7 ms·

Texas Hold'em Hand Strength, Visualized

179 points by alexcasalboni 12y ago

Aardwolf 12y ago
I thougt that a 2 and a 7 were the worst, but here a 2 and 3 has -35.4% win%-lose% while 2 and 7 has a slightly better -30.8% win%-lose% (second diagram). Any idea what the difference is?
- soupboy 12y ago
  Not a poker expert by far, but this could be because 2/3 can only participate in three straight draws (A2345,23456,34567) whereas 2/7 can be in seven straight hands. (A2345, 23456, 34567, 45678, 56789, 6789T, 789TJ)
  - swang 12y ago
    But you need 4 cards on the board vs using both in your hand for all those 2-7 "straights"
  - mod 12y ago
    Suffice it to say that no, that's not the reason. The reason is entirely high-card strength vs all the hands including cards less than 7, like 65, 64, etc. EDIT: And the value of pairing the 7 vs pairing the 3, again where high card strength matters--6X pairing the 6 won't matter as often. Given there's only two hands, they are somewhat likely to showdown unimproved, not making a pair or better. If you analyzed ten handed tables, 2-7 fares the worst.
- davis_m 12y ago
  72 offsuit is the worst hand against a table of opponents because it is the lowest hand that has no straight or flush opportunities (using both hole cards) 23 offsuit is the worst hand against a single opponent because the likelihood of needing a straight or flush to win the hand decreases and the penalty for having a lower highcard is higher.
- deleted 12y ago
  [deleted]
- swang 12y ago
  27 has high card value compared to 23 vs all other hands. Vs a hand someone would actually play, like JT, you don't have any high card value.
- jdmichal 12y ago
  I'm going to guess that, after you enumerate all possible opposing hands along with all possible table cards, that the 7 gets paired up for a win more often than the straight lines up for a win. Also, due to their low card value, a 2-3 will pretty much automatically lose with no pairings, straights, or flushes. (And even then, it's a weak flush.) If you look at the final diagram, they are almost equal in average expected payout, with the 2-7 dropping value much more than the 2-3. I would expect this is due to the paired 7 not being very strong against more commonly played hands with higher average card value. (In other words, 7 doesn't mean as much when the average opponent has cards higher than a 7.)
regularfry 12y ago
Fascinating. I've seen a few remarkable showdowns between AQ and KK, I never would have guessed the odds were quite so tilted in KK's favour.
- davis_m 12y ago
  AQ vs KK is no different than A10 or AJ vs KK. Even middling Aces like A7 or A8 aren't much worse off because of the lost straight possibilities due to two K's being taken. I haven't looked at the numbers but I would bet that A2-A5 are better off against KK than AQ.
  - GCA10 12y ago
    The numbers have arrived -- thanks to www.sliceeq.com -- and your hunch is right. After 100,000 Monte Carlo simulations, Slice declares that winning odds are as follows KK (70.9%) vs AQsuited or AQoffsuit (29.1%) KK (69.2%) vs A5suited or A5offsuit (30.8%) The reason why the seemingly inferior hand of A5 does better is that it has about a 4.6% chance of making a straight. (Usually A2345, but conceivably something else) The AQ hand has only about a 2.7% chance of making a straight, because the deck is depleted of Kings, making the desired AKQJT straight hard to catch. In general, Slice is a wonderful -- and free -- tool for testing various hand outcomes, particularly against likely ranges that an opponent might have. NOTE: The simulation percentages above will vary slightly each time Slice goes hunting for 100,000 random hands. But the disparity between AQ and A5 generally is somewhere between 1.5 and 2 percentage points.
    - cleatsupkeep 12y ago
      Additionally, if AQ has made a straight with a K, KK now has a redraw to a full house. If the flop was KJT, KK still has a 34% chance to win the hand.
- Devthrowaway80 12y ago
  Yup, it's a classic poker situation referred to as a "race". Interesting visualization!
mod 12y ago
Looks like you used heads up hand strengths, and then full-table hand frequencies, and computed weighted hand strenghts based on that--which is a mistake, of course.
- asselinpaul 12y ago
  Could you expand on this please.
  - pg_is_a_butt 12y ago
    if you don't understand yet, you'll never understand.
  - mod 12y ago
    Yeah, so he's using heads-up hand strengths. So how well QQ fares vs all other hands, for instance, but only one hand at a time. (It's not QQ in a pot with 9 other hands.) Then he computing weighted hand strengths based on frequency of opponents playing a hand. So now QQ doesn't win as often, given that the opponent doesn't play 27o very often so you don't actually get to win hands vs 27o (they don't occur). BUT to determine how often an opponent plays a given hand, he's using the IRC poker database, which is full-table hands. So we see that people play K7 infrequently, for instance, which is true at a full ring game. But in a heads-up match, K7 is played very, very frequently. 85%+ I'd say, but easily 75%. [1] [1] I played heads-up poker professionally for a few years. I logged millions of hands.
  - defen 12y ago
    Hand strength isn't all that matters - you also need to account for "expected winnings". A-9 has a strong hand strength, but in a real game you'll probably win small pots with that and lose large ones. 9-10 has a middling hand strength but when you win, you'll probably win big.
    - mod 12y ago
      That's not what this blog post was about at all, and certainly not what GP asked me to expand upon. You're not wrong, it's just off-topic, essentially. You could say that you need to account for talent, too--but at some point you're going beyond what can be analyzed, at least currently. NLHUHE is not a solved game.
juggernautq 12y ago
This doesn't consider the post-flop strategy though. Since heads-up limit holdem has been solved, it would be interesting to look at the EV of the hand with the optimal play. I would think hands like 22 should have been weaker than show down value suggested..
- TylerE 12y ago
  weakly solved. What was shown is essentially a optimal solution for playing against itself. Not an abitrary villain of unknown tendencies. Besides, limit is dead.
  - mod 12y ago
    This is mostly untrue, to my understanding. It was not solved to maximally exploit a bad villain, but it was solved to be unexploitable, and therefore unbeatable. It would make mountains of chips vs 99%+ of players, probably at 80%+ of the possible rate.
  - iopq 12y ago
    No, it's a solution that loses very slowly vs. an opponent that plays optimally against its strategy. Meaning bluffs and calls at optimal percentages KNOWING the percentages of the program.
- mod 12y ago
  Good luck with that decision tree for 1.3 trillion hands!
  - juggernautq 12y ago
    Actually this EV is the by-product of solving the game. U of Alberta must save it somewhere.. only if we can access it..
jchendy 12y ago
Repost of my reddit rant about UX issues in this visualization: This is interesting, but the layout is pretty confusing for somebody who studies poker. Every other hand chart has AA in the top left, like these from Poker Stove: http://i.imgur.com/cvgvh.png http://i.imgur.com/cvgvh.png Also, yours have duplicate information. As2d and 2dAs are both displayed, when in fact they are equal in poker. Most charts split up suited and offsuit hands, like the ones above or this one from Equilab which lists out every combo separately like you do: http://i.imgur.com/BqS1iht.png http://i.imgur.com/BqS1iht.png Your input format is also confusing. On your site, users have to type '6C 6D' but everywhere else on the internet, that would be written '6c6d'
- cbeaumont 12y ago
  OP here. Your points about the convention for lowercase and aces on the upper-left are fair. However, the duplicated information in these grids was intentional. I was less interested in building a standard suited/off suited odds table (those are readily available) or a strategy tool, and more interested in a visually clear way to explore the "probability landscape". I find these grids with duplicate information easier to grasp on first viewing. I also like seeing how subtle suit-based patterns in panel 1 cancel out when you average all slices into panel 2. You can't show that unless you retain all suit information
encoderer 12y ago
I don't mean to be OT, but does anybody play online anymore? I paid my (cheap) rent in college playing on FullTilt and PokerStars, but the (unreasonable) gov't crackdown seems to have put an end to online play. Or is there a secret I'm not in on?
- jonathansizz 12y ago
  The shutdown was only in the United States. You could cross the border to play freely. Inside the US, there's a couple of states that have legalized within-state play; other than that there are a few minor online venues (several hundred to a couple of thousand players online at any given time). I believe that r/poker has an updated list if you're interested.
- slb 12y ago
  Online poker for real money is still strong outside the USA, and there are a few state-specific enterprises (for example, wsop.com and--until recently--the unfortunately named Ultimate Poker in Nevada) starting for USA states that have specifically legalized online poker.
- mod 12y ago
  There's a couple of sites where you can still play, however there's not much action and you essentially can't cash out. I have a buddy with ~200k on some network, I think merge, and he has a pending cashout for 2k (the max) that has been pending for months. He can't initiate another until the last has been resolved. Until about a year ago (post-black-friday, of course) he was able to remove about 12-14k/year, given the wait time per check and the maximum of 2k per check. You need to move or play in a casino.
- rjett 12y ago
  bovada... not nearly as many players as there used to be when it operated as bodog, but it'll still scratch your itch.
- pcthrowaway 12y ago
  Seals With Clubs. Deposit bitcoin. Play with bitcoin. Withdraw bitcoin. Probably not legal in the U.S. but it's a great site (and they don't check your IP origin afaik, though you could probably get in with TOR or a proxy anyway).
cju 12y ago
It reminds me the "Temperature Maps" by Sho Sengoku in backgammon [1]. For a given move, it syntheses the equities (somehow probability of winning) for each possible dice roll of the opponent on next turn. It allows among other thing to see how dependent of luck you be after a move (by looking at the contrast between squares). [1] http://www.bkgm.com/articles/Sengoku/TemperatureMap/index.html http://www.bkgm.com/articles/Sengoku/TemperatureMap/index.ht...
ChuckMcM 12y ago
I enjoyed this analysis, I went through something similar when designing a bridge hand 'strength' estimator as part of an automated bidding program. I'm going to have to go back and see if I can borrow some of these ideas :-)

179 points by alexcasalboni 12y ago

I thougt that a 2 and a 7 were the worst, but here a 2 and 3 has -35.4% win%-lose% while 2 and 7 has a slightly better -30.8% win%-lose% (second diagram). Any idea what the difference is?

soupboy 12y ago
Not a poker expert by far, but this could be because 2/3 can only participate in three straight draws (A2345,23456,34567) whereas 2/7 can be in seven straight hands. (A2345, 23456, 34567, 45678, 56789, 6789T, 789TJ)
- swang 12y ago
  But you need 4 cards on the board vs using both in your hand for all those 2-7 "straights"
- mod 12y ago
  Suffice it to say that no, that's not the reason. The reason is entirely high-card strength vs all the hands including cards less than 7, like 65, 64, etc. EDIT: And the value of pairing the 7 vs pairing the 3, again where high card strength matters--6X pairing the 6 won't matter as often. Given there's only two hands, they are somewhat likely to showdown unimproved, not making a pair or better. If you analyzed ten handed tables, 2-7 fares the worst.
davis_m 12y ago
72 offsuit is the worst hand against a table of opponents because it is the lowest hand that has no straight or flush opportunities (using both hole cards) 23 offsuit is the worst hand against a single opponent because the likelihood of needing a straight or flush to win the hand decreases and the penalty for having a lower highcard is higher.
deleted 12y ago
[deleted]
swang 12y ago
27 has high card value compared to 23 vs all other hands. Vs a hand someone would actually play, like JT, you don't have any high card value.
jdmichal 12y ago
I'm going to guess that, after you enumerate all possible opposing hands along with all possible table cards, that the 7 gets paired up for a win more often than the straight lines up for a win. Also, due to their low card value, a 2-3 will pretty much automatically lose with no pairings, straights, or flushes. (And even then, it's a weak flush.) If you look at the final diagram, they are almost equal in average expected payout, with the 2-7 dropping value much more than the 2-3. I would expect this is due to the paired 7 not being very strong against more commonly played hands with higher average card value. (In other words, 7 doesn't mean as much when the average opponent has cards higher than a 7.)

regularfry 12y ago

Fascinating. I've seen a few remarkable showdowns between AQ and KK, I never would have guessed the odds were quite so tilted in KK's favour.

davis_m 12y ago
AQ vs KK is no different than A10 or AJ vs KK. Even middling Aces like A7 or A8 aren't much worse off because of the lost straight possibilities due to two K's being taken. I haven't looked at the numbers but I would bet that A2-A5 are better off against KK than AQ.
- GCA10 12y ago
  The numbers have arrived -- thanks to www.sliceeq.com -- and your hunch is right. After 100,000 Monte Carlo simulations, Slice declares that winning odds are as follows KK (70.9%) vs AQsuited or AQoffsuit (29.1%) KK (69.2%) vs A5suited or A5offsuit (30.8%) The reason why the seemingly inferior hand of A5 does better is that it has about a 4.6% chance of making a straight. (Usually A2345, but conceivably something else) The AQ hand has only about a 2.7% chance of making a straight, because the deck is depleted of Kings, making the desired AKQJT straight hard to catch. In general, Slice is a wonderful -- and free -- tool for testing various hand outcomes, particularly against likely ranges that an opponent might have. NOTE: The simulation percentages above will vary slightly each time Slice goes hunting for 100,000 random hands. But the disparity between AQ and A5 generally is somewhere between 1.5 and 2 percentage points.
  - cleatsupkeep 12y ago
    Additionally, if AQ has made a straight with a K, KK now has a redraw to a full house. If the flop was KJT, KK still has a 34% chance to win the hand.
Devthrowaway80 12y ago
Yup, it's a classic poker situation referred to as a "race". Interesting visualization!

mod 12y ago

Looks like you used heads up hand strengths, and then full-table hand frequencies, and computed weighted hand strenghts based on that--which is a mistake, of course.

asselinpaul 12y ago
Could you expand on this please.
- pg_is_a_butt 12y ago
  if you don't understand yet, you'll never understand.
- mod 12y ago
  Yeah, so he's using heads-up hand strengths. So how well QQ fares vs all other hands, for instance, but only one hand at a time. (It's not QQ in a pot with 9 other hands.) Then he computing weighted hand strengths based on frequency of opponents playing a hand. So now QQ doesn't win as often, given that the opponent doesn't play 27o very often so you don't actually get to win hands vs 27o (they don't occur). BUT to determine how often an opponent plays a given hand, he's using the IRC poker database, which is full-table hands. So we see that people play K7 infrequently, for instance, which is true at a full ring game. But in a heads-up match, K7 is played very, very frequently. 85%+ I'd say, but easily 75%. [1] [1] I played heads-up poker professionally for a few years. I logged millions of hands.
- defen 12y ago
  Hand strength isn't all that matters - you also need to account for "expected winnings". A-9 has a strong hand strength, but in a real game you'll probably win small pots with that and lose large ones. 9-10 has a middling hand strength but when you win, you'll probably win big.
  - mod 12y ago
    That's not what this blog post was about at all, and certainly not what GP asked me to expand upon. You're not wrong, it's just off-topic, essentially. You could say that you need to account for talent, too--but at some point you're going beyond what can be analyzed, at least currently. NLHUHE is not a solved game.

juggernautq 12y ago

This doesn't consider the post-flop strategy though. Since heads-up limit holdem has been solved, it would be interesting to look at the EV of the hand with the optimal play. I would think hands like 22 should have been weaker than show down value suggested..

TylerE 12y ago
weakly solved. What was shown is essentially a optimal solution for playing against itself. Not an abitrary villain of unknown tendencies. Besides, limit is dead.
- mod 12y ago
  This is mostly untrue, to my understanding. It was not solved to maximally exploit a bad villain, but it was solved to be unexploitable, and therefore unbeatable. It would make mountains of chips vs 99%+ of players, probably at 80%+ of the possible rate.
- iopq 12y ago
  No, it's a solution that loses very slowly vs. an opponent that plays optimally against its strategy. Meaning bluffs and calls at optimal percentages KNOWING the percentages of the program.
mod 12y ago
Good luck with that decision tree for 1.3 trillion hands!
- juggernautq 12y ago
  Actually this EV is the by-product of solving the game. U of Alberta must save it somewhere.. only if we can access it..

jchendy 12y ago

Repost of my reddit rant about UX issues in this visualization: This is interesting, but the layout is pretty confusing for somebody who studies poker. Every other hand chart has AA in the top left, like these from Poker Stove: http://i.imgur.com/cvgvh.png http://i.imgur.com/cvgvh.png Also, yours have duplicate information. As2d and 2dAs are both displayed, when in fact they are equal in poker. Most charts split up suited and offsuit hands, like the ones above or this one from Equilab which lists out every combo separately like you do: http://i.imgur.com/BqS1iht.png http://i.imgur.com/BqS1iht.png Your input format is also confusing. On your site, users have to type '6C 6D' but everywhere else on the internet, that would be written '6c6d'

cbeaumont 12y ago
OP here. Your points about the convention for lowercase and aces on the upper-left are fair. However, the duplicated information in these grids was intentional. I was less interested in building a standard suited/off suited odds table (those are readily available) or a strategy tool, and more interested in a visually clear way to explore the "probability landscape". I find these grids with duplicate information easier to grasp on first viewing. I also like seeing how subtle suit-based patterns in panel 1 cancel out when you average all slices into panel 2. You can't show that unless you retain all suit information

encoderer 12y ago

I don't mean to be OT, but does anybody play online anymore? I paid my (cheap) rent in college playing on FullTilt and PokerStars, but the (unreasonable) gov't crackdown seems to have put an end to online play. Or is there a secret I'm not in on?

jonathansizz 12y ago
The shutdown was only in the United States. You could cross the border to play freely. Inside the US, there's a couple of states that have legalized within-state play; other than that there are a few minor online venues (several hundred to a couple of thousand players online at any given time). I believe that r/poker has an updated list if you're interested.
slb 12y ago
Online poker for real money is still strong outside the USA, and there are a few state-specific enterprises (for example, wsop.com and--until recently--the unfortunately named Ultimate Poker in Nevada) starting for USA states that have specifically legalized online poker.
mod 12y ago
There's a couple of sites where you can still play, however there's not much action and you essentially can't cash out. I have a buddy with ~200k on some network, I think merge, and he has a pending cashout for 2k (the max) that has been pending for months. He can't initiate another until the last has been resolved. Until about a year ago (post-black-friday, of course) he was able to remove about 12-14k/year, given the wait time per check and the maximum of 2k per check. You need to move or play in a casino.
rjett 12y ago
bovada... not nearly as many players as there used to be when it operated as bodog, but it'll still scratch your itch.
pcthrowaway 12y ago
Seals With Clubs. Deposit bitcoin. Play with bitcoin. Withdraw bitcoin. Probably not legal in the U.S. but it's a great site (and they don't check your IP origin afaik, though you could probably get in with TOR or a proxy anyway).

cju 12y ago

It reminds me the "Temperature Maps" by Sho Sengoku in backgammon [1]. For a given move, it syntheses the equities (somehow probability of winning) for each possible dice roll of the opponent on next turn. It allows among other thing to see how dependent of luck you be after a move (by looking at the contrast between squares). [1] http://www.bkgm.com/articles/Sengoku/TemperatureMap/index.html http://www.bkgm.com/articles/Sengoku/TemperatureMap/index.ht...

ChuckMcM 12y ago

I enjoyed this analysis, I went through something similar when designing a bridge hand 'strength' estimator as part of an automated bidding program. I'm going to have to go back and see if I can borrow some of these ideas :-)