poker chip sets

Poker Tourney Draws Full House

2005-06-23T22:46:00.000-07:00

Poker Tourney Draws Full House
The tension in the room was unbearable, so when College sophomore Harry Crane finally pushed all his poker chips set to the center and said, "call," it was like a weight had been lifted. Soon it was all over.
Engineering junior Dave Thornton won this year's Penn Poker Championship, the largest free-entry poker tournament ever at Penn with over 175 students and faculty participating.
Sponsored by the Penn Poker Club, the Texas Hold'em-style poker tournament took place in Houston Hall on Saturday and Irvine Auditorium on Sunday. Participants varied from novices to mavericks and freshmen to faculty.
"I figured there would be mostly rookies here, but people know what they're doing," said College and Wharton senior Brian Lempel, who was wearing a tie with $100 bills on it, a giant necklace with a dollar sign, gold watches and other assorted "bling."
"I thought maybe I could intimidate some kids," he said.
Most of the players at the final table were serious poker players.
The Penn Poker Club "brought good players here," said Thornton, who walked away with the grand prize: $1,000 and an engraved metal bracelet. "The final table was solid."
Smaller cash prizes were awarded to the top 20 finishers.
"It's pretty good to get 75 bucks for nothing," said sixth-place finisher and College freshman Dave Verbofsky, who is known in poker circles as "The Yellow Dart."
Last year, only 20 students participated in the Penn Poker Championship, while over 350 people requested to play in this year's 200-seat tournament.
Penn Poker Club President and College senior Alex Graeffe attributed the success of this year's tournament to significant cash prizes made available.
Graeffe cited poker's recent increase in popularity and visibility, mostly due to online poker gaming and televised pro tournaments.
Whatever the reason, this year's Penn Poker Championship was able to draw students.
"What other organization can bring that many people together at 10 a.m. on a Saturday?" said Penn Poker Club Co-Vice President and Wharton sophomore Dan Kline.
Despite the large turnout, only eight females participated.
"Girls should get more into poker," said 10th-place finisher and College sophomore Teresa Fallon. "I think they are underestimated at a poker table."
The tournament lasted a grueling 12 hours split between Saturday and Sunday. Participants were given chips worth an imaginary $500 and continued to play until they had none left.
The end of the tournament was a nailbiter as Thornton and Crane, with equal chip totals, both went all in before the flop. Thornton's pair of threes and Crane's ace-king had exactly equal odds to win, but Thornton came out on top. "The last hand was basically a coin flip, and it just came out my way," Thornton said.
Despite the lack of an entrance fee, a $3 donation to the Chess in Philadelphia Schools Initiative was suggested by Graeffe, president of the Penn Chess Club.

Poker Tour Cards-Chip Sets

2005-06-09T01:19:00.000-07:00

Official World Poker Tour Cards-Chip Sets A Sure BetNovember 5, 2004
Dreaming of going “All-In” with pocket aces and two more on the flop? If so, then your dream wouldn’t be complete without a stack of “Official World Poker Tour” (WPT) chips to push into the center of the table. This holiday season every poker enthusiast has a chance to feel like he or she is playing a Final Table at a WPT tournament, because U.S. Playing Card Company is introducing “Official World Poker Tour” Cards, Chips and Poker Sets—complete with high quality casino cards and those heavyweight professional clay chips like the pros shuffle around their fingers.
Home game poker players will have a choice of a range of $3.49 stocking stuffer single WPT Deck of Cards to a $149.99 top-of-the-line Official World Poker Tour Poker Set.
The World Poker Tour, which airs every Wednesday at 9 p.m. E/PT on the Travel Channel is the television show that has launched the poker boom. With its groundbreaking cameras on the hole cards, fans have been caught up in the drama and many have learned to play the game by watching the pros.
“Playing on the World Poker Tour is a dream come true for some, but now you can simulate that exciting experience in your own home with the U.S. Playing Card Official World Poker Tour products,” said Scott Kling, Vice President of Sales and Marketing for U.S. Playing Card Company, the world’s leader in playing cards for more than 100 years and the manufacturer of the best-selling Bicycle and Bee brand cards. “Just as the WPT is the leader and innovator in the field of televised poker, we feel our products are the premium items in the category -- the ones real serious poker enthusiasts will want for the holidays.”
U.S. Playing Card, the exclusive licensee of the World Poker Tour-branded cards, chips, and poker sets, is the world's largest manufacturer and distributor of playing cards and a leader in the marketing of poker chips and sets, collectible tins, puzzles, and card accessories for the North American market. The company has developed a full line of WPT-branded merchandise available this holiday season at retailers. These are the same cards and chip sets awarded to World Poker Tour tournament winners and WPT Hollywood Home Game celebrity players

world poker tour chip sets

2005-05-25T23:06:00.000-07:00

Home Poker Chips Sets

2005-05-13T12:23:00.000-07:00

Home Poker Chip Sets:

You can choose one of twelve different stock designs or you can order something completely unique with your own artwork on it. When choosing one of the stock designs, you can add things such as your name or the name of your casino. Pro-Tech chips are 39mm and weigh 10 grams. The color chart shows the various colors available for your chip and the edge spot page lists the various options available for the rolling edge.

Bugsy's Hideaway Poker Chip Sets

2005-05-12T10:44:00.000-07:00

This are plastic poker chip sets. The full inlay has a multi-colored holographic inlay that is quite unique and flashy. It looks great under the lights.
I could easily peel the label off this chip on the first try. I could then stick the same label back on the chip without any noticeable change. The labels are very high quality and are difficult to pierce or tear. Once you take the label off, there are slightly visible, but untellable, injection mold marks under the sticker. There are no edge ridges on this chip.
These chips have a good sound and feel and balance. They stack well but are a bit slippery.

Why I Chose Horse Head Chip sets?

2005-05-11T07:20:00.000-07:00

Why I Chose Horse Head Chip sets?

I wanted a traditional 'clay' poker chip with traditional colors. I wanted the colors to be somewhat bland, I did not want fluorescent colors. I wanted traditional edge spots, multi-colored if possible. I wanted my HPT logo and internet address printed on the label because I will be using these poker chips as business cards. I did not want denominations printed on the label. I was willing to pay $1-$2 per chip. I did not mind waiting for delivery of my chips. These basic requirements left me with about a half dozen options as to where I could order my chips ...

All-in Poker Chips Sets

2005-05-07T13:40:00.000-07:00

These poker chips sets look similar to an 11.5 gram composite chip but feel and sound a bit more like a clay chip. They definitely do not have the metallic ring of a metal insert chip, nor the slippery feeling. In fact, these are some of the least slippery chips that I have reviewed, they have an almost chalky or rubbery feel to them, somewhat similar to a 13 gram "Casino' composite chip, but much more rubbery.
The chips are quite soft, I could gouge into these chips with my fingernail. They will get marked-up over time but marks did seem to blend in fairly well, perhaps a bit better than a 13 gram 'Casino' composite. The colors are deep so there is no worry about any of the graphics wearing away but they did seem to fade rather easily. There are no edge ridges on this chip. There is a small injection mold spot in a trench on one side of the chip but it is hardly noticeable. There are no edge spots, instead, there is a series of circles, similar to a suited 11.5 gram composite chip. The lack of edge spots is a big negative.
I received numerous samples of this chip. The diameters were 39.85, 39.77, 39.74 and 39.83mm. Three chips stacked together was 10.31mm thick.
I could break a chip in half with my hands but it was not quite as easy to do as I had heard on some other websites. I don't really see this as being a big problem. The softness of the clay was a bigger problem. I also didn't find the advertising "All-in Chips" on the graphics to be too overbearing. The chipset itself is fairly good looking so I guess it's a question of how brittle the chips will be ... time will tell. If you want something a bit better than a 11.5 or 13 gram composite, but you can't spend the money on casino quality chips, these may not be a bad deal.

Poker Chip Sets Info

2005-05-06T13:15:00.000-07:00

Add class to your poker game with custom poker chips and poker chip sets, complete with a handsome rugged aluminum storage and carrying case.

Usually poker chip sets advertised on the Internet feature:

- 500 or 700 poker chips in the set
- 11.5 gram clay composite construction
- 39mm - 40mm casino size poker chips
- 5 colors
- Playing cards and 5 poker dice

New face of poker!

2005-05-04T13:53:00.000-07:00

Effie Shaffer and her adult daughter found themselves in the poker lounge of Strawbridge's in Oxford Valley Mall. Casino tables and authentic betting chips were matched by decorative posters of dogs playing poker - all of it just off the men's department near the escalator to housewares.

Shaffer of Penndel was buying a Texas Hold'em poker deck and chip sets for her 14-year-old and 16-year-old grandsons. Mom had already gotten the boys a poker table, despite objections from their dad. "This will keep them out of trouble," Shaffer said, without a laugh. "The boys both put poker on their Christmas list." And apparently they aren't the only ones. Retailers are betting high on casino game sales this holiday season. Nearly every major department store in Bucks has a casino section. Even the toy stores are selling card games and tables.

Macy's mannequins are dressed as poker dealers.Celebrity poker tournaments on television and star-studded casino capers in theaters have fueled a growing interest in games of chance that, some gambling addiction counselors warn, could spell trouble. While the lottery and horse racing remain the only legal games of chance in Pennsylvania, anyone with a credit card can log onto the seemingly countless Internet casinos. There, rounds are dealt and thousands of dollars waged from a computer-generated poker table. The average hand lasts about 40 seconds as players match up with faceless opponents from around the world. But who needs a computer when you can bet right in the street?Susan Meo has a gambling problem - people treat the parking lot of her Pennsbury Court Apartments in Morrisville like a craps parlor.

Since she began working there as apartment manager in 2002, tenants have complained about teens in the parking lot laying bets and shooting dice. Despite her complaints to local cops, the gambling continues, she said. "A couple times the police came by, but unless they caught them with money changing hands, they couldn't do anything," Meo said.And, while the cold has probably driven the gamblers inside, Meo sees it as a safe bet her problems are not over."It's not so much anymore but only because of the winter. I'm sure they'll be back," she said.AN EASY BET For poker players like Bernie Woodford of Bristol Township, the sudden interest in card games was greeted with mixed emotion. “It’s a good thing for your more experienced players ’cause we can always take advantage of the new guys,” Woodford said. “But then sometimes, too, they don’t know how to play the game or the rules.” Woodford said he’s played poker once a week with buddies since he was 18, about eight years ago. For him, Texas Hold’em was a way to meet new friends. With a cup of coffee in his hand and seven friends at the table, he bluffs, raises and deals.

They started playing after seeing “Rounders,” in which actor Matt Damon starred as a retired poker player who returns to the game to help his friend pay off loan sharks. The film is not a favorite for Jim Pappas, executive director of the Council on Compulsive Gambling of Pennsylvania. The excitement of horse track betting led him to rack up debts that took more than a decade to pay off, he estimated. “[Gambling] gave me a form of excitement that I had never experienced before. There was such a feeling of importance and a feeling of power,” Pappas said. Now he answers phones for the Pennsylvania Compulsive Gambling Hotline. Listening to the stories of other addicts, Pappas believes, has strengthened his convictions never to return to the track. The hot line gets an estimated 25 calls a day. Most of them are from men. Betting illegally on sports and in the casinos leads most people to call, Pappas said. In recent months, more parents have called with concerns about their children gambling in schools and in the street, and Pappas said celebrity poker shows are partly to blame. “It’s a new form of entertainment that the media is capitalizing on.

They are marketing it and targeting new audiences and they’re making it look much more fun and exciting,” he said. The CCGP expects to see a “spike” in addiction as the first slots casinos open in Pennsylvania, including one in Bensalem, in the next two years, Pappas added. And online, the speed of the game makes it even more addictive, Pappas warned. While the casino table is computer-generated imagery, the money is very real. PartyPoker.com is one of the more popular online casinos on the Web. With little more than a name, an address and a credit card number, you can wage thousands of dollars at 7-card stud or Texas Hold’em. Players use a credit card or check routing number to deposit cash in an account.

The Web site often gives players "bonus codes" and free money for continued play. As the electronic dealer dealt another round, at Table No. 420789, "Hightop243," holding a pair of jacks, raised the stakes from $5 to $10. "Bigdreamer24," with two pairs, called. "HomerBigTime" checked with a pair of queens and a flush of diamonds, winning the jackpot of $36.

Poker Chip Sets Facts!

2005-04-29T10:13:00.000-07:00

There’s really no such thing as a “real” clay chip anymore. Certainly not in use in the casinos—they break too easily and cost too much to use everyday! Usually they weigh 10 grams each, and have about 15% clay. Their average life is short, for they wear out in 3 years or less due to the softness. They cost about $1.08 each to make, plus art work. Casinos use them because they have many built in security measures, like the embossed hat/cane you see around the edge that our home use imports don't have.
We are pleased to offer a variety of chips-from the 11.5 gram clay and acrylic composition chip, to "real clay" chips like the old-time casino chips...to the lighter, plastic composition chips that are just right for the beginner-or the large casino night you're planning!
The best Poker Chips sets are 11.5 grams, unless otherwise marked. Casino Chips are 10 grams.
How many chips do you need? It really depends on what you play and how many people are in your game. A decent rule-of-thumb is 40 to 50 chips per person. From the lowest denomination to the highest, you'll want to use a 4,3,2, 1 mix. For example: 200 white, 150 red, 100 blue, 50 green. But this is just a guide. ..you can mix and match anyway you'd like to have them!"

poker chip sets thats what you are looking for

2005-04-28T09:51:00.000-07:00

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Poker Chip Sets Tricks!

2005-04-27T08:14:00.000-07:00

Poker chips sets tricks are fun to know and can show people that you have been playing the game for a while. Here are some basic ones...

Chip Shuffle: Line up to stacks of equal chips. 3-5 high is a good number of chips to have in each pile. Put your hand over both sides of ths chips. Put your thumb in the middle gap of the chips. No push all the chips together and slightly lift your thumb up. Great effect, and easy to do!

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Poker Chip Sets Info

2005-04-26T17:46:00.000-07:00

Usually Poker chip sets comes in an aluminum case with reinforced edges to protect your valuable chips. There are currently a few types of poker chip cases. The chip sets includes 2 decks of playing cards, 5 red transparent dice and 1 dealer button. (except 100 piece sets) You can fully customize your color preference for any poker chip set.

Playing With Poker Chip Sets poker chip sets

2005-04-22T18:44:00.000-07:00

poker chip sets are an interesting test-bed for artificial intelligence research. It is a game of imperfect knowledge, where multiple competing agents must deal with risk management, opponent modeling, unreliable information, and deception, much like decision-making applications in the real world. Opponent modeling is one of the most difficult problems in decision-making applications and in poker chip sets it is essential to achieving high performance. This paper describes and evaluates the implicit and explicit learning in the poker chip sets program Loki. Loki implicitly "learns" sophisticated strategies by selectively sampling likely cards for the opponents and then simulating the remainder of the game. The program has explicit learning for observing its opponents, constructing opponent models and dynamically adapting its play to exploit patterns in the opponents’ play. The result is a program capable of playing reasonably strong poker chip sets, but there remains considerable research to be done to play at a world-class level. 

1. INTRODUCTION

The artificial intelligence community has recently benefited from the tremendous publicity generated by the development of chess, checkers and Othello programs that are capable of defeating the best human players. However, there is an important difference between these board games and popular card games like bridge and poker chip sets. In the board games, players always have complete knowledge of the entire game state since it is visible to both participants. This property allows high performance to be achieved by a brute-force search of the game tree. In contrast, bridge and poker chip sets involve imperfect information since the other players’ cards are not known, and search alone is insufficient to play these games well. Dealing with imperfect information is the main reason why progress on developing strong bridge and poker chip sets programs has lagged behind the advances in other games. However, it is also the reason why these games promise higher potential research benefits.

poker chip sets has a rich history of scientific investigation. Economists and mathematicians have applied a variety of analytical techniques to certain poker chip sets-related problems. However, since "real" poker chip sets is too complex for this approach, they have studied simplified variants ([15] for example). Other individuals, including expert players with a penchant for mathematics, have gained considerable insight about "real" poker chip sets by using partial mathematical analyses, simulation, and ad-hoc expert experience ([18] is a popular example).

Until recently, the computing science community has largely ignored poker chip sets. However, poker chip sets has a number of attributes that make it an interesting domain for artificial intelligence (AI) research. These attributes include imperfect knowledge, multiple competing agents, risk management, opponent modeling, deception, and dealing with unreliable information. All of these are challenging dimensions to a difficult problem.

There are two ways that poker chip sets can be used as an interesting testbed for artificial intelligence research. One approach is to use simplified variants that are easier to analyze. For example, Findler worked on and off for 20 years on a poker chip sets-playing program for simplified 5-card draw poker chip sets [7]. He modeled human cognitive processes and built a program that could learn. The danger with this approach is that simplification can remove the challenging components of the problem that are of interest to AI researchers. A variant of this approach is to look at a subset of the game, and try to address each component in isolation. Several attempts have been made to apply machine learning techniques to individual aspects of poker chip sets (some examples include [19,21,6]).

The second approach, and the one that we advocate, is to tackle the entire problem: choose a real variant of poker chip sets and address all the considerations necessary to build a program that performs at a level comparable to that of the best human players. Clearly this is the most ambitious approach, but also the one that promises the most exciting research opportunities.

Recently, Koller and Pfeffer have been investigating poker chip sets from a theoretical point of view [13]. They present a new algorithm for finding optimal randomized strategies in two-player imperfect information competitive games. This is done in their Gala system, a tool for specifying and solving problems of imperfect information. Their system builds decision trees to find the optimal game-theoretic strategy. However the tree sizes prompted the authors to state that "...we are nowhere close to being able to solve huge games such as full-scale poker chip sets, and it is unlikely that we will ever be able to do so." In theory, their approach could be used to build an optimal poker chip sets player for a real variant of poker chip sets. In practice, it will require too many computational resources unless further improvements are discovered.

We are attempting to build a program that is capable of playing world-class poker chip sets. We have chosen to study the game of Texas Hold'em, the poker chip sets variation used to determine the world champion in the annual World Series of poker chip sets. Hold’em is considered to be the most strategically complex poker chip sets variant that is widely played.

Our experiences with our first poker chip sets program, called Loki, were positive [1,14]. However, we quickly discovered two limitations to further performance gains:

1 The betting strategy—whether to fold, call, or raise in a given situation—was defined using expert knowledge. This became cumbersome, since it was awkward to define rules to cover all the possible scenarios. Furthermore, any static strategy is suspect. A successful strategy must depend on changing game conditions.

2 Initially, in games played over the Internet, Loki performed quite well. However, some opponents detected patterns and weaknesses in Loki’s play, and they altered their strategy to exploit them. An opponent can exploit any predictable strategy, both in theory and in practice. To be a strong poker chip sets player, one must model the opponent’s play and adjust to it.

This paper describes and evaluates two types of learning in Loki. First, its knowledge-based betting strategy can be viewed as a static evaluation function. In two-player games, such as chess, the quality of the evaluation can be improved through search. In poker chip sets, imperfect information makes a search of the full game tree impractical. Instead, a simulation that samples from the set of likely scenarios can be used to enhance an evaluation. We found that a simple evaluation function augmented by search can uncover sophisticated strategies, as has been observed in perfect-information games. In other words, search compensates for a lack of knowledge. In effect, Loki uses simulations to implicitly learn advanced strategies.

Second, Loki observes and records the actions of each opponent and uses this information to build a simple model of their play. This model is used to predict each opponent’s hidden cards. The program adapts to the style of each opponent and exploits any predictable actions.

We have experimentally assessed each of these styles of learning, both in the laboratory and in play with human opponents. To the best of our knowledge, Loki is the first successful demonstration of using real-time learning to improve performance in a high-performance game-playing program.

This paper describes our previous work on Loki [1,2,3,4,14] and outlines some of the future directions we are pursuing. Section 2 provides an overview of Texas Hold’em. Section 3 identifies the minimal set of requirements necessary to achieve world-class play. Loki’s architecture is described in Section 4. Section 5 discusses the implicit learning used in the betting strategy, while Section 6 addresses the explicit opponent modeling. The performance of the program is assessed in Section 7. Section 8 identifies future work, and Section 9 provides some conclusions.

2. Texas Hold’em

A hand of Texas Hold’em begins with the pre-flop, where each player is dealt two hole cards face down, followed by the first round of betting. Three community cards are then dealt face up on the table, called the flop, and the second round of betting occurs. On the turn, a fourth community card is dealt face up and another round of betting ensues. Finally, on the river, a fifth community card is dealt face up and the final round of betting occurs. All players still in the game reveal their two hole cards for the showdown. The best five-card poker chip sets hand formed from the two hole cards and the five community cards wins the pot. If a tie occurs, the pot is split. Texas Hold’em is typically played with 8 to 10 players.

Limit Texas Hold’em uses a structured betting system, where the order and amount of betting is strictly controlled on each betting round. There are two denominations of bets, called the small bet and the big bet ($2 and $4 in this paper). In the first two betting rounds, all bets and raises are $2, while in the last two rounds, they are $4. In general, when it is a player’s turn to act, one of five betting options is available: fold, call/check, or raise/bet. There is normally a maximum of three raises allowed per betting round. The betting option rotates clockwise until each player has matched the current bet or folded. If there is only one player remaining (all others having folded) that player is the winner and is awarded the pot without having to reveal their cards.

3. Requirements for a World-Class poker chip sets Player

We have identified several key components that address some of the required activities of a strong poker chip sets player. However, these components are not independent. They must be continually refined as new capabilities are added to the program.

Hand strength assesses the strength of a hand in relation to the other hands. The simplest hand strength computation is a function of the cards in the hand and the current community cards. A better hand strength computation takes into account the number of players still in the game, the position of the player at the table, and the history of betting for the current game. An even more accurate calculation considers the probabilities for each possible opponent hand, based on the likelihood of each hand being played to the current point in the game.

Hand potential computes the probability that a hand will improve to win, or that a leading hand will lose, as additional community cards appear. For example, a hand that contains four cards in the same suit may have a low hand strength, but has good potential to win with a flush as more community cards are dealt. Conversely, a hand with a high pair could decrease in strength and lose to a flush as many cards of a common suit appear on the board. At a minimum, hand potential is a function of the cards in the hand and the current community cards. However, a better calculation could use all of the additional factors described in the hand strength computation.

Betting strategy determines whether to fold, call/check, or bet/raise in any given situation. A minimum strategy is based on hand strength. Refinements consider hand potential, pot odds (your winning chances compared to the expected return from the pot), bluffing, opponent modeling and trying to play unpredictably.

Bluffing allows you to make a profit from weak hands, and can be used to create a false impression about your play to improve the profitability of subsequent hands. Bluffing is essential for successful play. Game theory can be used to compute a theoretically optimal bluffing frequency in certain situations. A minimal bluffing system merely bluffs this percentage of hands indiscriminately. In practice, you should also consider other factors (such as hand potential) and be able to predict the probability that your opponent will fold in order to identify profitable bluffing opportunities.

Unpredictability makes it difficult for opponents to form an accurate model of your strategy. By varying your playing strategy over time, opponents may be induced to make mistakes based on an incorrect model. 

Opponent modeling allows you to determine a likely probability distribution for your opponent’s hidden cards. A minimal opponent model might use a single model for all opponents in a given hand. Opponent modeling may be improved by modifying those probabilities based on the collected statistics and betting history of each opponent.

There are several other identifiable characteristics that may not be necessary to play reasonably strong poker chip sets, but may eventually be required for world-class play.

Opponent modeling is integral to successful poker chip sets play. Koller and Pfeffer have proposed a system for constructing a game-theoretic optimal player [13]. However, it is important to differentiate an optimal strategy from a maximizing strategy. The optimal player makes its decisions based on game-theoretic probabilities, without regard to specific context. The maximizing player takes into account the opponent’s sub-optimal tendencies and adjusts its play to exploit these weaknesses.

In poker chip sets, a player that detects and adjusts to opponent weaknesses will win more than a player who does not. For example, against a strong conservative player, it would be correct to fold the probable second-best hand. However, against a weaker player who bluffs too much, it would be an error to fold that same hand. In real poker chip sets it is very common for opponents to play sub-optimally. A player who fails to detect and exploit these weaknesses will not win as much as a better player who does. Thus, a maximizing program will out-perform an optimal program against sub-optimal players.

Although a game-theoretic optimal solution for Hold’em would be interesting and provide a good baseline for comparing program (and human) performance, it would in no way "solve the game." To produce a world-class poker chip sets program, strong opponent modeling is essential.

4. Loki’s Architecture

This section gives a brief overview of the important components of Loki’s architecture [4]. Figure 1 illustrates how these components interact. In the diagram, rectangles are major components, rounded rectangles are major data structures, and ovals are actions. The data follows the arrows between components. An annotated arrow indicates how many times data moves between the components for each of our betting actions.

The architecture revolves around generating and using probability triples. It is an ordered triple of values, PT = [f,c,r], such that f + c + r = 1.0, representing the probability distribution that the next betting action in a given context should be a fold, call, or raise, respectively. The Triple Generator contains our poker chip sets knowledge, and is analogous to an evaluation function in two-player games. The Triple Generator calls the Hand Evaluator to evaluate any two-card hand in the current context. It uses the resulting hand value, the current game state, and expert-defined betting rules to compute the triple. To evaluate a hand, the Hand Evaluator enumerates over all possible opponent hands and counts how many of them would win, lose or tie the given hand.

Each time it is Loki’s turn to bet, the Action Selector uses a single probability triple to decide what action to take. For example, if the triple [0.0,0.8,0.2] were generated, then the Action Selector would never fold, call 80% of the time and raise 20% of the time. A random number is generated to select one of these actions so that the program varies its play, even in identical situations. Although this is analogous to a mixed strategy in game theory, the probability triple implicitly contains contextual information.

After the flop, the probability for each possible opponent hand is different. For example, the probability that Ace-Ace hole cards are held is much higher than the cards 7-2, since most players will fold 7-2 before the flop. There is a weight table for each opponent. Each weight table contains one value for each possible two-card hand that the opponent could hold. The value is the probability that the hand would be played exactly as that opponent has played so far. For example, assume that an opponent called before the flop. The updated probability value for the hand 7-2 might be 2% since it normally should be folded. Similarly the probability of Ace-King might be 60% since it would seldom be folded before the flop, but is often raised. After an opponent action, the Opponent Modeler updates the Weight Table for that opponent in a process called re-weighting. The value for each hand is increased or decreased to be consistent with the opponent's action. The Hand Evaluator uses the Weight Table in assessing the strength of each possible hand, and these values are in turn used to update the Weight Table after each opponent action. The absolute values of these probabilities are of little consequence, since only the relative weights affect the later calculations. The details are discussed in Section 6.

 

 

Figure 1. The architecture of Loki.

Probability triples are used in three places in Loki. The Action Selector uses a probability triple to decide on a course of action (fold, call, raise) as previously described. The Simulator uses probability triples to choose actions for simulated opponent hands (see Section 5). The Opponent Modeler uses an array of probability triples to update the model of each opponent (see Section 6).

An important advantage of the probability triple representation is that imperfect information is restricted to the Triple Generator and does not affect the rest of the program. This is similar to the way that alpha-beta search restricts knowledge to the evaluation function. The probability triple framework allows the "messy" elements of the program to be amalgamated into one component, which can then be treated as a "black box" by the rest of the system. Thus, aspects like game-specific information, complex expert-defined rule systems, and knowledge of human behavior are all isolated from the engine that uses this input.

5. Implicit Learning

Loki's original Action Selector component consisted of expert-defined rules that used hand strength, hand potential, game conditions, and probabilities to decide on an action. A professional poker chip sets player defined the system as a first approximation of the return on investment for each betting decision. As other aspects of Loki improved, this simplistic betting strategy became the limiting factor to the playing strength of the program. Unfortunately, any rule-based system is inherently rigid, and even simple changes were difficult to implement and verify for correctness. A more flexible, computation-based approach was needed.

In effect, a knowledge-based betting strategy is equivalent to a static evaluation function. Given the current state of the game and the hole cards, it attempts to determine the action that yields the best result. If we use deterministic perfect information games as a model, the obvious extension is to add search to the evaluation function. While this is easy to achieve in a perfect-information game such as chess (consider all possible moves as deeply as resources permit), the same approach is not feasible for real imperfect information games because there are too many possibilities to consider [13].

Having an expert identify all the betting rules necessary to play world-class poker chip sets is time consuming and difficult. Decisions must be based on context, within a game and between games. Covering all eventualities is not practical. In such a system, the expert does the learning, transferring his knowledge into new or modified rules. We prefer a dynamic computation-based approach, where the program does the learning as it plays.

Loki’s improved betting strategy consists of playing out many likely scenarios to determine how much money each decision will win or lose. Every time it faces a decision, Loki invokes the Simulator to get an estimate of the expected value (EV) of each betting action (see the dashed box in Figure 1 with the Simulator replacing the Action Selector). A simulation consists of playing out the hand a specified number of times, from the current state of the game through to the end. Folding is considered to have a zero EV, because we do not make any future profit or loss. Each trial is played out twice—once to consider the consequences of a check/call and once to consider a bet/raise. In each trial, cards are dealt to each opponent (based on the probabilities maintained in the Weight Table), the resulting game is simulated to the end, and the amount of money won or lost is determined. Probability triples are used to simulate the actions of the opponents based on the two cards they are assigned for that trial. The average over all of the trials is taken as the EV of each action. In the current implementation we simply choose the action with the greatest expectation, folding if both expectations are negative. If two actions have the same expectation, we opt for the most aggressive one (call over fold and raise over call). To increase the programs unpredictability, we can randomize the selection of betting actions whose EVs are close in value.

Enumerating all possible opponent hands and future community cards is analogous to exhaustive game tree search and is impractical for poker chip sets. However, simulation is analogous to a selective expansion of some branches of a game tree. To get a good approximation of the expected value of each betting action, one must have a preference for expanding and evaluating the nodes that are most likely to occur. To obtain a correctly weighted average, all of the possibilities must be considered in proportion to the underlying probability distribution of the opponent hands and future community cards. The distribution of future community cards is uniform across unseen cards, but the probable opponent hands are not! We use selective sampling to select the most probable hands for each opponent. 

When simulating a hand, we have specific information that can be used to bias the selection of cards. For example, a player who has been raising is more likely to have a strong hand than a player who has just called every bet. For each opponent, Loki maintains a probability distribution over the entire set of possible hands (the Weight Table), and the random generation of each opponent’s hole cards is based on those probabilities. Thus, we are biasing our selection of hole cards for the opponent to the ones that are most likely to occur.

At each node in the decision tree, a player must choose between one of three alternatives. Since the choice is strongly correlated to the quality of the cards that they have, we can use the Triple Generator to compute the likelihood that the player will fold, check/call, or bet/raise in each instance. The player’s action is then randomly selected based on the probability distribution, and the simulation proceeds. As shown in Figure 1, the Simulator calls the TripleGenerator to obtain each of our betting actions and each of our opponent actions. Where two actions are equally viable, the resulting EVs should be nearly equal, so there is little consequence if the "wrong" action is chosen.

It should be obvious that the simulation approach must be better than the static approach, since it essentially uses a selective search to augment and refine a static evaluation function. Barring a serious misconception (or bad luck on a limited sample size), playing out relevant scenarios will improve the default values obtained by heuristics, resulting in a more accurate estimate.

As seen inother search domains, the search itself contains implicit knowledge. A simulation contains inherent information that improves the basic evaluation. For example, a simulation contains implicit knowledge such as:

• hand strength (fraction of trials where our hand is better than the one assigned to the opponent),

• hand potential (fraction of trials where our hand improves to the best, or is overtaken), and

• subtle implications not addressed in the simplistic betting strategy (e.g. "implied odds"—extra bets won after a successful draw).

It also allows complex strategies to be uncovered without providing additional expert knowledge. For example, simulations can result in the emergence of advanced betting tactics like a check-raise, even if the basic strategy without simulation is incapable of this play.

At the heart of the simulation is an evaluation function. The better the quality of the evaluation function, the better the simulation results will be. One of the interesting results of work on alpha-beta has been that even a simple evaluation function can result in a powerful program. We see a similar situation in poker chip sets. The implicit knowledge contained in the search improves the basic evaluation, magnifying the quality of the search. As with alpha-beta, there are tradeoffs to be made. A more sophisticated evaluation function can reduce the size of the tree, at the cost of more time spent on each node. In simulation analysis, we can improve the accuracy of each trial, but at the expense of the total number of trials performed in real-time.

Variations of selective sampling have been used in other games, including Scrabble [17], backgammon [20], and bridge [9]. Selective sampling is similar to the idea of likelihood weighting in stochastic simulation [8,16]. In our case, the goal is different because we need to differentiate between EVs (for call/check, bet/raise) instead of counting events. Also, poker chip sets complicates matters by imposing tight real-time constraints (typically a maximum of two seconds). This forces us to maximize the information gained from a limited number of samples. Further, the problem of handling unlikely events (which is a concern for any sampling-based result) is smoothly handled by our re-weighting system (Section 6), allowing Loki to dynamically adjust the likelihood of an event based on observed actions. An unlikely event with a big payoff figures naturally into the EV calculations.

6. Explicit Learning

In strategic games like chess, the performance loss by ignoring opponent modeling is small, and hence it is usually ignored (although it has been studied [5,11,12]). In contrast, not only does opponent modeling have tremendous value in poker chip sets, it can be the distinguishing feature between players at different skill levels. If a set of players all have a comparable knowledge of poker chip sets fundamentals, the ability to alter decisions based on an accurate model of the opponent may have a greater impact on success than any other strategic principle.

To assess a hand, the Hand Evaluator compares those cards against all possible opponent holdings. Naively, one could treat all opponent hands as equally likely, however this skews the hand evaluations compared to more realistic assumptions. Many weak hands are likely to have been folded before the flop, making them less likely to be held later in the hand. Similarly, a hand made strong by the turn and river cards may have been folded on the flop. Therefore, for each starting hand, we need to define a probability that our opponent would have played that hand in the observed manner. We call the probabilities for each of these (52 choose 2) = 1,326 subcases weights since they act as multipliers in the enumeration computations.

The use of these weights is the first step toward opponent modeling since we are changing our computations based on the relative probabilities our opponent’s possible hole cards. The simplest approach to determining these weights is to treat all opponents the same, calculating a single set of weights to reflect "reasonable" behavior, and use them for all opponents. An initial set of weights was determined by ranking the starting hands (as determined by off-line simulations) and assigning a probability commensurate with the average return on investment of each hand. These results closely approximate the ranking of hands by strong players [18]. 

In Loki, the Opponent Modeler uses probability triples to update the Weight Table after each opponent action. To accomplish this, the Triple Generator is called for each possible two-card hand. It then multiplies each weight in the Weight Table by the entry in the probability triple that corresponds to the opponent’s action. For example, suppose the previous weight for Ace-Ace is 0.7 (meaning that if it has been dealt, there is a 70% chance the opponent would have played it in exactly the manner observed so far), and the opponent now calls. If the probability triple for the current context is [0.0, 0.2, 0.8], then the updated weight for this case would be 0.7 x 0.2 = 0.14. The relative likelihood of the opponent holding Ace-Ace has decreased to 14% because they did not raise. The call value of 0.2 reflects the possibility that this particular opponent might deliberately try to mislead us by calling instead of raising. Using a probability distribution allows us to account for uncertainty in our beliefs. This process of updating the weight table is repeated for each entry.

The above corresponds to what we call Generic Opponent Modeling (GOM). Each hand is viewed in isolation and all opponents are treated as the same player. Each player’s Weight Table is initially identical, and gets modified based on their betting action. Although rather simplistic, this model is quite powerful in that it does a good job of skewing the hand evaluations to take into account the most likely opponent holdings.

Obviously, treating all opponents the same is clearly wrong. Each player has a different style, ranging from loose (plays most hands beyond the flop) to tight (usually plays the few hands that have a very high probability of winning), and from aggressive to conservative. Knowing the style of the opponents allows a player to adjust their betting decisions. For example, if a perceived tight player is betting aggressively, there is a good chance that they have a strong hand. A loose player will play many marginal hands or may bluff a lot. This is useful information and may allow you to fold a strong hand or call with a weak one when it is correct to do so. In general, a bet made by a loose or aggressive player should not be taken as seriously as one made by a tight or conservative player.

Specific Opponent Modeling (SOM) customizes the probability triple function to represent the playing style of each opponent. For a given game, the reweighting factor applied to the entries of the Weight table is adjusted by betting frequency statistics gathered on that opponent from previous hands. This results in a shift of the assumed call and raise thresholds for each player. In the case of a tight player, the call and raise thresholds will increase, indicating fewer hands that are likely to be played. Conversely, a loose player’s thresholds will be lowered. During each round of a game, the history of previous actions by the opponent is used to influence the probability triple generated for that opponent.

In competitive poker chip sets, opponent modeling is much more complex than portrayed here. For example, players can act to mislead their opponents into constructing an erroneous model. Early in a session a strong poker chip sets player may try to create the impression of being very conservative, only to exploit that image later in that session when the opponents are using an incorrect opponent model. A strong player has to adapt their model to the opponents varying their playing style.

7. Experiments

Self-play experiments offer a convenient method for the comparison of two or more versions of the program. Our experiments use a duplicate tournament system, based on the same principle as duplicate bridge. Since each hand can be played with no memory of preceding hands, it is possible to replay the same deal, but with the participants holding a different set of hole cards each time. Our tournament system simulates a ten-player game, where each deal is replayed ten times, shuffling the seating arrangement so that every participant has the opportunity to play each set of hole cards once. This arrangement greatly reduces the "luck element" of the game, since each player will have the same number of good and bad hands. The differences in the performance of players will therefore be based more strongly on the quality of the decisions made in each situation. This reduction in natural variance allows meaningful results to be obtained with a smaller number of trials than in a typical game setting. Nevertheless, it is important to not over-interpret the results of one experiment.

Experiments have been performed with Loki to measure the performance of generic opponent modeling (GOM), simulation (S), and both combined (GOM+S). The results were obtained by playing a self-play tournament containing two enhanced versions of Loki against 8 unenhanced versions. A tournament consisted of 2,500 different deals (i.e. 25,000 games). Each simulation consisted of 500 trials, since the results obtained after 500 trials were reasonably stable. 

The metric used to measure program performance is the average number of small bets won per hand (sb/hand), a metric that is sometimes used by human players. For example, in a game of $10/$20 Hold’em, an improvement of +0.10 sb/hand translates into an extra $30 per hour (based on 30 hands per hour). Anything above +0.05 small bets per hand is considered a large improvement. In play on an Internet poker chip sets server against human opponents, Loki has consistently performed at or above the +0.05 sb/hand level.

The experiments showed that GOM improved performance by 0.031 ± 0.019 sb/hand, simulations improved by 0.093 ± 0.04 sb/hand, and the combination was worth 0.095 ± 0.045 sb/hand (note that these are newer numbers than those appearing in [2,3,4]). The results reported here may be slightly misleading since each experiment used two similar programs. As has been shown in chess, one has to be careful about interpreting the results of these types of experiments.

GOM is a significant gain as expected. Given that all players in the tournaments were variants of Loki, the wide variety of play that is seen in human play is missing. Hence, GOM may be of greater benefit against typical human opponents. Simulations, on the other hand, are a huge win in self-play experiments against non-simulation opponents. As expected, they have a naturally occurring higher variance. The use of simulations represents a large improvement in the quality and variety of the betting strategies employed by Loki (or, possibly, overcome a serious weakness in the older version of the program). Whereas our initial knowledge-based betting strategy routine [1,14] was limited by the amount of knowledge we could code and tune, the simulation-based approach has no such restrictions. The simulations implicitly enable advanced betting strategies, with a degree of unpredictability that makes it harder for the opponents to model Loki.

Note that although each feature is a win by itself, the combination is not necessarily additive because there may be some interdependence between GOM and simulations (i.e. both ideas may exploit the same weaknesses). As well, the magnitude of the simulation improvement is such that it hides the effects of combining it with GOM. The larger the winning margin, the smaller the opportunity there is for demonstrating further improvement against the same opposition.

Each set of improvements reported over the past two years were measured against the previous strongest versions of Loki. As a result, the magnitude of the change may be dampened over time, simply because it is being tested against generally stronger opposition. For example, if you have three generations of poker chip sets-playing programs (A, B, and C) with B defeating A by 0.1 sb/hand and C is better than B by 0.1 sb/hand, it does not follow that C will be .2 sb/hand better than A.

Specific opponent modeling (SOM) is harder to measure, due in part to the nature of our self-play experiments. In previous work we demonstrated improvements for both GOM and SOM against a static default model [2]. However, since that time Loki has improved significantly (for example, with improved reweighting and simulations). A consequence is that our simplistic SOM model has not yet added significantly to the performance of the stronger version of Loki. Improving SOM is our current focus, and some of the ideas we are pursuing are discussed in the next section.

Loki has been tested in more realistic games against human opposition. For this purpose, the program participates in an on-line poker chip sets game, running on the Internet Relay Chat (IRC). Human players connect to IRC and participate in games conducted by dedicated server programs. No real money is at stake, but bankroll statistics on each player are maintained. The new versions of Loki using GOM and simulations win consistently when playing on the IRC server. Although there is a high level of variance in this environment, there is strong evidence that GOM is a major advance in the program’s playing strength against human opposition (as opposed to the self-play experiments where the advantage was not as significant). The performance of the program depends strongly on which players happen to be playing, and on IRC it ranges from novices to professional players. Consequently, it is dangerous to quantify the results of our recent improvements to Loki.

8. Ongoing Research

The work reported here is our first experience with a betting strategy based on simulations and opponent modeling. Each area has numerous opportunities for improvement, some of which are currently being addressed. Indeed, the poker chip sets project is rich in research opportunities. There is no shortage of good ideas to investigate; only a shortage of time and resources.

For the simulations, the major problem is variance (standard deviation) in the results. We have identified several ways in which the experiments could be conducted with less noise. Nevertheless, even with these enhancements, we expect the variance to still be high. Faster machines and parallel computations might be helpful since this will result in a larger sample of simulation data points. However, this has diminishing returns and our experiments suggest that beyond a critical minimum number of simulation data points (in the 100-500 range) the benefits may be small.

There are tradeoffs in doing the simulations. Currently, each data point contains a small amount of information about the expected value. Given the simplicity of the calculation, one can acquire numerous data points. Alternatively, one could do fewer simulations, but have each return a more accurate value. The quantity versus quality trade-off needs to be explored in more detail.

For the game of bridge, simulations have successfully allowed computers to play hands at a world-class level (GIB [9]). Nevertheless, limitations in the simulation-based approach and the high variance have prompted the author of GIB, Matt Ginsberg, to look at other solutions (including building the entire search tree) [10]. We too may have to look for new approaches to overcome the limits of simulations. 

In the area of opponent modeling, there are numerous avenues that can be explored. One serious limitation of our current work that needs to be address is the resistance to change that is built into our system. Our opponent modeling works well in some cases because most of the opponents have a fixed style that does not vary over time (certainly the computer opponents that we use in our self-play experiments have this property). However, it does not necessarily follow that opponent modeling will be as successful in games against human players as it is in the closed experiments. Humans are also very good at opponent modeling, and can be much less predictable than the players in our experiments. We have not yet investigated making our opponent models quickly responsive to perceived changes in an opponent’s style. For a strong human player, a single data point is often sufficient for them to set or alter their model of an opponent. Our models are far too slow to adapt. This must change!

A sampling of some of the ideas being investigated include:

• Use the simulations to refine the opponent modeling. Having done a simulation, record the expected reaction for each opponent. If their actions frequently differ from what is predicted, then Loki can adjust its opponent model.

• With opponent modeling, it is easy to gather lots of data. The problem is filtering it and attaching the appropriate importance to it. Without this, our modeling will be too slow to react, or base its decisions on irrelevant information. We are investigating condensing the data into simpler metrics that may be better predictors of an opponent’s style and future behavior. For example, measuring the amount of money that a player invests per game may be a good predictor of loose/tight play.

• Previous specific opponent modeling was hampered by the crude method used for collecting and applying observed statistics. Much of the relevant context was ignored for simplicity, such as combinations of actions within the same betting round. A more sophisticated method for observing and utilizing opponent behavior would allow for a more flexible and accurate opponent model. For example, we are currently experimenting with modifying our model based on sequences of opponent’s actions. A check followed by a raise (typically a show of strength) has more meaning than looking at these two actions in isolation. 

• Loki does not currently use showdown information. The cards seen at the showdown reveal clues about how that opponent perceived each decision during the hand. These hindsight observations can be used to adaptively measure important characteristics like aggressiveness, bluffing frequency, predictability, affinity for draws, and so forth.

• We have yet to fully explore the variety of techniques available in the literature for learning in a noisy domain where one must make inferences based on limited data.

9. Conclusions 

To master the game of poker chip sets, one must be adaptive. Any form of deterministic play can and will be exploited by a good opponent. A player must change their style based on the dynamic game conditions observed over a series of hands (looking at each hand in isolation is an artificial limitation). Our work has made some progress towards achieving a poker chip sets-playing program that can learn and adapt. Loki successfully uses opponent modeling to improve its play. However, it is abundantly clear that these are only the first steps, and there is considerable room for improvement.poker chip sets

poker chip sets is a complex game. Strong play requires the player to excel in many different aspects of the game. Developing Loki has been a cyclic process. We improve one aspect of the program until it becomes apparent that another aspect is the performance bottleneck. That problem is then tackled until it is no longer the limiting factor, and new weaknesses in the program’s play are revealed. We made our initial foray into opponent modeling and were pleased with the results. With the success of the new simulation-based betting strategy, opponent modeling is now back on the critical path since it will offer the biggest performance gains. We will now refocus our efforts on that topic, until it too moves off the critical path.poker chip sets

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