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If the casino has an edge, why is video poker so popular?

17 January 2011

Video poker games commonly give the house edges between roughly 2.5 and 5 percent – or, stated optimistically – offer players about 97.5 to 95 percent return. Euphemisms aside, the figures show that, on the average, funds flow freely from bettors' bankrolls to casinos' coffers.

Some people won't gamble in casinos at all – and wouldn't, even if the joints had no advantage. Others indulge, but enjoy certain games and avoid the rest. Reasons in differing instances vary widely – from considerations of anticipated gains or losses to the excitement of the contests, the need for or irrelevance of skills, or the "extras" that are part and parcel of the casino experience.

Much of the popularity of reel- or matrix-type slots as well as video poker machines derives from the large payoffs possible with only small bets at stake. Thousands of dollars for a buck a shot. "Utility theory"suggests that solid citizens find these games appealing because the bets seem like throw-away money while the jackpots are the otherwise impossible dream. "Cumulative prospect theory," a newer wrinkle in attempts to understand how decisions are made in the face of risk, carries explanations about avoidance of or participation in various games to a deeper level.

Picture single-hand jacks-or-better video poker machines on which full houses return 8-for-1 and flushes 5-for-1 (an "8-5" game), with all the other wins essentially standard. The laws of probability give the house 2.7 percent edge (97.3 percent player return). About 54.5 percent of all rounds are expected to lose, 21.5 percent to push and refund players' bets, and so forth – up to 0.00249 percent (one out of approximately 40,000) to pay 800-for-1 jackpot on a royal.

According to prospect theory, wagers may appeal to individuals despite their negative statistical "expected value." The extent to which they do or don't depends on three innate attributes. 1) Alpha: The marginal impact on people of increasing amounts of profit or loss; for instance a $500 profit may seem twice, more than twice, or less than twice as important as a $250 gain – and ditto for the perceived penalty of a $500 compared to a $250 loss. 2) Lambda: The relative sensitivity of players to losses in contrast to gains of the same magnitudes. 3) Delta: The degree to which individuals unconsciously raise their senses of low, moderate, and high probabilities.

Prospect theory gurus have used experimental data to assign values to the three parameters. Alpha is in the range from 0 to 1 with a median at 0.88; lower alpha implies that increasing amounts have decreasing marginal impact and conversely for higher alpha. Lambda runs from 1 to 4 – occasionally higher – with a median at 2.25; lower lambda connotes decreasing sensitivity to losses and vice-versa. Delta is from 0 to 1 with a median of 0.65; lower delta signifies greater subliminal elevation of low probabilities relative to their actual values and inversely.

To illustrate the effect, prospect theory parameters can be applied to the returns and probabilities in each round of a video poker game. These will be used to compute the "prospect value"of the coup to potential bettors. While nobody does this explicitly, it helps rationalize the thought processes involved in deciding intuitively whether or not to go for it.

Expected value for an 8-5 game is always a $0.027 loss per $1 bet. Using the median parameters cited previously, the same gamble has a prospect value somewhat over $0.25 profit for a $1 bet. Most players would perceive this to be an attractive proposition – a situation that goes a long way toward explaining why video poker has such a large and dedicated following.

What if a person's alpha goes up with the other two parameters kept constant? Marginal utility of bigger gains still declines, but less steeply than the median and increasing wins have stronger impact. At alpha = 0.95, the prospect value of the gamble rises to a $0.61 profit per $1 bet. Conversely when alpha goes down. At alpha = 0.75, the prospect value is a $0.15 loss per $1 bet.

Similarly for sensitivity to loss. To someone having lambda = 1 the prospect value of the gamble would seem like at a gain of $0.84 per $1 bet. Were sensitivity to loss greater than the median, the opposite would hold. For instance, at lambda = 4, prospect value is a loss of $0.56 per $1 bet.

Perceived probabilities also have a major influence. If delta drops to 0.5, low probabilities are greatly overestimated and the prospect value of the gamble escalates to $2.28 for a $1 bet. Going the other way, if the player's visceral notion of the chance of the jackpot was just about the actual probability, with delta = 0.85, the prospect value would be a loss of about $0.55 per $1 bet.

You can't clamp a meter on people's heads to measure alpha, lambda, and delta. So prospect theory can't predict whether they'll find a particular bet alluring. Rather, it explains why games with negative expectation seem attractive to some folks, and how different personality traits affect their interest in one or another gamble. As the beloved bard, Sumner A Ingmark, wrote:
Beauty's not the only thing in the eye of the beholder,
Judgement calls abound in which some are timid, others bolder.

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.