PROBLEMS with an INSTANT SCRATCH LOTTERY GAME: an Analysis of Why the OLGC's Tictactoe Game Was Exploitable

PROBLEMS WITH AN INSTANT SCRATCH LOTTERY GAME: An analysis of why the OLGC’s TicTacToe game was exploitable

A REPORT PREPARED BY FSS CANADA CONSULTANTS INC.

JULY 2003

froidevaux srivastava  2003, R. Mohan Srivastava NOT TO BE COPIED WITHOUT COPYRIGHT HOLDER’S PERMISSION schofield For further information on this report, or to obtain additional copies, please contact FSS Canada Consultants Inc. at either of the following locations:

In Canada: In the United States:

Address: R. Mohan Srivastava Address: Douglas R. Hartzell 2179 Danforth Avenue, #302 5782 Golden Eagle Drive Toronto, Ontario Reno, NV 89523 Canada M4C 1K4 U.S.A.

Phone: (416) 693-7739 Phone: (775) 846-5811 Fax: (416) 693-5968 Fax: (775) 201-0256 E-mail: [email protected] E-mail: [email protected]

froidevaux srivastava  2003, R. Mohan Srivastava NOT TO BE COPIED WITHOUT COPYRIGHT HOLDER’S PERMISSION schofield EXECUTIVE SUMMARY

This report presents an analysis of the flaw discovered by R. Mohan Srivastava in TicTacToe, one of the instant scratch games offered by the Ontario Lotteries and Gaming Corporation (OLGC) and printed by Pollard Banknote Limited. This flaw effectively resulted in all of the cards being “marked” in the sense that one could, prior to playing a ticket, quickly predict with a very high success rate whether or not that ticket was a winner, using information readily visible on the face of the card. Since lottery customers in Ontario are allowed to inspect the cards prior to purchase, and to pick and choose the cards that “feel lucky”, this flaw in the TicTacToe game entailed that anyone who knew the trick would have had a much higher chance of buying winning tickets than the general public. Furthermore, if the trick became widely known soon after the game was released to the market, its use by the general public could have caused the OLGC to incur a financial loss, paying out more in prize money than was recovered in ticket sales. After Mr. Srivastava contacted the Ontario Provincial Police and the OLGC and demonstrated to them the fact that winners could be separated from losers, the OLGC pulled the remaining unsold TicTacToe tickets from the market and began an internal review process to better understand how the problem occurred. This report is intended to aid this review process by providing a detailed analysis of the flaw, its likely causes and the factors that could have contributed to it being a much bigger problem if knowledge of the card-reading trick became widely known and practiced before the OLGC became aware of the problem.

Conclusions

The principal conclusions of this report are given below; further explanation and other supporting information can be found in the main text of the report.

1. The OLGC’s version of the TicTacToe game could have been exploited using a visual/mental procedure for separating unplayed cards into likely winners and likely losers. This procedure: • had a very high success rate, with 85 to 90% of its predictions being correct, • was able to accurately predict the likely prize amount, and • could be executed within two minutes or less for each card.

2. The ﬂaw in the TicTacToe game was likely the result of a problem with the software used to implement the creative concept. Even though the software did produce cards that conformed to the creative concept design, it did so in such a way that the information printed on the face of the card could be used to exploit the game.

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3. There are other instant scratch games that also have card-specific information printed on the face of their tickets and that could be exploited in a similar manner. Preliminary examinations of a few of these games indicates that some of them definitely do have a similar flaw. 4. Several aspects of the lottery industry contribute to the severity of the problem: • a growth in the type of instant scratch game that uses card-specific information to make the instant scratch games more engaging and interesting; • a game development process that often involves borrowing popular games from other jurisdictions and tweaking their game mechanics; • a lack of industry-wide standards needed to support statistical quality assurance and quality control; • a retail culture in which customers are allowed to pick and choose their own cards and that permits customers to trade in unplayed cards; • a distribution mechanism that could allow cards deemed to be likely losers to be returned to distributors along with unsold tickets; and, • a lack of legal remedies to prevent anyone from attempting such exploitation or from disseminating information about the flaws in instant scratch games.

While none of these, by itself, caused the problem, they would all have contributed to a significant loss of lottery revenue, and an even more significant loss of public trust in the lottery system if the trick for reading the cards had been widely used to skim the TicTacToe game’s prize money. 5. The specific problem with the TicTacToe can be corrected, as can similar problems with other instant scratch games that use similar game mechanics.

Recommendations

The principal recommendations of this report are given below; the final section of the main text of the report discusses the reasons for these recommendations in greater detail, and provides a brief summary of some other possible solutions that were considered but that did not form the basis for a specific recommendation. 1. The lottery industry should not treat the problem identified with the TicTacToe game as a fluke, a one-off problem that requires no specific response because it is never likely to occur again. Problems similar to the one revealed in the TicTacToe game have already been identified with other instant scratch games currently on the market. An entire category of instant scratch games is at risk: those that use tickets with unique information visible on their face prior to being played. With these types of game already being among the most popular and successful on the market, their market share is likely to increase. Problems similar to the one discovered in the TicTacToe game are likely to occur again unless the industry takes specific preventative steps.

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2. The lottery industry should continue to use instant scratch games that provide card- specific information, but should take steps to ensure that they are printed in such a way that this information cannot be used to significantly increase the odds of winning. 3. All instant scratch games with card-specific information should be accompanied by a report on the various aspects of the game mechanics — e.g. the number of playing boards, the range of numbers (or symbols) used and the size of the scratch list — and the tolerances within which these can be acceptably adjusted. Since there are few printers that serve a much larger number of lottery jurisdictions and since printers currently assume responsibility for the details of the actual implementation of the creative concepts, it makes sense that printers take the lead in doing the analysis of game mechanics and for preparing documentation that can be used to make sure that variations of the game do not become exploitable when key parameters are modified. 4. Lottery organizations should carefully consider the pro’s and con’s of changing the industry’s current practices regarding who determines what exactly gets printed on instant scratch tickets. Introducing a technical services group to handle this task, one that specializes in customized software development for lottery applications and that liaises with other contractors chosen to handle other aspects of bringing a game to market — the printers and the auditors, for example — would help to make sure that games cannot be exploited because of unforeseen probabilistic flaws. 5. State and provincial lottery jurisdictions should lead an effort to strengthen the lottery industry’s quality assurance procedures by establishing industry-wide standards for statistical properties that warranty the integrity and fairness of instant scratch games. 6. State and provincial lottery jurisdictions should lead an effort to introduce statistical quality control procedures for continuous, real-time verification that instant scratch games are not being exploited. 7. The lottery industry should study the advantages and drawbacks of removing the element of consumer choice from instant scratch lottery games. Specifically, they should gather data from any existing studies of consumer response to ticket dispensing machines and commission new studies if existing studies are inadequate or inconclusive. 8. State and provincial lottery organizations should focus on restoring and maintaining the integrity and fairness of all instant scratch lottery games. They should not duck the problem by imposing restrictions on retailers; such restrictions would correctly be perceived as a concession that some instant scratch games can be exploited. 9. In lottery jurisdictions where consumers are allowed to choose their instant scratch tickets, retailers should not be allowed to return unsold tickets that were not originally distributed to them. 10. State and provincial lottery jurisdictions should investigate the legal remedies at their disposal to mitigate or prevent exploitation of instant scratch games.

froidevaux srivastava  2003, R. Mohan Srivastava NOT TO BE COPIED WITHOUT COPYRIGHT HOLDER’S PERMISSION schofield CONTENTS

EXECUTIVE SUMMARY iii Conclusions ...... iii Recommendations ...... iv

INTRODUCTION 1 Background ...... 1 A cautionary note ...... 1 Organization of the report ...... 2

1 GAME MECHANICS 4 Diﬀerent types of instant scratch games ...... 6 Information unique to each card ...... 6 One scratch list with multiple playing boards ...... 9

2 THE TRICK 10 The trick explained ...... 10 Implementation with and without computer support ...... 11 Accelerating the visual/mental implementation ...... 11 Why the trick works ...... 12 How successful is the trick? ...... 13 False positives and false negatives ...... 13 False positive and negative rates based on empirical evidence ...... 15 False positive rate based on a probability argument ...... 16 Authoritative estimates of false positive and false negative rates ...... 16 A diﬀerent aspect of success: ...... 17 Making the trick work even better ...... 17 Including some of the doubles ...... 17 Bayesian analysis ...... 18

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3 PRIMARY CAUSE 20 An unintentional and unforeseen flaw ...... 20 Evidence of an unintentional flaw ...... 21 The possibility of intentional fraud ...... 21 Fraud limited to the TicTacToe game would have to be modest ...... 23 Difficulty of prosecution ...... 23

4 CONTRIBUTING FACTORS 24 Game design and creative concept ...... 24 Allowing cards to contain visible and unique information ...... 24 Tweaking the concepts of others ...... 24 Statistical quality assurance and quality control ...... 25 Software QA/QC ...... 25 Quality assurance and industry standards ...... 25 Quality control ...... 27 Distribution and marketing ...... 28 Allowing consumers to pick and choose their cards ...... 28 Allowing consumers to trade cards ...... 29 Allowing retailers to return unsold cards ...... 29 Lack of legal remedies ...... 30

5 SOLUTIONS AND RECOMMENDATIONS 32 The “steady-as-she-goes” non-solution ...... 32 Game design and creative concept ...... 33 Allowing cards to contain visible and unique information ...... 33 Tweaking the concepts of others ...... 34 Analysis of parameters controlling TicTacToe game mechanics ...... 35 Software quality assurance and quality control ...... 36 Quality assurance and industry standards ...... 38 Quality control ...... 39 Allowing consumers to pick and choose their cards ...... 40 Allowing consumers to trade cards ...... 41

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Allowing retailers to return unsold cards ...... 42 Lack of legal remedies ...... 43

APPENDIX A: USING BAYES’ THEOREM 44

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Table 1. Statistical summary of numbers on card in Figure 7a ...... 13 Table A.1. Number of cards with various prize values ...... 44 Table A.2. Prior probabilities ...... 45 Table A.3. P x(f) and P o(f) as calculated from 37 cards ...... 46

froidevaux srivastava  2003, R. Mohan Srivastava NOT TO BE COPIED WITHOUT COPYRIGHT HOLDER’S PERMISSION schofield LIST OF FIGURES

Figure 1. A TicTacToe card ...... 4 Figure 2. Playing the TicTacToe game ...... 5 Figure 3. The card is a winner! ...... 5 Figure 4. The card played to completion ...... 5 Figure 5. Examples of instant scratch games with unique cards ...... 7 Figure 6. Examples of instant scratch games with identical cards ...... 8 Figure 7. The trick for identifying winners in the TicTacToe game ...... 10 Figure 8. A false positive ...... 15 Figure 9. A false negative ...... 15 Figure 10. Predicted $5 winner ...... 17 Figure A.1. Board #1 from Fig. 7b with two possible conﬁgurations of Xs and Os . . . 45

froidevaux srivastava  2003, R. Mohan Srivastava NOT TO BE COPIED WITHOUT COPYRIGHT HOLDER’S PERMISSION schofield INTRODUCTION

Background

In June 2003, R. Mohan Srivastava, a geostatistical consultant with FSS Canada Consultants Inc. (FSS), recognized that the TicTacToe tickets being being sold by the Ontario Lotteries and Gaming Corporation (OLGC) were “marked” in the sense that winning tickets could be separated from losing ones with a high degree of success, prior to playing the ticket and using information readily visible on the face of the card. Realizing that this flaw severely compromised the integrity of the game, he reported his findings to the Ontario Provincial Police who referred him to Rob Zufelt, the person responsible for security issues at the OLGC. Mr. Srivastava provided a demonstration of the procedure by sending to Mr. Zufelt a set of twenty unplayed TicTacToe cards, separated into probable winners and probable losers. Of the seven cards that had been identified as probable winners, six of them were, in fact, winners; and of the thirteen that had been identified as probable losers, all were, in fact, losers. The success of this demonstration quickly set in motion a chain of events: the OLGC pulled the remaining unsold TicTacToe tickets from the market; they notified the printer, Pollard Banknote Limited, of the problem; and both the OLGC and Pollard began to review their procedures and protocols in an attempt to identify the root cause(s) of the problem and to take steps to prevent similar problems from occurring again in the future. This report is Mr. Srivastava’s own contribution to the exercise of improving the lottery industry by learning as much as possible from this incident. It provides a detailed analysis of the flaw, along with discussion on why it may have occurred and what steps can be taken to prevent its reccurrence.

A cautionary note

Neither Mr. Srivastava, nor FSS, the company he co-founded, have any experience in the lottery industry.1 Readers of this report are therefore well advised to keep in mind that it is not written by an industry insider. At the same time that this lack of experience is undoubtedly a limitation — parts of the report may be based on misconceptions or misun- derstandings about the way the industry operates — it may also be an advantage. With no preconceptions about how things “should” be done, and no vested interest in any aspect of 1Not even to the extent of occasionally buying lottery tickets: the ticket that triggered the discovery was given as a birthday present and was the ﬁrst instant scratch ticket that Mr. Srivastava had ever played.

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the industry, industry outsiders are able to raise, discuss and explore issues that industry insiders might ﬁnd awkward. It should also be made clear that a lack of experience in the lottery industry does not entail a lack of ability to analyze problems related to lottery games, to propose solutions and to make recommendations. Mr. Srivastava is one of the world’s leading experts on applied probability and statistics, having studied and practised as a geostatistician for 25 years. He has written the most popular textbook on geostatistics, has served on numerous occasions as an expert witness in cases that involve probability and statistics, and has been the sparkplug for one of the leading international geostatistical consulting organizations. He and his colleagues at FSS are routinely hired by private and public companies, as well as by governments and academic institutions, to solve problems that involve applied probability and statistics. They have worked with international standards development organizations to standardize and improve the practice of applied probability and statistics, and have established a strong reputation in statistical quality assurance and quality control (QA/QC). Healthy skepticism about commentary from industry outsiders should therefore be tempered by due consideration of the possibility that this report may well contain information that improves the lottery industry. It is, after all, written by the one person who, on seeing only a single TicTacToe card, suspected a serious problem that escaped the notice of many well qualiﬁed industry insiders.

Organization of the report

This report is divided into the following sections: 1. Game Mechanics, a section that explains how the TicTacToe game is played, that discusses the characteristics of the game that distinguish it from other instant scratch lottery games and that identifies other characteristics that make it similar to other games. 2. The Trick, a section that presents the procedure developed by Mr. Srivastava for identifying winning cards, that discusses why this procedure works, how well it works and how it could have been made to work even better. 3. Primary Cause, a section that offers an explanation of why the flaw in the TicTacToe game likely occurred. 4. Contributing Factors, a section that identifies various aspects of the lottery industry that contributed to the TicTacToe game being exploitable and that could have made the problem worse had the card-reading trick become widely known and practiced. 5. Solutions and Recommendations, a section that takes a look at various specific recommendations for various players in the lottery industry: the state and provincial agencies that operate the lotteries, the companies that print the tickets and the organizations that provide external auditing of the games being offered.

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Appendix A: Using Bayes’ Theorem, material left out of the main text in order to maintain its readability that details the use of Bayesian updating to improve the card- reading trick.

froidevaux srivastava  2003, R. Mohan Srivastava NOT TO BE COPIED WITHOUT COPYRIGHT HOLDER’S PERMISSION schofield 1 GAME MECHANICS

Figure 1 shows a TicTacToe card as it appears prior to being played. The right side of the card shows eight 3×3 grids of numbers; these will be referred to in this report as the eight “playing boards”. The numbers that appear on these eight playing boards go from 1 to 39. On the left side of the card, under the heading “YOUR NUMBERS” is a list of numbers that are concealed at the beginning of the game; this list, which will be referred to in this report as the “scratch list”2, contains 24 of the numbers from 1 to 39. The game is played by scratching off the opaque latex coating that conceals the numbers on the scratch list, as shown in Figure 2. As numbers are revealed on the scratch list, the player searches for the same number on the playing boards. By scratching off the translucent latex coating on the playing boards, the player can easily mark the cells that contain numbers from the scratch list, turning them from cyan to white. In Figure 2a, for example, the first number on the scratch list has been revealed to be 21, which appears on two of the playing boards. In Figure 2b, the second number Figure 1. A TicTacToe card. on the scratch list, 20, appears on only one of the playing boards. In Figure 2c, the third number on the scratch list, 30, also appears only once. If three of the numbers on the scratch list make any of the conventional tic-tac-toe alignments (a column, row or either diagonal) on any of the playing boards, then the card is a winner. The dollar amounts shown in red indicate the value of the prizes for the column and row alignments; these range from a low of $3 (for the first column) to $100 (for the last column or row). Diagonal alignments provide the prizes that are large enough that the player is obliged to cash in the winning ticket through the OLGC itself rather than through a lottery retailer3; the dollar amounts shown in greenish gold indicate the value of the various possible diagonal alignments, from $250 to $50,000. Much of a player’s sense of the fun in playing the game comes from the anticipation created by getting promising alignments of two numbers. Even if the ticket turns out eventually to be a loser, many players will have the sense that they “almost” won because it is inevitable that there will be a handful of such promising two-number alignments, even on losing cards. 2On many lottery websites, this list is referred to as the “caller’s card”, an analogy with the game of bingo. Even though this particular game is called “TicTacToe”, it is, in many ways, more like a game of bingo played on 3×3 boards rather than on 5×5 boards. 3As indicated by the instructions on the back of the card, prizes under $200 can be claimed from any lottery retailer while those greater than $200 must be claimed from the OLGC.

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a) First number revealed b) Second number revealed c) Third number revealed

Figure 2. Playing the TicTacToe game.

Figure 3. The card is a winner! Figure 4. The card played to completion.

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For the example shown in Figure 2, this sense of anticipation develops quickly. After only three numbers have been revealed on the scratch list (Figure 2c), there is already a promising two-number alignment on the sixth playing board, for which the 20 and the 30 in the ﬁrst column both appear on the scratch list . . . now if only we could get the 18 above them . . . ! Figures 3 and 4 show how the game unfolds. The 17th number to revealed is the number we were hoping for: 18. As shown in Figure 3, when this number appears on the scratch list, the sixth playing board has a winning alignment in the ﬁrst column: 18, 20 and 30. At this point, the card is at least a $3 winner. It may turn out to win more than $3 because, as indicated on the card, it is possible to win up to eight times on a single card. Figure 4 shows the card played to completion; it turns out that there are no other winning alignments, so this particular card is a $3 winner.

Diﬀerent types of instant scratch games

For the discussion that follows later in this report, it is useful to draw attention to certain characteristics of the TicTacToe game, some of which make it fundamentally diﬀerent from other instant scratch games, and others of which make it similar to some games.

Information unique to each card

The ﬁrst relevant characteristic of the TicTacToe game is that each card has a diﬀerent set of numbers that appear on the eight playing boards. Since these numbers are all visible at the outset, the cards provide information on their face that makes each one unique. The existence of such information creates the possibility that the game can be exploited. Figure 5 shows a few examples4 of similar games:

• Bingo, a game widely used by lottery organizations in North America that has a game mechanic very similar to TicTacToe; • Keno, an instant scratch version of the casino game, with numerical information that is card-specific; • Pharaoh’s Gold, another instant scratch game with card-specific numerical information; • Crossword, a popular instant scratch game in which the card-specific information is alphabetic; • Lucky Lines, a game in which the card-specific information is symbolic; and, • Poker Royale, another game with symbolic information that renders each card unique. 4Figure 5 is not comprehensive; there are many other games that share the essential characteristic of using cards that are easily distinguishable from each other because of some kind of information printed on the face of the card.

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Figure 5. Examples of instant scratch games with unique cards.

With the advent in recent years of instant scratch games that attempt to provide the player with a more engaging, interactive and fun experience, there has been a rapid increase in the number of instant scratch games whose cards are visibily unique. Though they are still in the minority, in terms of the total number of instant scratch games on the market, their popularity is growing as many lottery jurisdictions experiment with new types of games. In a recent survey of North America’s state and provincial lotteries5, several jurisdictions reported games shown in Figure 5 as their top sellers, both in terms of revenue generated 5Reported in the February 2003 edition of Public Gaming International magazine.

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and in terms of total tickets sold. Variations of Crossword were the top selling game in Kansas, New Jersey, Oregon, Rhode Island and Western Canada. Bingo was the top seller in Kentucky and Ontario, and many jurisdictions identiﬁed it as one of their most successful products, year in and year out. Keno was West Virginia’s top selling instant scratch game. When each card contains unique information, the task of identifying winners is similar in many ways to a cryptography problem. In the case of the TicTacToe game, the 72 numbers that are visible on the face of the card can be viewed as a kind of code and the prize value of the card can be viewed as a kind of original message, the content of which is supposed to be a secret. Cryptographers have developed tools for deciphering a coded message (what they call “ciphertext”) into a meaningful original message (what they call “plaintext”). These tools can be adapted to the problem of deciphering card-speciﬁc information (whether numerical, alphabetic or symbolic) in an attempt to learn the underlying prize value. It should not be surprising that there is an analogy between cryptography and certain instant scratch lottery games. Both make use of computer-generated random numbers; both involve deterministic procedures — computer software — whose inner workings are often revealed by incidental or ancillary output (such as the 72 numbers on the TicTacToe playing boards). Both are most successful when everything is properly and truly randomized; and both begin to fail when elements of predictability creep in.

Figure 6. Examples of instant scratch games with identical cards.

Figure 6 shows some examples of games that do not have the characteristic of using cards that are visibly unique prior to being played. For all of the games shown in this ﬁgure, every game ticket is identical to every other and, apart from the serial number (see discussion below), there is no card-speciﬁc information that can be leveraged to improve one’s odds of winning, no clues that might reveal the winning tickets.

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Serial numbers

The problem of card-specific information is already apparent to the lottery industry in a different form. All instant scratch tickets have a serial number that assists with distribution, tracking and authentication. For the vast majority of the instant scratch games currently on the market, this serial number is visible on the face of the card, sometimes as a barcode. Discussions with Nancy Thompson, a ticket security specialist with the OLGC, indicate that the industry’s normal audit practice for instant scratch games includes checking that there is no simple pattern of winning tickets when they are examined in their serial print order. The audit reports prepared by OLGC’s external auditor typically confirm that winning cards do not occur in a predictable order (e.g. every third card) and that the location of one winning card in the print sequence does not depend on others (e.g. winners do not occur back-to-back). These types of checks are essentially exercises in confirming that winning tickets cannot be easily identified simply by decoding the serial number.

One scratch list with multiple playing boards

The use of a single scratch list with multiple playing boards is another characteristic of the TicTacToe game that is shared by several of the other popular instant scratch games. Though most of the examples shown in Figure 5 use this style of play — Bingo, Keno, Lucky Lines, . . . — there are some games with unique cards that do not. The instant scratch game Slingo, for example, does have unique information visible on the face of the card, but uses only one playing board. The signiﬁcance of using a single playing board is that this (usually) limits the frequency of occurrence of a particular number (or symbol) from the scratch list. For instant scratch games like TicTacToe and Bingo, numbers can appear more than once on the playing boards; with games like Slingo, they cannot appear more than once. As discussed in the following section, the frequency of occurrence was the key to exploiting the TicTacToe game. Instant scratch games that use only a single playing board are therefore less susceptible to exploitation.

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The trick explained

The following procedure was developed by Mr. Srivastava for identifying winners and losers in the TicTacToe game:

Step 1: Replace each of the numbers on the playing boards with the number of times it occurs on all eight boards. Step 2: Locate all the “singletons”, the numbers that occur only once. Step 3: If the singletons form any winning alignments (row, column or diagonal), then the card is very likely a winner; if not, it is very likely a loser.

a) Original card b) Number of occurrences c) Singletons

Figure 7. The trick for identifying winners in the TicTacToe game.

An example of the use of this procedure is shown above in Figure 7. The original, unplayed card is shown in Figure 7a. In each of the 72 cells on the playing boards, Figure 7b shows how many times the number in that cell occurs on all eight boards. For example, the number

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24 that appears in the upper left corner of Board #1 appears a total of three times on the card: on Board #3 and again on Board #6; these locations where the number 24 appeared in Figure 7a have accordingly been recoded with the number 3 in Figure 7b. The number 11 that appears just to the right of the 24 on Board #1 appears only once on the entire card, so this cell has been recoded with the number 1 in Figure 7b. The number 07 that appears in the center of Board #1 appears once more on the card: on Board #8, so these cells have been recoded with the number 2 in Figure 7b. Figure 7c highlights the locations of the singletons; these are the cells that contained numbers that appeared only once on the entire card. On Board #6, the singletons form a tic-tac-toe alignment in the ﬁrst column, which indicates that this card is very likely a winner.

Implementation with and without computer support

Identifying the locations of the singletons and checking for possible tic-tac-toe alignments is extremely rapid on a computer. The 72 numbers can be entered manually in less than a minute, and the numerical computations take only a fraction of a second; replacing the tedious (and somewhat error-prone) task of manual data entry with automated electronic scanning technology would speed up the entire procedure to a few seconds. Even without the aid of a computer, the procedure is not diﬃcult. The visual/mental implementation requires a bit of concentration if one wants to do it rapidly, but if speed is not an issue, the vast majority of adults could perform the trick successfully. Many children, particularly those who retain eidetic memory6, have little diﬃculty performing the trick.

Accelerating the visual/mental implementation

If speed is critical, there is a way of visually scanning a card that considerably accelerates the procedure. On a single playing board there are eight winning tic-tac-toe alignments: the three columns, the three rows and the two diagonals. Every one of these eight winning alignments necessarily involves at least one of the numbers that falls on the main diagonal that passes from the upper left to the lower right of the 3×3 grid. One can therefore eliminate the playing boards one at a time simply by checking for repeated occurrences of the three numbers that occur on the main diagonal. 6Eidetic memory is the technical term for what is usually informally referred of as “photographic memory”, the ability to retain an accurate, detailed visual image of a complex scene or pattern. Many studies have conﬁrmed that most young children possess strong eidetic memory but that this weakens with age, a sad loss that appears to begin to occur around the time that children learn to read and to develop speciﬁc left-to-right patterns for scanning imagery.

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Using the example shown in Figure 7, the main diagonal on Board #1 is 24–07–14. With these three numbers committed to memory, one then scans the rest of the card to see if they all occur elsewhere. In about a second, most people find that 24 occurs on Board #3; a few seconds later, they have worked their way all the way to the last board, where they find that 07 and 14 both occur a second time. All told, it takes only four or five seconds to dismiss Board #1 as a possibility. One then moves onto Board #2, with a main diagonal of 03-09-02; the 03 occurs on Board #1, the 02 is found on Board #5, but 09 is nowhere else to be found. At this point, one knows that a winning alignment might pass through the center and a little bit more work is needed to check the various possibilities. There are three: the second column, the second row and the minor diagonal. All three of these must involve either the 13 in the upper right or the 25 that appears in the middle of column 1. Scanning the card for these two possibilities quickly finishes off Board #2 as a possibility; the 25 appears on Board #1 and the 13 appears on Board #3. When the main diagonal falls immediately, as it does for Boards #1, #3, #4 and #8 for the example shown in Figure 7, the board can be eliminated in a few seconds. When the main diagonal contains singletons, as it does for Boards #2, #5, #6 and #7, it takes a few more seconds to scan for an additional two numbers that might eliminate the card. The worst case scenario (in terms of time, not reward) occurs when there is a winning alignment of singletons. On Board #6, for example, it probably takes about 10 or 15 seconds of scanning the card for various possibilities to realize that the first column consists entirely of singletons. Experiments with this visual/mental implementation of the procedure have shown that it is not hard to get the scanning time for a single card down to about two minutes. The fastest time ever achieved with a visual scan was barely 20 seconds (for a card on which Board #2 was a winner). For losing cards, the fastest time ever achieved with a visual scan was just over one minute. Likely winners can generally be identified more quickly than likely losers because the confirmation of a likely losing card requires checking of all eight boards while the confirmation of a likely winner usually occurs before one reaches the last board.

Why the trick works

The reason that the Srivastava’s TicTacToe Trick works is that the frequency with which a particular number appears on the entire card is an indicator of whether or not that number is on the scratch list. Table 1 summarizes the frequencies of occurrence for the example shown in Figure 7; it also identiﬁes the numbers that appear on the scratch list7. The 16 singletons are 01, 04, 05, 09, 11, 12, 15, 16, 17, 18, 20, 27, 28, 30, 32 and 38. Of these 16 numbers, 15 of them are on the scratch list; the only one missing from the scratch list is 05. Analysis of a small group (about

7This is the same card used in earlier examples; its full scratch list can be seen on Figure 4.

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Table 1. Statistical summary of numbers on card in Figure 7a. # of Is it on # of Is it on # of Is it on times it scratch times it scratch times it scratch occurs list? occurs list? occurs list? 01 1 Yes 14 2 No 27 1 Yes 02 2 No 15 1 Yes 28 1 Yes 03 3 No 16 1 Yes 29 3 Yes 04 1 Yes 17 1 Yes 30 1 Yes 05 1 No 18 1 Yes 31 2 No 06 2 Yes 19 2 Yes 32 1 Yes 07 2 No 20 1 Yes 33 2 No 08 3 No 21 2 Yes 34 2 Yes 09 1 Yes 22 3 No 35 3 No 10 2 Yes 23 3 No 36 3 No 11 1 Yes 24 3 No 37 2 Yes 12 1 Yes 25 3 No 38 1 Yes 13 2 Yes 26 2 No 39 3 Yes

40) of other TicTacToe cards conﬁrms the relationship between frequency of occurrence and whether or not the number appears on the scratch list. When a number appears only once on all eight playing boards, the chance that it is on the scratch list is close to 95%. With each occurrence of a singleton being extremely likely to correspond to a number on the scratch list, a tic-tac-toe alignment of three singletons stands a very good chance of being a prize-winning alignment.

How successful is the trick?

False positives and false negatives

The statistical analysis of a procedure for exploiting a lottery game considers two diﬀerent aspects of success: the ability to correctly identify winners and the ability to correctly identify losers. For the purposes of this discussion, a “false positive” is a card that is deemed to be a likely winner but that is, in fact, a loser; and a “false negative” is the reverse: a likely loser that turns out to be a winner. With no ability to identify winners, if we simply treated every card as if it might be a winner (i.e. as a “positive”) the false positive rate would be the overall proportion of losing tickets 2 in the entire game. For the TicTacToe game, 3 of the tickets are losers, so the false positive rate for the simplistic strategy of treating everything as a winner would be 67% — of the cards purchased as “positives”, 67% would turn out not to be the winners we had hoped for. With no ability to identify losers, if we simply treated every card as if it might be a loser (i.e. as a “negative”) the false negative rate would be the overall proportion of winning

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tickets in the entire game. For the TicTacToe game, the false negative rate for the simplistic strategy of treating everything as a loser would be 33% — of the cards rejected as “negatives” 33% of them would, in fact, be winners. When developing a procedure for exploiting a lottery game, the ultimate (but practically unattainable8) goal is to drive both the false positive rate and the false negative rate to zero. A procedure that has both rates at 0% would never misidentify a card; every card it picked as a winner would, in fact, be a winner and every card it picked as a loser would, in fact, be a loser. In comparing two procedures for exploiting a lottery game, each with diﬀerent false positive and false negative rates, one needs to consider which of two mistakes is worse: occasionally buying losing tickets in the false belief that they are winners, or occasionally passing on winning tickets in the false belief that they are losers. Consider, for example, the following hypothetical case:

False positive rate False negative rate Procedure A 20% 10% Procedure B 25% 5%

Which is better? The answer depends on details such as the cost of tickets and the value of the prizes. If tickets are costly and prize money low, Procedure A looks better because it makes the costly mistake (buying losing tickets) less often. If tickets are cheap and prize money high, Procedure B looks better because the costly mistake is now missing out on valuable prizes, something that Procedure B does less often. Srivastava’s TicTacToe Trick focuses more on a low false positive rate than on a low false negative rate; it tries to make sure that the cards picked as winners are rarely losers and does not worry if this is achieved at the price of letting a few winners slip through into the pool of rejected cards. As discussed later in this report, it is possible to tweak the trick to lower the false negative rate (i.e. to make the mistake less often of passing on a winner), but this comes at the price of increasing the false positive rate (i.e. buying more losing cards).

Why do false positives and negatives occur with the TicTacToe trick?

The reason that false positives and negatives occur is that frequency of occurrence is not a perfect predictor of whether or not a particular number is on the scratch list. Even when a number shows up only once, it might not be on the scratch list; we saw an example of this in Figure 7 and Table 1, where 05 was a singleton but did not appear on the scratch list. Similarly, even when a number appears more than once, it might still be on the scratch list; Table 1 shows that nine of the scratch list numbers in that example did occur more than once.

8Unless the game is really badly messed up.

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Figure 8. A false positive. Figure 9. A false negative.

Figure 8 shows an example of a false positive, a card that is not a winner despite the fact that it does have an alignment of singletons: the 03, 27 and 19 in the middle column of Board #3 all appear only once on the entire card. Unfortunately, the 27 is not on the scratch list and, in this case, the alignment of singletons is misleading. Figure 9 shows an example of a false negative, a card that is a winner despite the fact that it does not have an alignment of singletons. On this card, the 29, 15 and 01 in the ﬁrst column of Board #6 are all on the scratch list, so the card is a $3 winner. Unfortunately, the 29 and 01 also appear elsewhere on the playing boards (the 29 shows up on Board #7 and the 01 on Board #2), and, in this case, the lack of an alignment of singletons is misleading.

False positive and negative rates based on empirical evidence

Of the 19 cards9 identified in June 2003 as likely winners using Srivastava’s TicTacToe Trick, only three were losers. Based on this limited empirical evidence, the false positive rate is close to 15%. In addition to checking the cards identified as likely winners, cards identified as likely losers were also checked in order to assess the false negative rate. Of the 38 cards identified as 9This includes the seven sent to Rob Zufelt at the OLGC plus twelve others that were identified earlier as the procedure was being studied and documented.

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losers using Srivastava’s TicTacToe Trick, only four were actually winners. Based on this limited empirical evidence, the false negative rate is roughly 10%.

False positive rate based on a probability argument

It is possible to check the empirical estimate of the false positive rate by comparing it to a ballpark estimate based on probabilistic reasoning. In studying the trick and getting a better sense for how and why it worked, a total of 37 cards were fully played. The information from these 37 cards forms a training data set that can be used to build a very reliable estimate of the probability that a singleton is on the scratch list. In the training set of 37 cards, there were a total of 606 singletons (an average of about 16 per card). Of these 606 singletons; 580 were on the scratch list. The probability of a number being on the scratch list given that it is a singleton is therefore very close to 95%. Knowing that 95% of the singletons are, in fact, on the scratch list, the probability that three singletons in a row will all be on the scratch list is 0.953 = 0.857. With three singletons in a row having roughly an 86% chance of being an actual winner, they have a 14% chance of being a loser. The estimate is only “ballpark” because there are some subtleties that have been left out of the argument. The actual false positive rate could be lower because even if the three singletons in a row turn out not to all be on the scratch list, the card might still be a winner because of some other unforeseen winning alignment. This kind of card is not strictly a false positive, but it is kind of unusual: a card picked as a likely winner that does turn out to be a winner, but not on the alignment identiﬁed by the singletons. Despite being only a ballpark estimate, the false positive rate from this probability argument is very close to the empirical estimate given above, so it seems reasonable to believe that if Srivastava’s TicTacToe Trick was used systematically on every TicTacToe card on the market, the actual false positive rate would end up being in the 10–15% range. Estimating the false negative rate in a similar manner is more diﬃcult, requiring additional information that cannot be reliably estimated from the training data set of 37 cards.

Authoritative estimates of false positive and false negative rates

Authoritative estimates of the false positive and false negative rates could be calculated from a comprehensive data base of the numerical information printed on each of the 4,000,000 cards in the TicTacToe game. Such a data base does exist, having been generated by Pollard Banknote Limited, the company that printed the tickets, to assist OLGC’s external auditors in their task of checking the integrity of the game.

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A different aspect of success: guessing the prize value One of the most intriguing features of the trick is that, in addition to identifying likely winners, it can also be used to predict the winning prize value. Since the prize amount depends on the location of the winning alignment, and since this is very likely to be exactly where three singletons line up, one can guess the prize value before scratching off anything. The vast majority10 of prizes in the TicTacToe game are $3 prizes for an alignment in the first column. Much rarer11 are $5 prizes for an alignment in the first row. Figure 10 shows an example of one card identified as a likely $5 winner because of the fact that three singletons were lined up in the first row of Board #7. Since the prize amount can be guessed, someone keen on exploiting the game need not buy many tickets. They can knowingly screen out the lower prize amounts, intentionally accepting a much higher false negative rate in order to avoid purchasing the $3 tickets that are revenue-neutral. Figure 10. Predicted $5 winner.

Making the trick work even better

Including some of the doubles

As noted earlier, many of the false negatives are the result of the winning alignment containing numbers that appear more than once on the playing boards. As indicated by Table 1 (and conﬁrmed with the much larger data set of 37 cards for which the scratch list is known), when a number occurs twice on the playing boards it has close to a 50-50 chance of being on the scratch list; but when it appears three times, it has only a 10% chance of being on the scratch list. Because “doubles” (numbers appearing twice on the playing boards) are much more likely to be on the scratch list than “triples” (numbers appearing three times on the playing boards), an alignment of two singletons and a double is considerably more promising than an alignment of two singletons and a triple. If one wants to catch some of the winner cards that previously were slipping through when only the singletons were being considered, one could modify the trick to include any alignment with two singletons and a double. The cost of this modiﬁcation is a slight increase in the 10According to information provided by the OLGC, 75% of the prize-winning tickets are $3 winners. 1115% of the prize-winning tickets are $5 winners.

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false positive rate (i.e. a few more losers are purchased); the beneﬁt is a decrease in the false negative rate.

Bayesian analysis

One of the most powerful theorems in probability is one developed by Reverend Bayes in the 1700’s. Appendix A contains an overview of Bayes’ Theorem and how it can be used to exploit the information printed on the face of the TicTacToe cards. For the current discussion, it is probably more useful to get a feel for what Bayes’ Theorem is all about rather than to bog down in the details of its mathematics. Bayes’ Theorem applies to situations where one has a “prior” guess about the likelihood that something will occur and then one acquires additional information that might permit one to modify one’s original guess. This is something that humans do all the time without even thinking about it. For example, we might hear on the morning radio that there is a 20% chance of rain today. As we head out the door, we notice that the sky is very dark and cloudy; rain seems imminent, so we head back to fetch the umbrella. In this example, we had prior knowledge that there was a 20% chance of rain. When we saw the sky, we acquired new information that caused us to believe that rain was actually much more likely than 20%, more like 90% judging by the dark clouds and gathering wind. In the language of Bayesian analysis, we “updated” the prior probability (20%) to create a “posterior” probability (90%). What Bayes’ Theorem provides is a specific formula for doing the updating. Bayes’ Theorem is particularly useful for a formal analysis of the TicTacToe game because there is a lot of ancillary information that impinges on the issue of whether or not a particular row, column or diagonal contains three numbers that are all on the scratch list. For example, the knowledge that $3 prizes are 50,000 times more common than the major prizes ($250 and greater) means that a card with a singleton alignment in the first column is much more likely to be a winner than a card with a singleton alignment on one of the diagonals.12 With all cards having a prior probability of 33% of being winners, Bayes’ Theorem allows us to update this in such a way that the location of the alignment is taken into account. For a singleton alignment in the first column, it turns out that the posterior probability of the card being a winner is close to 90%; for a singleton alignment on one of the diagonals, the posterior probability is higher than the prior (the singleton alignment is a promising sign), but it remains less than 40%. A Bayesian analysis of the TicTacToe game also allows one to use the information about doubles and triples in a rigorous manner. The implications of having two singletons and a double are intertwined with the previous issue regarding the location of the alignment. In some locations, the double is a promising sign (the card is more likely to be a winner); in 12There is still the issue of risk versus reward. Depending on how much lower the probability of the major prize was, many of us would still go ahead and buy the card, recognizing that the expected value of the card — probability of winning times the prize amount — might still be more than the cost of the ticket.

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others, it provides cautionary information (the card is less likely to be a winner). Another useful aspect of a Bayesian approach to the problem is that it allows one to incorpo- rate information on what is happening elsewhere on the playing boards. To avoid confusion on how much a player has won, each playing board can have only one winning alignment. On certain cards, treating a particular double as being on the scratch list will entail that one of the other boards has an illegitmate arrangement of two winning alignments. The whole set of eight playing boards has to hang together as a legitimate arrangment, and this provides constraints on which combinations of numbers can be together on the scratch list. The only drawback of trying to exploit the TicTacToe game through a formal Bayesian analysis is that one is now forced to use a computer; the numerical computations required cannot be done in one’s head, regardless of how much time is allowed. As discussed later, the reliance on a computer may not be much of a handicap to successfully exploiting an instant lottery game in actual practice. A program has been written to implement a full Bayesian updating of the prior probabilities (those reported by the OLGC as the breakdown of the number of tickets in diﬀerent prize categories) using the information printed on the face of an unplayed TicTacToe card. When tested on the training data set of 37 cards, it erred only twice: one false positive and one false negative. With the false positive and false negative rates both well below 10%, this approach to exploiting the game would, if widely practiced, result in a net loss of revenue for this one game. Applied to the entire set of 4,000,000 cards printed for this game, the Bayesian approach would accept only about 1.5 million as likely winners, rejecting the other 2.5 million as likely losers. The gross revenue would be $3 × 1,500,000 = $4,500,000. The total amount of prize money paid out would be roughly $5,500,00013.

13About 90% of the total prize money; some of the prizes would be left in the false negatives, the unpur- chased tickets that were actually winners.

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In order to prevent reccurrence of the same (or similar) problems, it is necessary to try to understand why the problem with the TicTacToe game occurred. Such an exercise inevitably involves speculation at this point; the author of this report is not in a position to thoroughly investigate all possibilities and cannot present ﬁndings supported by forensic evidence. Nev- ertheless, it is important to make sure that plausible explanations are tabled so that others can make sure they are properly explored.

An unintentional and unforeseen ﬂaw

The most likely explanation for the TicTacToe cards being eﬀectively marked is that those who wrote the software used in printing the cards did not recognize the diﬃculties created by a creative concept that called for visible and unique information to be printed on the face of each card. The most straightforward and obvious way to program the printing of such cards is as follows:14

1. For each card, decide if it is going to be a winner or a loser. 2. If it’s going to be a winner, lay down a pattern of X’s and O’s that has the desired alignment of X’s in exactly the correct position. This is easy to do since there are only a few hundred possible arrangments of X’s and O’s; these can be calculated beforehand, stored and then searched for an arrangement that has the X’s in a certain location. 3. If it’s going to be a loser, lay down a pattern of X’s and O’s that has no winning alignment in the X’s. As with setting up a winning card with its X’s in exactly the right position, ensuring that there are no winning alignments is a simple task of searching through a precalculated data base of all possible arrangments and selecting from one of the many that have no winning alignments. 4. Having fixed the location of the X’s and the O’s, the remaining tasks are to choose 24 numbers for the scratch list, assign the X locations numbers from this list and assign the O locations numbers that do not appear on the list. 14A less efficient, but perhaps simpler alternative is to replace Steps 2 through 4 with a trial and error procedure. Randomly assign 72 numbers to the playing boards and 24 numbers to the scratch list and then check to see if the result produces the kind of winner or loser chosen in Step 1. This type of trial and error procedure would also create the same kind of flaw because the configurations produced in the trial and error step are more likely to correspond to the desired outcome if the numbers on the scratch list are singletons or doubles on the playing boards.

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The problem with this procedure is in the very last step. With the X’s and O’s being roughly equal in number, there are somewhere around 36 cells that will be assigned numbers from the scratch list and 36 that will be assigned numbers not on the scratch list. Since the scratch list contains 24 possibilities, one ends up assigning 24 numbers to about 36 X-cells and only 15 numbers (there are 39 – 24 = 15 numbers not on the scratch list) to about 36 O-cells. With 15 numbers going into 36 cells, each number is going to be used, on average, twice or more. With 24 numbers going into 36 cells, each number is going to be used, on average, once or twice. The heart of the problem, therefore, is that there are far more numbers on the scratch list than off it (a result, in part, of someone’s decision that the choices should go from 1 to 39), but the number of cells that need to be populated with numbers from each group is roughly equal. The consequence of this is that the number of repetitions of a particular number on the eight playing boards ends up providing clues to whether or not that number is on the scratch list. It is possible to write a computer program that avoids this kind of problem, but it is a somewhat complicated task if one of the specifications of the creative design is that the scratch list have 24 numbers and that the game use only the numbers from 1 to 39. Two easier solutions, both of which require the original creative concept to be modified, would be: 1) use fewer numbers on the scratch list (about 19); or, 2) allow the numbers in play to span a greater range (as discussed below, 1 to 52 would be a better range).

Evidence of an unintentional ﬂaw

If the flaw in the TicTacToe game was due to unforeseen complexity in the task of writing software that implements the creative concept, then one would expect to see similar oversights in similar games. Preliminary investigations into some of the games shown in Figure 5 suggest that this is, indeed, the case. The card-specific information printed on the face of the cards often yields clues to the numbers (or letters, or symbols) hidden under the scratch list; Bayesian updating of the prior probabilities can be used to improve the odds of winning on several of the most popular instant scratch games. For certain variations of Bingo, the false positive and false negative rates can be reduced to the point where the profitability of the game would be compromised if the trick was widely known and practiced.

The possibility of intentional fraud

Even though a ﬂawed software implementation seems like the most plausible explanation for the problems with the TicTacToe game, it is also important to consider another possibility: that the ﬂaw was intentional. It is, unfortunately, entirely predictable that any attempt to explore this possibility will immediately encounter resistance. To those responsible for the

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game, from conception to printing to marketing and distribution, the mere suggestion of fraud will be offensive, interpreted as an insult to their genuine commitment to maintaining integrity. In the past fifteen years of consulting to various industries, FSS has often had to deal with this type of reaction; people generally don’t like the implications of a suggestion that we explore, together, whether or not somebody was knowingly doing something wrong. Yet our experience has taught us that this is one of the most important and useful roles that consultants play: raising the questions that no-one else wants to deal with and making sure they get properly addressed. Regrettably, fraud does exist in this world. It is especially common wherever large sums of money are available. With instant scratch lottery revenues running into the tens of billions of dollars annually across North America, the ability to skim even a fraction of 1% could earn someone millions of dollars. The most successful frauds are often those that go unchallenged for a long time because no-one bothered to make sure that someone was not doing what everyone assumed was undoable. One of the salves to the wounded pride of those insulted by allegations of fraud is this: successful frauds are often conducted by remarkably few people (ideally, only one) and do not implicate the integrity of the vast majority of their colleagues. What does cause colleagues and co-workers to get swept into the whirlwind created by the revelation of fraud is the failure to exercise due diligence and to explore with all of their professional talent and skill the suggestion that someone else has chosen not to play by the rules. The Bre-X saga, very familiar even outside the mining industry, is a good case in point. Even when something started to smell fishy at the Busang site, there was a chorus of industry professionals who said “No way! A salting scam couldn’t possibly have been perpetrated on that massive a scale without someone having noticed something before now”. In hindsight, it appears that perhaps as few as two people did, in fact, pull it off (and with truly enviable precision and skill) for several years, and under the noses of many of the industry’s best and brightest. Hindsight also reveals that there were several points in time where a specific investigation into the possibility of salting would likely have revealed the scam. The reason that litigation continues on this case, long after most of the principals have died or vanished, is that many people and organizations, from disgruntled shareholders to regulatory agencies, are angered by the failure of key players to exercise due diligence. Having given all these warnings in an attempt to overcome opposition to consideration of the fraud possibility, it is worth reiterating the earlier observation: even though there is no specific evidence at hand that either supports or refutes the possibility that the flaw was intentional, the author of this report personally takes the view that the more likely explanation is an entirely unintentional software mistake. This is offered not to discourage further investigation into the possibility of fraud, but simply to reassure those whose noses are a bit out of joint at this point that their trust in the integrity of others is shared. At the same time that further investigation is strongly recommended (because, without actual

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evidence, we’re all just guessing), it seems quite likely that it will demonstrate that the ﬂaw was nothing more than an unfortunate accident.

Fraud limited to the TicTacToe game would have to be modest

If an intentional attempt at fraud underlies the flaw in the TicTacToe game, the fraud would have been modest fraud in terms of the amount of money it targeted. The top prizes in the TicTacToe game are exceedingly rare: only four $50,000 prize winning tickets (out of a total of 4,000,000 tickets in the game) and only four $1,000 tickets. The perpetrator(s) of the scam, if it existed, would have to content themselves with skimming small prize money since it is extremely unlikely that they would stumble across one of the eight top prizes. With a few million dollars in small prize money available through this game, it seems unlikely that any single person could casually exploit the flaw for more than several thousand dollars. For most people intent on fraud, this is simply too small. But for someone who wasn’t planning on committing fraud, and who simply recognized an opportunity to make some small extra income by allowing a software flaw to persist, the scale of the potential reward might seem acceptable.

Diﬃculty of prosecution

The modesty of such a potential fraud would be compensated by the fact that it would be very difficult to prosecute, even if detected. The fact that the trick for reading the cards is not perfect — it sometimes produces an incorrect prediction — makes it seem less like intentional fraud. A “probabilistic fraud”, one that is successful over the long run but that may occasionally misfire in the short run, would open up new legal territory in the area of what constitutes “reasonable doubt” in criminal prosecutions. And even if an individual was identified as being responsible for designing the critical piece of software, it is very doubtful that poor software design would be regarded as compelling evidence of fraud in this day and age when people have come to accept that software always has problems. So a modest fraud could be perpetrated indefinitely and with little risk of prosecution, staying just below the OLGC’s radar by focusing on tickets worth less than $200 that can be redeemed for cash, without ID, at thousands of retail locations.

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This section outlines several factors that contributed to the exploitation of the TicTacToe game. These have been called “contributing factors” rather than “causes” because it would be hard to think of any one of these, by itself, as “the reason” for the problem. Most of these factors can be seen, individually, as positive aspects of the instant scratch game industry. But when other problems occur, such as a faulty software implementation, the interconnectedness of the issues quickly becomes apparent as previously benign aspects of the lottery industry start to interact in ways that makes the problem worse. The reason for drawing attention to these contributing factors is not to suggest that they all need to be changed or ﬁxed but only to help the process of thinking about changes that might, in future, help to prevent problems or, at least, mitigate their severity.

Game design and creative concept

Allowing cards to contain visible and unique information

State and provincial lottery organizations are always searching for new ways to attract lottery consumers, many of whom are more drawn to the kind of game shown in Figure 5 — one that contains card-specific information visible prior to playing the game — than to the kind of game shown in Figure 6 — one whose cards appear identical. Having card- specific information visible to the player definitely enables a style of play that is engaging and interesting; there is a sense of anticipation, created by discovering what is on the scratch list, checking for its occurrence on the playing boards, noticing the start of a pattern that might turn into a winner and hoping that a particular item soon turns up on the scratch list. As noted earlier, however, the existence of card-specific information that is visible to the player also opens up the possibility of this information being used to exploit the game.

Tweaking the concepts of others

State and provincial lottery organizations are quick to try to mimic the successes of each other. Popular games in one jurisdiction are quickly borrowed, adapted and tweaked to suit a new market. The fact that the entire market is serviced by a handful of companies that are capable of printing instant scratch games accelerates this process as a printing company will often draw a client’s attention to successful games that it prints for other clients. The OLGC is not the only lottery organization that runs a TicTacToe game; Colorado, for example, runs one with four playing boards and a scratch list containing 10 numbers;

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Kentucky runs one with six playing boards and a scratch list containing 24 numbers. The existence of similar games in other jurisdictions indicates that the OLGC’s TicTacToe game was one of the instant scratch games that was adapted from elsewhere and tweaked. Similar tweaking goes on with other games. Nebraska’s version of Crossword, for example, incorporates words with local significance. Some states run a game similar to Lucky Lines, but with additional symbols that have local relevance. The problem with tweaking games is this: if the original concept was sound (i.e. the original game could not be exploited), changing the game mechanics, even slightly, may introduce an unforeseen flaw. And if the original concept was not sound (i.e. the original game could be exploited, even if no-one every realized it), then changing the game mechanics may turn a minor flaw into a major problem.

Statistical quality assurance and quality control

Software QA/QC

There is a field of study, supported by a growing technical literature, that addresses the problems of applying QA/QC principles to software design. Beginning with functional specifications for each individual module in a program, and extending to functional specifications for an entire program, this field of study provides procedures for writing source code, doc- umenting it, testing it, controlling modifications and providing an auditable and traceable history. Unfortunately, very few industries have adopted software QA/QC principles as part of their standard practice. One of the few that has is the high-level radioactive waste industry, one of the areas where geostatistics is applied and where FSS has considerable experience. The lottery industry (along with most other industries) appears not to have any systematic QA/QC procedures and protocols that pertain to software.

Quality assurance and industry standards

QA/QC has two parts: assurance and control. The lottery industry does have procedures that pertain to quality assurance: the existence of an external audit report for the TicTacToe game (and for most other instant scratch games) demonstrates that there are procedures in place for checking quality. Unfortunately, “checking” (even double-checking or triple- checking) is not the same as “assuring”. In order to assure that something is going to happen (or that something else in not going to happen), one needs to have technical speciﬁcations that prescribe appropriate behavior. In many industrial processes that use statistical QA/QC, the industry has developed speciﬁc

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technical standards that serve as probabilistic targets; for example: “This machine will produce copper wire whose diameter is, 99.9% of the time, within one micron of one millimeter”. Without such specifications, the best an auditor can hope to do is “check” — to document what is and what is not happening. If an auditor is charged with the responsibility of quality assurance, they also need to be given technical specifications that define “acceptable quality”. National and international standards organizations such as the American Society for Testing and Materials (ASTM), the International Standards Organization (ISO) and the National Institute for Standards and Technology (NIST), to name but a few, exist to help governments and industries develop the standards needed to enable proper quality assurance. FSS has assisted these groups in developing standards, so we know that the task is slow and tedious15, but the reward is considerable: quality assurance that properly deserves the title “assurance”. The revelation of a flaw in the TicTacToe game raises important QA/QC issues for the entire lottery industry. With the OLGC having made the decision (a correct one, we believe) to pull the game from the market, one of the major lottery jurisdictions in North America has set a precedent: instant scratch games should not be left in circulation if someone (or some group of people) has substantially better odds of winning than others. This is a very good precedent to have, one that helps to demonstrate the industry’s unwavering commitment to the fairness and integrity of the games it provides. But what constitutes a “substantial” improvement in the odds? For the TicTacToe game, the original winning odds were 33% (1 in 3); by showing the OLGC that he could correctly predict 19 out of 20 cards, both winners and losers, Mr. Srivastava demonstrated that the odds could be “substantially” improved. This demonstration, convincing though it was, did not establish the exact improvement in the odds; there was a chance that he was simply lucky and that the success would not be as remarkable on a second attempt. What should state and provincial lottery organizations do in future to respond appropriately when similar problems arise? How do they quickly establish the technical basis for their decision? Should they require that the details of the procedure be revealed, or is a demonstration of the success rate (with no details provided) sufficient? How big a demonstration is needed? Does the entire data base of printed tickets need to be checked? How convincing does the demonstration need to be? If Mr. Srivastava had got only 14 out of 20 correct, would that merit further action? All of these are questions that currently don’t have answers. These questions do not become irrelevant even if the industry takes steps to ensure that software problems don’t recur. So long as there is some information on the face of the card — and a serial number will suffice — there will be some procedure that can be used to improve the winning odds. If the software and printing procedures are working as intended, then the improvement should be small, but it will never be zero. The auditor’s task then becomes one of assuring that there is no procedure that can change the winning odds by more than some small, prespecified amount. But what that small amount should be has not 15A lot of nitpicking over specific wording, with various parties trying to defend their vested interests.

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yet been decided. Should an auditor sign off on a game if they discover a procedure that can improve the winning odds from 33% to 35%? What if the winning odds were doubled to 66%, would that be enough to declare the game unmarketable? Or does the change in the winning odds have to approach the level of success of the Srivastava’s TicTacToe Trick, somewhere above 85%? Should the auditor focus only on the false positive and/or false negative rate, or should they also take into account information on the ticket price and the prize amounts to ensure that the game will not incur a financial loss? Until the lottery industry develops standards that prescribe the criteria for a game to be deemed acceptable, these types of questions will remain difficult to answer. It is fortunate, in a peculiar way, that the flaw discovered in TicTacToe enabled the game to be exploited with such a high success rate that the decision on whether or not to allow the game to remain on the market was a no-brainer. Had the flaw enabled a card-reading trick that had only modest success, the lack of industry-wide standards would have made the decision more difficult.

Quality control

The second part of QA/QC is the monitoring and feedback mechanisms that allow quality to be properly controlled. The lottery industry is now able to do real-time monitoring of instant scratch ticket sales and prize redemption. In Ontario, for example, even though winners are not obliged to redeem the smaller prize tickets through the OLGC, they still need to go to a lottery retailer who scans the bar-coded ticket and provides an authentication code to confirm that the ticket is genuine. This allows real-time data to be gathered on the redemption of all prize winning tickets. Some states, such as Virginia and Washington, provide Internet access to daily updated data bases of the remaining prizes in any game; many other jurisdictions track this information but do not make it publically available. The availability of this type of monitoring information makes it easier to detect when a particular instant scratch game is being exploited. Unfortunately, it appears that even though the monitoring data exists, few lottery jurisdictions have a feedback mechanism that sets off warning or alarm bells when the statistics of actual prize redemption no longer fall within predetermined quality control limits. Since the lottery organization knows the number of prize winning tickets in various amounts16, it could precalculate statistical quality control 16Even this is starting to become complicated by a style of instant scratch game known as a “probability game” in which a ticket is not predetermined to be a winner or loser. Instead, the amount that a player can win depends on what they choose to scratch off, and in what order. GTECH, among others, is promoting this type of game as an exciting and attractive new alternative to the more traditional style of instant scratch ticket. With this type of game, it becomes even more important that the industry improve its procedures for analyzing the possibility that card-specific information can be exploited. Even if all the cards initially appear identical, by virtue of the player’s control over what they scratch, and in what order, they may all succumb to computer-aided analysis of the possibility of Bayesian updating.

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limits for how many prize winning tickets of a certain value should be redeemed when a certain number of tickets have been sold. If the statistics of actual prize ticket redemption start to wander into a range that was determined at the outset of the game to be implausible, this could trigger some kind of remedial action — temporary suspension of distribution, perhaps, pending the results of a speciﬁc type of investigation. If the statistics continue to wander, reaching a range that was determined at the outset to be virtually impossible, this could trigger more drastic suspension — termination of the game, for example, and an immediate recall of all unsold tickets. The key to statistical quality control is the predetermination of the range of acceptable behavior and the creation of response protocols for deviations outside of this range. As things currently stand, without these predetermined ranges and response protocols, the lottery industry is in the position of depending on anyone who discovers a problem to report it immediately rather than use it. Human nature being what it is, this is a dangerous strategy. The entire industry would be on a much safer footing if it was able to detect anomalies in the prize winnings of any game and, equally important, to respond appropriately and swiftly to any anomalies that are detected.

Distribution and marketing

Allowing consumers to pick and choose their cards

The ability to exploit a game depends, in part, on the ability to inspect the cards prior to purchase. Some lottery jurisdictions allow retailers to use the “silent sellers”, the display tablet that usually sits beside the cash register and that allows lottery customers to see the faces of the speciﬁc tickets available for immediate purchase; other jurisdictions prefer to use ticket dispensing machines that do not allow buyers to pick and choose their lucky tickets. The advantage of the silent seller culture is that it allows customers to have a sense of control by following their instinct on the tickets that feel lucky; for certain games, particularly Bingo, this is likely part of the attraction of the game. In actual bingo halls, those smoke- ﬁlled auditoriums packed with hardcore players with their special daubing markers and their shrines of luck-enhancing paraphenalia, many avid players like to pick and choose the cards they play, feeling that they have a knack for spotting the good ones. The existence of a similar possibility in the instant scratch version of the game probably accounts for some of its popularity. Even for instant scratch games that are not based on a some other popular game of chance, there are many players who will always prefer being able to choose their ticket. This opportunity helps to reinforce the sense of the game’s integrity — the refusal to allow the customer to choose their own tickets risks raising the concern that the game might not be as fair as the lottery organization is pretending.

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Allowing consumers to trade cards

The perceived integrity and fairness of instant scratch games makes most lottery retailers willing to allow customers to trade unplayed cards that feel unlucky for others that have a better feeling. Though this practice is very likely not sanctioned by the lottery organizations, it is widepsread17. As long as the customer has not scratched anything off the card, why wouldn’t a lottery retailer allow someone to trade? If the trade-in has not been folded, spindled or mutilated, if it still looks like a new card, the retailer will have no difficulty finding another customer for it. And because everyone kind of chuckles at claims of the ability to spot lucky cards, believing all cards to be equally likely to be winners, there is no reason for the suggestion of a trade-in to be regarded with suspicion. Most lottery retailers who agree to allow trade-ins will agree to take tickets that were not actually purchased from them; none actually bothered to check the batch code in the serial number to see if the ticket was originally purchased at their store. They will also allow tickets from one game to be traded for tickets of the same value from a different game. All of these accommodations to the nervous customer make it far easier to exploit a flaw in an instant scratch ticket game. In particular, they make it possible to run a much more sophisticated computer-based analysis of a ticket. Batches of tickets can be purchased, with the likely winners being retained and the likely losers being returned as part of the purchase price of the next batch. In this scenario, the lottery organization is not the only “loser”. The retailers themselves would likely end up getting the short end of the stick as other lottery customers, ones who are playing instant scratch games without any awareness that others are able to selectively buy the winners, start to develop a sense that most of the tickets at particular retail outlets are losers. Those outlets then risk being regarded as “unlucky” and being passed over in favor of other outlets that are not so obviously cursed.

Allowing retailers to return unsold cards

Part of what makes the current lottery distribution and retail system work is the assurance that retailers can return unsold tickets to their distributor. Without this, retailers would be assuming some financial risk for unsold tickets. Unfortunately, this feature of the system also creates a return mechanism for someone exploiting an instant scratch game. One of the few problems with actually trying to exploit the TicTacToe game on a wide scale would have been the issue of what to do with tickets that were identified as likely losers. One solution would have been to avoid buying likely losers, 17In an actual experiment in downtown Toronto, 18 of 20 lottery retailers readily agreed to the suggestion (put to them by someone who was considering buying a few tickets for a supposedly bedridden grandmother) that the customer should be able to trade them in if they got home and granny berated then for buying “unlucky” tickets . . . “You see, she thinks she’s had a bit of luck with this game so she figures she can spot the winners; I just play along because it makes her feel good”.

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but this requires that the buy/don’t-buy decision be made rapidly and in front of the retailer. As discussed earlier, this was actually not very diﬃcult for the TicTacToe game; but other games might not be so amenable to visual/mental card-reading tricks. The exploitation of any game is made easier by any return mechanism (like trading in cards) that facilitates the disposal of unwanted tickets. Unlike trading tickets, which most retailers would do without any sense of wrongdoing, returning to the distributor tickets that had been winnowed out as likely losers would probably have to involve the retailer as a co-conspirator. As noted earlier in the section that discussed the possibility of intentional fraud, raising the possibility of collusion with retailers is not intended to suggest that retailers are dishonest. It is raised only to draw attention to the fact that the current retail and distribution system has a feature that makes it easier to commit fraud in a multi-billion dollar industry. The appropriate response to this fact is not to question the honesty of lottery retailers; it is, instead, to ensure that the distribution and return systems include protocols that make it extremely diﬃcult (if not outright impossible) for anyone to return tickets that someone has been able to analyze and reject.

Lack of legal remedies

Part of what enables instant scratch games to be exploited is that it is not specifically illegal to do so. While some would feel that there was something a bit dubious about raiding the prize money in a game that was supposed to be fair to all players, others would take the view that if a government sanctioned lottery organization marketed a flawed game, then it was ultimately the government’s problem that clever people were able to get rich at the expense of others. In Canada, there was recently an investigation into whether or not it was illegal for a math- ematician to exploit a weakness he recognized in a casino game. Having won the big prize twice, he immediately came under suspicion and it was initially assumed that he must have done something illegal. It eventually became apparent that his only “crime” was being smarter than those who designed the game and he was allowed to keep his substantial winnings. In Nevada, courts have ruled that blackjack players are free to use any information that is made available to them, provided that there is no collusion between a player and casino personnel. At the same time, it is illegal in Nevada to use any kind of computational support, even a small handheld mechanical device, to assist with card counting in blackjack. Even though it would likely be impossible to prevent an individual from exploiting a flaw they had discovered in an instant scratch game, it might be possible to prevent them from sharing this information with others. Even in jurisdictions with government sanctioned lotteries, there are public morality laws that may make it illegal to teach gambling or to disseminate information on how to improve one’s odds of winning. The Supreme Court of

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Michigan, for example, cited such laws in its 1996 decision to uphold the state’s decision to deny a gambling school permission to operate: “the purpose of the current scheme of criminal laws remains to suppress gambling as an activity injurious to public morals and welfare” (this in a state with a multibillion dollar casino and lottery system!). So it might be possible to seek a legal injunction preventing the dissemination of information on how to exploit an instant scratch game. Apart from the possible use of public morality laws to prevent dissemination of information on flawed games, the law does little to assist the lottery industry in preventing the exploitation of instant scratch games with unforeseen flaws. There are, of course, laws that deal with intentional fraud; but, as noted earlier, even if an intentional fraud was at the root of the flaw in the TicTacToe game, prosecuting it would be extremely difficult since reasonable doubt exists in a procedure that includes false positives and false negatives and in a culture that generally accepts that software may contain bugs.

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This section goes through each of the problems and contributing factors and discusses potential solutions. These possible solutions are offered in a “brainstorming” mode, with no preconceptions about whether or not they are good or bad. Indeed, the fact that we are industry outsiders makes it hard for us to assess all the pro’s and con’s, to consider issues related to the cost of implementation or to weigh the tradeoffs with other aspects of running a government sanctioned lottery. We feel it is important to tackle the exercise of identifying possible solutions in this mode; even though many (if not most) of the possibilities will eventually fall by the wayside, this exercise of getting ideas out on the table, both good and bad, is more likely to lead to genuine improvements in the lottery industry than would an exercise that quickly dismissed potential solutions as unworkable. Following the brainstorming dump of possible solutions, there is a brief discussion of the pro’s and con’s that even an industry outsider can recognize, followed by a specific recommendation.

The “steady-as-she-goes” non-solution

One response to the problems identified in this report is simply to do nothing, to cross our fingers and hope that similar problems won’t occur again or, if they do, that no-one will notice. Even if the spirit of genuine brainstorming requires that this be considered as one possible “solution”, common sense dictates that it be rejected. We know that at least one instant scratch game was exploitable despite the best efforts of the OLGC, its printer and its external auditor. Earlier in this report, the appearance of similar flaws in other games was discussed, so it is apparent that the TicTacToe problem was not a one-off fluke. If the lottery industry is sincere in its wish to maintain the integrity and fairness of its instant scratch games, then it has to undertake changes to prevent the reccurrence of this type of problem.

Recommendation No. 1 The lottery industry should not treat the problem identified with the TicTacToe game as a fluke, a one-off problem that requires no specific response because it is never likely to occur again. Problems similar to the one revealed in the TicTacToe game have already been identified with other instant scratch games currently on the market. An entire category of instant scratch games is at risk: those that use tickets with unique information visible on their face prior to being played. With these types of game already being among the most popular

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and successful on the market, their market share is likely to increase. Problems similar to the one discovered in the TicTacToe game are likely to occur again unless the industry takes speciﬁc preventative steps.

Game design and creative concept

Allowing cards to contain visible and unique information

Possible solutions:

1. Don’t use games that have card-specific information. Remove even the bar-coded serial number as a possible avenue for exploiting instant scratch games by concealing it beneath a latex coating marked “void-if-removed” that is to be scratched off only by the lottery retailer redeeming the card. 2. Continue to use them, but make sure they are implemented in such a way that does not allow the information visible on the face of the card to be exploited to significantly increase the winning odds. 3. Continue to use them, and popularize the fact that the TicTacToe game was broken in an attempt to attract a new type of consumer: people who think they can break the code.

The first of these is actually already being done by certain lottery jurisdictions. It is not clear if states like Georgia and Massachusetts have made a policy decision to avoid certain types of games, but it is interesting to note that the lotteries in these two states do not currently included any instant scratch games that have card-specific information (except the serial number). Some jurisdictions are also currently concealing the serial number beneath a “void-if-removed” scratch-off coating. Though this certainly helps prevent attempts to decode the serial number, there is currently no evidence to indicate that this has ever been a problem. Furthermore, the serial printing order remains obvious when ticket dispensing machines are used, regardless of whether or not the serial number is concealed. The second possibility requires a commitment to improving other aspects of the printing, distribution and marketing of instant scratch tickets (see further discussion below). The third possibility is risky, but might turn a problem into an opportunity. In a Break The Code instant scratch game, players would be able to see visible, card-specific information and would have to decide which of several spaces they will scratch off. If it was clearly explained that there is a way of figuring out from the visible information which space(s) are winners,

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the “unfairness” of being able to crack the code evaporates. The game does not pretend to oﬀer an equal chance of winning to everyone; instead, it oﬀers a kind of puzzle that, once solved, will give certain people a better chance of winning. By making the card-reading procedure successful in a probabilistic sense (like the TicTacToe card-reading trick was), one could control the game’s payout. The popularity of such a game would depend on marketing the game to those with a particular kind of puzzle-solving mentality, like computer hackers, cryptographers and people who actually do the puzzle page in the newspaper.

Recommendation No. 2 The lottery industry should continue to use instant scratch games that provide card-speciﬁc information, but should take steps to ensure that they are printed in such a way that this information cannot be used to signiﬁcantly increase the odds of winning.

Tweaking the concepts of others

Possible solutions: 1. Don’t tweak popular games; either use them as-is, with the same game mechanics used elsewhere, or don’t use them at all. 2. Allow tweaking of the game mechanics to occur only when there is an analysis of the probabilistic implications of the proposed modiﬁcations.

Though the first possibility may seem like the safer solution, it does not address the possibility that the game being copied already contains an exploitable flaw. The second caters more to the current reality of the industry — people do like to fine tune an already successful game to suit their knowledge of their own market — but requires only that such tweaking be accompanied by a specific analysis of how the proposed enhancements might affect the game.

Recommendation No. 3 All instant scratch games with card-speciﬁc information should be accompanied by a report on the various aspects of the game mechanics — e.g. the number of playing boards, the range of numbers (or symbols) used and the size of the scratch list — and the tolerances within which these can be acceptably adjusted. Since there are few printers that serve a much larger number of lottery jurisdictions and since printers currently assume responsibility for the details of the actual implementation of the creative concepts, it makes sense that printers take

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the lead in doing the analysis of game mechanics and for preparing documentation that can be used to make sure that variations of the game do not become exploitable when key parameters are modiﬁed.

Analysis of parameters controlling TicTacToe game mechanics

One remedy to the problem with the TicTacToe game is to maintain an appropriate balance between the total number of numbers used in the game, the number that appear on the scratch list and the proportion of X and O spaces18 used on the playing boards.

Let PX be the average proportion of X spaces on the playing boards of all cards and PO be the average proportion of O spaces. Since every space is either an X or an O, PX + PO = 1. For the OLGC’s version of the TicTacToe game, PX appears to have been about 0.46 and PO 19 about 0.54. Let Ntotal be the total number of numbers used in the game. For the OLGC’s version of TicTacToe, with the numbers going from 1 to 39, Ntotal = 39. Let Nscratch be the total number of numbers that appear on the scratch list. For the OLGC’s version of TicTacToe, Nscratch = 24. The TicTacToe would not have been exploitable if the ratio of the numbers on the scratch list, Nscratch, to the total number of numbers used in the game, Ntotal, had been acceptably 20 close to PX . In the case of the OLGC’s version of the game,

Nscratch = 0.62 but PX ≈ 0.46 Ntotal

A simple remedy to the problem would be to change the game mechanics so that the numbers used in the game go from 1 to 52 (one could use symbols corresponding to the 52 cards in a deck!). With this simple modiﬁcation,

N scratch = 0.462 Ntotal

which is acceptably close to the proportion of X spaces that need to use numbers from the scratch list. 18For this discussion, an “X” space is one that contains a number from the scratch list and an “O” space is one that contains a number that does not appear on the scratch list. 19It would not be diﬃcult to make the cards have exactly half the spaces be X’s and half the spaces be O’s, but this does not appear to have been a design constraint in the OLGC’s version of the TicTacToe game. 20How close is “acceptable” depends on the number of playing boards.

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Software quality assurance and quality control

Possible solutions:

1. Have printers institute a QA/QC program for software used in printing instant scratch lottery tickets. 2. Have external auditors review the computer source code and input parameters used to create instant scratch tickets. 3. Develop benchmark standards for validating new implementations of software modules used in the printing of instant scratch tickets — e.g. the random number generators, the procedures for creating winning/losing cards in speciﬁc games. 4. Reassign the responsibility for specifying the information to be printed on the instant scratch tickets. Rather than delegating this task to the printer, as is currently done, assign it to a group that specializes in applied probability, statistics and customized software development for lottery applications.

The first of these is a very broad solution that would require the lottery industry’s printers to commit significant resources to a new way of thinking about the software they develop and use. Many industries have mistakenly assumed that because computer programming can be done cheaply21 it should be done cheaply. Unfortunately, software development for lottery games requires a level of knowledge of applied probability and statistics that programmers often do not have. Regardless of whether or not a formal software QA/QC program is put in place, the second suggestion would help to avoid the kind of oversight that likely caused the problem with the TicTacToe game. If the industry is reluctant to commit the resources needed to ensure that its software is not creating exploitable flaws, then an intermediate solution is to spend a little bit more to make sure that the programs being used are externally reviewed by people qualified to comment on the reliability of the program for the intended application. The third suggestion borrows an idea that has been successful in several industries: create a standardized suite of tests that anyone can use to test that their programs are functioning properly. In situations where there is a reluctance to permit source code to be externally reviewed, it increases the sense of comfort in a newly developed piece of software if it can be demonstrated to pass well established benchmark tests. Though the final suggestion is the one least likely to win the endorsement of lottery ticket printers (because it takes work away from them), it is the solution most likely to prevent the reccurrence of the kind of problem that affected the TicTacToe game. As it currently stands, the division of responsibilities in bringing an instant scratch game to market usually leaves it up to the printer to determine the details of what gets printed on each card. This 21Either by entry-level junior staff or by offshore contract programming service bureaus.

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work is then checked, usually by an external auditor hired by the lottery organization that contracted the printer. What printers excel at is printing. Many of them have also established strong secondary lines of business in services and technologies that relate to their primary line of business. Judging by their own web-sites, none of the major instant scratch ticket printers places much stock in their expertise in applied probability and statistics programming. This may be because it seems less important than their many other qualifications; it could also be a sign that expertise in this area has simply never occurred to them as something that might be critical. Whatever the reason, it is clear that their business emphasis, their raison d’être as a company, is not software development. Any software development that they do end up doing is a sideline that supports other business activities that are perceived as being more central to their corporate mission. In this regard, the lottery industry is not very different from many others. Apart from companies that have made software development their main line of business, most companies view software development as a necessary but minor aspect of their business. Some companies have reached the point of concluding that software development is not their business and make it a matter of company policy that any software development required by their main line of business must be contracted out to qualified software developers. Though such policies often rankle those within the organization who have some talent or interest in programming, there are sound business reasons for placing responsibility for software (and liability for its results) in the hands of a group that has the skills and experience required to write, document and maintain software for specific applications. Though it is not currently structured this way, the lottery industry would likely work better if there existed a new kind of technical services group, one that took the creative concept from the lottery organization, determined exactly what needed to be printed on each and every ticket, and then provided the printer with a data base of ticket information. The introduction of this new kind of technical services group does not entail any loss of accountability or auditability; in the same way that lottery ticket printers currently provide information to external auditors, this new kind of technical services group could do the same. What the introduction of this new kind of technical services group does entail is an improvement in the quality of the final product. Freed from the constraints of the printer’s primary line of business, such a group could focus on the probability, statistics and computer science issues that arise when innovative creative concepts for instant scratch games are developed. Printers would be freed of the responsibility for choosing what goes on the tickets and from any liability that results from problems in this step of the process; they would simply print onto each ticket the information provided to them by the lottery organization’s chosen contractor.

Recommendation No. 4 Lottery organizations should carefully consider the pro’s and con’s of changing the industry’s current practices regarding who determines what exactly gets

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printed on instant scratch tickets. Introducing a technical services group to handle this task, one that specializes in customized software development for lottery applications and that liaises with other contractors chosen to handle other aspects of bringing a game to market — the printers and the auditors, for example — would help to make sure that games cannot be exploited because of unforeseen probabilistic ﬂaws.

Quality assurance and industry standards

Possible solutions:

1. Develop industry-wide consensus standards for what constitutes a “fair” game and what constitutes a “signiﬁcant” enhancement in the winning odds. 2. Provide to the auditors responsible for checking instant scratch lottery games the software (both executables and source code) that determines exactly what gets printed on each ticket, along with its input parameters. 3. Start using the tools and principles of applied cryptography to investigate the possibility that card-speciﬁc information can be used to exploit instant scratch games.

The first possible solution addresses the need for the lottery industry to start doing more than paying lip service to the notion of quality assurance; specifically, industry-wide standards are needed, ones that contain technical specifications for fairness and integrity. The lead- ership of this effort properly rests with the agencies that have been set up by various state and provincial governments to administer the government sanctioned lotteries. The term “consensus standards” refers to the approach used by standards development organizations like the ASTM. The most successful standards, those that are widely followed and that hold up under scrutiny in legal proceedings, are those developed using a consensus approach in which everyone affected by a proposed standard is invited to have a hand in its development; anyone can object to any detail of the proposed wording and the committee responsible for the standard is obliged to respond substantively, either by making appropriate changes or by providing specific explanations for declining to do so. According to the mission statement of the North American Association of State and Provin- cial Lotteries (NASPL), the function of this organization includes “facilitating communication among lottery organizations regarding development of industry standards and matters of mu- tual interest, particularly those which relate to the integrity, security and efficiency of state and provincial lottery jurisdictions”. This seems like the ideal organization to launch and coordinate the development of the standards needed to give meaning to “quality assurance”.

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Even if the industry is not able to cultivate the consensus needed to create widely used standards for quality assurance, the second suggestion above addresses one of the problems that exists with the current system. When an auditor is asked to verify the integrity of a particular instant scratch game, their hands are somewhat tied if they are not allowed to review the details of the software that generated the information printed on the tickets. It would be easier to identify flaws in a game if the auditor was given access to the software, including the original source code, and could check whether or not appropriate parameters were used. Problems with the game will often be more readily apparent when the auditor can examine the genesis of the information printed on the tickets (instead of only seeing the final product). The final possible solution raises the possibility that the auditing of instant scratch games would be more effective if auditors extended their normal menu of statistical checks to include procedures that view the game as a decryption problem. The frequency of occurrence statistics shown in Table 1, for example, are exactly the kind of thing that readily occurs to cryptographers.

Recommendation No. 5 State and provincial lottery jurisdictions should lead an eﬀort to strengthen the lottery industry’s quality assurance procedures by establishing industry-wide standards for statistical properties that warranty the integrity and fairness of instant scratch games.

Quality control

Possible solutions:

1. Start using existing data on ticket sales and prize redemption to provide feedback on the actual distribution and winning patterns of instant scratch games. 2. Establish industry-wide standards for what constitutes an “acceptable” distribution/ winning patterns and what constitutes appropriate responses to deviations from this norm. 3. Study the possibility that the TicTacToe game was being exploited in Ontario before the OLGC learned that the game was exploitable.

The ﬁrst two of these possible solutions address the need for quality control in the lottery industry. As noted earlier, monitoring data does exist but, by itself, this does not constitute a quality control program. The industry needs to get into the habit of using this data to

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keep its finger on the pulse of each game. Part of this is predetermining appropriate bands for the number of prize winning tickets that should be redeemed when a certain number of tickets have been sold. Another part is industry-wide consensus on what should count as an anomaly in the distribution/winning patterns. The final idea suggests a first exercise in using data from sales and prize redemption: inves- tigating the possibility that the TicTacToe game was being exploited in Ontario prior to late June 2003. Even though the prevailing feeling is currently that Mr. Srivastava was the first to recognize the flaw in the game, it is important for the OLGC to establish whether or not this was, indeed, the case. With no evidence one way or the other, it remains possible that Mr. Srivastava was only the first to reveal knowledge of a trick that many others had also discovered. If the card-reading trick was being widely used to exploit the game prior to June 2003, this should be apparent in the spatial and temporal patterns of ticket sales and prize ticket redemptions. Since the data for this exercise exists, there is an enviable opportunity to undertake a hindsight quality control study that has direct relevance to the security and integrity of the industry’s instant scratch games.

Recommendation No. 6 The State and provincial lottery jurisdictions should lead an eﬀort to introduce statistical quality control procedures for continuous, real-time veriﬁcation that instant scratch games are not being exploited.

Allowing consumers to pick and choose their cards

Possible solutions:

1. Use ticket dispensing machines. 2. Continue to allow consumers to choose.

There are not many choices here; either consumers are allowed to choose their cards or they are not. Though vendors of the ticket dispensing machines already in use in some jurisdictions report that many retailers prefer these machines to the silent sellers, there is little data on how consumers respond. As noted earlier, the appeal of certain instant scratch games like Bingo is based on their similarity to and resonance with another popular game of chance in which players typically are allowed an element of choice. It is possible that the loss of consumer choice at the retail outlet will result in a loss of sales of certain instant scratch games and that this loss will not be oﬀset by increased sales of other games (i.e. that certain consumers will not buy any tickets if they aren’t allowed to choose).

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One of the drawbacks of removing consumer choice in jurisdictions where it currently exists is that this large a change in the tradition and culture of buying lottery tickets will inevitably cause some to wonder if the lottery organization has reason to doubt the integrity and fairness of the current system. If there are such doubts, it would be preferable to solve the problems and restore integrity and fairness (by correcting underlying software problems, for example) rather than papering over the problem by removing consumer choice.

Recommendation No. 7 The lottery industry should study the advantages and drawbacks of removing the element of consumer choice from instant scratch lottery games. Speciﬁcally, they should gather data from any existing studies of consumer response to ticket dispensing machines and commission new studies if existing studies are inadequate or inconclusive.

Allowing consumers to trade cards

Possible solutions:

1. Disallow the informal practice of allowing unplayed instant scratch lottery tickets to be traded in. 2. Inform retailers about the potential disadvantages of this practice, and let each retailer to decide if they will allow it. 3. Make no changes to the current distribution and retail practices.

The diﬃculty of barring an existing practice that is based on a belief in the integrity of the lottery is that it will raise doubts about the lottery’s integrity. Nevertheless, this may be the best way to curtail the use of computer-aided attempts to exploit instant scratch games. The second choice tries to accommodate the fact that it is virtually impossible (or, at least, prohibitively costly) to enforce a ban on trading unplayed cards; if a new rule can’t be enforced, why rock the boat by insisting on the change? It might make more sense to educate retailers about the price that they may pay — becoming viewed as an “unlucky” retailer that sells mostly losing tickets — if they accommodate this practice. The third possible solution — let the status quo continue — aims to avoid widespread discussion of potential problems with the integrity and fairness of the instant scratch games. The more the problem gets discussed, the greater the likelihood that others will become aware of the possibility that these games are exploitable. If the industry addresses the root causes of the problem and is able to develop games that cannot be exploited in the way that

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the TicTacToe game was, then it may not be necessary to curtail the practice of allowing unplayed tickets to be traded in. One of the solutions from the previous section is also relevant here: the use of ticket dispensing machines. In jurisdictions where these are used, there is no way for retailers to accommodate a customer’s request for a trade-in.

Recommendation No. 8 State and provincial lottery organizations should focus on restoring and maintaining the integrity and fairness of all instant scratch lottery games. They should not duck the problem by imposing restrictions on retailers; such restrictions would correctly be perceived as a concession that some instant scratch games can be exploited.

Allowing retailers to return unsold cards

Possible solutions: 1. Make no changes to the current procedures for returning unsold tickets. 2. Allow retailers to return only unopened packets of tickets. 3. Do not allow retailers to include in their returns any tickets that were not originally distributed to their retail outlet.

Like the issue of allowing trade-ins, the issue of constraints on the return of unsold tickets risks raising the profile of the underlying concern that certain types of instant scratch games might be exploitable. The first suggestion takes the view that the responsibility for the problem lies with the lottery organizations and not with the retailers. The second takes the opposite view: that retailers assume some financial risk when they decide to break open a packet of tickets (which are usually sold in packets of 50–100). The third solution simply tries to limit the damage that can be done by a single retailer. As long as individual retailers can return, at most, the losing tickets that were distributed to their one outlet, then someone trying to exploit a flaw in an instant scratch game is severely constrained. The serial number typically includes a printing batch number or package number. In jurisdictions where instant scratch tickets show their serial number, it is therefore easy to quickly determine whether or not a particular ticket was originally distributed to a particular retail outlet. As with the trade-in issue, the return of unsold tickets would not be much of a problem if ticket dispensing machines were used. Even though the use of these machines may not

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be welcomed by certain consumers, the fact that it eﬃciently solves many of the potential problems with the existing distribution and retail systems makes it worthy of serious consideration.

Recommendation No. 9 In lottery jurisdictions where consumers are allowed to choose their instant scratch tickets, retailers should not be allowed to return unsold tickets that were not originally distributed to them.

Lack of legal remedies

Possible solutions:

1. Clarify the laws that pertain to exploiting lottery games. 2. Prepare boilerplate injunction applications.

The fairness of instant scratch games for the average player would be enhanced if all lottery jurisdictions had a law similar to the one in Nevada that outlaws the use of computational support for gambling. Even this may not be enough to prevent attempts at exploiting instant scratch games, because it is not clear that they would apply to using a computer to assist the process of developing a visual/manual technique. The analogy with card counting in blackjack is again useful. Even in Nevada, where you’re not allowed to lug a laptop computer with you into a casino, you are allowed to sit at home and use your computer to reﬁne a card counting system and to hone your skills. In addition to checking to make sure that existing laws are up to date and able to deal with aggressive attempts to exploit instant scratch games, it would be useful to ﬁnd out if there is a legal basis for applying for an injunction to prevent the dissemination of information. Rather than scrambling at the last minute to prevent the Toronto Star from publishing a how-to article in its weekend edition, it would be better to establish beforehand the legal basis for seeking such an injunction.

Recommendation No. 10 State and provincial lottery jurisdictions should investigate the legal remedies at their disposal to mitigate or prevent exploitation of instant scratch games.

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Bayes’ Theorem provides a way of updating the prior probabilities of a set of events, B1, . . . , Bn when another event, A has been observed:

P (A|Bi)P (Bi) P (Bi|A) = j=n (1) j=1 P (A|Bj)P (Bj) P For the TicTacToe game, the events B1, . . . , Bn are the 372 possible conﬁgurations of X’s and O’s that can be used on the playing boards.22

The prior probability of a particular conﬁguration, Bi, can be calculated from knowledge of the number of tickets with various prize values. This information is provided by the OLGC through a toll free phone number. For the TicTacToe game, the breakdown is shown in Table A.1:23

Table A.1. Number of cards with various prize values. Location of winning Number alignment Prize amount of tickets Probability None $0 2,663,180 0.6877692 First column $3 1,000,000 0.2582511 Second column $20 4,000 0.0010330 Third column $100 500 0.0001291 First row $5 200,000 0.0516502 Second row $10 4,000 0.0010330 Third row $100 500 0.0001291 Main diagonal $250, $1,000 or $50,000 10 0.0000026 Minor diagonal $250, $1,000 or $50,000 10 0.0000026 3,872,200 1.0000000

Table A.1 provides the total probability for several configurations that share the same kind of winning alignment. For example, 20 of the 372 possible configurations have an alignment of X’s in the first column. Table A.1 tells us that the sum of the probabilities for these 20 configurations is 0.258; the probability of any one of these 20 is therefore 0.013. 22As before, the X’s mark the cells with numbers on the scratch list and the O’s mark the cells with numbers that are not on the scratch list. 23Table A.1 omits the 127,800 cards that have winners on multiple playing boards. These are a bit awkward to handle in this analysis because they include combinations of winning alignments. Ignoring these makes the results somewhat approximate, but since they represent less than 2% of the total cards, the approximation is still very reliable. Rather than leaving them out of the count, an alternative would be to include these with the configurations that have a winning alignment in the first column, since the vast majority of them do have this particular winning alignment (along with the first row, from time to time).

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Table A.2 reports the number of conﬁgurations that fall in each of the categories reported in Table A.1 and also reports the prior probability of any one conﬁguration.

Table A.2. Prior probabilities. Location Number of Prior probability of winning Cumulative possible for any single alignment probability conﬁgurations conﬁguration None 0.6877692 220 0.0031262 First column 0.2582511 20 0.0129126 Second column 0.0010330 19 0.0000544 Third column 0.0001291 20 0.0000065 First row 0.0516502 20 0.0025825 Second row 0.0010330 19 0.0000544 Third row 0.0001291 20 0.0000065 Main diagonal 0.0000026 17 0.0000002 Minor diagonal 0.0000026 17 0.0000002 372

The event A is the frequency of occurrence of the numbers on the playing boards, an example of which was shown on Figure 7b. Given a particular arrangement of frequency of occurrence, certain configurations become more likely, others become less likely. Figure A.1 shows the example of the first playing board from Figure 7b, along with two possible configurations of X’s and O’s. Since both come from the first category in Table A.2, the prior probability of both configurations is the same; if we didn’t have the benefit of knowing the frequencies of occurrence, there would be no reason for Configuration A in Figure 7b to be more likely than Configuration B. After studying the frequencies of occurrence, however, it is clear that Configuration A, which has its X’s in locations where the frequency of occurrence is 3, is less likely than Configuration B, in which the X’s occur at locations with lower frequencies.

Conﬁguration A Conﬁguration B

Figure A.1. Board #1 from Figure 7b with two possible conﬁgurations of X’s and O’s.

The exact calculation of how much more likely Conﬁguration A is than Conﬁguration B is done using Bayes’ Theorem. The numerator in Equation 1 shows that the prior probability, P (Bi), increases by a factor of P (A|Bi) in the Bayesian updating. P (A|Bi) is the probability

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that the observed frequencies will occur given that the actual conﬁguration of X’s and O’s is Bi. This conditional probability is calculated as follows:

f=4 Nx i (f) No i (f) B NxBi (f) B NoBi (f) P (A|Bi) = g=f P x(f) g=f P o(f) (2) g=1 NxBi (g) ! g=1 NoBi (g) ! fY=1 P P where:

• NxBi (f) is the number of X cells on the playing board that have a frequency count of f when we use conﬁguration Bi as the template for determining which are the X cells;

• NoBi (f) is the number of O cells on the playing board that have a frequency count of f when we use conﬁguration Bi as the template for determining which are the O cells; • P x(f) is the probability that an X cell will contain a number with a frequency of f; • P o(f) is the probability that an O cell will contain a number with a frequency of f.

The probabilities P x(f) and P o(f) need to be compiled from a training data set, a group of TicTacToe tickets that have been played and that can be used to study the relationship between the frequency of occurrence and the presence/absence of a number on the scratch list. Table A.3 shows empirical values for P x(f) and P o(f) using a data base of 37 cards. The 0.6531 in the P x(f) column for f = 1 means that in our data base of 37 cards, 65.31% of the X spaces contained a number that appeared only once. Continuing on down the P x(f) column, the table records the fact that 30.43% of the X spaces contained a number that appeared twice, 4.26% of them contained a number that appeared three times, and none of them ever contained a number that appeared four times. In the P o(f) column, the table records the fact that 4.65% of the O spaces contained a number that appeared only once, 32.71% of them contained a number that appeared twice, 62.33% of them contained a number than appeared three times, and 0.16% of them contained a number that appeared four times.

Table A.3. P x(f) and P o(f) as calculated from 37 cards.

f P x(f) P o(f) 1 0.6531 0.0465 2 0.3043 0.3271 3 0.0426 0.6233 4 0.0000 0.0016

Continuing with the example shown in Figure A.1, the P (A|Bi) term for Conﬁguration A is:

3 3 2 2 4 4 P (A|Bi) = 0.0426 0.0465 0.3271 3 ! 6 ! 4 !

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3! 6! 4! = 0.04263 · 0.04652 · 0.32714 3!0! 2!4! 4!0! = 2.87 × 10−8

while the P (A|Bi) term for Conﬁguration B is:

2 2 1 1 3 3 3 3 P (A|Bi) = 0.6531 0.3043 0.3271 0.6233 3 ! 1 ! 6 ! 3 ! 3! 1! 6! 3! = 0.65312 · 0.30431 · 0.32713 · 0.62333 2!1! 1!0! 3!3! 3!0! = 6.60 × 10−2

For the frequencies of occurrence that appear on Board #1, Configuration B is more than 2,000,000 times more likely than Configuration A.24 Even though the two configurations were equally likely before the frequency information was taken into account, the information contained in the frequencies of occurrence allows us to make very clear distinctions about which configurations are likely and which ones are not. The updating of the prior probabilities is done one playing board at a time, considering all of the 372 possible configurations, resulting in posterior probabilities for each of these 372 configurations. Summing over the 152 configurations that are winners, we get an estimate of the probability for that particular board to be a winner. Once the probabilities for a winner on each playing board have been calculated, the probability for the entire card to be a winner is calculated as follows:

i=8 PCard is winner = 1 − 1 − PBoard #i is a winner (3) i Y=1

24The denominator in Equation 1 is the same for all configurations and serves as a rescaling term to ensure that all the posterior probabilities correctly sum to 1. The relative effect of the Bayesian updating can therefore be assessed simply by evaluating the first term in the numerator.

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