Certainty Equivalent Question for Mayor

[MJ]

Mayor,

I see this term from time to time and am curious to know exactly what it means. On CVCX it is defined as "the value of a wager". How would you explain the meaning of this term?

In the "Hot Shoe" DVD Andy Bloch from the MIT team mentions "certainty equivalent" when talking about how the players are paid. He says something along the lines of after a playing session a player would find the EV for various hands that they played(using a SIM), then subtract the variance to calculate the certainty equivalent. In this fashion players are paid based upon the CE that they generate. So this way players were encouraged to make money and reduce variance at the same time.

For some reason I still don't quite grasp this concept. Can you give an example as it pertains to the paragraph above? In other words can you show me how to use the CE to calculate a players earnings. If the CE is positive does that mean I am GUARANTEED to make money from a wager because there is no risk? Thanks for any help.

-MJ

 

[The Mayor]

We need someone from economics...

I will do my best, but my understanding of this is limited...

>I see this term from time to time and am curious to know exactly what it means. On CVCX it is defined as "the value of a wager". How would you explain the meaning of this term?

That is really how I would define it too. Here is an example. Suppose you make a wager with a 10% edge and bet $100. Sure it has an EV of $10, but it also has some variance associated with it. Someone comes along and offers you $110 to buy that bet from you. Would you take it? In a heartbeat! You get your EV with no variance! Now, someone comes along and offers you $1 for that wager. That is, they offer you $101. Now, the more certain you are of getting your EV, the less you want to accept this offer. If you are absolutely certain you will get your EV, then there is no way you will take it. If you want income with no risk, you will gladly sell your bet to earn a buck.

So, on the open market, this wager has a value based on its implied risk. It is not $110 and it is not $100, it is somewhere in between. That can only be determined by the current market forces. That value is the CE of the wager.

As for the MIT team, it goes the same way. Each team member is playing with an EV for his action. The member can live with the full risk and possibly not earn a thing (or even lose), or he can let market forces (competing team members) determine the true value of what he is doing, and be guaranteed to be paid that amount (which will be less than EV, but more than nothing).

This is my understanding, and I could be completely wrong! I hope someone else will come along and add something to this.

--Mayor

 

[MJ]

Another question

Thanks Mayor! In the case of the $100 wager with a 10%Expected value, if somebody offered $101 for it how much variance should there be in order to accept this offer? Mathematically there must be a point when accepting the wager becomes our advantage no? I would imagine the probability of winning the hand would have to be very low...maybe around 1%.

In the case where somebody buys my wager for $110 does that mean the CE becomes $110-$0 = $110? In other words EV - Variance = CE.

-MJ

 

[The Mayor]

From bjmath.com

Here is the discussion from bjmath.com (their Kelly.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.