Fred Renzey: Betting Over Your Max

[Fred Renzey]

Suppose your top bet is say $200 at +4 true, and you're properly bankrolled to do that. The player next to you takes out a $1000 marker and goes all in with it. He's dealt A/A vs. a 3 in a neutral count. He refuses to take another marker, and prepares to just hit his soft 12.

Now at +4 true, where you'd normally be willing to risk $200, you'd have about a +1.75% EV on the next blind hand coming up -- but -- his hand has a whopping +26% EV if you take one of his Aces!

Since Kelly advises that you bet in proportion to your advantage, are you warranted in betting 5 times your normal max when you have 15 times the advantage??

[Norm Wattenberger]

Is my bankroll replenishable?
What is my hand?
Will he hurt me if he loses?

[Chucke]

I?m not one of the math guys out here but I think it would be well worth the risk. My logic is this: if the AA has a 26% advantage then one A would have a 13% advantage (not sure of this). A $200. bet at +4 equates to about a 15K bankroll. 13% of the bank is $1,900. ? so ? you would be betting a little more then ? Kelly. Don (or any body else) please correct me if my logic and math are wrong.

In this case, I might ask the other player if he would like to partner on the hand (instead of he takes one card and I take the other). It would cut variance for both of us and it would be a better deal for the other player in terms of risk and ev (in the case of him hitting the AA and not splitting).

[Fred Renzey]

> I?m not one of the math guys out here but I think it
> would be well worth the risk. My logic is this: if the
> AA has a 26% advantage then one A would have a 13%
> advantage

snip: The 26% advantage was for taking just 1 Ace.

[SZhark]

I would jump on an opportunity to risk 5*max bet on such an advantage. I don't think that your ROR grows much with betting that much with such advantage. It is a nice boost to EV. Probably this hand worths more than an hour of your play.

> snip: The 26% advantage was for taking just 1 Ace.

[Chucke]

> snip: The 26% advantage was for taking just 1 Ace.

Thanks Fred - in that case, it makes it a much stronger play. The $1000. bet is now a little more then 1/4 kelly. Now I could get greedy and not offer the partnership unless it was the only way to get to play one of those As. Am I on the right track?

[Fred Renzey]

> Thanks Fred - in that case, it makes it a much
> stronger play. The $1000. bet is now a little more
> then 1/4 kelly. Am I on the right track?

Chucke, I think you are, but I'd like some input from Don here. I've run some sims betting 5 times max with 15 times the advantage while playing on the same bankroll, and going belly-up was virtually non-existent. It seems that having only 20 max bets with a 25% advantage is less risky than having 100 max bets with a 1.75% advantage.

I raise the question only because these opportunities do arise from time-to-time.

[Don Schlesinger]

> Chucke, I think you are, but I'd like some input from
> Don here. I've run some sims betting 5 times max with
> 15 times the advantage while playing on the same
> bankroll, and going belly-up was virtually
> non-existent. It seems that having only 20 max bets
> with a 25% advantage is less risky than having 100 max
> bets with a 1.75% advantage.

> I raise the question only because these opportunities
> do arise from time-to-time.

Having the huge advantage is always superior, and Kelly- optimal wagers are designed to reflect both edge AND the ratio of a winning bet to a losing one.

For example, you can get the exact same edge from having a huge payoff (longshot) just a few times, or from having a smaller one (favorite) but that pays off much more frequently. The Kelly wager, and hence the ultimate profits, are always bigger for the situation in which you win more frequently, even though the edges might be the same.

I understand that, here, the edges and the banks are different, but the principle is similar.

Don

[Chucke]

> Having the huge advantage is always superior, and
> Kelly- optimal wagers are designed to reflect both
> edge AND the ratio of a winning bet to a losing one.

> For example, you can get the exact same edge from
> having a huge payoff (longshot) just a few times, or
> from having a smaller one (favorite) but that pays off
> much more frequently. The Kelly wager, and hence the
> ultimate profits, are always bigger for the situation
> in which you win more frequently, even though the
> edges might be the same.

> I understand that, here, the edges and the banks are
> different, but the principle is similar.

> Don

Don - I've read your reply more then 25 times and I don't fully understand what you are saying. I'm sorry to be so thick. Are you saying that one would be better off not taking the opportunity (with a 26% edge) because it does not occur very frequently? That thought had crossed my mind. Should it be looked at like a one time prop bet? Sorry if I'm really off the mark here.

Also, would you please explain what you mean when you say "...reflect both edge AND the ratio of a winning bet to a losing one." It's the ..AND the ratio of a winning bet to a losing one." part of the statement that I don't understand.

Thanks for your help. I really appreciate your patience.
Chucke

[Don Schlesinger]

> Don - I've read your reply more then 25 times and I
> don't fully understand what you are saying. I'm sorry
> to be so thick. Are you saying that one would be
> better off not taking the opportunity (with a 26%
> edge) because it does not occur very frequently? That
> thought had crossed my mind. Should it be looked at
> like a one time prop bet? Sorry if I'm really off the
> mark here.

Sorry. It isn't an obvious concept. No, I'm not saying not to make the bet, but what I'm saying is that there simply is a great deal of variance, since the edge is so heavily predicated on the ace's turning into a natural. So, when you have the opportunity -- and only once, on top of it all -- you have to realize that it's still a risky proposition.

> Also, would you please explain what you mean when you
> say "...reflect both edge AND the ratio of a
> winning bet to a losing one." It's the ..AND the
> ratio of a winning bet to a losing one." part of
> the statement that I don't understand.

You can create the SAME edge for a wager but with various scenarios that have different payoffs. For example, you may know that an optimal wager, f*, can be expressed as e/a, where e is your positive expected return, and a is the ratio of a winning payoff to a losing one. That, in essence, is the Kelly criterion. So now, consider the following four cases, ALL of which have precisely the same +20% expectation, and all of which lose $1 when we lose.

1. Win $1.40 50% of the time; lose $1 50% of the time.
f* = 0.2/1.4 = 0.143.

2. Win $1 60% of the time; lose $1 40% of the time.
f* = 0.2/1 = 0.20.

3. Win $0.50 80% of the time; lose $1 20% of the time.
f* = 0.2/.5 = 0.40.

4. Win $5 20% of the time; lose $1 80% of the time.
f* = 0.2/5 = 0.04.

Now, for a FIXED bankroll, although all four cases return the same 20%, Case 3 permits one to wager, right from the outset, 40% of one's capital. In this case, the probability of losing even two bets in a row is just (.2)^2 = 0.04 (very small).

On the other hand, despite the same 20% expectation, Case 4 permits initial wagers of just 4% of the bank, because we lose so frequently. Here, there is a greater than 25% chance [(0.8)^6 = 0.262] that we could lose SIX times in a row. Hence, the need to bet a much smaller fraction of the initial bank on each coup.

Cases 1 and 2 represent intermediate positions, with Case 2 the special "even-money" payoff, where we wager our exact 20% edge each time.

While none of the above is directly analogous to your original proposition, it does point out how "expectation isn't everything."

Hope this helps a little.

Don

[Chucke]

> Sorry. It isn't an obvious concept. No, I'm not saying
> not to make the bet, but what I'm saying is that there
> simply is a great deal of variance, since the edge is
> so heavily predicated on the ace's turning into a
> natural. So, when you have the opportunity -- and only
> once, on top of it all -- you have to realize that
> it's still a risky proposition.

Yes, I now realize it is very risky. My initial reaction was to jump at the bet as the 26% edge seems so large as compared to the "normal" edges, even at very high counts, in BJ. When I thought about it, I realized it was much more of a gamble then a long term advantage play. I think I would still take the bet.

> You can create the SAME edge for a wager but with
> various scenarios that have different payoffs. For
> example, you may know that an optimal wager, f*, can
> be expressed as e/a, where e is your positive expected
> return, and a is the ratio of a winning payoff to a
> losing one. That, in essence, is the Kelly criterion.
> So now, consider the following four cases, ALL of
> which have precisely the same +20% expectation, and
> all of which lose $1 when we lose.

> 1. Win $1.40 50% of the time; lose $1 50% of the time.
> f* = 0.2/1.4 = 0.143.

> 2. Win $1 60% of the time; lose $1 40% of the time.
> f* = 0.2/1 = 0.20.

> 3. Win $0.50 80% of the time; lose $1 20% of the time.
> f* = 0.2/.5 = 0.40.

> 4. Win $5 20% of the time; lose $1 80% of the time.
> f* = 0.2/5 = 0.04.

> Now, for a FIXED bankroll, although all four cases
> return the same 20%, Case 3 permits one to wager,
> right from the outset, 40% of one's capital. In this
> case, the probability of losing even two bets in a row
> is just (.2)^2 = 0.04 (very small).

> On the other hand, despite the same 20% expectation,
> Case 4 permits initial wagers of just 4% of the bank,
> because we lose so frequently. Here, there is a
> greater than 25% chance [(0.8)^6 = 0.262] that we
> could lose SIX times in a row. Hence, the need to bet
> a much smaller fraction of the initial bank on each
> coup.

> Cases 1 and 2 represent intermediate positions, with
> Case 2 the special "even-money" payoff,
> where we wager our exact 20% edge each time.

> While none of the above is directly analogous to your
> original proposition, it does point out how
> "expectation isn't everything."

> Hope this helps a little.

> Don

Thanks Don. Your explanation helped a great deal. The examples really did it for me. I now understand it.

Thanks again.
Chucke

[Fred Renzey]

> Suppose your top bet is say $200 at +4 true, and
> you're properly bankrolled to do that. The player next
> to you takes out a $1000 marker and goes all in with
> it. He's dealt A/A vs. a 3 in a neutral count. He
> refuses to take another marker, and prepares to just
> hit his soft 12.

> Now at +4 true, where you'd normally be willing to
> risk $200, you'd have about a +1.75% EV on the next
> blind hand coming up -- but -- his hand has a whopping
> +26% EV if you take one of his Aces!

> Since Kelly advises that you bet in proportion to your
> advantage, are you warranted in betting 5 times your
> normal max when you have 15 times the advantage??

snip> Ran two sims, then checked their ROR's using BJRM: First, betting $20-to-$200 on a $20,000 bank, this Hi/Lo player's hourly win was $34 and his 1000 hour ROR was 12.4%.

Second, used the same $20,000 bank, played with an altered deck so that a flat betting player had a 26% advantage. Then proceeded to bet $1000 every hand. This player's hourly win was $26,000 and his 1000 hour ROR was 0.6%.

[kewljasond]

> snip> Ran two sims, then checked their ROR's using
> BJRM: First, betting $20-to-$200 on a $20,000 bank,
> this Hi/Lo player's hourly win was $34 and his 1000
> hour ROR was 12.4%.

> Second, used the same $20,000 bank, played with an
> altered deck so that a flat betting player had a 26%
> advantage. Then proceeded to bet $1000 every hand.
> This player's hourly win was $26,000 and his 1000 hour
> ROR was 0.6%.

The RoR might be .06, IF the player was carrying his entire $20,000 bankroll with him, but suppose he only had a trip BR of $4000 and had already lost $1000. You are now betting a third of your remaining stakes? Plus what if you draw a 6 or 7 and need to double. Now you are betting two thirds of your remaing BR.