Rather useless Deal or No Deal Kelly Calculator

vanzack · 06-21-07, 09:35 AM

Im not really sure how that calculator works, but isnt the calculation for deal or no deal a straight mathematical average of the remaining numbers?

If that average is higher than the offer, keep playing, if the average is lower than the offer, take offer?

I realize there is utility involved, but from a purely mathematical standpoint isnt that the formula?

Ganchrow · 06-21-07, 11:54 AM

Originally posted by vanzack

Im not really sure how that calculator works, but isnt the calculation for deal or no deal a straight mathematical average of the remaining numbers?

If that average is higher than the offer, keep playing, if the average is lower than the offer, take offer?

I realize there is utility involved, but from a purely mathematical standpoint isnt that the formula?

Sure. That would be the manner in which a risk neutral better should behave. But as often explained, risk neutrality is generally antithetical to bankroll growth.

Here's how this calculator works:

uncheck those briefcases that have already been revealed
enter your total $ worth (prior to starting the game)
enter your "Kelly multiplier" (e.g., 1 for full Kelly, 0.5 for half Kelly, 0.25 for quarter Kelly, etc.) A value of '0' implies complete risk aversion and a Kelly value equal to that of the lowest remaining case. A value of 'Inf' or 'Infinity' implies complete risk neutrality and a Kelly value equal to the average of the values of the remaining cases.

The first calculated field ("Average Value") is simply the "expected value" or arithmetic mean of the remaining cases, so if a player were to continue from that point an infinite number of times that's how much his case would contain on average.

The second calculated field is the "Kelly value" of the remaining cases, representing the offer level at which a Kelly bettor (of type specified in the above Kelly multiplier box) would be indifferent between accepting the banker's offer and continuing to play on. Were the banker's offer below that Kelly value, the player would strictly prefer to play, and were the banker's offer above the Kelly value, the player would strictly prefer to accept the offer.

The Kelly value will alys be strictly less than the average case value for all finite Kelly multipliers, and will be strictly greater than the lowest case value for all positive Kelly multipliers.

MrX · 06-21-07, 12:35 PM

But will they let me use this when I make it on the show?

Ganchrow · 06-21-07, 01:22 PM

Originally posted by MrX

But will they let me use this when I make it on the show?

No problem. Just take me on the show with you and I'll handle the rest.

rjp · 06-22-07, 11:23 AM

Pretty cool Ganchrow.

Ganchrow · 06-22-07, 02:02 PM

Originally posted by rjp

Pretty cool Ganchrow.

Thank you.

Next up ... Kelly optimal wagering in beer pong.

Arilou · 06-24-07, 10:29 AM

There's one problem, which is that continuing isn't worth what it's worth in Kelly - it's worth a lot more. The reason for this is that the banker gets more and more reasonable every offer. The first offer is a joke. The second is also a joke, just not quite as funny. By the last one, you could have a $1 case and a $1 million case and be offered $450,000, which is a fine offer. So in general, continuing is VERY strong because you'll get a good offer in terms of Kelly late compared to what you're offered now, which in turn leads to the basic principle that you should never accept a deal when you have multiple high-level cases remaining. Once you're down to just one, you can think about a deal because continuing has real risk.

RickySteve · 06-24-07, 01:04 PM

The banker will make offers with positive expectation late in the game.

imgv94 · 06-24-07, 01:31 PM

Ganchrow you would be a great person to bring on the show..

Ganchrow · 06-24-07, 06:27 PM

Originally posted by Arilou

There's one problem, which is that continuing isn't worth what it's worth in Kelly - it's worth a lot more. The reason for this is that the banker gets more and more reasonable every offer. The first offer is a joke. The second is also a joke, just not quite as funny. By the last one, you could have a $1 case and a $1 million case and be offered $450,000, which is a fine offer. So in general, continuing is VERY strong because you'll get a good offer in terms of Kelly late compared to what you're offered now, which in turn leads to the basic principle that you should never accept a deal when you have multiple high-level cases remaining. Once you're down to just one, you can think about a deal because continuing has real risk.

The next step would be including the implied distribution of future offers based on some user-defined starting point, which to produce exact results would require knowledge of the banker algorithm or at least a suitable approximation. With that in place you'd have to consider the n! possible offers as opposed to just the n remaining cases. (although in general fewer than n! offers are actually attainable as some would only appear after an offer were taken.)

But long as you have the banker algorithm, the full solution itself is still fairly trivial.

However, now that my fever has passed I've ceased production on this but you're certainly welcome to the source code if you care to improve upon it.

RickySteve · 06-25-07, 03:11 AM

Originally posted by imgv94

Ganchrow you would be a great person to bring on the show..

He's not eligible.