Ask the Wizard #196

I agree! As discussed in my section on the Kelly Criterion, there is an optimal bet size for any given bet with a player advantage, for purposes of balancing both risk and reward. Betting the exact Kelly amount will result in the greatest bankroll growth for the player with average luck.

For example, in full pay deuces wild, with a return of 100.76%, the optimal amount to bet every hand is 0.03419% of bankroll. These days if you can find full pay deuces wild it will probably be only at the 25-cent denomination, but if you could bet anything, 0.03419% of your total bankroll would be the optimal amount for long-term bankroll growth. For a player with a bankroll of $3,656, a quarter denomination game would be the perfect Kelly bet size.

As I discuss in my section on Kelly, the optimal bet amount is the one that maximizes the expected log of the bankroll after the bet, which I will call the Kelly Utility. Usually the Kelly Utility is maximized by making the optimal strategy play. However, one exception would be a five 3s to 9s, with three deuces. In particular, let’s look at 22266. The expected value of keeping the deuces only is 15.057354, and keeping the five of a kind is always worth 15 exactly.

The following table shows both the conventional expected value and the Kelly Utility holding the three deuces. The Kelly Utility for any given hand on the draw is p*log(1+0.0003419*w), where p is the probability, and w is the win.

Player Holds Three Deuces
Hand	Pays	Combinations	Probability	Return	Kelly Utility
Four deuces	200	46	0.042553	8.510638	0.001222
Wild royal	25	40	0.037003	0.925069	0.000137
Five of a kind	15	67	0.06198	0.929695	0.000138
Straight flush	9	108	0.099907	0.899167	0.000133
Four of a kind	5	820	0.758557	3.792784	0.000563
Total		1081	1	15.057354	0.002193

The next table shows the same figures for holding the five of a kind.

Player Holds Five of a Kind
Hand	Pays	Combinations	Probability	Return	Kelly Utility
Four deuces	200	0	0	0	0
Wild royal	25	0	0	0	0
Five of a kind	15	1	1	15	0.002222
Straight flush	9	0	0	0	0
Four of a kind	5	0	0	0	0
Total		1	1	15	0.002222

You can see that the Kelly Utility is higher keeping the pat five of a kind, at 0.002222 vs. 0.002193. For this particular hand, keeping the five of a kind will be the correct play under the Kelly Criterion for bankrolls up to 13,290 units, or for quarter players up to $16,613.

As I said, the optimal Kelly bet size for the optimal strategy player is 0.03419% of bankroll. The optimal bet size for the player playing optimal strategy, except keeping a dealt 22233 to 22299 is 0.03434% of bankroll. The bankroll growth of the optimal strategy player will be 0.0002605% per bet made. For the Kelly player it will be 0.0002615% per bet made. Every 40,000 hands the player following optimal straegy, and Kelly bet sizing, can expect bankroll growth of 10.98%. The conservative player keeping a dealt 22233 to 22299, and Kelly bet sizing based on that strategy, can expect growth of 11.03% per 40,000 hands.

So, I maintain that in some situations, indeed, you should go against optimal strategy and go for the more conservative play. I just hope Rob Singer doesn’t hear about this.

Before any cards are seen, the probability of any three ranks not appearing on the flop are combin(40,3)/combin(52,3) = 9880/22100 = 44.71%. So this guy had a 10.59% advantage.

I asked a source of mine who works at one of the casinos that utilize this technology. Besides the Treasure Island, this technology is also used at casinos in California, Michigan, and Mississippi. Here is what he said,

"Nothing can be changed if there are credits on the game. The slot machine will always reject any changes sent when there are credits on the meter. In Nevada, the machine also has to be idle for four minutes prior to and following any changes. It’s also not completely transparent to anyone standing in front of the slot machine. A black window pops up that says ’Remote Configuration In Progress’ (or something like that).

We mainly use ours to change the available denominations on our games. Similar to how table games will raise minimum bets on when the casino is busier, we will remove lower denominations on Friday morning and return them on Monday morning."

So, rest assured, that the slot manager can not tighten up a game on you just because he doesn’t like you. As long as you have credit in the game, nothing can be changed.

I forwarded this one to Brian, who is a former gaming regulator, and currently a casino manager. Here is what he said,

The casino would not know that someone was in the country illegally. If he had a valid passport, the jackpot would be honored. The illegal may not know this, be scared or they may not have a valid ID to show. Whenever someone wins $1,200 or more, ID is required for tax purposes. If someone doesn’t have his ID, the jackpot would be held in the cage waiting for them to claim it. In most cases, the person has legitimately forgotten their ID; however, sometimes you run into a problem, such as a minor who was gaming. If he doesn’t claim it, the money has to be added back into revenue because the deduction (jackpot) was never paid or there are abandoned property rules that prevail. Also, like the U.S., most countries tax worldwide income. To that end, the U.S. has tax treaties with several countries to withhold or notify the respective governments of monies won in the U.S. so Uncle Sam always gets his cut.

For readers who may not know, a "set" is a three of a kind after the flop, including a pocket pair. The probability of not making a set is (48+combin(48,3))/combin(50,3) = 17,344/19600 = 88.49%. So the probability of making a set is 11.51%. In 2,787 pairs you should have made a set 320.8 times. So you are 47.8 sets under expectations. The variance is n × p × (1-p), where n = number of hands, and p = probability of making the set. In this case the variance is 2,787 × .1176 × .8824 = 283.86. The standard deviation is the square root of that, or 16.85. So you are 47.8/16.85 = 2.84 standard deviations south of expectations. The probability of luck this bad or worse can be found in any Standard Normal table, or in Excel as norsdist(-2.84) = 0.002256, or 1 in 443.