Posted by pacomartin
Mar 24, 2011

Slot Machines: PARS

PARS == Paytable and Reel Strip
PARS == Performance and Appraisal Sheet

Quote: Kevin A. Harrigan and Mike Dixon, University of Waterloo, Waterloo, Ontario, Canada E-mail:

PAR Sheets, probabilities, and slot machine play: Implications for problem and non-problem gambling

Through the Freedom of Information and Protection of Privacy Act, we obtained design documents, called PAR Sheets, for slot machine games that are in use in Ontario, Canada. From our analysis of these PAR Sheets and observations from playing and watching others play these games, we report on the design of the structural characteristics of Ontario slots and their implications for problem gambling. We discuss characteristics such as speed of play, stop buttons, bonus modes, hand-pays, nudges, near misses, how some wins are in fact losses, and how two identical looking slot machines can have very different payback percentages. We then discuss how these characteristics can lead to multi-level reinforcement schedules (different reinforcement schedules for frequent and infrequent gamblers playing the same game) and how they may provide an illusion of control and contribute in other ways to irrational thinking, all of which are known risk factors for problem gambling.

Kevin Harrigan's website
Much more detailed studies on psychology of slot design

A PAR sheet is a very closely held document that is associated with a slot machine that reveals information about performance of the machine. The Wizard discusses in great detail about how slot machines are designed on his Wizard of Odds/Slots site. I don't want to reproduce his excellent work, as you can read it on his site. But he doesn't show you what a PAR sheet looks like as they are proprietary.

A blogger only identified as Ray Watts has posted a PARS on Blazing 7's machine. It looks like an old version of the machine. It is a "2 coin options buy a pay with 3rd coin multiplier" which is noted in the upper right hand corner of the PARS. The rest of this post will help explain that phrase.

This Blazing 7's slot machine is different than the one discussed on the Red White & Blue on the Wizard of Odds site. The older machine described on the WOO site encourages you to play more than one coin by giving you more bang for the extra coins by paying the jackpots at a higher rate (i.e. If you put in three coins the jackpot is $10,000 instead of $7,200). The Blazing 7's is a three coin machine with a very strong psychological manipulation to play more than one coin.

This older motivation of increasing the jackpot has largely been replaced in most slot machine design. Instead machine now deny you a class of payouts if you only use a single coin. Probably the best known slot machine with this option is the "Wheel of Fortune" that won't let you spin the wheel unless you put in multiple coins.

As most of you know, a slot machine determines your win or loss once you hit the button; everything else is window dressing for your amusement. So I have omitted the exact combination of bars and 7's that elicit each payout to discuss the core mathematics. The PARS on Ray Watts site is a document intended for professionals. For instance it does not show you % of total out for one or two coins since that is of little use.

I devised the table below to explain what happens when you put in 1, 2, or 3 coins into the machine described on PAR sheet. In order to make it more intuitive, I assume that you are playing 10 plays per minute, and put down the mean time between occurrences of each of the 14 wins.

The payout is for one, two or three $1 coins. The last three columns show you expected wins if you are playing 1,2, or 3 quarters for 4057 plays. The number of plays was chosen because the amount played was close to $1K, $2K, and $3K, and at 10 plays per minute it is for almost 7 hours (average day play of a machine, and thirdly because it is the mean number of plays from an important playout.

HH:MM:SS 1 $ 2 $ 3 $ 3 quarters 1 quarter 2 quarters 3 quarters
0:01:55 $2 <---- <---- $0.50 $106 $106 $106
0:02:09 $10 <---- <---- $2.50 $472 $472 $472
0:11:16 $20 <---- <---- $5.00 $180 $180 $180
0:17:06 Ring! $100 $200 $50.00 $593 $1,186
0:36:46 $40 <---- <---- $10.00 $110 $110 $110
0:45:58 $4 <---- <---- $1.00 $9 $9 $9
1:51:05 $150 $300 $75.00 $137 $274
5:08:28 $80 <---- <---- $20.00 $26 $26 $26
5:45:36 $60 <---- <---- $15.00 $18 $18 $18
6:45:42 $500 $1,000 $250.00 $125 $250 4057 plays
11:06:31 $300 $600 $150.00 $46 $91
17:16:48 $120 <---- <---- $30.00 $12 $12 $12
20:44:10 $200 $400 $100.00 $16 $33
11:31:12+6 days $2,500 $5,000 $1,250.00 $27 $54
$933 $1,877 $2,821 Won
$1,014 $2,029 $3,043 Played
$82 $152 $222 Casino
8.04% 7.48% 7.30% Casino

All times are "play time" not including "rest time" at 10 plays per minute. The paytable would be posted on the machine associated with various combinations of the triple wheel display. It is a 72 stop virtual real machine, and the jackpot is expected every once every 4*72^3 plays .


1) The payouts are not correlated directly with how hard it is to get them. I ordered the table by difficulty of getting a payout, not by the size of the payout. On this slot machine, the $2 and $10 payouts are at almost the same frequency, while $4 payout is so infrequent that you realize that it is just there as window dressing. The $10 payout is the re-inforcer as it is the constant feedback that keeps you playing.

For a single coin, eight payouts of $2, $2, $12, $40, $50, $300, $360, $2039 would result in the same house advantage, but would more closely line up with the :true odds" for each bet. The first two bets instead of paying $2 and $10 would now both pay $2 as they occur at about the same rate. The resultant game would be considerably less "fun" as it would mostly be two to one payouts with scattered higher payouts.

This mismatch of payouts and "true odds" is a feature of all slot machines and video poker paytables. For instance, straights, flushes, and full houses are all achieved at about the same frequency, but video poker machines pay off them at very different rates.

2) If you only play with one coin, then the high win settings will still happen at the given frequency, you just won't get any money for them. The crushing feeling that you would have won a lot of money if you had bet higher provides a powerful incentive to play more than one coin at a time.

3) Notice that if you play one coin, then half your money will come back on the $10 return. Even though possible winnings are posted from $2 to $120, you will seldom see much above $40.

4) In terms of percentage that the casino keeps, there is very little value in playing more than one coin. In terms of excitement (or variance in more mathematical terms) there is considerable incentive to play more than one coin. Also there is very little motivation to increase from two to three coins. But people are conditioned to play max coins. The company can honestly tell you that the overall return is better if you go from 2 to 3 coins, they don't tell you it is only better by 0.18%.

5) The "ringer" is the $100/$200 payout for 2 or 3 coins which comes every 171 times on average (17.1 minutes). It's enough to keep you excited, but not enough to make you quit to enjoy your winnings. The "ringer" is an almost universal feature of slot machines. It is considerably less often that you will get the $150/$300 payout even though it is not much more money. The return here is what will keep you playing for hours.

6) The "ringer" will show up at the same frequency if you are playing one coin. Since you will be paid nothing it will encourage you to increase your bet. You will feel like you have missed a big opportunity. You should note that their is only a slight improvement in the player edge as you increase the number of coins. That is because extra coins do not increase the lower paybacks.

7) The jackpot is relatively small considering how difficult it is to achieve. The probability is equivalent to calling the result of a throw of the dice perfectly six times in a row, and then calling odds or even for a 7th throw. It is possible to change the design as to minimize the other paybacks so you get a much bigger jackpot. However, networked slot machines that pay a huge multimillion dollar jackpot and have taken much of that market. The stand alone slot machine design favors paying you more often. For a three coin play the payout for the jackpot could be increased from $5000 to $11,426 if the casino reduced it's hold from 7.3% to 5.0%. It could be as high as $25,430 for a hypothetical machine that has a 100% return.

Posted by pacomartin
Apr 04, 2010

Mexican Total Fertility Rate

Temporarily skipping the mathematics of the curve fit, what follows is the US census department estimate of the total fertility rate (TFR) of Mexico. The international division of the US census bureau is charged with estimating the future population of every country in the world.

The total fertility rate (TFR) of a population is the average number of children that would be born to a woman over her lifetime if
(1) she were to experience the exact current age-specific fertility rates (ASFRs) through her lifetime, and
(2) she were to survive from birth through the end of her reproductive life.

In the early 1980's the Mexican fertility rate was higher than the American peak in the 1950's. By about 1987 it dropped below 3.8 (which was the highest TFR ever recorded in modern America). Colonial America was extremely fertile, but there was only a few million Europeans living in the continent at the time. By today (year 2010) the Mexican TFR is estimated to be roughly 2.3 children per woman, slightly above the 2.1 children per woman in the USA.

There is phenomena called population momentum. While the American birth rate started plunging in the mid 1960's, the Mexican birth rate only became fairly low three decades later. As a result, the Mexican median age is a decade younger than that of Americans. There are more Mexican woman at family rearing age.

However, the population momentum will eventually fade as the Mexicans age. At present rates of legal immigration, the USA will be growing faster than Mexico in only 11 years.

Within 40 years, the demographics of the western hemisphere will be completely changed. The USA will be the population growth leader with the exception of some small high growth countries. The latest projection is that by the year 2050 USA will growing faster than all countries in the Western hemisphere except for the following 5 countries:
Guatemala (Fertility Rate = USA); Haiti and Belize (Fertility Rate > USA); and Anguilla and Cayman Islands (Because of Migration). The five countries represent less than 5% of population of W. Hemisphere.

There is a traditional view of Mexican fertility that is now almost decades out of date. Even this Latino made comic video embraces this stereotype. Latino Comedy Project: Movie 300 trailer .
Simple Mathematics Explanation
This process is called Logistic regression with only one independent variable (the year).

F(x)= (U-L)/(1+1/exp(x)) + L; U and L are constants (U,L Upper and Lower): Logistic Function
f(y)= ln(y-L)-ln(U-y) ; The inverse of the Logistic Function (or the Logit Function)

Take the databases of known TFR, and calculate the logit of the TFR values.
Use U=10, L=2 (U,L are discussed below)

1980.50 4.57 -0.7480
1982.00 4.26 -0.9321
1989.00 3.44 -1.5163
1991.00 3.24 -1.6978
1997.62 2.85 -2.1296

Do a linear fit on the Logit values above against the year. Use the slope() and intercept() functions in Excel.
You will get slope=-0.0807, and intercept=159.0879 (for U=10, L=2). If you leave these functions in your spreadsheet the slope and intercept will recalculate for other values of U and L.

Now recalculate the Linear fit of the Logit using the slope and intercept.
Insert the linear fit values into the Logistic function.

YearTFR-Based on linear fitLogit(TFR) linear fit
1980.50 4.48 -0.7974
1982.00 4.28 -0.9185
1989.00 3.48 -1.4836
1991.00 3.29 -1.6450
1997.62 2.81 -2.1799

Using the Linear fit you can interpolate the Logistic function between 1980 and 1997, and then extrapolate it for years beyond 1997 (graph above goes to the year 2015). Now you can calculate the TFR for interpolated and extrapolated values for years. The graph above is result for year values from 1980 to 2015.

L (or lower asymptote) is the value that you think that the TFR is headed toward in the distant future. Census bureau uses L=2.0 for developing countries, and L=1.7 for highly developed countries that already have a TFR below 2.0 .
U ( or upper asymptote) is chosen for individual countries. The process is a little arbitrary. If you program this equation into a spreadsheet you will also see that various nearby value of U, do not dramatically change the curve. U is always at least 1 higher than the largest data point or at least 6. The researcher selected U=10 for Mexico based on some internal reasoning. I suspect he/she had some partial data that he didn't include in the database, but it dictated how the curve should look. For U=10, TFR(year 2010.5)=2.31 ; for U=6, TFR(year 2010.5)=2.22 which the researcher may have felt was too low based on some other data.

In any case using this fertility rate, the predicted death rates, and migration rate, the population of Mexico will stabilize somewhere just over 150 million after the year 2050.

The USA growth rate will increase for the known future without stabilizing(simply because we now have enough children to replace ourselves and we continue to have immigrants). In the mid 1970's when modern immigration began, the TFR of the United States was not anywhere near high enough to replace the population.

I don't want this blog to come off xenophobic. Strictly speaking it is simply explaining the mathematical basis for the census department projections and it does not offer opinions,

However, I think that with birth rates stabilizing it seems almost dumb to have the USA growth rate exceed all the Latino countries. Maybe 30 years ago with high birth rates, it seemed like immigration helped Latino populations who couldn't create jobs fast enough for their growing populations. It also helped provide labor for the USA. But now with fertility rates headed towards Zero Population Growth, the immigration can suck away the best and the brightest from countries that badly need these young people. It divides families for decades.


Wizard Apr 08, 2010

As a former Social Security actuary, this topic is right up my alley. Until your post, I didn't know much about fertility rates in Mexico. I would have guessed them to be higher, so it is good to learn something. Your blog entry was so good I can't think of any good questions about it, so will leave it with just my applause.

pacomartin Apr 09, 2010

Thanks Wizard.

The fitted curve is the Logistic Function .

One of the questions that I had for the census bureau was how do you know what is the value that TFR approaches (i.e. the asymptote) for each country? The answer is that there is really no way to know where society is eventually headed. For Mexico (and most developing countries they use a TFR=2.0). Now if the data points are very high TFR (like Haiti) the curve won't reach 2.0 for many decades.

For highly developed countries the asymptote is TFR=1.7 (even for countries that are presently below that number). The assumption is that there will be some backlash in those countries where fertility is almost non-existent. There are government programs to encourage fertility, and their will be some social pressure. Extremely low fertility rates are really not much of an issue in the Western Hemisphere, but more in Europe.

In reality I think they use either 1.7 or 2.0 as the asymptote for all the countries. However, since they are only projecting to the year 2050, some countries never get near the asymptote. They did projections out to the year 2100 for the United States in the year 1999, but it was more of an exercise to show that it is nearly impossible to say anything remotely reliable in a 100 year projection. Very small variations in parameters led to huge variation in final population. The report became a bell-ringer for many social activist groups since the upper bound was 1.2 billion in the year 2100 for the population of the USA.

For example if you read the link at a sustainable USA you will see a reference to this census report in the 3rd paragraph from the bottom.

Wizard Apr 14, 2010

Good points there. When I was with SSA I was in the short-range office, meaning I only cared about the next 10 years. The long-range actuaries would have been the ones to assign a long-term fertility rate, although critical statistics like that come from elsewhere. I tend to think important statistics like this tend to be influenced by political pressure. Much like a Ponzi scheme, an economy tends to flourish as long as the population keeps going up, because of a high worker to retiree ratio, up to some point like in China were shortages of recourses become an issue. So if you're trying to make a case for lowering taxes based on a future robust economy, it helps to make your case if you get your actuaries to use a high fertility rate.

I know this isn't a very mathematical reply. You did mention logistic regression. Did you see my tables for run lines in baseball. Those were all done via logistic regression.

pacomartin Apr 15, 2010

The mathematics of logistic regression is reasonably sophisticated, but you still have to pick the asymptotes. As you try to project further into the future, the values you pick for U and L affect you more and more. There is no mathematical technique for choosing the value L for the future asymptote (i.e low bound).

But the main points are:

(1) That most countries in the Western Hemisphere (WH) are headed toward stability except the US.

(2) Just a decade ago most countries in the WH were growing faster than the US, but we have passed a number of them recently. In the next decade we will pass some of the larger countries in Latin America.

Neither of these two points is commonly known to people. In fact most public policy is based on the belief that Latin America will continue growing much faster than the USA.

Posted by pacomartin
Mar 25, 2010

Media coverage

Some news articles that I am quoted. My given name is Frank Martin, but I use the Spanish name, pacomartin, for blogging. If you want to contact me e-mail

#TitleDateAuthorOriginal Periodical
13Billionaire's hedge fund buys into MGM Mirage, Boyd18-May-10 Howard StutzLas Vegas Review Journal
12Riviera posts first-quarter loss: Adds to casino woes on older end of Strip17-May-10 Arnold M. KnightlyLas Vegas Review Journal
11SkyJump,' rel='nofollow' target='_blank'>]SkyJump, renovations seen as boost for Stratosphere parent12-May-10 Arnold M. KnightlyLas Vegas Review Journal
10' rel='nofollow' target='_blank'>] Hard Rock posts $26.5 million loss in first quarter 10-May-10 Arnold M. KnightlyLas Vegas Review Journal
9Empty lots hurt nearby casinos on the Strip's north end 05-Apr-10 Liz BenstonLas Vegas Sun
8 ' rel='nofollow' target='_blank'> New owners bring money, optimism to downtown04-Apr-10 Arnold M. Knightly Las Vegas Review Journal
7 ' rel='nofollow' target='_blank'> Downtown hopes for upturn 29-Mar-10 Arnold M. Knightly Las Vegas Business Press
6 ' rel='nofollow' target='_blank'> Slots have gotten looser, not that the average gambler can tell 16-Mar-10 Liz Benston Las Vegas Sun
5 ' rel='nofollow' target='_blank'> Hard Rock has Asia on mind 18-Feb-10 Arnold M. Knightly Las Vegas Review Journal
4 ' rel='nofollow' target='_blank'> Golden Nugget owner sees revenue fall in LV, Laughlin 23-Jul-09 Benjamin Spillman Las Vegas Review Journal
3 ' rel='nofollow' target='_blank'> Danger signs for Binion's 6-May-09 Benjamin Spillman Las Vegas Review Journal
2 ' rel='nofollow' target='_blank'> Twenty Thousand Fees Under a C 29-Mar-09 Benjamin Spillman Las Vegas Review Journal
1 ' rel='nofollow' target='_blank'> Downtown may lose a casino or two, analyst says 3-Dec-08 Benjamin Spillman Las Vegas Business Press

For article #1 I predicted a $71 million gaming loss for fiscal year 2009. The actual result was $66 million for the year. There was also a $62 million loss in non-gaming revenue. At the time looking at current data the median downtown casinos made $62.9 million in total revenue. I think I made a fairly common analytic mistake, that one casino would close rather than all of them losing money at this incredible rate. Instead of actually closing the casinos have cut back services massively by not serving hot food at night, Binion's shutting down their hotel and coffee shop. Fitzgerald's closed Don B's Steakhouse except for Friday and Saturday night. The Plaza property temporarily closed a hotel tower and parts of its casino, including its restaurants. Hundreds of slot machines have been taken off the casino floor and partitioned off near the recently closed Aqua Lounge. The lounge, Lombardi's Italian, the Omelet House, and 2 days of the Stuft buffet have also been temporarily shuttered as part of the "minimizing of the casino". El Cortez closed it's Chinese buffet, and stopped serving food at night in the coffee shop. Cutting back services had the cyclical effect of further cutting revenue. However, it is still not enough as the casinos still lost $54 million last fiscal year, and revenue is still dropping as of Jan 2010.

For article #2 Ben asked me for some information on room rates. Casino operators were talking about how room rates were killing the business. Downtown is accustomed to running food as loss leader, but they lose less and less every year as they limit hours and limit the loss leaders to one or two entrees on the menu. Beverages are the same way, the casinos lose less and less money every year. They cut back on cocktail waitresses and quality of drinks. But generally downtown they sell enough drinks to cover the comps. When the room department stops making money, the casino stops making money. In most cases the casino must close a hotel tower (especially since they own other rooms nearby). With gaming revenue at historic lows, the hotel must also make money.

For article #3 I made a general comment about that cost cutting would have to unprecedented to make up for the revenue shortfall (based on a 20 year analysis of cost data). Binion's with it's recent $32 million purchase was saddled with interest debt, and was trying to re-negotiate leases with the owners of property under the casino. Because downtown is built on lots that were laid out in 1905, the lots were designed for store sizes of the period. Many businesses lease several tiny lots under modern buildings. You can see the tiny lots all along Fremont Street.

For article #4 I made a short comment. I expected Golden Nugget to build the shell of their new building and delay finishing the inside. Several casinos like Ceasars and Venetian are choosing this tactic. Golden Nugget opened the building anyway, and was rewarded with having it's bond ratings demoted by Moody's who felt that they would not generate enough income and was in danger of defaulting. I regret the curtness of my comment, because the improvements look beautiful, but the bottom line every quarter is terrible. Landry's stockholders are likely to revolt at some point, because a money making restaurant business is supporting a money losing gaming division. Two days after this article Moody's downgraded Golden Nugget debt

For article #5, Arnold was doing a story on the remodel of the Hard Rock Casino gaming area. They were announcing their intention to look for mid-level Asian gamblers (below the high rollers being hunted by the Wynn). They re-designed their casino with this new customer in mind. What I actually said to Arnold was that the old Hard Rock had shockingly low gaming revenue (less than many locals casino) especially given that they had a huge number of gaming tables and baccarat tables. Their return per machine or per table was on par with downtown Las Vegas, not with the strip. All the more surprising since they had such a huge non-gaming revenue from their expensive restaurants and clubs. He chose a softer phrasing, by simply saying that a very low percentage of their revenue was from gaming. Technically it means the same thing, but it sounds less harsh.

Since fiscal year 1999, on the strip non-gaming revenue has exceeded gaming revenue. This statistic is often mentioned as proof the the Vegas economy had permanently changed. However, resorts where non gaming revenue is more than twice gaming revenue are relatively rare. Gaming is still significant part of revenue stream. In Hard Rock's case the resort was sold, and the new owners built two new hotel towers in the middle of the recession. The new owners are hoping for a double payback by not only increasing non-gaming revenue, but offering the luxuries so that the casino will start having rates of gaming return consistent with other strip casinos.

For article #6, Liz Benston took up my theme about the crash in blackjack. Blackjack was the only significant game that was dropping in revenue for the first 6 months of the recession, and has now approached the biggest percentage loss. Revenue is back to 1990's levels. The comment that degraded versions of blackjack is partly responsible for the revenue loss is mine, but without detailed data sheets from each version of blackjack it is difficult to prove. There is a widespread anger towards Harrah's in particular for degrading the game. Blackjack led off the recession much earlier than slots, and has not veered from it's steady downward trend. The game is now 34% off it's peak in October 2007.

The continued crash in most games and slots while baccarat hits new highs, is producing a side story about certain traditional casinos which are now losing money. Circus Circus in particular is the largest resort that is operating in the red. Although many resorts have been imploded if you actually look at the size of those resorts they have all been less than 1000 rooms with the exception of Stardust. Plus most were imploded with the intention of building new resorts (a few became parking lots). No one has ever closed a resort the size of Circus Circus simply because they can't make money. The fate of Circus Circus, Riviera, Sahara, and Stratosphere are somewhat intertwined. They are all losing money (presumably) at this time.

For article #7,#8, Arnold Knightly did a review of downtown Las Vegas for the LV Business Press. He cited me four times, and then included the ranking that I prepared of the casinos (without dollar amounts) of the 16 licenses that made over $1 million last fiscal year. The same article ran in two different publications.

Article #9, was based on a memorandum I sent to Liz. The memorandum highlighted the two different properties with similar operating expenses (Circus Circus and Excalibur). Even though they were both budget properties one is among the most profitable (by percentage) of MGM-MIRAGE properties and the other is the only one actually losing money. Liz used the memorandum and cited me, but she changed the focus to the problems caused to the extreme north strip by imploding the Stardust and the New Frontier.