Fred Renzey: Betting Over Your Max

[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