PROBLEMS WITH AN INSTANT SCRATCH LOTTERY GAME: An

Transcription

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
 2003, R. Mohan Srivastava
NOT TO BE COPIED WITHOUT COPYRIGHT HOLDER’S PERMISSION
froidevaux
srivastava
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:
Address:
R. Mohan Srivastava
2179 Danforth Avenue, #302
Toronto, Ontario
Canada M4C 1K4
Douglas R. Hartzell
5782 Golden Eagle Drive
Reno, NV 89523
U.S.A.
Phone: (416) 693-7739
Fax: (416) 693-5968
E-mail: [email protected]
Phone:
Fax:
E-mail:
(775) 846-5811
(775) 201-0256
[email protected]
ii
froidevaux
srivastava
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.
iii
froidevaux
srivastava
schofield
EXECUTIVE SUMMARY
iv
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.
froidevaux
srivastava
schofield
EXECUTIVE SUMMARY
v
2. The lottery industry should continue to use instant scratch games that provide cardspecific 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
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
vi
froidevaux
srivastava
schofield
EXECUTIVE SUMMARY
vii
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
froidevaux
srivastava
schofield
EXECUTIVE SUMMARY
viii
Allowing retailers to return unsold cards . . . . . . . . . . . . . . . . . . . . . . .
42
Lack of legal remedies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
APPENDIX A: USING BAYES’ THEOREM
44
froidevaux
srivastava
schofield
LIST OF TABLES
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
ix
froidevaux
srivastava
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
x
froidevaux
srivastava
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 misunderstandings 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
1
Not 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.
1
froidevaux
srivastava
schofield
2
INTRODUCTION
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.
froidevaux
srivastava
schofield
3
INTRODUCTION
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 cardreading trick.
froidevaux
srivastava
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.
2
On 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.
3
As 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.
4
froidevaux
srivastava
schofield
5
GAME MECHANICS
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.
froidevaux
srivastava
schofield
6
GAME MECHANICS
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.
4
Figure 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.
froidevaux
srivastava
schofield
7
GAME MECHANICS
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
5
Reported in the February 2003 edition of Public Gaming International magazine.
froidevaux
srivastava
schofield
8
GAME MECHANICS
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.
froidevaux
srivastava
schofield
9
GAME MECHANICS
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.
froidevaux
srivastava
schofield
2
THE TRICK
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
10
froidevaux
srivastava
schofield
11
THE TRICK
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.
6
Eidetic 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.
froidevaux
srivastava
schofield
12
THE TRICK
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
7
This is the same card used in earlier examples; its full scratch list can be seen on Figure 4.
froidevaux
srivastava
schofield
13
THE TRICK
01
02
03
04
05
06
07
08
09
10
11
12
13
Table 1. Statistical summary of numbers on card in Figure 7a.
# of
Is it on
# of
Is it on
# of
times it scratch
times it scratch
times it
occurs
list?
occurs
list?
occurs
1
Yes
14
2
No
27
1
2
No
15
1
Yes
28
1
3
No
16
1
Yes
29
3
1
Yes
17
1
Yes
30
1
1
No
18
1
Yes
31
2
2
Yes
19
2
Yes
32
1
2
No
20
1
Yes
33
2
3
No
21
2
Yes
34
2
1
Yes
22
3
No
35
3
2
Yes
23
3
No
36
3
1
Yes
24
3
No
37
2
1
Yes
25
3
No
38
1
2
Yes
26
2
No
39
3
Is it on
scratch
list?
Yes
Yes
Yes
Yes
No
Yes
No
Yes
No
No
Yes
Yes
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
in the entire game. For the TicTacToe game, 32 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
froidevaux
srivastava
schofield
14
THE TRICK
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
Procedure A
20%
Procedure B
25%
False negative rate
10%
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.
8
Unless the game is really badly messed up.
froidevaux
srivastava
schofield
15
THE TRICK
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
9
This 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.
froidevaux
srivastava
schofield
16
THE TRICK
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.
froidevaux
srivastava
schofield
17
THE TRICK
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
10
11
According to information provided by the OLGC, 75% of the prize-winning tickets are $3 winners.
15% of the prize-winning tickets are $5 winners.
froidevaux
srivastava
schofield
18
THE TRICK
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
12
There 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.
froidevaux
srivastava
schofield
19
THE TRICK
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 incorporate 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 .
13
About 90% of the total prize money; some of the prizes would be left in the false negatives, the unpurchased tickets that were actually winners.
froidevaux
srivastava
schofield
3
PRIMARY CAUSE
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. Nevertheless, 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.
14
A 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.
20
froidevaux
srivastava
schofield
21
PRIMARY CAUSE
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
froidevaux
srivastava
schofield
22
PRIMARY CAUSE
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
froidevaux
srivastava
schofield
23
PRIMARY CAUSE
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.
froidevaux
srivastava
schofield
4
CONTRIBUTING FACTORS
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 cardspecific 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;
24
froidevaux
srivastava
schofield
25
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, documenting 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.
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 triplechecking) 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
froidevaux
srivastava
schofield
26
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 tedious 15 ,
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
15
A lot of nitpicking over specific wording, with various parties trying to defend their vested interests.
froidevaux
srivastava
schofield
27
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
16
Even 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.
froidevaux
srivastava
schofield
28
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 smokefilled 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.
froidevaux
srivastava
schofield
29
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,
17
In 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”.
froidevaux
srivastava
schofield
30
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 mathematician 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
froidevaux
srivastava
schofield
31
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.
froidevaux
srivastava
schofield
5
SOLUTIONS AND RECOMMENDATIONS
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
32
froidevaux
srivastava
schofield
33
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,
froidevaux
srivastava
schofield
34
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.
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.
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
froidevaux
srivastava
schofield
35
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
about 0.54.19 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
close20 to PX . In the case of the OLGC’s version of the game,
Nscratch
= 0.62
Ntotal
but
PX ≈ 0.46
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,
Nscratch
= 0.462
Ntotal
which is acceptably close to the proportion of X spaces that need to use numbers from the
scratch list.
18
For 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.
19
It 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.
20
How close is “acceptable” depends on the number of playing boards.
froidevaux
srivastava
schofield
36
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
21
Either by entry-level junior staff or by offshore contract programming service bureaus.
froidevaux
srivastava
schofield
37
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.
Lottery organizations should carefully consider the pro’s and con’s of changing the industry’s current practices regarding who determines what exactly gets
froidevaux
srivastava
schofield
38
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.
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 leadership 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 Provincial Lotteries (NASPL), the function of this organization includes “facilitating communication
among lottery organizations regarding development of industry standards and matters of mutual 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”.
froidevaux
srivastava
schofield
39
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.
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
froidevaux
srivastava
schofield
40
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: investigating 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.
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).
froidevaux
srivastava
schofield
41
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.
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.
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
froidevaux
srivastava
schofield
42
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.
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
froidevaux
srivastava
schofield
43
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.
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.
State and provincial lottery jurisdictions should investigate the legal remedies
at their disposal to mitigate or prevent exploitation of instant scratch games.
froidevaux
srivastava
schofield
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) = Pj=n
j=1 P (A|Bj )P (Bj )
(1)
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.
22
As 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.
23
Table 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).
44
froidevaux
srivastava
schofield
45
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.
Location
of winning
alignment
None
First column
Second column
Third column
First row
Second row
Third row
Main diagonal
Minor diagonal
Table A.2. Prior probabilities.
Number of
Cumulative
possible
probability
configurations
0.6877692
220
0.2582511
20
0.0010330
19
0.0001291
20
0.0516502
20
0.0010330
19
0.0001291
20
0.0000026
17
0.0000026
17
372
Prior probability
for any single
configuration
0.0031262
0.0129126
0.0000544
0.0000065
0.0025825
0.0000544
0.0000065
0.0000002
0.0000002
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
froidevaux
srivastava
schofield
46
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:
P (A|Bi ) =
fY
=4
f =1
where:
N xB (f )
Pg=f i
g=1 N xBi (g)
!
P x(f )
N xBi (f )
N oB (f )
Pg=f i
g=1 N oBi (g)
!
P o(f )N oBi (f )
(2)
• N xBi (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;
• N oBi (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.
P x(f )
0.6531
0.3043
0.0426
0.0000
f
1
2
3
4
P o(f )
0.0465
0.3271
0.6233
0.0016
Continuing with the example shown in Figure A.1, the P (A|Bi ) term for Configuration A is:
P (A|Bi ) =
3
3
!
0.0426
3
2
6
!
0.0465
2
4
4
!
0.32714
froidevaux
srivastava
schofield
47
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
1
3
3
P (A|Bi ) =
0.65312
0.30431
0.32713
0.62333
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:
PCard is winner = 1 −
i=8
Y
1 − PBoard #i is a winner
i=1
(3)
24
The 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.
froidevaux
srivastava
schofield

PROBLEMS WITH AN INSTANT SCRATCH LOTTERY GAME: An

Transcription

Similar documents

Hoosier Lotto Prize Pool Sheet

A Retailer`s Guide to Selling Maine State Lottery Tickets

2nd Annual Hospital Cash Lotto Poster

thin lizzy down hollywood undead the gaslight

PlayCentral - Scientific Games

N11 0805 Movie Ticket Prize Draw.cdr - C

BOY SCOUTS NIGHT OUT AT BLUE MAN GROUP!

April 24 News - Chautauqua County Community USD 286

Lottery Group Play Form

Wednesday February 4 2009