Certainty Equivalence
- Thread starter NewToTheGame
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From the BJ FAQ
Q3: What is "Certainty Equivalent"?
A3: Would you rather make a bet of $200 on a coin flip with an average profit of $20 or accept $5 risk-free? Would $10 risk-free persuade you not to make the bet? How about $15? Your "certainty equivalent" (or risk-free equivalent) is the amount that participation in the bet is worth to you. -- perhaps $5, $10, or $15 in this example.
The Kelly criterion with Kelly number 0.3 advises you to maximize the expected value of u(x) = x^(1-1/k) / (1-1/k), where k = 0.3 and x is your resulting bankroll. If your bankroll is $10,000 then the $200 bet gives an average value of u(x) of
55% * u(10200) + 45% * u(9800) = some number
If instead you were offered an amount "CE" risk-free the average value of u(x) would be
100% * u(10000 + CE) = some other number
These two expressions are equal when CE = $13.38. This is the "certainty equivalent" of the above bet for you if you are a Kelly better with the Kelly Number 0.3 and with a $10,000 bankroll. This amount, $13.38, is how much participation in the bet is worth to you. In particular, if the CE for this bet were negative the bet would be worth a negative amount to you and you should avoid it if possible.
Q4: How can certainty equivalents be used in a practical setting?
A4: The need for the use of logarithms and exponentiation makes the calculations quite difficult when analyzing a complex game such as blackjack. A formula for approximating the certainty equivalent (that is very accurate when your advantage or disadvantage is 10% or less) is
CE = E - V/2kB
where CE is the certainty equivalent, E is the expected winnings, V is the variance of those winnings (i.e. the square of the standard deviation), B is your bankroll and k is your Kelly Number, a measure of the amount of risk you wish to take. The Kelly criterion corresponds to k = 1.0 and in this situation this formula closely approximates calculations based upon the log(x) utility function. When k is not 1, the utility function that you are approximating is x^(1-1/k) / (1-1/k).
For the $200 coin flip above which has E = $20 and V = $$39600 (the standard deviation is $198.997) the formula gives a CE = $13.40 which is quite close to the exact value of $13.38 derived above.
Q3: What is "Certainty Equivalent"?
A3: Would you rather make a bet of $200 on a coin flip with an average profit of $20 or accept $5 risk-free? Would $10 risk-free persuade you not to make the bet? How about $15? Your "certainty equivalent" (or risk-free equivalent) is the amount that participation in the bet is worth to you. -- perhaps $5, $10, or $15 in this example.
The Kelly criterion with Kelly number 0.3 advises you to maximize the expected value of u(x) = x^(1-1/k) / (1-1/k), where k = 0.3 and x is your resulting bankroll. If your bankroll is $10,000 then the $200 bet gives an average value of u(x) of
55% * u(10200) + 45% * u(9800) = some number
If instead you were offered an amount "CE" risk-free the average value of u(x) would be
100% * u(10000 + CE) = some other number
These two expressions are equal when CE = $13.38. This is the "certainty equivalent" of the above bet for you if you are a Kelly better with the Kelly Number 0.3 and with a $10,000 bankroll. This amount, $13.38, is how much participation in the bet is worth to you. In particular, if the CE for this bet were negative the bet would be worth a negative amount to you and you should avoid it if possible.
Q4: How can certainty equivalents be used in a practical setting?
A4: The need for the use of logarithms and exponentiation makes the calculations quite difficult when analyzing a complex game such as blackjack. A formula for approximating the certainty equivalent (that is very accurate when your advantage or disadvantage is 10% or less) is
CE = E - V/2kB
where CE is the certainty equivalent, E is the expected winnings, V is the variance of those winnings (i.e. the square of the standard deviation), B is your bankroll and k is your Kelly Number, a measure of the amount of risk you wish to take. The Kelly criterion corresponds to k = 1.0 and in this situation this formula closely approximates calculations based upon the log(x) utility function. When k is not 1, the utility function that you are approximating is x^(1-1/k) / (1-1/k).
For the $200 coin flip above which has E = $20 and V = $$39600 (the standard deviation is $198.997) the formula gives a CE = $13.40 which is quite close to the exact value of $13.38 derived above.
Mayor, you are great and certainly very educated in this field
Mayor, you are great and certainly very educated in this field. Now, would you please go over step by step in simple terms with a real blackjack example of how to go about the CE and how to use the CE number before we get involved in any game. Please do that.
I, personally, use three numbers to determine if I'm getting myself or not into any game. They are: EV, StDev and the DI ( desirability index). I would love to know how to use the CE related to BJ. What is a "risk free" bet in BJ? - I don't think is one. No bet or no game regardless of the rules is risk free.
Please go over with a real BJ example like this one:
2 Decks, S17, DAS, DA2, Sp3, SpA1, LS pen 66% betting limits $100-$10,000
Bankroll = $100,000
Total units = 1,000
Unit = $100
Spread $100 to $700
EV = $170 per hour
StDev/hour = +/- $2,575
DI = 6.6
In the above setup I will get involved in this game because my DI is above 6. From my experience is worth to get into it if the bank is or above 100K
How to use the CE in this case?
Best Regards,
AlexD30
Mayor, you are great and certainly very educated in this field. Now, would you please go over step by step in simple terms with a real blackjack example of how to go about the CE and how to use the CE number before we get involved in any game. Please do that.
I, personally, use three numbers to determine if I'm getting myself or not into any game. They are: EV, StDev and the DI ( desirability index). I would love to know how to use the CE related to BJ. What is a "risk free" bet in BJ? - I don't think is one. No bet or no game regardless of the rules is risk free.
Please go over with a real BJ example like this one:
2 Decks, S17, DAS, DA2, Sp3, SpA1, LS pen 66% betting limits $100-$10,000
Bankroll = $100,000
Total units = 1,000
Unit = $100
Spread $100 to $700
EV = $170 per hour
StDev/hour = +/- $2,575
DI = 6.6
In the above setup I will get involved in this game because my DI is above 6. From my experience is worth to get into it if the bank is or above 100K
How to use the CE in this case?
Best Regards,
AlexD30