Ask the Wizard #229
Frank from Copenhagen
If I ran a casino, then I would apply the Kelly Criterion to setting maximum bets. What I would do is equalize m*v/h for all games, where m is the max bet, v is the variance, and h is the house edge. Let’s call this the risk quotient. For example, suppose I’m comfortable taking $150,000 on the Banker bet in baccarat, which is about what a big Strip casino will take. The house edge is 1.06%, and the variance is 0.932. The risk quotient is thus 150,000*0.932/0.0106 = 12,239,150.
Next, let’s solve for m to equalize the risk quotient in blackjack. With liberal Strip rules, the house edge is 0.29%. Let’s say the player can bet up to three spots. The standard deviation per hand, given three bet spots, is 1.51957, so the variance is 1.519572 = 2.3091. Solving for m...
m×2.3091/0.0029 = 12,239,150
m = $15,371.
As a practical matter, few players play perfect basic strategy, so I might bump that up to $20,000. That is about what the big Strip casinos will take in blackjack, so I think there is proportionality there. It is the novelty games where I think they should be taking bigger bets.
David T. from Montego Bay
Normally I'm sick of bad beat questions, but this one was too painful to ignore. Before the first card is dealt, the probability of four kings being beaten by four aces, in a two-player game, with both players having pocket pairs, is 2*combin(4,2)*combin(4,2)*44/(combin(52,2)*combin(50,2)*combin(48,5)) = 2*6*6*44/(1326*1225*1712304) = 1 in 877,961,175. This was a six-player game, so there are combin(6,2) = 15 different player pairs. In a six-player game, the probability is 15 times higher, or 1 in 58,530,745. After the indicated hole cards are dealt, and before the flop, the probability is 1 in 38,916 that the hand will finish as it did.
Caesars Palace here in Vegas changed their house way in the last few years. Previously, they never split up the gee joon pair, and now they do. A dealer at the Mandalay Bay recently commented to me about a rule change to play 0-7 over 3-4. About the reason behind the changes, I have no idea. I would be proud if I had something to do with it.
I recently added a house way for the Casino Canberra. In the future I may also add house ways for the Claridge in Atlantic City and Treasure Island in Las Vegas. It is tedious and time consuming to put them into HTML, which is why I have not done those yet.
My assistant JB is working on an optimal strategy against the Foxwoods house way. It should be up on the site shortly.
Victor M. from Genoa City, WI
I hope you’re happy; I spent all day on this. The answer and solution can be found on my other site mathproblems.info, problem 203, or the academic paper Game Theory and Poker by Jason Swanson.