Ivars Peterson's MathTrek

March 5, 2001

# Quirky Video Poker

The lure of easy money brings gullible bettors back again and again to the game of video poker--an immensely popular pastime in casinos and other gambling venues throughout the United States.

Most players are bound to lose money, says Todd D. Mateer, a recent graduate of Clemson University, who has studied video poker machines in South Carolina. Moreover, imposing limits on how much a gambler may win per machine increases potential losses, even when the gambler plays a long time and makes the best possible choices in each game.

In video poker, a player receives five cards, displayed on a video monitor. The player has the option of discarding and replacing any or all of these cards to try to achieve a winning poker hand. A built-in payoff table sets the amount a player would get paid for each possible winning combination.

The payoff table can vary from machine to machine, depending on the odds set by the manufacturer. For example, a given machine may have an expected payoff level of 99.54 percent, meaning that for every \$100 put into the machine, a player would get \$99.54 back--if he or she were to make all the optimal choices. Without such perfect play, the expected return is much lower, and a bettor typically loses money very quickly.

Interestingly, a player with a perfect strategy can, over the long term, actually make a modest profit on certain video poker games in Las Vegas casinos and elsewhere, which typically have no limits on play and feature expected returns of more than 100 percent. Jackpots occur, on average, about once every 40,000 games. So, in the short term, a player must be prepared to go deep into debt before earning back any initial losses. However, even using a flawless strategy and playing continuously for 40 hours per week, a player would earn no more than minimum wage. It's possible to boost the return, however, by playing on a machine that offers several hands simultaneously.

This isn't news to professional gamblers and skilled amateurs. Indeed, a variety of books and online reference sources offer statistical analyses of various forms of video poker and optimal strategies for playing each variant. Expert players also check the payoff tables displayed on each machine and shun games that offer unfavorable odds.

Video poker with cash betting debuted in South Carolina in 1986. By 1999, there were more than 33,000 video poker machines scattered throughout the state--at convenience stores, gas stations, restaurants, bars, bowling alleys, and many other sites. It was a lucrative business generating billions of dollars in revenue.

South Carolina video poker machines, which are now banned from the state, generally had unfavorable payoff rates, often less than 98 percent. Furthermore, state law mandated that the maximum cash payment per machine in a 24-hour period would be \$125. Mateer's 1999 mathematical investigation of video poker, which was the subject of his master's thesis, focused on the effect of this payoff limit.

To develop a strategy for perfect play, one has to consider every hand that can be dealt--precisely 2,598,960 of them in all when the order in which the cards are dealt is ignored--then picking the best hold-discard possibility for each hand.

"This best possibility is determined by calculating the expected return for each of the 32 actions one can take with a given hand and selecting the action which offers the highest expected return for that hand," Mateer and Clemson mathematician Joel V. Brawley state in a paper summarizing Mateer's results. "One can then sum the 2,598,960 best-action expected returns and arrive at the expected return for the machine under perfect play."

Using a combination of probability theory and computer simulation, Mateer confirmed that the "Jacks-or-Better" machine found at many casinos has an expected payoff of 99.54 percent when played perfectly. Machines in South Carolina generally had lower expected payoffs. Mateer went on to demonstrate that the state's payoff limit made playing video poker a losing venture for any gambler, no matter how expert he or she might be or how favorable the odds were.

"The mathematical effect of the payoff limit [was] that it. . .reduced the expected return on nearly all South Carolina machines and. . .transformed the perfect-play winning games on the fairest machines into losing games," Mateer concluded in his thesis. In fact, once the payoff restrictions were imposed in South Carolina, knowledgeable video poker players quickly stopped playing the games and went elsewhere.

The question of whether video poker would remain legal in South Carolina became the subject of a state referendum, scheduled for Nov. 2, 1999. The vote was cancelled just weeks before it was to take place when the state Supreme Court ruled the referendum unconstitutional. At the same time, the court upheld an earlier vote by the state legislature to ban video poker. The ban went into effect on July 1, 2000.

That was the end of video poker for cash payment in South Carolina. The machines didn't just disappear, however. They were shipped to other states where video poker remains legal.

References:

Brawley, J.V., and T.D. Mateer. 1999. Video poker in South Carolina: A mathematical study. Available at http://www.strom.clemson.edu/publications/gambling/pokermath.pdf.

Mateer, T.D. 2001. Video poker in South Carolina. Abstracts of Papers Presented to the American Mathematical Society 22(No. 1):265.

Peterson, I. 2001. Quirks of video poker. Science News 159(Jan. 27):56.

Ivars Peterson is the mathematics/computer writer and online editor at Science News (http://www.sciencenews.org). He is the author of The Mathematical Tourist, Islands of Truth, Newton's Clock, Fatal Defect, and The Jungles of Randomness. He also writes for the children's magazine Muse (http://www.musemag.com) and is working on a book about math and art.

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Math Trek 2: A Mathematical Space Odyssey by Ivars Peterson and Nancy Henderson. For children ages 10 and up. New York: Wiley, 2001. ISBN 0-471-31571-0. \$12.95 USA (paper).