Two-Player Collusion for Ultimate Texas Hold’em
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 |
Discount Gambling 
i like your articles,could you show all of the hands,and note the reasons ,good job,thanks.
Hi stephen,
thank you so much for posting all this awesome work here.
Please, I would like to ask you something:
How would the EV look like if we could see the cards of all other 4 or 5 players?
Thank you very much,
George
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
What about copy’s they are not mentioned at all in collusion info.
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 ?
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?
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?
This great stuff
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
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.