Lexington, Virginia • June 16, 2009
For Adam Schwartz, the problem was simply irresistible.
An associate professor of business administration at Washington and Lee University, Schwartz has played poker on occasion but does not consider himself a great poker player and certainly doesn't endorse gambling, especially for the students he teaches.
But when a former student asked Schwartz to design a new payout system for the World Series of Poker, the challenge of applying his expertise to such a real world problem was too much.
It started in 2005, when Jack Effel, tournament director of the World Series of Poker and one of Schwartz's students at the University of Mississippi called. Effel now manages more than 1,000 employees, and his tournament schedule from May through July grosses more than $200 million and has many different events. The goal for the players in any event is not only to win one of the coveted bracelets, but also a share of the prize money. The problem was designing an equitable payout structure for a single event which could have over 6,000 players.
Schwartz said that many of his former students call him when they are stuck with financial computing problems because they remember his class. "I've done implied binomial option models, executive option valuation models, real estate options, corporate performance measures, but this particular problem was from left field. In finance we would say it's an application of a constrained capital budgeting problem," Schwartz said, "I liked the problem. The World Series has access to many great mathematicians, and it was a pleasure to work with them. In fact, a lot of people become interested in math by learning about games of chance. I found myself talking payouts with many poker celebrities including Barry Greenstein and Andy Bloch, a member of the original MIT blackjack team."
Schwartz designed an optimization using Newton's method, a solution well known in numerical mathematics. "In 2005, the first year that I consulted, we did the payouts on the fly. The tournament does not know how many people will play until the third day. So, I plugged in the numbers, got a quick solution and e-mailed it to them. The people at the tournament looked at the payout structure, moved some dollars around among the winners, and then announced on the floor of the Rio (Rio All-Suite Hotel & Casino in Las Vegas) what everybody would be getting paid about 20 minutes after the number of players was determined," said Schwartz.
The tournament amended Schwartz's framework slightly in 2006 and 2007. Then, in 2009, they decided to go back to Schwartz and ask for further changes. The top winners were winning too much prize money and organizers wanted a "stimulus special." In the new payouts, lower prizes will be slightly higher, with a smoother transition between the players.
So Schwartz adapted his model incorporating suggestions from some of the world's top players. In an article on cardplayer.com recently, Effel said that although he may tweak the new model in the future, he believes he now has a sound payout system thanks to Schwartz's model.
"If 6,000 people play, that's almost $60 million in prize money," said Schwarz. "How do you divide that among the winners? They try to pay about 10 percent of the players, which would be 600 people. In the old days, the top person would get 50 percent, second place 30 percent and third place 20 percent, or something like that. Now the winner gets about 13 percent, second place around seven percent, and you don't have to go too far before it becomes a pretty neat problem trying to get all the numbers to add up to 100 percent. We want the percentages to transition smoothly from the top players down to the bottom players. The approach is about the same as finance guys use every morning to make a smooth yield curve from a limited number of traded bonds."
Schwartz said that making the payout appealing and logical to the players is important because the poker tournament "is a business like any other for the people who run it. What they are selling is the chance to play against the world's best players and they want the structure to be attractive. I can make countless different solutions which divide the pot among 600 people, but it needs to look right to the players and meet their expectations."
Schwartz used a mathematical formula to smooth out what he calls the second derivative, or the rate of change between the payouts. He said it took him an hour or two to write the program and then he consulted with Effel and Greenstein via e-mail to get it just right.
"My area is financial modeling and derivative pricing," said Schwartz. "This was a real world application of a constrained budgeting problem with over 20 constraints. I think it is interesting that the payouts in poker tournaments are not that different from compensation schemes in corporate America. The drop from the amount of money the CEO makes to the amount the second person makes, on down to the least compensated employee, is about the same as the drops we see in poker tournaments."
Schwartz said he was aware of the research possibilities when he took the problem on. "I knew there would be the possibility for me to get some really good data for a research paper. I have been thinking about how compensation works in corporate America. I am working on a survey to pin down the preference of the players. The World Series of Poker has more than 50,000 people register for the different events each year. That is some serious untapped survey data. What do poker player preferences tell us about how rewards in the private sector might best be structured? Maybe paying the field in the World Series (over $12 million to one player in 2006) will help teach us something about the thorny issue of incentives needed in the private sectors."
As he has become involved in the World Series project, Schwartz has been keenly aware of the fact that college students are more and more involved with sports betting and online poker. His experience has led him to offer his own advice on gambling.
"I can't tell students, or others, not to play, and hope to have them take my advice to heart," he said. "So if they do play, I want to be sure that they assume their online opponents know all the odds and will have more information than they do many times."
Schwartz said that there is a formula for how much to bet, which may keep people from ruin. The Kelly Criterion works for stock investors the same as it works for sports bettors, he said.
"There is a good book by William Poundstone called Fortunes Formula that does a nice job of explaining the Kelly system," said Schwartz. "In summary, that system holds that if you don't have an edge (as is common with the lottery, sports bets, and online poker), you should bet nothing or consider any bet a form of costly entertainment (as opposed to an investment). If you do perceive an edge (as sometimes with undervalued stocks), be careful to consider not only the edge, but also the risk. The optimal wager is usually a small bet as opposed to a large one."