vpFREE2 Forums

Certainty Equivalent

#1

I understand (conceptually, if not mathematically) the Kelly principle, but have not heard about “certainty equivalent” before. Can someone explain that to me (again, conceptually). If one’s bankroll is NOT large enough to properly qualify as an adequate “Kelly” bankroll, I understand that you have less than optimal chance of doubling bankroll, but does that situation also change the way you should play, or are plays of higher expectation still preferred over those of lower expectation in every situation?

Instinctively, it seems to me that even if you are playing (and I understand that long-term, it’s not “correct” play) above your bankroll, that the correct decision for each hand should still be the
same.

Thanks!

–BG

···

===============

1.1. Re: Proper hold JOB 3 card royal vs. 4 card flush???
Posted by: “007” …@…com mdmgyn
Date: Wed Sep 17, 2014 12:29 am ((PDT))

nightoftheiguana2…@…com wrote:

In case anyone wants to double check the numbers:

hand: AJT9s8o

hold AJTs:
outs: 1rf,35fl,15st,9-3k,27-2p,240hp
EV=1.2867715 VAR=592.17308

hold AJT9s:
outs: 9fl,6hp
EV=1.2765957 VAR=5.3915799

Certainty Equivalent = EV - VAR/2xBankroll

For Bankroll < 28,832 bets (about 36 royals), AJT9s has a higher Certainty Equivalent

I
compared the Kelly formulas and got 28,298 units as the bankroll
that equates them.

#2

I don't understand the difference between the "certainty equivalent"
and the Kelly Criterion. Just based on the example that
nightoftheiguana did, I suspect the "certainty equivalent" is an
approximation to the Kelly Criterion. The principle in deciding which
hold to make and deciding which game to play is the same, so the
correct decision for each hand should not necessarily still be the
same. Theoretically, just as each game as a whole should be analyzed
from a "Kelly" point of view, so should each strategy decision.
Sometimes, there are hands for which the hold with the highest
expected value takes a bigger bankroll to make than the game as a
whole takes to play.

···

I understand (conceptually, if not mathematically) the Kelly principle, but have not heard about "certainty equivalent" before. Can someone explain that to me (again, conceptually). If one's bankroll is NOT large enough to properly qualify as an adequate "Kelly" bankroll, I understand that you have less than optimal chance of doubling bankroll, but does that situation also change the way you should play, or are plays of higher expectation still preferred over those of lower expectation in every situation?

Instinctively, it seems to me that even if you are playing (and I understand that long-term, it's not "correct" play) above your bankroll, that the correct decision for each hand should still be the same.

Thanks!

--BG

1.1. Re: Proper hold JOB 3 card royal vs. 4 card flush???
   Posted by: "007" 007@embarqmail.com mdmgyn
   Date: Wed Sep 17, 2014 12:29 am ((PDT))

nightoftheiguana2000@yahoo.com wrote:

In case anyone wants to double check the numbers:

hand: AJT9s8o

hold AJTs:
outs: 1rf,35fl,15st,9-3k,27-2p,240hp
EV=1.2867715 VAR=592.17308

hold AJT9s:
outs: 9fl,6hp
EV=1.2765957 VAR=5.3915799

Certainty Equivalent = EV - VAR/2xBankroll

For Bankroll < 28,832 bets (about 36 royals), AJT9s has a higher Certainty Equivalent

I compared the Kelly formulas and got 28,298 units as the bankroll
that equates them.

#3

To elaborate a bit on my previous message, analyzing the game as a
whole and each play in that game are somewhat related, since changing
how any hand is played will change the frequency of each ending hand
overall, which changes the bankroll requirement for playing the game.
It's conceivable to not be able to afford a game if played by max-EV,
but be able to afford it if each hand is played according to the Kelly
Criterion, due to reducing fluctuation by more than EV has been
reduced.

···

I understand (conceptually, if not mathematically) the Kelly principle, but have not heard about "certainty equivalent" before. Can someone explain that to me (again, conceptually). If one's bankroll is NOT large enough to properly qualify as an adequate "Kelly" bankroll, I understand that you have less than optimal chance of doubling bankroll, but does that situation also change the way you should play, or are plays of higher expectation still preferred over those of lower expectation in every situation?

Instinctively, it seems to me that even if you are playing (and I understand that long-term, it's not "correct" play) above your bankroll, that the correct decision for each hand should still be the same.

Thanks!

--BG

1.1. Re: Proper hold JOB 3 card royal vs. 4 card flush???
   Posted by: "007" 007@embarqmail.com mdmgyn
   Date: Wed Sep 17, 2014 12:29 am ((PDT))

nightoftheiguana2000@yahoo.com wrote:

In case anyone wants to double check the numbers:

hand: AJT9s8o

hold AJTs:
outs: 1rf,35fl,15st,9-3k,27-2p,240hp
EV=1.2867715 VAR=592.17308

hold AJT9s:
outs: 9fl,6hp
EV=1.2765957 VAR=5.3915799

Certainty Equivalent = EV - VAR/2xBankroll

For Bankroll < 28,832 bets (about 36 royals), AJT9s has a higher Certainty Equivalent

I compared the Kelly formulas and got 28,298 units as the bankroll
that equates them.

#4

Suppose you come to me for a job and we sign an employment contract stating
I pay you $500 per day.

You have an EV of $500 per day. Simple enough.

I then point out the clause in the contract saying that after I pay you the
$500 per day, we flip a coin for $1000.

You still have an EV of $500 per day. But suddenly, it's very risky for
you.

So I point out the next clause in the contract saying you can negotiate a
lower amount per day to be paid so you can avoid the risk. Suppose we then
agree I'll pay you $400 per day instead of $500 and we'll skip the coin
toss.

$400 is your certainty equivalent. You have a $500 EV but you'll accept
less to avoid the risk. Depending on how big a bankroll you have, you may
settle for more or less of a difference.

···

On Thu, Sep 18, 2014 at 7:19 AM, Barry Glazer b.glazer@att.net [vpFREE] < vpFREE@yahoogroups.com> wrote:

I understand (conceptually, if not mathematically) the Kelly principle,
but have not heard about "certainty equivalent" before. Can someone
explain that to me (again, conceptually). If one's bankroll is NOT large
enough to properly qualify as an adequate "Kelly" bankroll, I understand
that you have less than optimal chance of doubling bankroll, but does that
situation also change the way you should play, or are plays of higher
expectation still preferred over those of lower expectation in every
situation?

#5

007 wrote: “I don’t understand the difference between the “certainty equivalent”
and the Kelly Criterion.”

The Certainty Equivalent (CE) is derived from the Kelly Criterion, I believe Kelly introduced it in his paper on the subject. You can google it and find tons of information. It’s big, very big, any serious gambler should get to know it. I don’t claim to know everything there is to know about CE.

007 wrote: “Theoretically, just as each game as a whole should be analyzed
from a “Kelly” point of view, so should each strategy decision.
Sometimes, there are hands for which the hold with the highest
expected value takes a bigger bankroll to make than the game as a
whole takes to play.”

Yes. One way to find the optimal Kelly strategy is to calculate the EV and VAR of each possible draw. To get maxEV you simply rank by EV. To get Kelly, you rank by CE.

#6

I should probably clarify a bit: The notion of a “Certainty Equivalent” precedes Kelly, but the particular formula I was using (EV-VAR/2Bankroll) was derived by Kelly and is the CE under the Kelly system.

The Certainty Equivalent itself is relatively simple. Given a gamble, for what fixed amount would a contestant be indifferent between chosing the gamble or the fixed amount? That indifference point is also the game theoretical optimal point.

A maxEV player would be indifferent between a 50% chance at $100 or a fixed $50.

A Kelly player would consider the variance and their current bankroll. If the Kelly CE of the gamble exceeded $50, they would gamble.

#7

nightoftheiguana wrote:

A maxEV player would be indifferent between a 50% chance at $100 or a fixed $50.

A Kelly player would consider the variance and their current bankroll. If the Kelly CE of the gamble exceeded $50, they would gamble.

I believe the Kelly CE is always less than EV so that the Kelly player
would always choose the fixed $50. The amount that's fixed is the
amount to be determined. The question is at what fixed payout,
necessarily being less than $50, would the Kelly player equate a 50%
chance at $100.