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Tuesday, June 21, 2005

How to Pick a Price Per Head Provider

Featuring Dalton Wagner, Founder V.O. Group, S.A.

The number of Price Per Head providers has grown exponentially over the last twelve months with entrants to the marketplace almost daily. With so many choices, the big question is: “How do I know which Price Per Head provider or sports book to choose?”

When evaluating an offshore operation, it is often difficult to sort through the facts, for this reason, you must rely on the following to make a sound choice:

Years In Business: Make sure you choose a Price Per Head operator that has been in business for at least 6 years. A proven track record is a must. You can not afford to risk your clients or your sports season to a fly by night operator or ‘wanna be’ operator. Avoid new sportsbooks. Avoid anyone with an unreasonably low price (<$25/head/week) who may be attempting to gain entry into the market (sign of a new company).

Dalton Wagner Commentary: I invented the concept of PPH in 1999. I have the largest economies of scale and I write more PPH clients than anyone offshore. And, this is in addition to my sportsbook business, my casino business, my racebook business and my poker business. The fact is I make very little money at $25/head/week. It scares me to see new operators entering the market with prices as low as $25. I can tell you that unless they are writing over 15,000 head, they can not make it. Let me re-phrase that…if they are quoting prices as low as $25/head/week there is no way that they have enough employees to service an agent, there is no way they have tri-dundant phone service, there is no way they have tri-dundant internet service – in short, there is no way they can service an agent’s business well enough to help the agent grow. Secondly, PPH companies will lie to you about their company’s age. You must be able to verify the companies age. This is a must. Use Internic, use OSGA.com, and use ThePrescription.com. You must verify the facts.

Location: Check to make sure that the PPH provider you will outsource your clients to is licensed and legally able to service clients world-wide.

Dalton Wagner Commentary: There are only two locations that we know of that allow the outsourcing of bookmaking. The two locations are Costa Rica and Panama. The UK and Dutch Antilles to NOT allow this activity in their licenses. The only one you can trust at the current time is Costa Rica (as evidenced by the recent blow up of www.BetPanAm.com in Panama).

Reputation/Integrity: One of the best ways to make sure that your clients and your business will be safe with a provider is by checking their reputation. Use verifiable and known sources of information like the Off Shore Gaming Association, The Prescription, Bettors World, etc.

Dalton Wagner Commentary: If your outsourcing your business to someone, you expect them to treat it as their own. The person accepting the outsourcing better have a reputation as strong as or stronger than yours. This is the only way you should trust your business to them. One additional note, if the PPH Company also has a post-up division, check the reputation of the post-up side of the company. This will tell you volumes about the company. Rather, if the PPH company is tied to a loser post-up sportsbook or one that has a negative reputation, you know what to expect for your clients.

Service and Offerings: Reputable PPH companies are open 24-hours a day, 7-days a week, 365-days a year. They offer quarter lines, half time lines, game lines, money lines, propositions, multiple casinos, horse betting, and more. You are outsourcing to better service your clients – therefore service is one of the key deciding factors on who to go with.

Dalton Wagner Commentary: Simply put, the more accessible the service is and the more wagering options that exist, the more money the agent will make. You should consider the service mentioned above to be the minimum acceptable. Rather, if they have one casino – you should NOT work with them. If they do not have horses – you should NOT work with them. If they are not open 24-hours a day – you should NOT work with them, PERIOD!

Security: A PPH company must have safeguards and must keep your information and clients secure and anonymous If the company you are considering is publicly held, then your security is compromised (it CAN BE accessed via the government). If the company you are considering has integrity issues, you can assume your data will be passed on to the wrong people.

Dalton Wagner Commentary: Security is the ultimate issue. Our company has had the opportunity to be purchased or taken public on multiple occasions. We decided not to do so because it would put our clients/agents/bookmakers in a compromised situation. Partner agents and bookmakers use our service because our integrity and the security of their data is complete.

Additional Information on a Price Per Head Agency Relationship (also known as outsourcing, outsourced bookmaking, sportsbook agent, sportsbook service, call center outsourcing, etc.):

The concept of Price Per Head Agency or Outsourcing Bookmaking to an Offshore Sports book has become popular recently. By using software written by offshore sports books, agents or illegal credit bookmakers have the ability to outsource the writing of their business to companies like www.BettorsNet.com (owned by V.O. Group, S.A. one of the largest and most respected offshore operators in the world – contact is 1-877-512-1001). Therefore, the agent or bookmaker simply points their clients to a phone number and/or website and allows them to bet with a simple Pin/Password system. The agent or bookmaker then pulls reports at the end of the week, and pays or collects what the client won or lost. The only money that trades hands between the agent or bookmaker and the service provider is a simple Price Per Head service fee.

Obvious benefits of the Price Per Head Agency relationship are (i) improved earning potential for the agent, (ii) decreased legal exposure for the agent, (iii) increased customer service for the agent’s clients, and (iv) ability for the agent to do something other than answer wagering calls (they actually get time to enjoy the money they are earning).

Dalton Wagner’s commentary is found throughout the above. Mr. Wagner has the reputation of being the most innovative operator in offshore gaming and the largest Price Per Head operator offshore. His most recent innovation has allowed small sportsbooks to outsource their complete operation to V.O. Group, S.A. while focusing completely on marketing. With formidable competition, Mr. Wagner founded V.O. Group, S.A. in 1998. While most companies formed that late failed, V.O. Group, S.A. has become one of the top 10 operators offshore (as measured by AnteUp; Gambling Online Magazine; Poker Player Magazine; and Many other Publications).

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BetCris.com Opinion on Price Per Head Explored

Featuring Dalton Wagner, Founder V.O. Group, S.A.

The concept of Price Per Head Agency or Outsourcing Bookmaking to an Offshore Sports book has become popular recently. By using software written by offshore sports books, agents or illegal credit bookmakers have the ability to outsource the writing of their business to companies like www.BettorsNet.com. Therefore, the agent or bookmaker simply points their clients to a phone number and/or website and allows them to bet with a simple Pin/Password system. The agent or bookmaker then pulls reports at the end of the week, and pays or collects what the client won or lost. The only money that trades hands between the agent or bookmaker and the service provider is a simple Price Per Head service fee.

This new system is a departure from the way it used to work in which the agent would split the winnings or losings with the offshore sportsbook.

Obvious benefits of the Price Per Head Agency relationship are (i) improved earning potential for the agent, (ii) decreased legal exposure for the agent, (iii) increased customer service for the agent’s clients, and (iv) ability for the agent to do something other than answer wagering calls (they actually get time to enjoy the money they are earning).

There has been much debate over which of the two methods is better for the partner agent or bookmaker. In an effort to get to the bottom of this, we attempted to contact the dominant player in each of the models:

Revenue Split Model: The owners of www.BetCris.com (Costa Rica International Sportsbook is a well known offshore credit bookmaker who operates only on a revenue split model) were contacted but refused to return our call or comment;

Price Per Head Agency Model: Dalton Wagner, proprietor of V.O. Group, S.A. and BettorsNet.com (one of the largest offshore gambling operators who only operates on a Price Per Head basis when dealing with credit agents) was contacted and agreed to be interviewed.

Unfortunately, since only one of the persons contacted agreed to participate, our interview will show mostly his view. However, we did our best to probe the positives and negatives of each model.

Post-Up Newsletter: The Price Per Head model appears to be the most appropriate model for an agent or bookmaker to maximize his revenue. Why then are companies like www.BetCris.com so against them?
D. Wagner: The answer is simple dollars and cents. If you have a client that loses $10,000 in a week and you have a Price Per Head Agency relationship with www.BettorsNet.com, you will simply need to pay your service fee of $25-$30 for that client using our service. However, if you are using a revenue split relationship, you will have to fork over $5,000. So, in the example you provided, companies like the one you mentioned are in danger of losing $4,970 in revenue if their agents find out about PPH.

Post-Up Newsletter: Yes, I understand the math, but when speaking to companies like www.BetCris.com they act as if the PPH industry is putting the offshore industry in danger. They speak of people like you as if you had the plague. Why?
D. Wagner: I know the gentleman that owns Bet Cris. I know the guys who run Bet Cris. They are the best in the world at what they do. There is no way I would dispute that and no way anyone would dispute that. But, BetCris.com can not exist by taking all of their revenue split relationships and having them converted to PPH. So, if I had to guess, they act that way because I am endangering their way of life. I have actually had conversations with the owner of BetCris.com and he firmly feels that he provides his agents a better opportunity to earn more, therefore he feels his much higher price is warranted. I don’t know if I agree, but I think it best for my reputation to speak for itself. The fact is that I have NEVER had an agent or bookmaker come to me who switched to PPH from revenue split EVER GO BACK. 100% of the people who have tried my service are still with me from day 1. That shows me that agents do not agree with BetCris.com. They think the service is comparable and they think the earning potential is greater with www.BettorsNet.com.

Post-Up Newsletter: I am lost. Why does the owner of BetCris.com feel he provides his agents a better opportunity to earn more? The math seems obvious.
D. Wagner: Actually, I agree with him in one way and one way only. If an agent is weak and is booking ‘over his head’ (an industry term that implies the bookmaker or agent is taking more risk than they can afford in the long term), then a partnership with www.BetCris.com can be a positive thing. Not only do you share 50% of your revenue with them on a revenue split relationship, but you also share 50% of the risk. So, if you are an agent that has a bettor that wants to bet $5,000 a game, and you can only afford to book $2,500, then a partnership with Bet Cris is a good thing and will allow you to stay in business longer. You should never ever book over your head.

Post-Up Newsletter: Oh, I see. So that is an example of an agent you would tell to use a revenue split model, right?
D. Wagner: Yes. And, No. Yes, if the agent plans on booking over his head for the long term. No, if he can cut his client down to an acceptable level and keep all the profit for himself. Then, when the agent or partner bookmaker has a large enough bank roll, he can open the client up to the larger wagers. I still believe, and I believe the math proves that PPH is always bettor for the agent.

Post-Up Newsletter: Are there any other Price Per Head agency companies that you would recommend?
D. Wagner: Absolutely NOT. Please print that in big bold letters. It is difficult for any company in our industry to survive. The overhead is huge. Most Price Per Head companies have come to Costa Rica or Panama or Antigua to try and compete with just Price Per Head revenue. Trust me, I have been doing this for 7 years, and PPH does NOT generate enough for survival. You have to build a relationship with a company that has a huge staff, technical know-how, tri-dundant phones, tri-dundant internet service, millions of dollars in investment in hardware and software – and, we are the only ones that have it at the current time. Why? Because we have been operating major post-up brands for years. We are one of the most successful operators offshore and we are able to write PPH as a sideline business. We could NEVER survive on PPH alone. And, anyone who says they can, is lying to you. The guys who say they can write business at $17or $20/head are on their way out – they just don’t know it yet.

Post-Up Newsletter: So how does a bookmaker or agent learn more about this?
D. Wagner: Simply go to any of the V.O. Group, S.A. sites to learn more. I suggest starting at www.BettorsNet.com or calling 1-877-512-1001 or emailing agents@bettorsnet.com to get more information.



Additional Information on the Companies Mentioned Above:

As Founder and Proprietor of V.O. Group, S.A., Dalton Wagner has survived multiple buyout attempts and maintains one of the largest privately held offshore gaming companies in the industry. While the rest of the industry is going public and being required to provide more and more information in a public format, Mr. Wagner’s privately held company is a favorite associate and partner for bookmakers and agents onshore. Privacy is one thing you never have to worry about when dealing with Dalton and V.O. Group, S.A. With formidable competition, Mr. Wagner founded V.O. Group, S.A. in 1998. While most companies formed that late failed, V.O. Group, S.A. has become one of the top 10 most admired offshore companies (as measured by Online Gambling Magazine; Post-Up Newsletter; Offshore Operator Industry Magazine; and Many other Publications). If you are looking for a Price Per Head Agency relationship, there is no place better online or offshore.

BetCris.com is considered the undisputed leader in offshore credit bookmaking. The owner of the company has been called the Godfather of the Offshore Bookmaking Industry in many publications. It is estimated that 90% of Costa Rica International Sportsbooks revenue is generated via revenue splits with Partner Agents and Bookmakers. If you are looking for a Revenue Split relationship, there is no place better online or offshore.

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Benefits of Price Per Head (PPH) Exploration & Quantification

Q & A with
Dalton Wagner, Founder V.O. Group, S.A.

Dalton Wagner, Proprietor of V.O. Group, S.A. has found a great deal of success by tripling his bottom line profit numbers with the addition of what has been called the Greatest Price Per Head Offering online. While his Board of Directors and Investors are thrilled, we wanted more than vague statements – we wanted facts. We recently caught up with Dalton in San Juan, Puerto Rico while he was on vacation at El Conquistador Hotel and Resort. Our goal was to Explore and Quantify the actual benefits that a Price Per Head (PPH) Agency Relationship offers the bookmaker, agent or entrepreneur.

As Founder and Proprietor of V.O. Group, S.A., Dalton Wagner has survived multiple buyout attempts and maintains one of the largest privately held offshore gaming companies in the industry. While the rest of the industry is going public and being required to provide more and more information in a public format, Mr. Wagner’s privately held company is a favorite associate and partner for bookmakers and agents onshore. Privacy is one thing you never have to worry about when dealing with Dalton and V.O. Group, S.A. With formidable competition, Mr. Wagner founded V.O. Group, S.A. in 1998. While most companies formed that late failed, V.O. Group, S.A. has become one of the top 10 most admired offshore companies (as measured by Online Gambling Magazine; Post-Up Newsletter; Offshore Operator Industry Magazine; and Many other Publications).

The Offshore Operator: We have heard about all of the success you have found with the best Price Per Head (PPH) offering in the offshore gambling industry. However, we are not hearing actual numbers. Can you tell us how many clients you are now writing business for?
Dalton Wagner: As you know, the answer really depends on the season you are speaking about. In football season, we are now writing about 7,500 PPH clients a week (7,500 clients of our partner agents which outsourced the writing of their bets via our service and software). And, in baseball season we have historically written only half of that number. But, there is a very favorable trend occurring for our Agent and Bookmaker Partners. Since the addition of our Horse Wagering at over 300+ tracks and Two Casino Platforms with over 35 games, we are now seeing an increasing number of clients playing year-round. This, as you know, makes our Agent and Bookmaker Partners very, very happy.

The Offshore Operator: 7,500 PPH clients clearly makes you the largest PPH provider offshore. What do you attribute your success to?
Dalton Wagner: I am not sure we can find one distinct reason for our success. But, I can tell you what our Agent and Bookmaker Partners tell us. On average an Agent or Bookmaker partner doubles his revenue when he switches his clients to our service. The reason for this is access to 24-hour wagering online or via our call center, about 700% more wagering options when you look at our period wagering and propositions, and the addition of two casinos and horses has not hurt either. So, if you are a bookmaker or agent making $200k a year, you can reasonably expect to make $400k a year using our service. It is virtually guaranteed.

The Offshore Operator: Dalton, you are famous for making that kind of statement. But, can you back it up with quantifiable numbers?
Dalton Wagner: Absolutely I can. Ask me what you would like.

The Offshore Operator: OK. Let’s start with examples of how you can guarantee a 200% increase in earnings for Partner bookmakers or agents?
Dalton Wagner: Easy enough. In the United States, Mexico and Canada the typical credit bookmaker offers lines on games and totals. Sometimes they offer money lines too – but, rarely. And, these same bookmakers do not offer period betting or propositions. Furthermore, they take wagers only over the phone and only 2-4 hours a day. If you use our Price Per Head Software, you offer spreads, money lines, totals, half-time spreads, half-time money lines, half-time totals, quarter lines, propositions, and we allow access to betting 24-hours a day; 7-days a week; 365 days a year. In summary, I can safely say we offer 600% of the access time a normal credit bookmaker would provide as we are opened 24 hours compared to 4 hours. And, I can safely say we offer 700% more wagering options. These increases of 600% in access and 700% in wagering options easily equate to 200% more revenue for the partner bookmaker or agent.

The Offshore Operator: I understand, but I don’t necessarily agree. For example, if I am a credit bettor and I have $200 to lose, does it matter that I can access your site 24-hours a day?
Dalton Wagner: That question shows that you are not a bookmaker or agent. The reason is, the answer is ABSOLUTELY YES. If you compare a post-up client to a credit client, the credit client will typically lose 3-5x what the post-up client will. The reason is access. If you give a client access to credit, and access to use the credit, he will use it. Heck, forget about that example…if you are married and your wife has a credit card, you know that. The fact is as you allow more access to betting options and credit, the client will lose more. And, my current partner bookmakers and agents report to me that they typically double their earnings within weeks after joining my service.

The Offshore Operator: OK. That makes more sense to me. How about the casino, how does that affect the partner bookmaker or agents earnings?
Dalton Wagner: The online casino is relatively new to partner bookmakers and agents. The fact is that in the past, agents were scared to open up their clients to online casinos. The reason is that when clients lose in sports – they know they were treated fairly. However, when they lose in an online casino, for some reason their first reaction is “I was cheated”. However, as integrity in online gaming has risen over the years, so has trust in online casinos. For this reason, you are starting to see partner agents and bookmakers open their clients up to our two online casinos. And, boy, when they do – they are very happily surprised. An online casino allows wagering 24-hours a day. This means revenue potential for the agent or bookmaker 24-hours a day. It’s nice to run your reports in the morning and realize you made an extra $20,000 while you were sleeping! We estimate that when a partner agent or bookmaker opens up his clients to our casino, they will make an extra 15%-20% a year. This is purely earned off casino winnings.

The Offshore Operator: Fifteen to Twenty percent a year is nothing to laugh at. What about the horses?
Dalton Wagner: The horse product we have offers 300+ tracks including harness, thoroughbred and dogs. It is not just horses. Typically when the partner bookmaker or agent opens it up, adoption is slow. But, people who bet horses and dogs, love to bet horses and dogs. It is not rare to see a client who likes the ponies to double his daily action. And, as any bookmaker knows, in sports you hope to hold 6% on turnover. Well, in horses you hope to hold 20%. The difference in expected hold is where the value is in horses.

The Offshore Operator: Wow! So, having all the offerings really does matter doesn’t it?
Dalton Wagner: What I am selling is access – pure and simple. My goal is to take a partner agent or bookmakers clients and put them in the candy store 24-hours a day surrounded by all of their favorite treats. And, as you know, when any of us are put in that situation, we will indulge. And, they do.

The Offshore Operator: Have you ever had anyone become a partner Agent or Bookmaker and leave because they did not like what you had to offer?
Dalton Wagner: That is a very interesting question. The fact is, and I should have used this in the past know that I think about it, the answer is NO. As a matter of fact, every single person to use our service or try our service has been with us since initiating trial. We really are, without exaggeration, at 100%. Thank you for pointing that out to me.

The Offshore Operator: How could you not have known that?
Dalton Wagner: Embarrassingly, I never thought about it. I have always said “try it, you’ll love it”. But, I guess I never looked at the quantifiable fact – NOBODY HAS EVER TRIED OUR SERVICE AND LEFT OUR SERVICE they’re clients love it, the agents love it, and the agents make more than ever before – so they stay around!

The Offshore Operator: So how does a bookmaker or agent learn more about this profitable product?
Dalton Wagner: Simply go to any of the V.O. Group, S.A. sites to learn more. I suggest starting at www.BettorsNet.com or calling 1-877-512-1001 or emailing agents@bettorsnet.com to get more information. V.O. Group, S.A. sites include www.BettorsNet.com; www.MVPsportsbook.com; www.V-Wager.com; www.Players-SB.com; etc.

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Success With Price Per Head (Small Sportsbook Success)

Case Study in Small Sportsbook Success via Use of Price Per Head (PPH) Software Written and Owned by V.O. Group, S.A., the most successful offshore sportsbook in Costa Rica.

Price Per Head vs. Small Sportsbook Operation

Featuring Dalton Wagner, Founder V.O. Group, S.A.

As Founder and Proprietor of V.O. Group, S.A., Dalton Wagner has the reputation of being the most innovative operator in offshore gaming. His most recent innovation has allowed small sportsbooks to outsource their complete operation to V.O. Group, S.A. while focusing completely on marketing. With formidable competition, Mr. Wagner founded V.O. Group, S.A. in 1998. While most companies formed that late failed, V.O. Group, S.A. has become one of the top 10 operators offshore (as measured by AnteUp; Gambling Online Magazine; Poker Player Magazine; and Many other Publications).

Playing Poker Online Magazine: Recently we have been told that V.O. Group, S.A. is now allowing small sportsbook operations to outsource the complete writing of their business to your organization. How is this possible?
Wagner: The situation you speak about is 100% true. However, has only happened one time in our history.

Playing Poker Online Magazine: Can you be more specific?
Wagner: When an offshore sportsbook starts, in 99% of all cases it will fail in the first 24 months of operations given the current market conditions. Examples are everywhere from The Dunes, Platinum, Super Sports Book, etc. It happens every single year. In the past, large sportsbooks like MVP Sportsbook or V-Wager.com would swoop in and try to save the books via financial help or simply buy them at a low price. Well, V.O. Group, S.A. now offers another more innovative solution.

Playing Poker Online Magazine: OK. You’ve got me, explain!
Wagner: We write business for onshore bookmakers and agents charging them a simple price per head per week. This price takes into consideration our considerable economies of scale and a small profit margin. In the case you are speaking of, instead of buying the sportsbook for a low cost that basically sent the owners home broken hearted and with empty pockets, we cut them a deal where they could focus on marketing and leverage off of our economies of scale. Simply put, we took over writing their business for them, and they now focus their limited resources on marketing 100% of the time.

Playing Poker Online Magazine: That sounds like snake oil. How exactly would that work?
Wagner: We don’t sell snake oil. The facts speak for themselves. When V.O. Group, S.A. opened it’s doors in 1998 the market was different. You could recruit players, write their business, and make a profit in years one, two and three. With the current market conditions, there is NO WAY a start up credit or post-up book can be profitable in year one, two or three without 2,000 head a week in business. That is the theoretical breakeven point. If you are writing less than 2,000 head a week, you can double or triple your net earn by letting V.O. Group, S.A. write your business.

Playing Poker Online Magazine: OK. That makes sense because in 1998 and 1999, it was cheaper to find a new client. Is that where the difference in earnings comes from?
Wagner: The fact is yes, that is where the difference is. But, that is not our selling point. Our selling point is that if you stop worrying about writing business and let us do it, you can focus on marketing – which is the key to the success of an online sportsbook. And, that is the simple fact for credit bookmaking, post-up bookmaking and future success in this industry. As long as you think ‘beating the clients’ is the key, you are limiting your business. Let our company worry about that – and, you find clients. Spend your time finding clients and you will make money.

Playing Poker Online Magazine: The theory is sound, but what about the facts?
Wagner: The owners of the sports operation in question do not want me to be specific about whom they are. But, let me simply say that when we took over their operations, they could not even make the post-up money good. Now they have $1.4 million in the bank. So, yes, the facts do follow the theory.

Playing Poker Online Magazine: Wow! That is incredible. Can any poker room, casino, racebook or sports book do this with V.O. Group, S.A.
Wagner: Yes. We have programs for all online products except for bingo. And, by 2006 we should be able to offer options for even bingo operators..

Playing Poker Online Magazine: So how does a bookmaker or current operation learn more about outsourcing their operations to V.O. Group, S.A.?
Wagner: Simply go to any of the V.O. Group, S.A. sites to learn more. I suggest starting at www.BettorsNet.com or calling 1-877-512-1001 or emailing agents@bettorsnet.com to get more information.



Article Note: The situation in this Case Study is a true case, however may not be representative of your situation. All situations are different and therefore may result differently.

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Monday, June 20, 2005

Success With Price Per Head (One Agents Story)

Case Study demonstrating one agents success via Use of Price Per Head (PPH) Software Written and Owned by V.O. Group, S.A., the most successful offshore sportsbook in Costa Rica.

Price Per Head vs. One Agents Old Operation

Featuring Dalton Wagner, Founder V.O. Group, S.A.

As Founder and Proprietor of V.O. Group, S.A., Dalton Wagner has the reputation of being the most innovative operator in offshore gaming. His most recent innovation involves the direct attack upon the Onshore Bookmaking market via the unique value proposition of actually partnering with Bookmakers and Agents onshore (as opposed to attempting to steal their business). With formidable competition, Mr. Wagner founded V.O. Group, S.A. in 1998. While most companies formed that late failed, V.O. Group, S.A. has become one of the top 10 operators offshore (as measured by AnteUp; Gambling Online Magazine; Poker Player Magazine; and Many other Publications).

Sports Betting Business: The ‘buzz’ in offshore gambling appears to be focusing on only two subjects as of late, including: Poker and Your Price Per Head Offering. What exactly is Price Per Head and Price Per Head Agency?
Wagner: As you know, one market that a post-up sportsbook has historically been unable to attack is the huge market of bettors that have been betting with their local guy for years (local guy = agent or corner bookie). The reason this market exists is the personal relationship that an onshore bookie or agent has with his clients. Sportsbooks have attempted to educate the player and lure him away from the local bookie, however V.O. Group, S.A. has decided to take another tact. Via our Price Per Head software written by Matthew Wilson, we have been able to PARTNER with onshore bookies or agents with our Price Per Head Agency Relationship.

Sports Betting Business: You called us with a specific Success Story that you said Had To Be Told Immediately! What in the heck could be so important?
Wagner: As the industry knows, we have been pushing Price Per Head Agency for a while now. We have a story that has to be told…it is that simple!

Sports Betting Business: OK. Go ahead and tell the story!
Wagner: The story starts with a gentleman named Paul Harris in Austin, Texas who is was a small town bookmaker making $250,000 a year in income.

Sports Betting Business: I already see where this is going. Now you are going to tell me of how V.O. Group, S.A. leveraged your PPH software and relationship with Paul to make him 30% or 50% more. Right?
Wagner: No, that is not the story at all. While 50% would truly be something to celebrate, it does not compare to this story!

Sports Betting Business: Stop already. Tell the story.
Wagner: Mr. Harris contacted V.O. Group, S.A. through our BettorsNet.com brand in August 2004, right before football season. After three phone calls we switched his 75 players from playing through his cell phone to playing via our website and call center. And, with the increased availability to the product, more offerings, and all the other items BettorsNet.com brings, guess how much Paul made?

Sports Betting Business: Well, since you did not bite on 50% more, I will say 100% more. But I doubt it!
Wagner: Yea, I doubt it too. How about 350% more. That’s right! I just ran the reports for Paul Harris and up to February 1, 2005, Paul has made $750,000.00.

Sports Betting Business: Wait, $750,000 is a hell of a lot more than 350% more. Isn’t it?
Wagner: Actually, if you take into consideration the decline we will see after the end of March Madness, he should easily make $1,200,000 this year. But, I didn’t want to overstate the facts!

Sports Betting Business: Are you telling me that by having you take the phone calls and do the accounting for Mr. Harris you made this difference in his life?
Wagner: That is exactly what I am saying!

Sports Betting Business: Was he that bad of a bookmaker?
Wagner: Well, I think we are better – but, he was fine. The increase doesn’t come from us beating his players more than he did. The increase comes from the product being available 24 hours a day and our company having more offerings. And, the secret ingredient helps too!

Sports Betting Business: Oh, I see. Instead of having his cell on for a few hours a day, you put the product in front of his clients all day and night for the buying. I see. But, what is the ‘secret ingredient’?
Wagner: The availability will make a person 100% more over night. But, the secret ingredient made the difference for Mr. Harris. And, the secret ingredient is TIME.

Sports Betting Business: OK, I am lost again. What does that mean?
Wagner: Paul Harris used to set lines, answer phone calls, watch games, grade wagers and keep figures. He did it so well, and so often he already lost his girlfriend. He simply spent all of his time operating as a bookie. When we took over, he became a RECRUITER. All Paul does now is sit in bars and recruit business. We write the business, and Paul grows his business daily. That is the secret ingredient.

Sports Betting Business: Ah, I see. He no longer spends time on the items that cost time and money…he spends his time on what makes him money – recruiting players!
Wagner: Christ! Your brilliant! It typically takes a bookmaker or agent about 3 months to figure that out. But, you, like Paul Harris already figured it out. You recruit, and I will beat them. Therefore the only thing that matters is getting more clients. The more you get, the more you will make.

Sports Betting Business: Of course I get it. That’s the Las Vegas model. Build in an edge and then find as many people to bet against the edge as possible. It’s simple!
Wagner: You sure make it sound simple. But, most bookies are stuck in the mode of trying to ‘beat the players’. The fact is that all you have to do is get a new client or two every day and you will be a millionaire over night. Simply ask Paul Harris who is on campus at University Of Texas recruiting players on 6th Street right now.

Sports Betting Business: So how does a bookmaker or agent learn more about this?
Wagner: Simply go to any of the V.O. Group, S.A. sites to learn more. I suggest starting at www.BettorsNet.com or calling 1-877-512-1001 or emailing agents@bettorsnet.com to get more information.

0 comments

Thursday, June 16, 2005

How Much Should a Bookmaker Expect to Make with PPH

Q & A with
Dalton Wagner, Founder V.O. Group, S.A.

Dalton Wagner, Proprietor of V.O. Group, S.A. has recently found himself one of the most popular interviews in the offshore sportsbook industry. The reason is his new Price Per Head Agency offering for Onshore Credit Bookmakers. We had an opportunity to catch up with Dalton Wagner and discuss the issue on his last visit to St. Kitts.

Vegas Offshore: Last time we talked you mentioned that your Price Per Head offering was growing beyond your wildest dreams. How is it doing now?
Wagner: Our Price Per Head software allows onshore bookmakers to make more than they have ever dreamed. For this reason, the product is growing faster than we could have ever imagined.

Vegas Offshore: Can you be a little more explicit?
Wagner: We wrote the software offering with input from two bookmakers 24 months ago. Today, we have over 500 bookmakers using the software writing business for over 20,000 clients a week, generating over $500,000 in revenue on a weekly basis in football season.

Vegas Offshore: That is incredible. For you to grow that fast, bookmakers must simply love the product.
Wagner: Yes, bookmakers love the product. Not only does it take the burden of answering phone calls and doing accounting away, but it also allows bookmakers to insulate themselves from legal issues and market their product more aggressively. We have literally 50+ examples bookmakers that are writing hundreds of clients with our software. Can you imagine how they could have done that without us?

Vegas Offshore: The largest onshore bookmaker I have ever met wrote 300 clients and had a staff of 10 clerks on football Sunday. Who is your largest onshore bookmaker?
Wagner: I obviously can’t reveal identities, but I can tell you that she is writing 750 head a week on average and employees nobody.

Vegas Offshore: You have a bookmaker working with you that writes over 750 head a week, what do you think she makes annually?
Wagner: I know exactly what she makes. Let’s just say it is well over a million a year.

Vegas Offshore: How many bookmakers do you currently work with?
Wagner: We are writing business for over 250 bookmakers now. Some represent groups as small as 2 or 3; and others represent groups as large as 750.

Vegas Offshore: Anything else you would like to add?
Wagner: Sure. If you’re a bookmaker onshore, call us or visit us online to learn how to save money and/or grow your business beyond your wildest dreams. The service is modestly priced, and we have never had a disgruntled user. The value proposition is simply huge for any bookmaker.

Vegas Offshore: So how does a bookmaker learn more about this?
Wagner: Simply go to any of the V.O. Group, S.A. sites to learn more. I suggest starting at www.BettorsNet.com or calling 1-877-512-1001 or emailing agents@bettorsnet.com to get more information.

0 comments

Tuesday, June 14, 2005

Innovation In Bookmaking

Q & A with
Dalton Wagner, Founder V.O. Group, S.A.

As Founder and Proprietor of V.O. Group, S.A., Dalton Wagner has the reputation of ferreting out new markets and attacking while other companies sit on their hindquarters and watch. One market that has always eluded offshore bookmakers is the ‘guy who likes dealing with his credit bookmaker’. Typically this market has been untouchable for the offshore bookmaker, but NOT for Dalton and V.O. Group, S.A. We had an opportunity to catch up with Dalton Wagner on his last visit to Panama.

Vegas Offshore: Tell us about this new product that you are calling your Price Per Head Agency Relationship.
Wagner: As you know, one market that a post-up sportsbook has historically been unable to attack is the huge market of bettors that have been betting with their local guy for years. The reason this market exists is the personal relationship that an onshore bookie has with his clients. Sportsbooks have attempted to educate the player and lure him away from the local bookie, however we have taken another tact. We have partnered with local bookies around the world to make their offering better and more valuable.

Vegas Offshore: So instead of attacking the local guy, you actually approach him with your offer?
Wagner: Exactly. The fact is that the relationship is the last thread holding clients to a local guy. Yes, there is the fact that the local guy extends credit, but with bonuses and promotions we can overcome that offering. The key is the relationship.

Vegas Offshore: I’m confused, please explain.
Wagner: As you know, the onshore bookmaking market has lost steam to the offshore providers like V.O. Group, S.A. The reason is that the offerings that I can provide are far superior to those that any onshore bookmaker can offer. The PPH concept is based on allowing the onshore bookmaker to provide his clients my offerings on a purely anonymous basis via an online automated tool. Rather, by signing up with my PPH service, an onshore bookmaker can compete with all offshore sportsbooks on equal footing offering 24-hour wagering, 365 days a year. All the bookmaker does is use an online system to assign PINs and PASSWORDs and pay me a small fee per week for the service.

Vegas Offshore: Wait a minute. So, this means that a small time operator in Cleveland, Ohio can offer all the same things that V.O. Group, S.A. does without making the multi-million dollar investment?
Wagner: Now your catching on. For as little as $25 per head per week, a guy in Ohio can compete with the biggest sportsbooks in the world. And, better yet, never answer another wagering phone call again.

Vegas Offshore: OK, I get it. They use your automated tool to drive their clients to your software online, and they simply pick up the figures at the end of the week.
Wagner: Exactly correct with one exception. Their clients can bet on the phone via our call center staffed with 250 people OR bet online. Again, they have access to all of the offerings V.O. Group, S.A. provides including call center access, client services and technical support.

Vegas Offshore: Is the PPH product only based on sports wagering?
Wagner: Very good question. The answer is no. We actually have the ability to offer clients two (2) casino products including blackjack, baccarat, slots, 3 card poker, roulette, etc. And, clients may also bet on horses at over 350 tracks worldwide.

Vegas Offshore: Wow! That truly is incredible. So, a bookie that has one client can compete with the likes of MVPsportsbook.com on equal footing?
Wagner: Exactly correct. But, there are two keys to the success of the product launch. The first is being able to compete head to head with the biggest sportsbooks in the world. But, arguably the most important factor is that since the onshore bookie no longer has to answer phones or track plays, he can spend all of his time picking up new clients, NOT writing bets.

Vegas Offshore: I remember that in a prior interview you had stated that the key is not winning, but finding more losers.
Wagner: An embarrassing quote, but true. Amateur bookmakers think that the key is to beat the clients you have. That is a huge mistake. The key to getting rich as a bookmaker is finding a new client every single day of the year. The numbers take care of themselves, the key is to get more and more people playing. That is the TRUE BEAUTY of the PPH software and agency relationship.

Vegas Offshore: Do you have any success stories to tell?
Wagner: I have many, but will tell you one that emphasizes what the PPH program can do. I have a bookmaker that has been working out of San Antonio, Texas for 10 years. The guy has historically made $200,000 a year from his 25 or 30 players. He was introduced to our product in July 2004, and today he is writing over 185 clients. The ability to outsource answering calls and writing tickets to us has allowed him to grow his business 600% and his revenue over 1000%.

Vegas Offshore: Why did his revenue grow more than his business? Is it because your lines are sharper than his?
Wagner: There you go again. That is the obvious mistake that everyone makes. Again, don’t worry about beating the clients. Worry about getting more clients to play. But, to answer your question directly, the ability for his clients to play in the casino, play 24 hours a day, play games/halves/quarters, props, etc. is where the extra revenue comes from. Our lines might be sharper, but the advantage is more clients betting more often.

Vegas Offshore: How many bookmakers do you currently work with?
Wagner: We are writing business for over 250 bookmakers now. Some represent groups as small as 2 or 3; and others represent groups as large as 500. Again, remember the key is finding as many clients as possible. If you get to where you are writing 15 clients or more, it is almost impossible to lose for an extended period of time.


Vegas Offshore: So how does a bookmaker learn more about this?
Wagner: Simply go to any of the V.O. Group, S.A. sites to learn more. I suggest starting at www.BettorsNet.com or calling 1-877-512-1001 or emailing agents@bettorsnet.com to get more information.

0 comments

Monday, June 13, 2005

State of the Offshore Gaming Industry

Q & A with
Dalton Wagner, Founder V.O. Group, S.A.

As Founder and Proprietor of V.O. Group, S.A., Dalton Wagner has the reputation of making one-dollar work like three in online marketing. With formidable competition, Mr. Wagner founded V.O. Group, S.A. in 1998 and has quickly risen to be one of the largest offshore operators in the world with over 50,000 active clients. We had an opportunity to catch up with Dalton Wagner on his last visit to Antigua.

Vegas Offshore: This year has been interesting for V.O. Group, S.A. and the offshore industry as a whole (advertising changes, etc.). How do you feel about the industry at the current time?
Wagner: I feel the industry is as strong as it has ever been. We are still finding that our advertising is bringing us new clients at an ever-growing rate. Some competitors are complaining that the crimp on advertising is affecting the industry, but we are not seeing this trend. Consolidation within the industry has begun, and I am sure you will continue to see smaller competitors gobbled up, and medium to larger companies merging. The weaker will go by the wayside.

Vegas Offshore: What about the rumors of V.O. Group, S.A. being bought?
Wagner: I am glad you used the term 'rumors'. The fact is that in July 2004 three suitors approached V.O. Group, S.A. because of our record-breaking profit year. In two of the cases we did see 'value' in a merger/relationship. However, at the end of the day, we could not come to terms that were agreeable on both sides. At this time V.O. Group, S.A. stands independent. And, we are happy being independent, profitable and growing.

Vegas Offshore: It is rumored that Bet On Sports (PLC; BSS.L) was one of the suitors. Is that true?
Wagner: There were three suitors that approached V.O. Group, S.A. And, in all cases confidentiality agreements were signed. For this reason, I cannot confirm or deny who any of the potential suitors were. What I can tell you is that one was a major sportsbook player, one was a major casino player and two were PLCs. I doubt that helps.

Vegas Offshore: Well, if Bet On Sports was involved, I bet you are happy you weren't involved in their November 24, 2004 stock debacle.
Wagner: The Bet On Sports stock debacle was an over-reaction by the market in my opinion. In one day you saw their stock drop 50% in value. And, the reason was poor performance in a very short period. If you looked at the other publicly held offshore gaming companies, you saw the same thing, just not as drastic as with BSS.L. Simply put, I know the founder of Bet On Sports, I know the management of Bet On Sports, I know the staff at Bet On Sports and I would not hesitate to invest in Bet On Sports. They were, and still are, undervalued in my opinion. They are a great competitor and a great company. I wish I had some of their stock at the new adjusted price. I just don't see how you can lose.

Vegas Offshore: It is rare to hear the 'competition' speaking so highly of a major competitor. Why would you do so?
Wagner: Simple. Bet On Sports is a great company. And, I am sure they would tell you the same about V.O. Group, S.A. The fact is, that there are companies out there that make our industry stronger. And, ones that make it weaker. Bet On Sports and V.O.Group, S.A. strive to make our industry stronger and more legitimate on a daily basis (as do BetCris.com; BoDog.com; etc.). What kind of ambassador for the industry would I be if I downplayed a 'good' competitor like Bet On Sports for my own benefit?

Vegas Offshore: If Bet On Sports did approach you, would you consider a merger?
Wagner: I like being independent. But, simply put, NEVER say NEVER.

Vegas Offshore: What of the rumors that you are looking at Panama as a potential relocation site?
Wagner: Our operation currently has offices in Costa Rica, Antigua and Belize. In Costa Rica we currently have over 400 employees. With employment costs in Costa Rica on the rise, the burden of Caja and the ever-changing political climate, we have looked at several alternative locations. One of the most desirable is/was Panama. However, due to the massive failure of BetPanAm.com and the political debacle involving their past gaming commission, we have decided to stay put for the current time. I think it is very safe to say that our primary location will be Costa Rica for the next 5 years. Pending any crazy licensing or governmental changes.

Vegas Offshore: I have visited your office in Antigua, however had no idea that you employed 400 individuals in Costa Rica. Why do you need such a large staff?
Wagner: Our industry has a reputation for being lazy and getting by on a shoestring budget. Most sportsbooks, casinos, racebooks and poker rooms answer phones when it is convenient and buy computers when the ones they are operating blow-up. Most are simply embarrassing to the legitimate operators. To run an operation like ours, servicing 50,000 bettors, you need lots of space, a 1st rate phone system, fiber and satellite phone and Internet backups, IT professionals, accounting professionals, etc. V.O. Group, S.A. is currently located in 35,000 square feet of space, operating 24-hours a day, 365 days a year, and we are busting at the seems. At the current time, we are negotiating to buy the building next to us for future expansion. I think we will employ 600 people come this same time next year.

Vegas Offshore: I have noticed that you are still branching into other products. How has this strategy worked for V.O. Group, S.A.?
Wagner: In 1998 we started with our sports product. We then branched into the casino product. In 2001 and 2002 we opened several major race books and poker rooms. The fact is that we are profitable in all products. However, our core product, and hub for our advertising, is focused on the sports product. The strategy has worked, but we will not desert our core product. You can look for our re-launch of interactive betting in early 2005 and bingo in mid 2005.

Vegas Offshore: What other changes do you see in 2005 for V.O. Group, S.A.
Wagner: I think you will see V.O. Group, S.A. work more on our Internet interface to make it more gamer-friendly; you will see our company move into land-based gaming; and you will see us acquire two or three smaller competitors. Rather, you will see what you have always seen, expansion.

Vegas Offshore: What other changes do you see in 2005 for the industry as a whole?
Wagner: The strong will get stronger. The weak will get weaker. And, the acquisition race will be on.

0 comments

Tuesday, March 15, 2005

Bookmakers!

Read Below To Earn More Cash Than Ever

Are you a onshore bookmaker that wants to Minimize Legal Exposure, Increase Potential Profits, Spend More Time Enjoying the Money You Are Making, and Grow Your Business Beyond Your Wildest Dreams? If you answered ‘YES’ to one or all of these questions, then you need to look at a Price Per Head Agent Relationship with our company.

The concept of a Price Per Head Agent Relationship is based on you, the onshore bookmaker, paying our company a service fee to offer the same services that we offer to thousands of clients (on an anonymous basis).

For the first time, you will be able to offer Internet and Phone access to Sports, Horse and Casino (2 casinos) wagering to all your current clients (via a simple PIN and PASSWORD betting system). No longer do you have to compete with offshore sportsbooks and lose clients. YOU WILL BE ABLE TO OFFER EVERYTHING THAT THEY OFFER!

We are Licensed to offer bookmaking services.

We have been in business for 7 years.

We write business for 15,000 clients daily.

We are open 24-hours a day, 7-days a week, 365 days a year.

We have over 500 operators standing by to make you money!

Take advantage of our $5,000,000 USD investment to make your services better than ever. No longer will you have to answer calls all Saturday and Sunday. No longer will you have to lose clients to the offshore bookmakers. No longer will you have to worry about having your door kicked in by the authorities. Now you can compete with the Big Boys! You will simply sit back, watch the games, and earn more than ever before.

To learn more, go to our website, and click on ‘Agents’. If you prefer to use a white-labeled site without bonus and promotional text, visit www.BettorsNet.com for more information.

We look forward to being your service provider.

Call 1-888-333-6914 for more information.

0 comments

Monday, March 14, 2005

Innovation In Bookmaking

Q & A with
Dalton Wagner, Founder V.O. Group, S.A.

As Founder and Proprietor of V.O. Group, S.A., Dalton Wagner has the reputation of ferreting out new markets and attacking while other companies sit on their hindquarters and watch. One market that has always eluded offshore bookmakers is the ‘guy who likes dealing with his credit bookmaker’. Typically this market has been untouchable for the offshore bookmaker, but NOT for Dalton and V.O. Group, S.A. We had an opportunity to catch up with Dalton Wagner on his last visit to Panama.

Vegas Offshore: Tell us about this new product that you are calling your Price Per Head Agency Relationship.

Wagner: As you know, one market that a post-up sportsbook has historically been unable to attack is the huge market of bettors that have been betting with their local guy for years. The reason this market exists is the personal relationship that an onshore bookie has with his clients. Sportsbooks have attempted to educate the player and lure him away from the local bookie, however we have taken another tact. We have partnered with local bookies around the world to make their offering better and more valuable.

Vegas Offshore: So instead of attacking the local guy, you actually approach him with your offer?

Wagner: Exactly. The fact is that the relationship is the last thread holding clients to a local guy. Yes, there is the fact that the local guy extends credit, but with bonuses and promotions we can overcome that offering. The key is the relationship.

Vegas Offshore: I’m confused, please explain.

Wagner: As you know, the onshore bookmaking market has lost steam to the offshore providers like V.O. Group, S.A. The reason is that the offerings that I can provide are far superior to those that any onshore bookmaker can offer. The PPH concept is based on allowing the onshore bookmaker to provide his clients my offerings on a purely anonymous basis via an online automated tool. Rather, by signing up with my PPH service, an onshore bookmaker can compete with all offshore sportsbooks on equal footing offering 24-hour wagering, 365 days a year. All the bookmaker does is use an online system to assign PINs and PASSWORDs and pay me a small fee per week for the service.

Vegas Offshore: Wait a minute. So, this means that a small time operator in Cleveland, Ohio can offer all the same things that V.O. Group, S.A. does without making the multi-million dollar investment?

Wagner: Now your catching on. For as little as $25 per head per week, a guy in Ohio can compete with the biggest sportsbooks in the world. And, better yet, never answer another wagering phone call again.

Vegas Offshore: OK, I get it. They use your automated tool to drive their clients to your software online, and they simply pick up the figures at the end of the week.

Wagner: Exactly correct with one exception. Their clients can bet on the phone via our call center staffed with 250 people OR bet online. Again, they have access to all of the offerings V.O. Group, S.A. provides including call center access, client services and technical support.

Vegas Offshore: Is the PPH product only based on sports wagering?

Wagner: Very good question. The answer is no. We actually have the ability to offer clients two (2) casino products including blackjack, baccarat, slots, 3 card poker, roulette, etc. And, clients may also bet on horses at over 350 tracks worldwide.

Vegas Offshore: Wow! That truly is incredible. So, a bookie that has one client can compete with the likes of MVPsportsbook.com on equal footing?

Wagner: Exactly correct. But, there are two keys to the success of the product launch. The first is being able to compete head to head with the biggest sportsbooks in the world. But, arguably the most important factor is that since the onshore bookie no longer has to answer phones or track plays, he can spend all of his time picking up new clients, NOT writing bets.

Vegas Offshore: I remember that in a prior interview you had stated that the key is not winning, but finding more losers.

Wagner: An embarrassing quote, but true. Amateur bookmakers think that the key is to beat the clients you have. That is a huge mistake. The key to getting rich as a bookmaker is finding a new client every single day of the year. The numbers take care of themselves, the key is to get more and more people playing. That is the TRUE BEAUTY of the PPH software and agency relationship.

Vegas Offshore: Do you have any success stories to tell?

Wagner: I have many, but will tell you one that emphasizes what the PPH program can do. I have a bookmaker that has been working out of San Antonio, Texas for 10 years. The guy has historically made $200,000 a year from his 25 or 30 players. He was introduced to our product in July 2004, and today he is writing over 185 clients. The ability to outsource answering calls and writing tickets to us has allowed him to grow his business 600% and his revenue over 1000%.

Vegas Offshore: Why did his revenue grow more than his business? Is it because your lines are sharper than his?

Wagner: There you go again. That is the obvious mistake that everyone makes. Again, don’t worry about beating the clients. Worry about getting more clients to play. But, to answer your question directly, the ability for his clients to play in the casino, play 24 hours a day, play games/halves/quarters, props, etc. is where the extra revenue comes from. Our lines might be sharper, but the advantage is more clients betting more often.

Vegas Offshore: How many bookmakers do you currently work with?

Wagner: We are writing business for over 250 bookmakers now. Some represent groups as small as 2 or 3; and others represent groups as large as 500. Again, remember the key is finding as many clients as possible. If you get to where you are writing 15 clients or more, it is almost impossible to lose for an extended period of time.

Vegas Offshore: So how does a bookmaker learn more about this?

Wagner: Simply go to any of the V.O. Group, S.A. sites to learn more. I suggest starting at www.BettorsNet.com or calling 1-877-512-1001 or emailing agents@bettorsnet.com to get more information.

0 comments

Poker as a Testbed for Machine Intelligence Research Online Poker

Darse Billings, Denis Papp, Jonathan Schaeffer, Duane Szafron{darse, dpapp, jonathan, duane}@cs.ualberta.ca

Department of Computing ScienceUniversity of Alberta

Edmonton, Alberta

Canada

T6G 2H1

ABSTRACT

For years, games researchers have used chess, checkers and other boardgames as a testbed for machine intelligence research. The success of world-championship-caliber programs for these games has resulted in a number ofinteresting games being overlooked. Specifically, we show that poker canserve as a better testbed for machine intelligence research related to decisionmaking problems. Poker is a game of imperfect knowledge, where multiplecompeting agents must deal with risk management, agent modeling,unreliable information and deception, much like decision-makingapplications in the real world. The heuristic search and evaluation methodssuccessfully employed in chess are not helpful here. This paper outlines thedifficulty of playing strong poker, and describes our first steps towardsbuilding a world-class poker-playing program.Keywords: poker, imperfect information, opponent modeling, computer games1. IntroductionWhy study computer games? By writing programs that play games, some insightscan be gained about machine intelligence. These lessons can then be used to develop usefulnon-game programs. Researchers have spent a lot of time and effort on board games suchas chess and checkers. These games all share the property that high performance can beachieved by brute-force search. This emphasis on search was taken to the extreme by theDeep Blue chess machine, which analyzed 200 million positions per second in its May1997 match against World Chess Champion Garry Kasparov. This achievement only confirmed the effectiveness of brute-force search for some application domains. However,

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2this result has been anticipated for several decades. Can computer games give us any freshinsights into machine intelligence, beyond brute-force search?We believe that the answer to this question is yes. However, real progress can onlybe made if we study games in which search is not the major criteria for success. Instead,we need to mimic real-world applications that are perceived to require intelligent behaviour.Activities such as financial trading, business negotiations, and forecasting (from weather topolitics) meet this criteria. The first column of Table 1 shows some of the characteristics ofthese applications from the AI point of view. We are not claiming that these are the onlyactivities of interest, just that they are important considerations for a wide range ofinteresting problem domains. Unfortunately, games like chess and checkers do not havethese characteristics, or involve them only in limited ways. Can these activities be studiedin the context of computer games, and if so, what games?General Application Problem Problem Realization in Pokerimperfect knowledgeopponents’ hands are hiddenmultiple competing agentsmany competing playersrisk managementbetting strategies and their consequencesagent modelingidentifying patterns in opponent’s play and exploiting themdeceptionbluffing and varying style of playunreliable informationtaking into account your opponents’ deceptive playsTable 1. Characteristics of AI problems and how they are exhibited by poker.We are currently studying the game of poker and are attempting to build a high-performance poker program that is capable of beating the best human players. As shown inthe second column of Table 1, poker exhibits all of the activities we are interested instudying on at least some level.Certain aspects of poker have been extensively studied by mathematicians andeconomists but, surprisingly, very little work has been done by computing scientists.Online Poker. Thereare two main approaches to poker research. One approach is to use simplified artificialvariants [vNM44] or simplified real variants [Ank81, SS92] that are easier to analyze. Forexample, one could use only two players or constrain the betting rules. The other approachis to pick a real variant, but to combine mathematical analysis, simulation and ad-hoc expertexperience. Expert players with a penchant for mathematics are usually involved in thisapproach (for example, [SM94]).

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3Simplification is a common technique for solving difficult problems. However, wemust be careful that simplification does not remove the complex activities that we areinterested in studying. For example, Findler worked on and off for 20 years on a poker-playing program for 5-card draw poker [Fin77]. His approach was to model humancognitive processes and build a program that could learn. Unfortunately his simplifiedapproach nullified many of the potential benefits of his research [Bil95].Recently, Koller and Pfeffer have been investigating poker [KoP97] from atheoretical point of view. They implement the first practical algorithm for finding optimalrandomized strategies in two-player imperfect information competitive games. This is donein their Gala system, a tool for specifying and solving problems of imperfect information.Their system builds trees to find the game-theoretic optimal (but not maximal) strategy,however only vastly simplified versions of poker can be solved due to the size of treesbeing built. The authors state that “...we are nowhere close to being able to solve hugegames such as full-scale poker, and it is unlikely that we will ever be able to do so.”We have chosen to study the game of Texas Hold'em, the poker variation used inthe annual World Series of Poker Championships. It is considered to be the moststrategically difficult poker variant that is widely played, and requires all of the complexactivities listed in Table 1. For example, the best risk management strategy in the worldcannot compensate for a lack of deception, since human opponents are quick to exploitpredictable players, no matter how strong they might otherwise be. Our objective is to builda program which handles all aspects of poker well enough to play at world-championshipcaliber. If we are successful, then the insights we gain should have wide applicability toreal applications that require similar activities.This paper describes our first steps towards building a strong poker program, calledLokibot. Section 2 gives the rules of Texas Hold’em. Section 3 discusses the requirementsof a strong Hold’em program and provides evidence that all of the activities listed in Table1 are necessary to play strong poker. Section 4 describes the Lokibot program and Section5 gives some initial performance assessments. Section 6 discusses ongoing work on thisproject.The research contributions of this paper include:• showing that poker can be a testbed of real-world decision making,• identifying the major requirements of high-performance poker,• presenting new enumeration techniques for hand-strength and potential, and• demonstrating a working program that successfully plays “real” poker.

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42. Texas Hold’emA hand of Texas Hold’em begins with the pre-flop, where each player is dealt twohole cards, face down, followed by the first round of betting. Then three community cardsare dealt face up on the table, called the flop, and the second round of betting occurs. Onthe 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 fourth (final) round ofbetting occurs. All players still in the game turn over their two hidden cards for theshowdown. The best five card poker hand formed from the two hole cards and the fivecommunity cards wins the pot. If a tie occurs, the pot is split. Typically Texas Hold’em isplayed with 8 to 10 players.Limit Texas Hold’em uses a structured betting system, where the order and amountof betting is strictly controlled on each betting round1. There are two denominations ofbets, called the small bet and the big bet. For simplicity, we will use a value of $10 for thesmall bet and $20 for the big bet. In the first and second betting rounds (pre-flop and flop),all bets and raises are $10, while in rounds three and four (turn and river), they are $20. Ingeneral, when it is a player’s turn to bet, one of five options is available: fold (withdrawfrom the hand, leaving all previously wagered money in the pot), call (match the currentoutstanding bet; if there is no current bet, one is said to check), or raise one bet (put thecurrent bet plus one into the pot; if there is no current bet, one is said to bet). There isusually a maximum of three raises allowed per betting round. The betting option rotatesclockwise until each player that has not folded has put the same amount of money into thepot for the current round, or until there is only one player remaining. In the latter case, thisplayer is the winner and is awarded the pot without having to reveal their cards.There is a strategic advantage to being the last bettor in any given round, so tomaintain fairness, the order of betting is rotated clockwise after each hand.3. Requirements for a World-Class Poker PlayerWe have identified several key components (modules) that incorporate some of therequired activities of a strong poker player and address most of the six characteristics listedin Table 1. However, these components are not independent. They must be continuallyrefined as new activities are supported.• Hand strength: assesses how strong your hand is in relation to what other playersmay hold. Hand strength is computed on the flop, turn and river. At a minimum, hand1In No-limit Texas Hold’em, there are no restrictions on the size of bets.

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5strength is a function of your cards and the community cards that have been dealt. Abetter hand strength computation takes into account the number of players still in thegame, position at the table, and history of betting in the hand. An even better modelconsiders different probabilities for each hidden hand, based on the relative chance ofeach hand being played to the current point in the game. This model may be improvedby varying the hidden hand probabilities for each player depending on the opponent’smodel of play for that player.• Hand potential: assesses the probability of the hand improving (or being overtaken)as additional community cards appear. For example, having four cards in the same suitdoes not count toward hand strength, but has good potential to become a winning flushas more community cards are dealt. At a minimum, hand potential is a function of yourcards and the community cards that have already been dealt. However, a better modelcan be evolved as capabilities are added to the program, similar to the hand strengthcomputation described above.• Betting strategy: determines whether to fold, call/check, or bet/raise in any givensituation. A minimum model is based on hand strength. Refinements consider handpotential, pot odds, bluffing, opponent modeling and unpredictability. Pot odds is animportant concept that differentiates poker from many other games and contributes to itsusefulness as a testbed for concepts in the real world. Pot odds is the comparison ofyour winning chances to the expected return from the pot. For example, if there is onlya 20% chance that we have the best hand on the river, should we fold, call or bet? Thecorrect answer is that we have not given enough information to answer the question.Assume the pot contains $100 after the only opponent bets $20. If you call in thissituation, you will lose 4 times out of 5, for an additional cost of $80. However, youwill win 1 time out of 5 for a profit of $100. Therefore, under these assumptions, youshould call, resulting in an average profit of $4 per hand. However, if the pot onlycontained $60, you should fold, since calling would yield an average loss of $4 perhand. Notice that an accurate computation of your winning chances is necessary. Sucha computation requires a sophisticated assessment of hand strength, as describedabove. Even with an accurate hand strength computation, the game theoretic optimalfolding/calling strategy may not be the best decision in practice, where bluffing,opponent modeling and unpredictability may be used to improve your betting strategy.• Bluffing2: allows you to make a profit from weak hands. Even if you only break evenon the bluffing plays, the false impression created about your play may improve the2Other forms of bluffing (semi-bluffing and betting a strong hand weakly) are not considered here.

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6profitability of subsequent hands. Thus, bluffing is critical to successful play. Gametheory can be used to compute a theoretical optimal bluffing frequency in certainsituations. The minimal bluffing system merely bluffs this percentage of hands. Inpractice, you need to be able to predict the probability that your opponent will call inorder to identify profitable bluffing opportunities. The better your opponent models are,the better your bluffing strategy will be.• Opponent modeling: allows you to determine a likely probability distribution foryour opponent’s hidden cards or betting strategy. A minimal opponent model might usea single model for all opponents in a given hand. Before the flop, a weighting systemmay be used to estimate the probability of possible holdings for all players who do notfold. After the flop, a second set of probabilities may be used for all opponents who donot fold, based on the three community cards that have been dealt. Opponent modelingmay be improved by modifying the probabilities based on a classification of eachopponent (e.g. weak/strong, passive/aggressive), betting history, and collectedstatistics. Opponent modeling has been attempted in two-player games but with limitedsuccess [CM95]. In poker, however, it is essential to success.• Unpredictability: makes it difficult for opponents to form an accurate model of yourstrategy. By varying playing strategy over time (e.g. pre-flop hand selection, variablebluffing rate), opponents may be induced to make mistakes based on an incorrectmodel.In addition, there is a number of less immediate concerns which may not be necessary toplay reasonably strong poker, but may be required for world-class play.This paper focuses on the issues of hand strength, hand potential and bettingstrategy. Other issues are the subject of on-going research.4. LokibotLokibot handles its play differently at the pre-flop, flop, turn and river. The play iscontrolled by two components: an evaluation of the hand and a betting strategy. Thestrategy is influenced both by the pot odds and our model of the opponent.4.1. Pre-flop EvaluationThe hand strength for pre-flop play has been extensively studied in the pokerliterature (for example, [SM94]). These works attempt to explain the play in humanunderstandable terms, by classifying all the initial two-card pre-flop combinations into ninebetting categories. For each hand category, a suggested betting strategy is given, based onthe strength of the hand, the number of players in the game, the position at the table, and

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7the type of opponents. For a poker program, these ideas could be implemented as an expertsystem, but a more general approach would be preferable.For the initial two cards, there are {52 choose 2} = 1326 possible combinations,but only 169 distinct hand types. For each one of the 169 possible hand types, a simulationof 1,000,000 poker games was done against nine random opponents. This produced astatistical measure of the approximate income rate for each starting hand. A pair of aces hadthe highest income rate; a 2 and 7 (of different suits) had the lowest income rate for a 10-player simulation. There is a strong correlation between our simulation results and the pre-flop card ordering given in [SM94] (although there are a few interesting differences).4.2. Hand EvaluationCritical to the program’s performance on the flop, turn and river is an assessment ofthe current strength of the program’s hand. Enumeration techniques can provide an accurateestimate of the probability of currently holding the strongest hand.For example, suppose our starting hand is A♦-Q♣and the flop is 3♥-4♣-J♥.Thereare 47 remaining unknown cards and there are {47 choose 2} = 1,081 possible hands anopponent might hold. To estimate hand strength, the enumeration technique gives apercentile ranking of our hand. We simply count the number of possible hands that arebetter than ours (any pair, two pair, A-K, or three of a kind: 444 hands), how many handsare equal to ours (9 possible remaining A-Q combinations), and how many hands areworse than ours (628). Counting ties as half, this corresponds to a percentile ranking, orhand strength (HS), of 0.585. In other words there is a 58.5% chance that our hand isbetter than a random hand. This measure is with respect to one opponent but can beextrapolated to multiple opponents by raising it to the power of the number of activeopponents. Against five opponents with random hands, the adjusted hand strength (HS5) is.5855= .069. Hence, the presence of additional opponents has reduced the likelihood ofour having the best hand to only 6.9%.In practice, hand strength alone is insufficient to assess the quality of a hand.Consider the hand 8♦-7♦with a flop of9♦-6♣ -2♦. The probability of having the strongesthand is very low, even against one random opponent.On the other hand, there is tremendouspotential for improvement. With two cards yet to come, any ♦, 10, or 5 will give us astraight or a flush. Hence there is a high probability that this hand will improvesubstantially in strength, so the hand has a lot of value. We need to be aware of thepotential changes of hand strength.In addition to this positive potential (Ppot) of pulling ahead when we are behind,enumeration can also compute the negative potential (Npot) of falling behind if we are

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8ahead. For each of the possible 1,081 opposing hands, we consider the {45 choose 2} =990 combinations of the next two cards. For each subcase we count how manycombinations of upcoming cards result in us being ahead, behind or tied.The potential for A♦-Q♣ / 3♥-4♣-J♥is shown in Table 2. The table shows, forcases where we were ahead, tied or behind after five cards, what the result would be afterseven cards. For example, if we did not have the best hand after five cards, then there are91,981 combinations of cards (pre-flop and two cards to come) for the opponents that willgive us the best hand. Of the remaining hands, 1,036 will leave us tied with the best hand,and 346,543 will leave us behind. In other words, if we are behind we have roughly a 21%chance of winning against one opponent.5 Cards7 CardsAheadTiedBehindSumAhead449,0053,211169,504621,720 = 628x990Tied08,3705408,910 = 9x990Behind91,9811,036346,543439,560 = 444x990Sum540,98612,617516,587 1,070,190 = 1,081x990Table 2. A♦-Q♣ / 3♥-4♣-J♥ potential.We use these values to generate Ppot and Npot. If T{row,col}refers to the values inthe table (for brevity we use B, T, A, and S for Behind, Tied, Ahead, and Sum) then Ppotand Npot are calculated by:Ppot = ( T{B,A}+ T{B,T}/2 + T{T,A}/2 ) / ( T{B,S}+ T{T,S}/2 )Npot = ( T{A,B}+ T{A,T}/2 + T{T,B}/2 ) / ( T{A,S}+ T{T,S}/2 )In the example Ppot is .208 and Npot is .274. The calculation for one cardlookahead is exactly the same as the above calculation, except there are only 45 possibleupcoming cards instead of 990 (or 44 if we are on the turn). With only one card to come onthe turn, Ppot is .108 and Npot is .145.By enumerating all possible card combinations, the program uses a brute-forceapproach to calculating hand strength and potential. The calculations are easily done in real-time and provide accurate probabilities that take into account every possible scenario.Hence the calculation gives smooth and robust results.4.3 Weighting the EnumerationSo far our calculations assume that all opponent hands are equally likely. In reality,this is not the case. Many weak hands like 4♥-J♣would have been folded before the flop.

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9However, with the example flop of 3♥-4♣-J♥, these hidden cards make a strong hand thatskews the hand evaluations.Accuracy of the estimates also depend strongly on models of our opponents.Ultimately, we want a different set of weights for each possible starting hand for eachopponent. These weights could then be adjusted depending on the opponent’s playingstyle. For example, raising on the flop probably indicates a strong hand that should bereflected in the weightings. We would then apply the appropriate weight for each of the1,081 possible subcases when calculating hand strength and potential.Although Lokibot treats all opponents the same, it was designed to supportgeneralized opponent modeling. Currently the common weights are based on thesimulations of the 169 different starting hand types, providing a reasonable startingtemplate for unknown players.4.4 Betting StrategyHand strength and potential are combined into effective hand strength (EHS):EHS = HSn+ (1 - HSn) x Ppotwhere HSnis the adjusted hand strength for n opponents and Ppot is the positive potential.This formula means that EHS is the probability that we are ahead, and in those cases wherewe are behind there is a Ppot chance that we will pull ahead. Currently, EHS is comparedto some thresholds to determine when to bet. For example, with an EHS greater than 0.5we can say there is a reasonable chance we are ahead of our opponents and will bet if noother opponent has bet. This is an optimistic estimate because we only consider positivepotential. Npot is not considered for two reasons. First, we do not know if our opponentwill play. Second, in many situations where we calculate a high Npot, it is often a betterstrategy to bet/raise to scare the opponent out of the hand.Determining if the pot is large enough and whether we have enough equity towarrant calling a bet is different than deciding when to bet. This decision is made bycomparing Ppot against the pot odds, wherepot_odds = bets_to_us / ( bets_in_pot + bets_to_us ) .We call when Ppot ≥ pot_odds. Note that even on the flop we use only one card look aheadfor Ppot. If we examine the situation two cards in the future we must consider whether wewill face another bet (or more) after the first card. Pot_odds is based on the immediatesituation. In the original example, if there are five opponents and we are first to act, EHS is0.15 so we check. If the first opponent behind us bets $10, two others call, and the fourthraises $10, then it is $20 to us and the pot is $175. Therefore pot_odds is 20 / (175+20) =0.103, so we call (Ppot is 0.108).

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105. ExperimentsA variety of different experimental methods have been used to measure thedevelopment of Lokibot. These include self-play simulations, play against humanopposition, and play against other computer programs. Each type of evaluation haslimitations, reinforcing the need for a wide range of experiments and testing methods.Self-play simulations offer a convenient method for the comparison of two or moreversions of the program. In addition to verifying that a certain enhancement has a beneficialeffect, it is possible to quantify the contribution made by each new component to thesystem. Since all participants in the simulated game are versions of the program, play canproceed at a rapid pace, and results can be based on large, statistically significant, samplesizes. Moreover, these closed experiments can be used as a vehicle for exploring theinterdependencies of program features. A combination of competing factors can producedifferent results than might be expected from looking at each variable in isolation.Exploring these results can help identify weaknesses in the current system and suggestareas to focus on, providing some direction for future work.The self-play simulations use a duplicate tournament system, based on the sameprinciple as duplicate bridge. Since each hand can be played with no memory of precedinghands, it is possible to replay the same deal, but with the participants holding a different setof hole cards. Our tournament system simulates a ten-player game, where each deal isreplayed ten times, shuffling the seating arrangement each time so that every participant hasthe 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 badhands. The differences in the performance of players will therefore be based more stronglyon the quality of the decisions made in each situation. This large reduction in naturalvariance means that meaningful results can be obtained with a much smaller number oftrials than a typical game setting.One simple application of a self-play simulation would be to play five copies of anew version against five copies of an older version, differing only in the addition of onenew feature. If the new component has improved the program, then the newer version willwin against the older version. The average margin of victory, in terms of expected numberof bets per hand, can also give a preliminary indication of the relative value of the newenhancement.However, there are limitations to how much can be concluded from a singleexperiment, since it is representative of only one particular type of game and style ofopponent. It is quite possible that the same feature will perform much worse (or muchbetter) in a game against human opposition, for example. A wider variety of testing is

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11necessary to get an accurate assessment of the new feature. One approach is to change thecontext of the simulated game. The next self-play experiment might include a number ofplayers who employ a different style of play, such as a more liberal selection of startinghands. If the new feature is successful over a wide variety of game types, we will have amore reliable indication of the value of that concept, with a metric to quantify itscontribution.Figure 1. Experiments with different versions of Lokibot.As an example of a self-play experiment, Figure 1 shows the results of atournament with five different versions of Lokibot. The average bankroll size (profit) isplotted against the number of hands played. Player A is the most advanced version of theprogram, including three major components that the most basic player does not have.Player E is a basic player, having no advanced features. The other three versions are thesame as Player A, but with one of the major components removed. Player B lacks anappropriate weighting of subcases, using a uniform distribution for all possible opponenthands. Player C uses a simplistic pre-flop hand selection method, rather than the advancedsystem which accounts for player position and number of opponents. Player D lacks the

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12computation of hand potential, which is used in modifying the effective hand strength andcalling with proper pot odds.As expected, the complete system performs the best, while the basic system losesthe most. The best program earned approximately +0.08 small bets per hand, while theworst lost at a rate of -0.11 small bets per hand.Within the context of this particular experiment, the use of hand potential had thegreatest impact on the strength of the program, since Player D, which lacked thatcomponent, performed poorly. Player B, missing the appropriate weighting of subcases,was still able to win against this field of opponents, but did not perform nearly as well asthe version having this feature. Player C, differing only in the use of the advanced pre-flophand selection method, did not lose much compared to the other weakened versions.It is important not to over-interpret the results of a single experiment. In thisparticular tournament, all of the participants are computer players with fairly conservativestyles. It is quite possible that the consequences of each change would be different against afield of opponents who employ different playing styles. For example, against severalhuman players, the weighting function may have a much bigger impact than the use of handpotential.Lokibot must also be tested in more realistic games against human opposition. Forthis purpose, the program participates in an on-line poker game, running on IRC (InternetRelay Chat). Human players connect to the server and participate in games. No real moneyis at stake, but statistics on each player are maintained. Certain games are reserved forplayers who have earned enough virtual dollars to qualify, and those games are usuallytaken more seriously than the games open to all players. This provides an environment withseveral games, differing in styles of play and skill level.Early versions of Lokibot had mixed results on the IRC server, but played at aboutthe same level as the average human participant in the open games, roughly breaking evenover the course of about 12,000 hands. When it qualified for the stronger game it lostslowly, averaging about -0.05 small bets per hand dealt, based on roughly 2,000 hands.This is not a large enough sample size for conclusive results, but strongly suggests it was alosing player overall in these games.The most recent versions of Lokibot have performed much better in the opengames, averaging about +0.20 small bets per hand over 3,500 hands dealt, which iscomparable to a solid human player (probably ranking in the top 10% of IRC players).A third form of competition was introduced against other computer programs on theIRC server. Four programs participated, using three copies of each in a 12-player game.Two programs, R00lbot and Lokibot, were clearly dominant over the other two, Xbot and

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13Replicat, with the more established R00lbot winning overall. Over 10,000 hands, Lokibotaveraged about +0.03 small bets per hand. It should be noted, however, that thiscompetition is representative of only one type of game, where all the players are quiteconservative. Replicat in particular performed much better in the open games against humanopposition than in this closed experiment.A final important method of evaluation is the critique of expert human players.Experts can review the play of the computer and determine if certain decisions are“reasonable” under the circumstances, or are indicative of a serious weakness ormisconception. Based on this opinion, it appears to be feasible to write a program that isstronger than the average human player in a casino game, although Lokibot has not yetachieved that level. Whether it will be possible to design a system capable of world-champion-caliber play remains an open question.6. Work in ProgressLokibot still suffers from some obvious problems. Most importantly, it is apredictable player that reacts the same in a given situation irrespective of any historicalinformation. This leaves it open to exploitation by an opponent who has deduced it’ssimplistic playing style. The two major areas requiring improvement are opponentmodeling and betting strategy. Both of these topics are open-ended, and will provideinteresting challenges for future work.The most important foreseeable advance is opponent modeling. When Lokibot isbetter able to infer likely holdings for the opponent, it will be capable of much betterdecisions. The hand strength and potential calculations will use a different table of weightsfor each particular opponent. Now the specific actions of that opponent can be taken intoconsideration, as well as historical and statistical information gathered on this opponentfrom previous games. A wide variety of properties can be measured and applied, such asbetting frequencies, known bluffs, recent trends, etc. Additionally for each opponent wewill also compute statistics to measure the betting strategy and thresholds for the variousbetting actions.Betting strategy is similarly a very broad concept. The current system is simplisticand predictable (it will always act the same in a given situation). A significantly betterbetting system would bluff with high potential hands and occasionally bet a strong handweakly. It would also predict opponent responses in order to choose the best practicalaction. Unpredictability and other advanced betting strategies can be incorporated.The infrastructure is in place to incorporate these features. Lokibot is changing on adaily basis. It is only six months old and already at a level that exceeds our initial

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14expectations. We understand many of the weaknesses in the program, but do not yet knowif all of them can be addressed sufficiently to produce a world-class poker player.ReferencesAnk81N. Ankeny, Poker Strategy: Winning with Game Theory, Basic Books,Inc., 1981.Bil95 D. Billings, Computer Poker, M.Sc. thesis, Dept. of Computing Science,University of Alberta, 1995.CM95 D. Carmel and S. Markovitch, Incorporating Opponent Models into AdversarySearch, AAAI, 1995, 120-125.Fin77 N. Findler, Studies in Machine Cognition Using the Game of Poker,Communications of the ACM 20, 4 (1977), 230-245.KoP97D. Koller and A. Pfeffer, Representations and Solutions for Game-Theoretic Problems, Artificial Intelligence 94, 1-2(1997), 167-215.SM94 D. Sklansky and M. Malmuth, Hold’em Poker for Advanced Players, Two PlusTwo Publishing, 1994.SS92 M. Sakaguchi and S. Sakai, Solutions of Some Three-person Stud and DrawPoker, Mathematics Japonica 37, 6(1992), 1147-1160.vNM44 J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior,Princeton University Press, 1944.

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