(This article was originally published at Statistical Modeling, Causal Inference, and Social Science, and syndicated at StatsBlogs.)
The thread of our recent discussion of quantifying luck vs. skill in sports turned to poker, motivating the present post.
1. Can good poker players really “read” my cards and figure out what’s in my hand?
For a couple years in grad school a group of us had a regular Thursday-night poker game, nickel-dime-quarter with a maximum bet of $2, I believe it was. I did ok, it wasn’t hard to be a steady winner by just laying low most of the time and raising when I had a good hand. Since then I’ve played only very rarely (one time was a memorable experience with some journalists and a foul-mouthed old-school politico—I got out of that one a couple hundred dollars up but with no real desire to return), but I did have a friend who was really good. I played a couple of times with him and some others, and it was like the kind of thing you hear about: he seemed to be able to tell what cards I was holding. Don’t get me wrong here, I’m not saying that he was cheating or that it was uncanny or anything, and it’s not like he was taking my money every hand. As always in a limit game, the outcomes had a lot of randomness. But from time to time, it big hands, it really did seem like he was figuring me out. I didn’t think to ask him how he was doing it but I was impressed.
Upon recent reflection, though (many years later), it seems to me that I was slightly missing the point. The key is that my friend didn’t need to “read” me or know what I had; all he needed to do was make the right bets (or, to be more precise, make betting decisions that would perform well on average). He could well have made some educated guesses about my holdings based on my betting patterns (or even my “tells”) and used that updated probability distribution to make more effective betting decisions. The point is that, in many many settings, he doesn’t need to guess my cards; he just needs a reasonable probability distribution (which might be implicit). For example, in some particular situation in a particular hand, perhaps it would be wise for him to fold if he the probability is more than 30% that a particular hole card of mine is an ace. With no information, he’d assess this event as having an (approximate) 2% probability. So do I have that ace? He just needs to judge whether the probability is greater or less than 30%, an assessment that he can do using lots of information available to him. But once he makes that call, if he does it right (as he will, often enough; that’s part of what it means to be a good poker player), it’ll seem to me like he was reading my hand.
2. Some references on luck vs. skill in poker
Louis Raes pointed to three papers:
Ben van der Genugten and Peter Borm wrote quite a bit on Poker and the extent to which skill or luck is important. This work is mainly geared towards Dutch regulation but interesting nonetheless.
See:
http://link.springer.com/article/10.1007/s001860400347#page-1
http://link.springer.com/chapter/10.1007/978-1-4615-4627-6_3#page-1
http://link.springer.com/article/10.1007/BF02579073#page-1
3. Rick Schoenberg’s definition
Rick Schoenberg sent along an excerpt from his book, Probability with Texas Holdem Applications. Rick writes:
Surprisingly, a lot of books on game theory do not define the words “luck” or “skill”, maybe because it is very hard to do so. . . . in poker I [Rick] define skill as equity gained during the betting rounds and luck as equity gained during the deal of the cards. I then go through a televised twenty-something hand battle between Howard Lederer and Dario Minieri, two players with about as opposite styles as you can get, and try to quantify how much of Lederer’s win was due to luck and how much was due to skill.
I’ll go through Rick’s material and intersperse some comments.
Here’s section 4.4 of Rick’s book:
The determination of whether Texas Hold’em is primarily a game of luck or skill has recently become the subject of intense legal debate. Complicating things is the fact that the terms luck and skill are extremely difficult to define. Surprisingly, rigorous definitions of these terms appear to have eluded most books and journal articles on game theory. A few articles have defined skill in terms of the variance in results among different players, with the idea that players should perform more similarly if a game is mostly based on luck, but their results might differ more substantially if the game is based on skill. Another definition of skill is the extent to which players can improve, and it has been shown that poker does indeed possess a significant amount of this feature (e.g. Dedonno and Detterman, 2008). Others have defined skill in terms of the variation in a given player’s results, since less variation would indicate that fewer repetitions are necessary in order to determine the statistical significance of one’s long-term edge in the game, and hence the sooner one can establish that one’s average profits or losses are primarily due to skill rather than short term luck.
These definitions above are obviously extremely problematic for
various reasons. One is that they rely on the game in
question being played repeatedly before even a crude assessment of
luck or skill could be made. More importantly, there are many contests
of skill wherein the differences between players are small, or where one’s results vary wildly. For instance, in Olympic trials of 100-meter sprints, the differences between finishers are typically quite
small, often just hundredths of a second. This hardly implies that the
results are based on luck. There are also sporting events where an
individual’s results might vary widely from one day to another, e.g. pitching in baseball, but that hardly means that luck plays a major
role.
Regarding the quantification of the amount of luck or skill in a particular game of poker, a possibility might be to define luck as equity
gained when cards are dealt by the dealer, and skill as equity gained
by one’s actions during the betting rounds. (Recall that equity was defined in Section 4.3 as the product cp.) That is, there are several
reasons you might gain equity during a hand:
* The cards dealt by the dealer (whether the players’ hole cards or
the flop, turn, or river) give you a greater chance of winning the
hand in a showdown,
* The size of the pot is increased while your chance to win the hand
in a showdown is better than that of your opponent(s).
* By betting, you get others to fold and thus win a pot that you might otherwise have lost.
Certainly, anyone would characterize the first case as luck, unless
perhaps one believes in ESP or time travel.
Uh oh. Daryl Bem’s getting off the bus right here, I think!
Thus [Rick continues], a possible way to
estimate the skill in poker can be obtained by looking at the
second and third cases above. That is, we may view one’s skill as
being comprised of the equity that one gains during the betting
rounds, whereas luck is equity gained by the deal of the cards. The
nice thing about this is that it is easily quantifiable, and one may
dissect a particular poker game and analyze how much equity each
player gained due to luck or skill.
There are obvious objections to this. First, why equity? One’s equity (which is sometimes called express equity) in a pot is defined as one’s expected
return from the pot given no future betting, and the assumption of no
future betting may seem absurdly simplistic and unrealistic. On the other
hand, unlike implied equity which would account for betting on future betting rounds, express equity is unambiguously defined and easy to compute.
Second, situations can occur where one would expect a terrible player
to gain equity during the betting rounds against even the greatest
player in the world, so to attribute such equity gains to skill
might be objectionable. For instance, in heads up Texas Hold’em, if the two players are dealt AA and KK, one would expect the player with KK to
put a great deal of chips in while way behind, and this situation
seems more like bad luck for the player with KK than any deficit in
skill. One possible response to this objection is that skill is difficult to define, and in fact most poker players, probably due to
their huge and fragile egos, tend to chalk up losses for virtually any
reason as being solely due to bad luck. In some sense, anything can be
attributed to luck if one has a general enough definition of the word.
Even if a player does an amazingly skillful poker play, such as
folding a very strong hand because of an observed tell or betting
pattern, one could argue that it was lucky that the player happened
to observe that tell, or even that the player was lucky to have been
born with the ability to discern the tell. On the other hand, situations like the AA versus KK example truly do seem like bad luck.
It is difficult to think of any remedy to this problem. It may be that
the word skill is too strong a word, and that while it may be of
interest to analyze hands in terms of equity, one should instead use
the term equity gained during the betting rounds rather than skill in what follows.
We should distinguish between the two concerns that Rick is noting here. First, the division between luck and skill is inherently arbitrary (in a similar way to the arbitrariness of the division between prior and likelihood in a hierarchical model). To take things to extremes, you could say that a skillful player is lucky to have been born with such skill. As Rick explains, some things are definitely luck, but nothing can really be defined as definitely being skill.
Rick’s second concern, which I share, is that in his example it does not seem like skill that, if you happen to get the pair of aces, your equity just keeps going up because the other player doesn’t know what you have. So I agree with him that his definition has real problems.
Rick continues:
Below is an extended example intended to illustrate the division of luck and skill in a given game of Texas Hold’em. The example involves the end of a tournament on Poker After Dark televised during the first week of October 2009. Dario Minieri and Howard Lederer were the final two players. Since this portion of the tournament involves only these two players, and since all hands (or virtually all) were televised, this example provides us with an opportunity to attempt to parse out how much of Lederer’s win was due to skill and how much was due to luck.
Technical side note: Before we begin, we need to clarify a couple of potential ambiguities. There is some ambiguity in the definition of equity before the flop,
since the small and big blind have put in different amounts of chips.
The definition used here is that preflop equity is the expected
profit (equity one would have in the pot after calling minus cost),
assuming the big blind and small blind call as well, or the equity one
would have by folding, whichever happens to be greater. For example,
in heads up Texas Hold’em with blinds of 800 and 1600, the preflop equity
for the big blind is 2bp – 1600, and max{2bp – 1600, -800} for the
small blind, where p is the probability of winning the pot in a
showdown, and b is the amount of the big blind. Define increases in
the size of the pot as relative to the big blind, i.e. increasing the
pot size by calling preflop does not count as skill. The probability p of winning the hand in a showdown was obtained using the odds calculator at cardplayer.com, and
the probability of a tie is divided equally among the two players in determining p.
Example 4.4.1. Below are summaries of all 27 hands shown on Poker After Dark in October 2009 between
Dario Minieri and Howard Lederer in the Heads Up portion of the
tournament, with each hand’s equity gains and losses broken down as
luck or skill. Each hand is analyzed from Minieri’s perspective, i.e. “skill -100” refers to 100 chips of equity
gained by Lederer during a betting round. The question we seek to address is: how much of Lederer’s win was due to skill, and how much of it was due to luck?
Based on the concerns discussed above, “skill” doesn’t seem to be the right word here. On the other hand, maybe this is no big deal because the “luck” aspects will ultimately average out.
For example [Rick continues], here is a detailed breakdown of hand 4, where the blinds were 800/1600, Minieri was dealt A♣ J♣, Lederer had A♥ 9♥, Minieri raised to 4300 and Lederer called. The flop was 6♣ 10♠ 10♣, Lederer checked, Minieri bet 6500, and Lederer folded.
a) Preflop dealing (luck): Minieri +642.08.
Minieri was dealt a 70.065% probability of winning the pot in a showdown, so Minieri’s
increase in equity is 70.065 x 3200 – 1600 = 642.08 chips.
Lederer was dealt a 29.935% probability to win the pot in a showdown, so his increase in equity is 29.935% * 3200
- 1600 = -642.08 chips.
b) Preflop betting (skill): Minieri +1083.51.
The pot was increased to 8600. 8600-3200=5400.
Minieri paid an additional 2700 but had 70.065% x 5400 = 3783.51
additional equity, so Minieri’s expected profit due to betting was 3783.51 –
2700 = 1083.51 chips.
Correspondingly, Lederer’s expected profit due to betting was -1083.51 chips, since
29.935% x 5400 – 2700 = -1083.51.
c) Flop dealing (luck): Minieri +1362.67.
After the flop was dealt, Minieri’s probability of winning the 8600 chip pot in a showdown went from 70.065% to 85.91%. So by luck, Minieri increased his equity by (85.91% –
70.065%) x 8600 = +1362.67 chips.
d) Flop betting (skill): Minieri +1211.74.
Because of the betting on the flop, Minieri’s equity went from 85.91%
of the 8600 pot to 100% of the pot, so Minieri increased his equity by
(100% – 85.91%)x8600 = 1211.74 chips.
So during the hand, by luck, Minieri increased his equity by 642.08 + 1362.67 = 2004.75 chips.
By skill, Minieri increased his equity by 1083.51 + 1211.74 = 2295.25 chips.
Notice that the total = 2004.75 + 2295.25 = 4300, which is the number
of chips Minieri won from Lederer in the hand.
Note that [Never use the expression "note that"! Also avoid "very" and "obviously" --- ed.] before the heads-up battle began, the broadcast reported that Minieri had 72,000 chips, and
Lederer 48,000. Minieri must have won some chips in hands they did not televise, because the grand
total has Minieri losing about 74,500 chips.
(Blinds 800 and 1600.)
Hand 1. Lederer A♣ 7♠, Minieri 6♠ 6•. Lederer 43.535%, Minieri 56.465%.
Lederer raises to 4300. Minieri raises to 47800. Lederer folds.
Luck +206.88.
Skill +4093.12.
Hand 2. Minieri 4♠ 2•, Lederer K♠ 7♥. Minieri 34.36%, Lederer 65.64%.
Minieri raises to 4300, Lederer raises all in for 43500, Minieri folds.
Luck -500.48.
Skill -3799.52.
Hand 3. Lederer 6♥ 3•, Minieri A• 9♣. Lederer 34.965%, Minieri 65.035%.
Lederer folds in the small blind.
Luck +481.12.
Skill +318.88.
Hand 4. Minieri A♣ J♣, Lederer A♥ 9♥. Minieri 70.065%, Lederer 29.935%.
Minieri raises to 4300, Lederer calls 2700.
Flop 6♣ 10♠ 10♣. Minieri 85.91%, Lederer 14.09%.
Lederer checks, Minieri bets 6500, Lederer folds.
Luck +2004.75.
Skill +2295.25.
Hand 5. Lederer 5♠ 3♥, Minieri 7• 6♠. Lederer 35.765%, Minieri 64.235%.
Lederer folds in the small blind.
Luck +455.52.
Skill +344.48.
Hand 6. Minieri K♥ 10♣, Lederer 5¨ 2¨.
Minieri 61.41%, Lederer 38.59%
Minieri raises to 3200, Lederer raises to 9700, Minieri folds.
Luck +365.12.
Skill -3565.12
Hand 7. Minieri 10• 7♠, Lederer Q♣ 2♥.
Minieri 43.57%, Lederer 56.43%.
Minieri raises to 3200, Lederer calls 1600.
Flop 8♠ 2♠ Q♥.
Minieri 7.27%, Lederer 92.73%.
Lederer checks, Minieri bets 3200, Lederer calls.
Turn 4•.
Minieri 0%, Lederer 100%.
Lederer checks, Minieri bets 10,000, Lederer calls.
River A♥.
Lederer checks, Minieri checks.
Luck -205.76 – 2323.20 – 930.56 = -3459.52.
Skill -205.76 – 2734.72 – 10000 = -12940.48.
Hand 8. Lederer 7♣ 2•, Minieri 9♣ 4•.
Minieri 64.28%, Lederer 35.72%.
Lederer folds.
Luck +456.96.
Skill +343.04.
Hand 9. Minieri 4♠ 2♣, Lederer 8♥ 7•.
Minieri 34.345%, Lederer 65.655%.
Minieri raises to 3200, Lederer calls 1600.
Flop 3• 9♥ J♥.
Minieri 22.025%, Lederer 77.975%.
Lederer checks, Minieri bets 4800, Lederer folds.
Luck -500.96 – 788.48 = -1289.44.
Skill -500.96 + 4990.40 = +4489.44.
Hand 10. Lederer K♠ 5♠, Minieri K♥ 7♣.
Minieri 59.15%, Lederer 40.85%.
Lederer calls 800, Minieri raises to 6400, Lederer folds.
Luck +292.80.
Skill +1307.20.
Hand 11. Minieri A♥ 8♥, Lederer 6♥ 3♠.
Minieri 66.85%, Lederer 33.15%.
Minieri raises to 3200. Lederer folds.
Luck +539.20.
Skill +1060.80.
Hand 12. Lederer A• 4•. Minieri 7• 3♥.
Minieri 34.655%, Lederer 65.345%.
Lederer raises to 4300, Minieri raises to 11500, Lederer folds.
Luck -491.04.
Skill +4791.04.
Hand 13. Minieri 6♣ 3♣, Lederer K♠ 6♠.
Minieri 29.825%, Lederer 70.175%.
Minieri raises to 4800, Lederer calls 3200.
Flop 5♥ J♣ 5♣.
Minieri 47.425%, Lederer 52.575%.
Lederer checks, Minieri bets 6000, Lederer folds.
Luck -645.60 + 1689.60 = +1044.
Skill -1291.20 + 5047.20 = +3756.
Hand 14. Lederer 7• 5♠, Minieri 8• 5•.
Minieri 69.44%, Lederer 30.56%.
Lederer calls 800, Minieri checks.
Flop K♥ 10♠ 8♣.
Minieri 94.395%, Lederer 5.605%.
Minieri checks, Lederer bets 1800, Minieri calls.
Turn 7♠.
Minieri 95.45%, Lederer 4.55%.
Minieri checks, Lederer checks.
River 6♥.
Check, check.
Luck +622.08 + 798.56 + 71.74 + 309.40 = 1801.78.
Skill 0 + 1598.22 + 0 + 0 = 1598.22.
Blinds 1000/2000.
Hand 15. Minieri 9• 5♠, Lederer A♥ 5•.
Minieri 26.755%, Lederer 73.245%.
Minieri calls 1000, Lederer raises to 7000, Minieri raises to 14000,
Lederer calls 7000.
Flop 10♠ Q• 6♥.
Minieri 15.35%, Lederer 84.65%.
Lederer checks, Minieri bets 14000, Lederer folds.
Luck -929.80 – 3193.40 = -4123.20.
Skill -5578.80 + 23702 = 18123.20.
Hand 16. Lederer 5♠ 5♥, Minieri A♣ J•.
Minieri 46.085%, Lederer 53.915%.
Lederer calls 1000, Minieri raises to 26800, Lederer calls all in. The board is 3♠ 9♠ K♠ 10• 9•.
Luck -156.60 – 24701.56 = -24858.16.
Skill -1941.84.
Hand 17. Minieri K♣ 10♣, Lederer 7• 5•.
Minieri 62.22%, Lederer 37.78%.
Minieri raises to 5000, Lederer calls 3000.
Flop J♠ J• 4♠.
Minieri 69.90%, Lederer 30.10%.
Check check.
Turn 8♠.
Minieri 77.27%, Lederer 22.73%.
Lederer bets 6000, Minieri folds.
Luck +488.80 + 768 + 737 = 1993.80.
Skill +733.20 + 0 – 7727 = -6993.80.
Hand 18. Lederer 5♠ 5♣, Minieri 10♠ 6♥.
Minieri 46.12%, Lederer 53.88%.
Lederer calls 1000, Minieri checks.
Flop 7♣ 8♣ Q♥.
Minieri 38.235%, Lederer 61.765%.
Minieri checks, Lederer bets 2000, Minieri calls.
Turn J♥.
Minieri 22.73%, Lederer 77.27%.
Minieri bets 4000, Lederer folds.
Luck -155.20 – 315.40 – 1240.40 = -1711.
Skill 0 – 470.60 + 6181.60 = +5711.
Hand 19. Lederer K♥ 5♠, Minieri K♣ 10•.
Minieri 73.175%, Lederer 26.825%.
Lederer raises to 5000, Minieri calls 3000.
Flop J• 8♥ 10♥.
Minieri 92.575%, Lederer 7.425%.
Check, check.
Turn 5•.
Minieri 95.45%, Lederer 4.55%.
Minieri bets 6000, Lederer folds.
Luck +927 + 1940 + 287.50 = 3154.50.
Skill +1390.50 + 0 + 455 = 1845.50.
Hand 20. Minieri 7♣ 2♠, Lederer Q♠ 9♠.
Minieri 30.205%, Lederer 69.795%.
Minieri raises to 6000. Lederer calls 4000.
Flop A• A♠ Q•.
Minieri 1.165%, Lederer 98.835%.
Lederer checks, Minieri bets 6000, Lederer calls.
Turn J♣.
Minieri 0%, Lederer 100%.
Lederer checks, Minieri bets 14000, Lederer raises to 35800, Minieri
folds.
Luck -791.80 – 3484.80 – 279.60 = -4556.20.
Skill -1583.60 – 5860.20 – 14000 = -21443.80.
Hand 21. Minieri 10♥ 3•, Lederer Q♥ J♠.
Minieri 30.00%, Lederer 70.00%.
Minieri calls 1000, Lederer checks.
Flop 8♠ 4♥ J♣.
Minieri 4.34%, Lederer 95.66%.
Lederer checks, Minieri bets 2000, Lederer raises to 7500, Minieri
raises to 18500, Lederer raises all-in, Minieri folds.
Luck -800 – 1026.40 = -1826.40.
Skill 0 – 18673.60 = -18673.60.
Hand 22. Lederer A♠ 2•, Minieri 5♣ 3♥.
Minieri 42.345%, Lederer 57.655%.
Lederer calls 1000. Minieri checks.
Flop K♠ 10♣ 3♠.
Minieri 80.10%, Lederer 19.90%.
Check check.
Turn Q♠.
Minieri 65.91%, Lederer 34.09%.
Check, Lederer bets 2000, Minieri folds.
Luck -306.20 + 1510.20 – 567.60 = 636.40.
Skill 0 + 0 – 2636.40 = -2636.40.
(Blinds 1500/3000.)
Hand 23. Minieri 7♥ 7♣, Lederer 8• 3•.
Minieri 68.175%, Lederer 31.825%.
Minieri all-in for 21,700, Lederer folds.
Luck +1090.50.
Skill +1909.50.
Hand 24. Minieri Q♥ 5♥, Lederer 8• 5•.
Minieri 68.37%, Lederer 31.63%.
Minieri all-in for 26,200, Lederer folds.
Luck +1102.20.
Skill +1897.80.
Hand 25. Lederer 9♣ 3♣, Minieri 5• 2•.
Minieri 40.63%, Lederer 59.37%.
Lederer folds.
Luck -562.20.
Skill +2060.20.
Hand 26. Minieri 10♣ 2♠, Lederer 7♣ 7♥.
Minieri 29.04%, Lederer 70.96%.
Minieri folds.
Luck -1257.60.
Skill -242.40.
Hand 27. Lederer Q♣ 9♣, Minieri A♣ 5♠.
Minieri 55.37%, Lederer 44.63%.
Lederer all-in for 29,200. Minieri calls.
Board 7♣ 6♣ 10♠ Q♠ 6•.
Luck +322.20 – 32336.08 = -32013.88.
Skill +2813.88.
Grand Totals:
Luck -61023.59.
Skill -13478.41.
Overall, although Lederer’s gains were primarily (about 81.9%) due
to luck, Lederer also gained more equity due to skill than
Minieri. On the first 19 hands, Minieri actually gained 20,836.41 in equity due to skill,
and it appeared that Minieri was outplaying Lederer.
On hands 20 and 21, however, Minieri tried two huge unsuccessful bluffs, both
on hands (especially hand 20) where he should probably have strongly suspected
that Lederer would be likely to call, and on those two hands combined,
Minieri lost 40,117.40 in equity due to skill. Although Minieri played
very well on every other hand, all of Minieri’s good plays on other
hands could not overcome the huge loss of skill equity from just
those two hands.
It is important to note that the player who gains the most equity due to skill does not always win. In the first 19 hands of this example, for instance, Minieri gained 20836.41 in equity attributed to skill, but because of bad luck, Minieri actually lost a total of 2800 chips over these same 19 hands. The bad luck Minieri suffered on hand 16 negated most of his gains due to skillful play. A common misconception is that one’s luck will ultimately balance out, i.e. that one’s total good luck will eventually exactly equal one’s total bad luck, but this is not true. Assuming one plays the same game repeatedly and independently, and assuming the expected value of one’s equity due to luck is 0 which seems reasonable, then one’s average equity per hand gained by luck will ultimately converge to zero. This is the law of large numbers, and is discussed further in Section 7.4. It does not imply that one’s total equity gained by luck will converge to zero, however. Potential misconceptions about the laws of large numbers and arguments about possible overemphasis on equity are discussed in Section 7.4.
To conclude this Section, a nice [I'd avoid the word "nice" too! --- AG] illustration of the potential pitfalls of analyzing a hand purely based on equity is a recent hand from Season 7 of High Stakes Poker. In this hand, with blinds of $400 and $800 plus $100 antes from each of the 8 players, after Bill Klein straddled for $1600, Phil Galfond raised to $3500 with Q♠ 10♥, Robert Croak [now that's a great poker name. --- ed.] called in the big blind with A♣ J♣, Klein called with 10♠ 6♠, and the other players folded. The flop came J♠ 9♥ 2♠, giving Croak top pair, Klein a flush draw, and Galfond an open ended straight draw. Croak bet $5500, Klein raised to $17500, and Galfond and Croak called. At this point, it is tempting to compute Klein’s probability of winning the hand by computing the probability of exactly one more spade coming on the turn and river without making a full house for Croak, or the turn and river including two 6s, or a 10 and a 6. This would yield a probability of [(8 x 35 - 4 - 4) + C(3,2) + 2x3] ÷ C(43,2) = 281/903 ~ 31.12%, and Klein could also split the pot with a straight if the turn and river were KQ or Q8 or 78, without a spade, which has a probability of [3x3 + 3x3 + 3x3] ÷ C(43,2) = 27/903 ~ 2.99%. These seem to be the combinations Klein needs, and one would not expect Klein to win the pot with a random turn and river combination not on this list, and especially not if the turn and river contain a king and a jack with no spades. But look at what actually happened in the hand. The turn was the K♣, giving Galfond a straight, and Croak checked, Klein bet $28000, Galfond raised to $67000, Croak folded, and Klein called. The river was the J♥, Klein bluffed $150000, and Galfond folded, giving Klein the $348,200 pot!
P.S. Image above from Barney Townshend shows some playing cards that Schoenberg designed a few years earlier.
The post Luck vs. skill in poker appeared first on Statistical Modeling, Causal Inference, and Social Science.
Please comment on the article here: Statistical Modeling, Causal Inference, and Social Science
The post Luck vs. skill in poker appeared first on All About Statistics.