1. Slot Operations The Myth And The Mathematics
2. Slot Operations The Myth And The Math Book
3. Slot Operations The Myth And The Math Worksheets
4. Slot Operations The Myth And The Math Game

It's a well-worn mantra that when it comes to casinos, the house always wins, at least in the long term. That's because casinos maintain a small house 'edge'—not enough to scare gamblers away, but sufficient to ensure that the house ultimately comes out ahead. Some gamblers think they can get around this by jumping from slot machine to slot machine, for instance, in hopes of hitting one at just the right time to win a big payout.

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There's a corresponding long-held belief among casino operators that experienced players can actually sense shifts in how much and how often a particular machine pays out—that is, they can detect subtle differences in the house edge between machines. But the math says otherwise, according to a recent paper in the International Journal of Contemporary Hospitality Management.

Slot machines are the source of most of a casino's revenue. It's all about manipulating the payback percentage: the percentage of the 'coin in' that a player gets back when they're done with the game. 'When slot machines are made, the manufacturer will license multiple pay tables (usually around five),' said co-author Anthony Lucas, a professor at the University of Nevada, Las Vegas. 'After a casino operator decides to buy the game, they must then decide which of the five licensed pars (aka pay tables) to install. This is where our work becomes helpful, as most operators do not know which par will optimize revenues.'

Lucas worked in Lake Tahoe as an operations and financial analyst in gaming right out of college for some 34 years, so he has a longstanding interest in the types of challenges facing the industry. He has been conducting research at UNLV for the last 20 years. 'Most of the research questions I've pursued came out of the problems we faced in the industry,' he said. 'We never had time to solve them, because the decision horizon is so short in business, and you don't have the chance to do that longer-term academic thinking.'

The price is right

Slot Operations The Myth And The Mathematics

The house edge for slot machines typically falls between 5% and 10%, with most machines delivering a payback percentage in the 90% to 97% range. (If it's 90%, the casino's take—and the player's loss—is 10% of the coin in, for example.) But how can casino operators determine the best house advantage for their bottom line within that range? 'It's really a pricing issue, because it's a unique product,' said Lucas. 'With real slots, the price isn't marked anywhere.'

Conventional wisdom among casino operators is that even though there's no expressly marked price, the law of demand kicks in. They think that regular players can sense intuitively when a game they're playing comes with a higher 'price'—that is, has too large of a house advantage—and thus, if that house edge is too large, the players will take their business to another casino with a more generous approach.

'It's basically how you respond to a price shock,' said Lucas. If Whole Foods doubles the price of your favorite brand of almond milk, for example, you might switch to a cheaper brand. That's the law of demand, which holds that if you raise a product's price, you should expect demand to decrease. The big question for slot machines is: 'how much will that demand decline?'

Slot Operations The Myth And The Math Book

'We might see a decrease in coin in, but if it's less than the product of the coin in at the higher par game, times the par, then we don't care,' said Lucas.

With Katherine Spilde of San Diego State University, Lucas set out to test this long-held belief by observing the behavior of slot machine players in local casinos in the suburbs of Sydney, Australia, where all the gaming is done electronically. Many past studies have relied on simulations or brought subjects into the lab to play slots. Lucas thinks field studies are a much better indicator, citing two primary advantages. 'Number one, you attenuate to the games differently, because you have the capacity for real loss of your own money,' he said. 'Plus, when people go into a casino and gamble, they don't play the games like they do in a lab. The advantage of the field study is you're observing behavior as it occurs in the setting you're trying to understand.'

For instance, virtual players in a simulation will not react how a human being would react when, say, they lose six or seven times in a row. And a typical lab setup will ask subjects to play a game for 500 spins with a constant wager. In a real-world scenario, that simply doesn't happen. Players are not going to play one slot machine for exactly 500 spins, never varying their wager. They're apt to mix up the wagers and switch machines if they don't get the result they want.

This creates significant variance in the outcome distribution, according to Lucas. 'You're jumping games, so that makes the identification of the true population parameter, which is par, even more difficult,' he said. 'You have unequal sample sizes. You have non-constant wagers.' For instance, if a study includes a 3% game and a 9% game, there is a wide range of outcome variance in both, with a great deal of overlap between them. 'As an engineer would say, the signal to noise ratio is not good,' said Lucas. 'You get this tiny weak signal beeping in a sea of variance.' That's before factoring in such cognitive biases as the gambler's fallacy and hot hand fallacy.

Myth: busted

Slot Operations The Myth And The Math Worksheets

For their own study, Lucas and Spilde tracked the daily performance of two identical slot machines (pairs of either 'Tokyo Rose' or 'Dragon's Fortune X') in similar positions on the casino floor for nine months. Each machine in a pairing was set to a different par (the percentage of total coin-in that the machines keep over time), ranging from 7.98% to 14.93%. They measured the daily coin-in as well as its theoretical win ('T-win'), determined by multiplying coin-in and par to get a machine's expected value. According to Lucas, such a comparison would demonstrate whether players really were migrating from higher par to lower par games over time.

Unfortunately, the math doesn't support the conventional wisdom. The differences between the high and low par games were stable for the entire nine-month period, plus the high-par games posted significantly larger revenues. But that conventional wisdom is so strong and enduring that Lucas has struggled to get operators to listen when his research challenges their favored operating paradigms.

'We take so much flak from industry because no one likes to have that mental model challenged,' he said. 'I get it. But the pie is growing, and everyone's getting a smaller piece, so they are going to have to get a little more technical. Eventually, I think the power of the research will overtake the resistance because there's so much pressure on them to optimize revenues.'

Lucas recently completed a similar study in Mexico involving two slot machines placed right next to each other. 'One was a 15% game and one was a 4% game, eleven percentage points apart—almost a 300% increase in price,' he said. Yet he found there was the same amount of play, and the same amount of win, on both games over a 180-day period. Players didn't switch to the machine with the smaller house edge. 'This place doesn't even have a hotel, so all the players are frequent players, which has been argued to be the population that would have the best chance in detecting price,' he said.

The message to casino operators is clear: they should stop fretting so much about whether they will lose customers by opting for higher-par machines, which in the long run are better for a casino's bottom line. 'Many believe that lower pars will attract more revenue by appealing to the customer's appreciation of gaming value (i.e., lower 'prices'), but our work suggests otherwise,' said Lucas. 'Our results are most likely due to the inability of players to detect par, of course. Still, knowing that the higher pars perform better is most helpful in the revenue optimization game.'

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Slot Operations The Myth And The Math Game

DOI: International Journal of Contemporary Hospitality Management, 2019. 10.1108/IJCHM-07-2018-0577 (About DOIs).