My cold from last week has shifted into a bit of bronchitis (and here I thought my virus-fighting strategy of staying up really late drinking beer would clear everything up), so I'm kind of groggy and lethargic. And I have book edits to work on, which precludes taking a long time to write blog posts about science, so here's some babble about card games.
Saturday night at the Tor party, a bunch of people started playing poker using Tor.com buttons as chips. They were playing the obligatory Texas Hold 'Em, and there were a couple of guys standing off to the side (one of whom reminded me of Razib), complaining about the choice of game.
Now, while I suspect that the dominance of Hold 'Em has more to do with the fact that it can be played with 10 people than any features of the game play, their objection surprised me. The complaint was, basically, that Hold 'Em lets you fold too quickly, after only one round of betting. The games being held up as superior to Hold 'Em were Follow the Queen and Baseball, because the plethora of wild cards force players to stick with bad hands in the hopes that they'll turn good.
This is an argument I have literally never heard before. Whenever I've encountered poker purists talking about the merits of different games, the argument has always been that games with fewer wild cards were better, and required more skill. These guys on Saturday were arguing that the extra random element of introducing lots of wild cards (and changing wild cards, at that) required more skill.
My immediate reaction was that this was the rare conversation about cards that made me want to play poker with the speaker, for money. It's possible, though, that this is a widely held opinion that I have somehow never encountered before. So I'll throw this out to the ScienceBlogs readership:
Which do you prefer, a game with a large random element, or one that minimizes the randomness? Which do you think requires more skill?
(NOTE: I'm not sure about the status of the spam filters since the upgrade, but in the past the words "Poker" and "Hold 'Em" have tended to trip the spam filters, and get comments blocked or held for moderation. I recommend trying to avoid those words, using acronyms or something.)
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Which do you prefer, a game with a large random element, or one that minimizes the randomness? Which do you think requires more skill?
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It depends. If it is a Friday night and I am playing to unwind with friends in a relaxed setting then I want a game that has random elements. Anyone can win. It minimizes the influence of the one person that wants to win no matter what and lets the rest of us relax still and have fun. If it is game for stakes to win then I want skill to be in play.
A straight game requires involves more skill in each hand, but keeping track of what's going on when you're playing lots of different games with lots of different wilds is a whole other skill. Especially when there's drink involved...
There's nothing quite like calling Omaha Hi/Low Usual Suspects (that's twos, one-eyed jacks, and the lowest valued non-wild card(s) in your hand wild) after an evening of whisky drinking to sort the men from the boys. Texas is a fine game, but it tends to become a bit of a grind after a while.
(1) Wild cards in poker suck, because:
Ivars Peterson's MathTrek
September 9, 1996
Trouble with Wild-Card Poker
Poker originated in the Louisiana territory around the year 1800. Ever since, this addictive card game has occupied the time and teased the minds of generations of gamblers. It has also attracted the attention of mathematicians and statisticians.
The standard game and its many variants involve a curious mixture of luck and skill. Given a deck of 52 cards, there are 2,598,960 ways to select a subset of five cards. So, the probability of getting any one hand is 1 in 2,598,960.
One of the first things a novice player learns is the relative value of different sets of five cards. At the top of the heap is the straight flush, which consists of any sequence of five cards of the same suit. There are 40 ways of getting such a hand, so the probability of being dealt a straight flush is 40/2,598,960, or .000015.
The next most valuable type of hand is four of a kind, and so on. The table below lists the number of possible ways that different types of hands can arise and their probability of occurrence.
Rankings of Poker Hands and Their Frequencies of Occurrence:
Hand No. of Ways Probability
Straight flush
40
.000015
Four of a kind
624
.000240
Full house
3,744
.001441
Flush
5,108
.001965
Straight
10,200
.003925
Three of a kind
54,912
.021129
Two pair
123,552
.047539
One pair
1,098,240
.422569
The rules of poker specify that a straight flush beats four of a kind, which tops a full house, which bests a flush, and so on through a straight, three of kind, two pair, and one pair. Whatever your hand, you can still bet and bluff your way into winning the pot, but the ranking (and value) of the hands truly reflects the probabilities of obtaining various combinations by random selections of five cards from
a deck.
Many people, however, play a livelier version of poker. They salt the deck with wild cards -- deuces, jokers, one-eyed jacks, or whatever. The presence of wild cards brings a new element into the game, allowing a player to use such a card to stand for any card of the player's choosing. It increases the chances of drawing more valuable
hands.
It also potentially alters the ranking of different hands. One can even draw a five of a kind, which typically goes to the top of the rankings.
Just how much wild cards alter the game is recounted in an article in the current issue of Chance, written by mathematician John Emert and statistician Dale Umbach of Ball State University in Muncie, Ind. They analyze wild-card poker and conclude, "When wild cards are allowed,
there is no ranking of the hands that can be formed for which more valuable hands occur less frequently."
That's a striking result. Wild cards increase the number of ways in which each type of hand can occur. The amount of that increase depends on which cards are designated as wild. For example, with deuces wild, four of a kind occurs more than twice as often as a full house. So, modifying the rules to rank a full house higher than four of a kind might produce a more consistent result.
A player, however, often has a choice in how to declare a hand, and that choice will invariably produce the strongest possible combination according to the declared rules. Thus, if a full house ranks higher than four of a kind, and a player has a wild card allowing him or her to choose either a full house or four of a kind, the full house will inevitably come up more often than four of a kind!
"There is no possible ranking of hands in wild-card poker that is based solely on frequency of occurrence," Emert and Umbach convincingly demonstrate. The authors examined several wild-card options and found that the standard ranking proves to have fewer inconsistencies than other possible ranking schemes.
They then went on to see if there exists a better way of ranking the hands, proposing a scheme that takes into account the fact that certain hands can be labeled in several ways. For example, any wild-card hand declared as a full house can also be considered as two pair, three of a kind, or even one pair or four of a kind.
The authors define a quantity called the inclusion frequency, which gives the number of five-card hands that can be declared as such for each type of hand. Rankings based on this number give hands with smaller inclusion frequencies a higher position in the list. In standard poker, this method leads to the traditional rankings.
Wild-card variants show a slightly different order. Interestingly, one result of this new ranking criterion is that the greater the number of wild cards, the more valuable a flush becomes.
"We believe that the use of the 'inclusion' ranking of the hands presents a more consistent game than deferring to ordinary ranking," Emert and Umbach declare.
Of course, this analysis doesn't really take into account the complexity of what actually happens in a poker game. You're not likely to be computing probabilities as you play. It may be much more advantageous for you to put on your best poker face and bluff as much as you think you can get away with.
In an analysis of simple games that involve bluffing, John Beasley, in The Mathematics of Games, wryly counsels: "Do not think that a reading of this chapter has equipped you to take the pants off your local poker school. Three assumptions have been made: that you can bluff without giving any indication, that nobody is cheating, and that the
winner actually gets paid. You will not necessarily be well advised to make these assumptions in practice."
There are some aspects of poker that are beyond the reach of mathematics.
Copyright (c) 1996 by Ivars Peterson.
References:
Beasley, John D. 1989. The Mathematics of Games. Oxford, England: Oxford University Press.
Emert, John, and Dale Umbach. 1996. Inconsistencies of "wild-card" poker. Chance 9(No. 3):17-22.
Packel, Edward W. 1981. The Mathematics of Games and Gambling. Washington, D.C.:Mathematical Association of America.
(2) I strongly recommend this book:
The Mathematics of Games And Gambling (2nd ed.), Edward Packel, 2006,
Mathematical Association of America, 190 pages, ISBN 0-883-85646-8
The new edition of a favorite introduces and develops some of the important and beautiful elementary mathematics needed for rational analysis of various gambling and game activities.
Most of the standard casino games (roulette, craps, blackjack, keno), some social games (backgammon, poker, bridge) and various other activities (state lotteries, horse racing) are treated in ways that bring out their mathematical aspects.
The mathematics developed ranges from the predictable concepts of probability, expectation, and binomial coefficients to some less well-known ideas of elementary game theory.
The second edition includes new material on:
⢠Sports betting and the mathematics behind it
⢠Game theory applied to bluffing in poker and related to the "Texas Holdem phenomenon"
⢠The Nash equilibrium concept and its emergence in popular culture
⢠Internet links to games and Java applets for practice and classroom use. Game-related exercises are included and solutions to some appear at the end of the book.
(3) I apologize for the length of this definitive data-dump. But it would have been SO much more obnoxious to say this at (say) the Tor party. Teresa would have laughed, and Patrick would have glared. And they'd both have been right...
I had never heard of THE until a few years ago. I grew up on dealer's choice sessions, which consisted mostly of draw and variations on seven-stud (of which Follow the Queen and Baseball are but two), occasionally spiced with a game called Guts.
I don't have a strong preference either way on the question, but there is definitely a different dynamic when there are wild cards involved. If you are looking for a game with a patsy, you are better off having wild cards in the game precisely because the patsy is likely to stay in too long with a mediocre hand. With straight games like draw or THE, you are dealing with pure percentages--the patsies in such games are the ones who don't know percentages, and I expect you won't find too many of those in a crowd like Boskone.
My limited experience with THE is that you often do find people folding immediately (or staying in only because it's costing them nothing to do so) because it's bloody obvious they hold a losing hand. Straight five-stud should suffer a bit more from this dynamic, but there seems to be more of a psychological factor involved.
I think people folding early in hold'em doesn't have much to do with dynamics of the game itself. It's just that people who have learned how to play poker in the last few years have learned how to play in an environment where hold'em is by the far the most exposed game, and frequently they have at least some idea about what constitutes good starting hand selection. Put a lot of these people in a game like Omaha, or 7 stud hi/lo, and it will be a completely different story, even with no wild cards, simply because few of them are likely to have good tools for evaluating their starting hands.
Being able to adjust to different games is definitely a skill that a more theoretical player is likely to have. That said, I find it very difficult to figure how a game like Follow the Queen specifically favors a more theoretically grounded player, as changing wild cards mean that you don't really have a lot of evaluative tools on early streets. It could well be the case that in games with static wild cards the unaware player will be wildly loose relative to what the game demands. I haven't played crazy dealer's choice games since I learned a bit about poker, so I can't say for sure.
For me, the games where luck is a bigger factor than skill are the way to go. But that is because I have little skill at cards when it comes to betting, mostly because I can't bluff worth a hoot. So luck suits me better than skills because I have no 'bluff face'. :)
Large randomness. I don't play for want of showing off with my brain skills. If I want to think, I will work on my next paper.
They were playing the obligatory Texas Hold 'Em, and there were a couple of guys standing off to the side (one of whom reminded me of Razib), complaining about the choice of game.
y? wuz he brown? or just super awesome?
y? wuz he brown? or just super awesome?
Short, South Asian, and hyperactive.
I think it's part of the same issue that motivates Fisher Chess---Bobby Fisher's proposal that competitive chess games should start with random reorderings of the back rank.
I tend to enjoy Rummy-like games, for the following reason:
1) The basic mental calculation, "What should I discard next", is rich and complex enough that I keep running into problems and tradeoffs I haven't seen before. My feeling is that this calculation gets richer with more wild cards.
2) It's not so complicated that the game feels random. I can tell whether I lost due to a bad judgement call, or because a "correct" high-expected-value play didn't pay off this time.
3) It is complicated and random enough that "keeping track of the state of all player's hands" is well into the noise. My brain doesn't store information like "Bob discarded a king three turns ago, but no red tens have gone by yet"; if I try to play games where that's the key to winning, I lose.
With a large group, "Liars Poker" (where you pass around a single hand, played by 5-card draw rules, and must declare a higher hand than the one you accepted) can be a blast. Hearts or Euchre plus beer fall in the same category.
I personally prefer some of the skill games (board version of Civilization or Diplomacy played over weeks in college) that have no external random element at all. With cards, the only truly pure skill game is duplicate bridge, where each half of your team plays the identical hand from opposite sides of the table.
Re #11:
At a deeper level (the former Women's US Champion confirmed this) there is considerable luck in Chess, as one gambles that the opponent will run out of computational capability before understanding one's clever move. And my professional cardplaying friends insist that, over a day or two, there's no luck involved in Draw poker, or 5-hand stud. All about reading the other player, even if both know the probabilities cold.
It would be rude to go deeper into Game Theory here, except to note that weird things happen if the players can be quantum entangled. Really. New Nash equilibria suddenly appear...
So I'm thinking... Quantum Civilization or Diagonalized Diplomacy?
I'd much rather play poker (or most games) with a minimal random element. Obviously the more randomness, the less impact skill has, by definition. Although bcooper (#5) is right about there being a different skill in adjusting to unfamiliar games.
My main hobby is boardgaming and I tend to prefer games with a little luck over no-luck and high-luck games. I prefer some luck because (1) it reduces "analysis paralysis", (2) it provides variety vs. something like chess with a few fixed openings, and (3) I enjoy the risk management aspect. But if a game is too luck-based there is just not enough reward for skillful play.
Clay