# Discount Gambling

## Betting Bottom Pair on Flop (UTH)

Posted in collusion, ultimate texas hold'em by stephenhow on June 16, 2010

In Ultimate Texas Hold’Em (UTH), I wondered if it was ever worth checking bottom pair on the flop (basic strategy says to 2x bet), given other player hand info. E.g., if none of the other 5 players paired the board, should you check bottom pair? I ran simulations to compare the value of 2x betting bottom pair, vs. checking on a 6 player table.

2x Bet Bottom Pair vs. Checking With 6 Players
Total table board hits (excluding bottom pair)
0 1 2 3 4 5 6
+10% +15% +20% +25% +30% +35% +40%

This shows that even if no one else hits middle or top pair on the board (i.e., players are holding 10 cards that don’t hit middle or top board cards), it’s still worth 10% of the Ante to 2x bet bottom pair on the flop, compared to checking. It was useful to see these results, because it shows the value of the other players hitting the board, as expected. I wasn’t sure about betting bottom pair when everyone else missed, but the simulations show its still a good bet.

## Collusion Performance for Ultimate Texas Hold’Em

Posted in collusion, ultimate texas hold'em by stephenhow on June 14, 2010

I finally got around to simulating the effects of player collusion in Ultimate Texas Hold’Em. Implementing preflop collusion, and the improved 2nd nut kicker and 3rd nut kicker strategies, the house edge is reduced from 2.2% to 1.6% for 6 players. While this is a small average gain, the real benefit comes from the reduced variance gained by not raising marginal hands when copied with other players.

By checking marginal copied hands, the session outcome distribution improves as shown below. Assuming a bankroll of 200 Antes, a goal of 20 Antes, and a maximum session length of 250 hands, the improved strategy reduces the probability of busting out from .20% down to .13%. The probability of reaching your 20 Ante win is reduced from 69% to 68%, but the distribution for the in-between cases improves as shown.

Session outcome probabilities for 200 Ante bankroll, 20 Ante goal, 250 max hands.

[Note: the low-variance strategy used for the distribution does not raise any basic strategy preflop checking hands; it is designed to minimize variance, and simulates at 2.2% house edge.]

## 3rd Nut Kicker on Paired Board (UTH)

Posted in collusion, ultimate texas hold'em by stephenhow on June 12, 2010

I looked a little deeper into the case of calling (1x bet) on the river with 3rd nut kicker against a paired board. Basic strategy (i.e., no other info known) says to call this hand. However, by taking into account how much the other players have “hit” the board, you can improve this decision. Similar to the 2nd nut kicker on an unpaired board analysis, the decision comes down to how many total “good” cards the players hold, relative to the table size. The table below provides an advanced strategy of when to call with 3rd nut kicker given easily obtained table info. A “good” player card is anything that hits the board, or 2nd nut kicker or better.

Calling vs. Folding with 3rd Nut Kicker on River Against Paired Board
players total player board hits & kickers notes
0 1 2 3 4 5 6
6 -25% -15% -5% +5% +20% +30% +45% fold if ≤ 3 good player cards
5 -12% -5% +6% +17% +28% +42% +54% fold if ≤ 2 good player cards
4 -4% +5% +17% +27% +38% +49% fold if ≤ 1 good player cards
3 +5% +15% +25% +35% +45% always call
2 +13% +23% +33% always call
1 +22% always call

Practically, this advanced strategy is very intuitive, and you’ll find it agrees with your experience. When no one hits the board, or even has a playable kicker, your chances of winning with a minimum kicker go down considerably. However, when everyone hits the board, sometimes multiply (e.g., full house), your chances increase considerably. The table above quantifies this effect, and shows you the relative strength of your hand for the range of scenarios.

Overall, you’ll find the 3rd nut kicker on a paired board decision comes up about 2.5% of the time. While it’s not really frequent, it’s a decision that most players mull over. Knowing the advanced strategy will help you make your final decision quicker, and with more confidence.

## Mississippi Stud Folding Strategy

Posted in +EV, mississippi stud by stephenhow on June 9, 2010

I’ve been playing +EV Mississippi Stud lately at my local Barona Casino, where all player hands are shown. I’ve found the key to playing the +EV strategy quickly is knowing the folding points, and looking ahead a street or two. I remember the folding points by street and high outs, which is pretty easy. I still play with the strategy table in my hand. The stronger your starting hand, the least amount of help (outs) you’ll need from the community cards. This means that you’ll never fold the strongest hands (5 high outs or better), and strong hands (3 or 4 high outs) will typically go to 4th street before you need to make a decision. I’ve made a flow diagram showing this folding decision. It highlights the key folding decision points, arranged by street and high outs.

Folding strategy for Mississippi Stud, all 6 player hands known.

Don’t use this chart at the tables. Just use it to understand the folding points better, and to learn to play faster. Understand that you’ll never fold a hand with 5 high outs. And with 3 high and 2 mid outs, you won’t have to think until 4th Street. Furthermore, you’ll never fold a hand with 3 high and 4 mid outs. However, if your starting hand has less than 3 high and 2 mid outs, you’ll need immediate help on 3th Street. Chances are, you’ll need more help on 4th Street. And you’ll probably need at least 3 high and 4 mid outs to see 5th Street. It’s uncommon when you end up calling with 2 high and 6 mid outs on 5th Street, and very rare to call then with 1 high and 8 mid outs.