# Discount Gambling

## Two-Player Collusion for Ultimate Texas Hold’em

Posted in collusion, ultimate texas hold'em by stephenhow on October 16, 2009

After completing basic strategy for Ultimate Texas Hold’em (UTH), I thought I’d look into the effect of knowing your neighbor’s cards. My program computes the EV for a players down cards, the board, and any known cards, so it’s really easy to analyze the effect of collusion. I didn’t think there’d be much EV gain by sharing info with a partner, because the Ante bet requires dealer to qualify, and the Blind bet pays only for straights and better. Furthermore, you’re only getting even money return on your Play bet; the effect of collusion might be greater for a paytable with odds, like in Mississippi Stud.

I broke the rough analysis into the three decision points, preflop, flop, and river. I kept the analysis to two players, since it’s hard to manage and rely on more than one other player. Also, its harder to formulate a collusion strategy for multiple players.

Overall, collusion only helps when you have a borderline decision, and your partner also happens to have cards that helps your decision. In these relatively rare cases, you’ll pick up an EV gain of about 10% of your Ante bet. So, if you experience these conditions in 1% of your hands, you’ll only gain an overall EV boost of (.01)(.10) = 0.1%, which is negligible. However, it’ll make you feel better when you don’t raise K6o when your friend has a K or 6, and you end up saving 3 Ante bets (the hand is likely to call 1x anyway).

## Preflop

The following table shows some examples of using partner info to change your preflop decision. Each table entry takes me around 16 hours of compute time to analyze. For pairs, partner info can only slow down your raise for pocket 3’s. Otherwise, pocket pair decisions are not affected. Borderline raising cases are influenced by partner cards if you’re within 2 kicker levels of the raising threshold. E.g., K5o +/- 2 means consider raising K3o if partner has an Ace; conversely, consider checking K7o if your partner has a K or 7.

hand partner EV(check) EV(4x) Notes
Kh 4d Ac 7h -0.1285 -0.0870 raise K4o when Ace seen
Ks 2c Ad 7h -0.2168 -0.2615 check K2o even if Ace seen
Qh 5d Ad Kc -0.1817 -0.2378 two overs aren’t enough to make Q5o a raising hand
Qh 6d Ad Kc -0.1243 -0.1238 two overs make Q6o a raising hand
Ks 7d 7h 6s -0.2019 -0.2020 kicker copy increases raising threshold
3s 3h 3d 8c -0.4135 -0.5515 check pocket 3’s if your friend has a 3
4s 4h 4c 8s -0.2033 -0.1467 raise with pocket 4’s, even if your friend has a 4
Ks 7d Kh 6s -0.2257 -0.2218 raise with K7o, even if your friend has a K
Ac 2s Ah 5d -0.2397 -0.2129 still raise A2o if Ace seen

## Flop

There are very few situations where knowledge of your partner’s hand will change the way you play your hand on the flop. Such cases will rarely come up in actual play (e.g., minimum flush draw bets).

hand partner board EV(check) EV(2x) Notes
2s 3d 2h 9h Kh -0.8182 -0.9389 check bottom pair no kicker/draw on a suited board
2s 3d Ah Qh 2h 9h Kh -0.6946 -0.7530 still check even if partner has two overcards and flush
2s 3d Kd 9s 2h 9h Kh -0.7569 -0.8189 still check even if partner has top two pairs
Ts 3s 2s 9s Kh +0.1865 +0.2069 bet 5th nut flush draw with board undercard
Ts 3s 4s 7h 2s 9s Kh +0.0895 +0.0600 check minimum raising draw if partner has one flush out
Js 3s 4s 7s 2s 9s Kh +0.0102 +0.0435 still bet 4th nut draw if partner has two flush outs
2d 3s 2h 3d 2s 9s Kh -0.5499 -0.5163 still bet bottom pair if partner has identical hand
2d 3s 2h 2c 2s 9s Kh -0.5499 -0.4871 still bet bottom pair if partner has two of your outs
Jd Th Qh 6d 8s 9h 2s -0.1535 -0.1725 check JT98 if partner has one of your outs
Jd Qh Kh 8d Ts 9h 2s -0.1730 -0.1352 still bet QJT9 if partner has two of your outs

## River

Partner information may occasionally help your minimum kicker requirements on the river. If the board is paired (i.e., qualifies the Ante bet), when your partner has 2 keys cards, you may decrease your kicker requirements by one.

hand partner board EV(fold) EV(call) Notes
Qs Jd Ts 9h 7c 2d 3h -2.0000 -2.1162 fold without 2nd nut kicker on unpaired board
Qs Jd As Kd Ts 9h 7c 2d 3h -2.0000 -2.0875 fold 3rd nut kicker even if partner has 2 overs
Qs Jd 2c 3d Ts 9h 7c 2d 3h -2.0000 -2.0310 fold 3rd nut kicker even if partner has 2 pairs
Ts Jd 9s 9h 7c 2d 3h -2.0000 -2.0030 fold 4th nut kicker on paired board
Ts Jd As Kd 9s 9h 7c 2d 3h -2.0000 -1.9070 call 4th nut kicker if partner has 2 overs
Ts Jd 2c 3s 9s 9h 7c 2d 3h -2.0000 -1.9070 call 4th nut kicker if partner has 2 dealer outs
Ts 8d 9s 9h 9c 2d 3h -2.0000 -2.0030 fold 5th nut kicker on trips board
Ts 8d As Kd 9s 9h 9c 2d 3h -2.0000 -1.9070 call 5th nut kicker on trips board if partner has 2 overs
Ts 8d As 2c 9s 9h 9c 2d 3h -2.0000 -1.9070 call 5th nut kicker on trips board if partner has 2 key cards
6s 7d Js Jd Kh Ks 2c -2.0000 -2.0697 fold 6th nut kicker on 2 pair board
6s 7d Ad Qh Js Jd Kh Ks 2c -2.0000 -1.9801 call 6th nut kicker on 2 pair board if partner has 2 overs

### 10 Responses

1. steven said, on February 7, 2010 at 2:18 am

i like your articles,could you show all of the hands,and note the reasons ,good job,thanks.

2. george said, on May 24, 2012 at 6:12 pm

Hi stephen,

thank you so much for posting all this awesome work here.

How would the EV look like if we could see the cards of all other 4 or 5 players?

Thank you very much,

George

• stephenhow said, on May 24, 2012 at 6:54 pm

Knowledge of even 6 players cards doesn’t help all that much (on average, it only helps by about 0.5%). For the cases when collusion helps you make a decision counter to basic strategy, you save a lot (10-20%). But these opportunities don’t happen often enough to swing the advantage over to the player. Still, it’s worth looking around before 4x raising (or checking) a marginal hand. You might make a better decision with the collusion info.

Steve

• ray said, on February 21, 2017 at 1:08 am

What about copy’s they are not mentioned at all in collusion info.

• ray said, on February 21, 2017 at 1:50 am

Sorry about the last dumb reply. What I was trying to say poorly, was what is the effect on your partner or partners having 2 of your card. I see this often, as I like to sit in the middle of the table too see as many of my partners cards as I can. the difference must be exponential, right ?

3. Rob said, on January 15, 2017 at 7:48 pm

Question. I know whenever you have a raising hand the book says to raise 4bb. Since there is an option of raising 3x the big blind or 4x the big blind, aren’t there some instances where raising only 3 BB is a better play than raising 4bb on some marginal raising hands? For example, lets say I have Q8 offsuit. I glance at my partners cards and see he has Q4. Even if it is still profitable to raise, what would be higher EV….3 or 4x the BB?

• Rob said, on January 15, 2017 at 7:57 pm

Well after reading my question again I feel quite stupid (Lol). I guess if it is profitable to raise, then you should raise the higher amount. But to rephrase the question with the same example, would this instance (or any similar situations) make a 3x bb raise profitable and a 4xbb raise not profitable?

4. ray said, on February 21, 2017 at 1:51 am

This great stuff

5. ray said, on March 21, 2017 at 3:08 am

What do you do before the flop when you see your partner has a pocket pair and you have one of the 4 as well. example

Your cards Qh,Td Partners cards Qd,Qs

• gordonmichaels said, on May 29, 2017 at 7:28 am

I believe you should check in this specific instance. I estimate that when you have QTo and you glance at one neighbor hand and see 2 of your outs (QQ or QT or TT) then Check is (roughly) about 8% better in EV than Bet 4X. If you glance at 2 neighbor hands with QT and see 2 of your outs (QQ or QT or TT) and 2 other random cards, then Checking is still better by about 5-6% then Bet 4X – and, of course, you will find 2 of your outs more frequently by glancing at 2 hands, lol.