

In the first case, you're putting up 30k of your own money that you won't have to pay taxes on. Uncle Sam is putting up the other 15k.

In the second case, you're putting up 30k of your own money, and uncle Sam is putting up 0. But if you win, 10k of that goes to uncle Sam.

In other words: option 1 means Sam is betting his own money. In 2, you're free rolling Sam.

You should not bet it all.

Quote: RS

You have to pay taxes on that extra 15k you bet when you win.

I factored that in. I suppose I could have written more clearly. Thus:

$45K taxable (but only $30K at risk) wins yielding $90K. I now have to pay taxes on the entire amount so subtract 1/3rd netting $60K.

In the second case where I don't owe taxes on the bankroll I put up $30K (the same risk) and it wins yielding $60K. I have to pay taxes on the winnings only ($30K) so deduct 1/3rd of the winnings ($10K) netting $30K+$20K=$50K.

$60K after taxes versus $50K after taxes, both with the same risk ($30K).

Where have I gone wrong here?

In scenario 1 (bet 45k), you're risking 30k but still betting 45k. The additional 15k on the bet takes care of the taxes.

In scenario 2 (bet 30K), there is no "extra" or "additional" money that will take care of the taxes.


Or rather, compare the amount you bet (not risked) vs how much you win (not what you end up with).

Bet 45K, win 30K. (you win 2/3 the bet).
Bet 30K, win 20K. (you win 2/3 the bet).

Both are the same.

Quote: RS

In scenario 1 (bet 45k), you're risking 30k but still betting 45k. The additional 15k on the bet takes care of the taxes.

In scenario 2 (bet 30K), there is no "extra" or "additional" money that will take care of the taxes.


Or rather, compare the amount you bet (not risked) vs how much you win (not what you end up with).

Bet 45K, win 30K. (you win 2/3 the bet).
Bet 30K, win 20K. (you win 2/3 the bet).

Both are the same.

Except that in the first scenario I wind up with $60K (at the end of the year; presuming I win).

In the second scenario I wind up with $50K.

In the first scenario I've won a net $30K and in the second a net $20K. A huge difference.

If I lose then at the end of the year I've lost the same amount I've risked ($30K) for each scenario.

I think it's because of the before-and-after. Uncle Sam is unknowingly betting with you on the first, because he won't get the W2G for what happened prior to this bet until the end of the year. On the second, he won't get the W2G at all on the prior amount, only on part of it if you win today. And if you lose, in either case he'll have to return the money/won't get paid if they didn't withhold it.

From the casino's viewpoint, you've won substantially more (90K in the first scenario, only 30K in the second). It's between you and Uncle Sam at that point; they're out the gross amount, however it's accounted for. And probably where the perspective should start, though of course your net is ultimately what matters. But what's throwing it off is that you're betting larger in the first scenario than the second, even though the first amount was winnings, not starting BR.

Quote: beachbumbabs

I think it's because of the before-and-after. Uncle Sam is unknowingly betting with you on the first, because he won't get the W2G for what happened prior to this bet until the end of the year. On the second, he won't get the W2G at all on the prior amount, only on part of it if you win today. And if you lose, in either case he'll have to return the money/won't get paid if they didn't withhold it.

Are W-2G's really that prevalent at table games that might even run into four- and five-digit wagers and payoffs?

Quote: SanchoPanza

Are W-2G's really that prevalent at table games that might even run into four- and five-digit wagers and payoffs?

Nothing on my tax return except W-2G's from table games and I am still pissed at the way Uncle Sam unfairly treats winnings vs. losses.

Quote: beachbumbabs

I think it's because of the before-and-after. Uncle Sam is unknowingly betting with you on the first, because he won't get the W2G for what happened prior to this bet until the end of the year. On the second, he won't get the W2G at all on the prior amount, only on part of it if you win today. And if you lose, in either case he'll have to return the money/won't get paid if they didn't withhold it.

From the casino's viewpoint, you've won substantially more (90K in the first scenario, only 30K in the second). It's between you and Uncle Sam at that point; they're out the gross amount, however it's accounted for. And probably where the perspective should start, though of course your net is ultimately what matters. But what's throwing it off is that you're betting larger in the first scenario than the second, even though the first amount was winnings, not starting BR.

Yes, I think you've given a very good explanation! In English even. :) Thanks.

I need to visit Las Vegas in December I think. :)

By the way, in my field we call things like this examples of the "secure composition problem."