Proof

PROOF

By Jerry Tuttle, FCAS
Society of Actuaries 2000 Actuarial Speculative Fiction Contest

    Economists named the first half decade of the new millennium "The Flood Years", referring to the Biblical story of Noah and the flood. Noah took two of each animal aboard his ark, and all the rest died in the flood. The early 2000ís were flood years for insurance companies: a period of intense merger and acquisition in the insurance industry, culminating in only two surviving insurance animals. By 2005 the brokers, life insurers and property-casualty insurers put aside their differences and combined forces into two giant competing insurance organizations. Most of the organizations supporting the insurance industry Ė consultants, rating and statistical organizations, and so on Ė either were bought up or quietly died. The two remaining organizations handled all possible insurance needs with immense cost efficiencies. Finally, the Federal government warned that this is where the merger activity would stop; it would allow these two competing organizations, but it would not permit a final merger into a single monopolistic organization.

    "Actuarial Unemployment Reaches 65%!" screamed the headline of the Society newsletter. By 2005 the CAS and SOA had also merged, not due to membersí demand, but rather because the two mega-employers had become multi-line and balked at supporting two actuarial societies. With the massive downsizing of employees of the new insurance organizations, the new single actuarial society spent much of its time calculating and forecasting the new actuarial unemployment index.

    The majority of Society educational efforts were about non-traditional actuarial careers. Thousands of actuaries clamored to get into fields which had previously been ignored by actuaries Ė sports bookmaking and airline pricing were two popular ones Ė anywhere where actuaries could use their actuarial skills to forecast the financial consequences of future contingent events.

    My name is Andrea Morgan, and I am one of those 65% unemployed actuaries. I had an idea for a non-traditional application of my actuarial skills. But I knew I would need a partner. I searched my e-mail directory of colleagues and found the one person who had the skills, the personality and the deviousness I needed: Don Burnett.

    I invited Don to have a coffee at The Mall at Short Hills and to ponder our future careers. Don is a computer expert in both hardware and software, who has an odd sense of humor. Don is the sort of person who once secretly hooked up a digital camera aimed at the company presidentís office waste paper basket, and then broadcast the results in real time over the company Intranet. He and I were among the unemployed, both divorced and in our late forties, and our careers had overlapped at several firms over the years. We were people-watching at the mall as we sipped our over-priced coffees.

    The Mall at Short Hills (TMASH), in Short Hills, New Jersey, is an upscale mall; even its name shouts that it is upper-class. It has stores like Nordstromís and Nieman Marcus. This is a mall you might go to if you were in the market for a fur coat. The last time I bought something here, I bought a gift certificate for a friendís wedding. I canít remember when I last bought something here for myself. A large portion of the people walking around are tanned and toned, and even in casual clothes they are expensively dressed. Don and I looked out of place here among the beautiful people with their Gucci shopping bags, but we didnít seem to mind.

    "Don, I have an idea for a business we could start that would combine your computer skills with my actuarial skills. I think we can make some decent money. Unfortunately, Iím not sure it is entirely legal."

    Don perked up. He was clearly interested.

    "This is an expensive mall where people buy expensive things," I began as I looked around at the stores. "If we could predict that a particular person was going to buy something today, I think that is useful information that a store manager might pay us for. Of course, I am only in the early stages of thinking about this."

    Don was silent for a long time as he thought about this.

    An attractive 30-year old woman walked by and Don watched her. "Did you see that woman?" Don asked me. "Suppose I had a way of knowing that she was married and her husbandís birthday was coming up. With the right technology, I could find that out. Then would you be able to actuarially forecast what she might buy her husband?" he asked me.

    Don had taken the beginning of my idea much further than I thought was possible. I didnít think I could forecast what someone might buy as a birthday present for her husband; the possibilities were limitless. But I was intrigued that Don thought he could acquire some basic facts about a random shopper. I asked him how he proposed to do that.

    "Easy," he replied with a twinkle in his eye. "Suppose I had a scanning device at each mall entrance. I scan each personís wallet for their credit cards. I download the name and number off the credit card into a database. I match the name against another database of census information of marriage and birthday data, and Iím done. Or QED, as you actuaries like to say."

    "Close your mouth, Andrea," Don said. I was shocked that such a thing was so easy. "In fact, I have a prototype of such a system at my front door now."

    "Go on," I said.

    "I really hate strangers who ring my doorbell, especially during dinner," Don explained, "so I installed a scanner on my front door. As soon as someone steps on my doormat, I scan his credit cards and display his name on my computer. If itís someone I donít already know, Iíll assume itís someone asking for contributions and Iíll just yell out the window, ĎNo thanks.í Of course the system isnít perfect Ė for some reason it didnít identify the girls selling Girl Scout cookies."

    I was very impressed! "How legal do you think this is, Don?" I asked.

    "Actually I did ask some lawyer friends about it. The lawyers were not in complete agreement. The laws have not caught up to technology. Although this sounds like it ought to be invasion of privacy, it isnít. Plus the scanning is not physically harmful, and itís not like Iím taking someoneís credit card information and selling it to a mailing list company."

    My mind raced with the possibilities Don suggested.

    "Suppose I was only interested in women who work and who earn salaries above $150,000," I continued. "Suppose I could construct a mathematical model that will predict whether such a woman was likely to buy something today. Could you identify those women in the mall from scanning and matching them against the right databases?" I asked.

    "Yes I could," Don replied. "I have a friend Paul at the Census Bureau who can get me Census and income tax data. I could also find out where someone worked. But every woman in this mall today is probably going to buy something. Maybe itís a cup of coffee, like you bought, or maybe itís a $10,000 diamond ring. Are you going to make separate mathematical models for everything?"

    Don had brought me back to earth. I didnít think I could model buying a cup of coffee. People buy their coffee with cash, so there would be no database history on that. Besides, what would a coffee shop do differently if I could model it? Now the diamond ring was different. People buy jewelry with credit cards. What if I had someoneís jewelry purchase history? If I thought a particular person had an 80% probability of buying a diamond ring, I bet a salesman would pay good money for that information because he would give that customer plenty of extra attention.

    Don and I talked about various things people buy at malls. We were searching for a high-ticket item with some homogeneity that people bought with some frequency. Jewelry wouldnít work Ė different kinds of jewelry were not homogeneous enough to model. Musical instruments, furniture, furs, and so on were bought too infrequently.

    "What about womenís clothes?" I asked. "Expensive clothes, not just T-shirts or jeans?"

    This seemed plausible to both of us. Although the thought process in deciding to buy clothes is complicated, this seemed like something I could model if I put my mind to it.

    "Oh really?" asked Don skeptically. "I have an idea. Letís pick some women and follow them around the mall. You predict for me in advance if you think they are in a buying mood," Don challenged me.

    This was exactly the sort of off-the-wall idea Don would have, but it sounded good to me. We followed a number of women around. Interestingly, once a woman got into a shopping trance, we could be standing very close to her without her noticing us. I had a few theories on womenís buying habits such as the better dressed someone is, or the more time she has spent in the mall without buying something, the more likely she is to buy, but none of my theories held. Plus, the enormous variety of womenís clothing seemed overwhelming from a modeling viewpoint.

    "Although I do enjoy following attractive women around, I donít think this is going to work," Don admitted.

    "How about menís clothing?" I suggested.

    We both liked this idea. Menís clothing, limited to suits, had more of a finite nature to it than womenís clothing. We decided to limit the project to men buying menís clothing; women who buy a manís suit for a man struck us as impossible to model. We walked around the mall until we found a high class menís store. We peered at the window, not wanting to go in, until we caught the managerís eye and he came out to talk to us.

    "Can I help you?" he asked politely, looking us up and down and no doubt making his own probabilistic judgment that we were dressed too shabbily to buy something from him.

    I decided to level with him and explain our plan. He introduced himself to us as Sam Adamson, who was not merely the manager of the store but also the owner. He seemed mildly interested. We asked him some questions about how people buy menís suits. Sam seemed to think he could tell fairly quickly who was there to buy and who was there just to look. He looked at a manís watch, for example, to judge a customerís affluence, even though the customer was casually dressed. He would ask customers casual questions, and we knew Sam was good. But as we pressed on, we discovered that Sam could still be fooled. Some men will spend 45 minutes in his store trying various suits on, and walk away without buying. Sam explained that there are at least two kinds of male customers he never read correctly: recreational shoppers who consider trying on clothes a fun thing to do on a Saturday, and men who were simply killing time waiting for a wife or girlfriend to shop somewhere else.

    "Suppose I could tell you how many suits a particular customer bought over the past five years, and when he bought his last one," I asked. "Would you pay for that information?"

    Sam smiled at me. "I already know that," he replied. "As soon as I ask his name, we check our store database for exactly that point-of-sale information."

    "Ah, but what if I could tell you that information for any store, not just yours?" Don countered. "And what if I already know his name without asking?"

    Samís eyes lit up. "Yes, that would be useful."

    Don and Sam smiled at each other. I wondered if Sam shared Donís trait of deviousness. Sam struck me as the kind of person who would seize any opportunity to make a sale; he was just the client we needed.

    Don and I took Samís business card, and we told him we would get back to him in a few weeks.

    Now Don and I had something to work on. We would create a menís suit scoring program, not unlike credit scoring, for use specifically at the Short Hills Mall. We would sell the output to a small number of menís clothing stores there. By the time each customer entered one of the stores, we would supply a one to ten score on the customerís buying probability. This would alert the staff to be sure to spend plenty of time with that customer, even perhaps to the exclusion of another customer with a lower score. Of course, this would always be at the staffís option; the staff could overrule the score if necessary.

    And so Don went to work on the hardware end. One of my questions had been how to install the scanning cameras in the mall without authority. Don explained that one of his philosophies is that people can do just about anything in public, no matter how outrageous, if they are appropriately dressed and act with seriousness of purpose. Somehow Don managed to get us some workersí uniforms. We returned to the mall one morning, and I held the ladder while Don calmly installed the cameras over each mall entrance. Nobody questioned us Ė we looked like we were supposed to be doing what we were doing. We got Samís permission to install one at his store.

    Meanwhile I went to work on the modeling. I began with a database of every male residing within a two-hundred mile radius of Short Hills who has a listed phone number. Don, with his Census Bureau friend Paulís help, showed me thousands of government databases that are either free or available to the public at nominal cost. Paul explained that the various departments of the Federal government never talk to each other, and never assume that their various databases would be merged together. Although I assumed that the credit card companies would guard their data, even if the government didnít, even they were willing to sell their data.

    Soon for every male within two hundred miles we knew his salary, his occupation, his employer, and his credit card history for male clothing stores. We decided we needed to know each employerís policy on business casual and how often people wear suits even in a business casual office, so we hired some college students to make phone calls.

    I now had lots of variables and lots of data to play with. What more can an actuary want? Don had to help me increase my computer power to handle all the data, especially since the databases were always being updated. It took me several months working nearly full-time to come up with my first attempt at the model. It took another month to test it and revise it.

    We gave the model to Sam and his store for free while he helped us test and improve it. We made some hilarious errors, such as the time we scanned some up-and-coming rock musicians who had very little historical income or prior suit purchases. The model had scored them close to a zero, and Sam chuckled at me when they bought five or six suits each in preparation for a tour they were about to go on. I realized I had underestimated the potential of the self-employed variable. Eventually we tweaked the model into its final form, and we signed a contract with Sam.

    Each person who entered the mall was scanned. We were not interested in women, and we were not interested in males without credit cards. Of the remaining men, either he was in our database, or he was not. If he was, we calculated his model score. Then if he entered Samís store, we flashed the summary information and the score on Samís computer. Sam of course could act on this information or not. But we did identify customers for Sam that were in a buying mood although Sam had guessed wrong, or that were not in a buying mood although Sam had guessed yes. Sam was happy with our model.

    We then licensed the model to a few of Samís competitors in the mall, and we installed scanners over their entrances. We had not told Sam we were doing this, but we hadnít told him we wouldnít, either. This is the old consultantís trick: once you solve a problem for one client, you sell the solution again to another client.

    As time went on, the model required a small amount of Donís and my time to update and tweak, but we moved on to other things. Specifically, we went on to other upscale malls to license our model there.

    Don and I were doing OK financially. We werenít making a fortune, but we were collecting some license fees. I happily checked the box for "Non-traditional" when the annual Society survey asked for my employment status.

    I was in a Los Angeles hotel room one night, getting ready to pitch the model to the South Coast Plaza (far too upscale to be called a mere Ďmall") the next day. I was quite excited about this, as the income levels in Beverly Hills were a step up even for Short Hills. An item on the television news caught my eye. There had been a robbery and murder in Glassboro, New Jersey.

    Glassboro is a small college town in the southwest corner of the state. There are plenty of murders in the news in New York City each year, or even in Newark, New Jersey, but I guessed there were not too many in Glassboro.

    The murdered woman was Catherine Purvis. I did not know her, but I did recognize the last name as the same name of someone who had once been president of a little insurance company in Glassboro, some years ago when there were little insurance companies. For a moment I wondered if they had been related, but then I got back to planning tomorrowís meeting.

    Over the next few months I earned plenty of frequent flier miles, as I pitched the model to some of the nicer malls in the country Ė Denver, Dallas, Atlanta, and Chicago. It was in my Chicago hotel room when I again saw something about the Purvis murder on television.

    The news said that the police had arrested a suspect in the Purvis murder. His name was Larry Davis, and he had been a waiter at a catered party at the Purvis home several weeks before the murder. Apparently he had cased the house during the party, returned a few weeks later to rob it, unexpectedly found Catherine Purvis at home, and murdered her. His fingerprints were found on some household items, including the kitchen knife that she was stabbed with. He was arrested a few hours after the murder occurred, and the police seemed satisfied they had arrested the right person.

    Davisís lawyer was on television in a press conference. The lawyer proclaimed his client was innocent because Davis was nowhere near the Purvis house at the time of the murder. Davis insisted he was Christmas shopping Ė at the Short Hills Mall!

    I followed the case over the next few weeks over the Internet. The police investigated Davisís claim that he was at the mall by viewing the mallís security videotapes and by interviewing salespeople at the mall. Nobody remembered him, nobody could find him on the tape, and Davis could not come up with any credit card receipts.

    A thought hit me like a lightning bolt. My model, and the scanned data underneath it, could provide proof of his alibi if he were telling the truth. We had to contact the police.

    I called Don to tell him about this. Don told me that although there were backup tapes of each dayís scanned data, the tapes were not saved indefinitely; this murder had occurred nearly six months ago. Further, he was not sure how much detail was on these backup dates anyway, especially at the individual customer level, because the majority of old data was saved at a summarized level. I told Don not to do anything with the tapes. I flew back to New Jersey to discuss this with Don in person.

    Don and I met. He suggested that if we were going to examine the tape, that we do it in the presence of the police so that there is no question about us tampering with the tape. Even I knew it wouldnít take much effort to tamper with a tape and then adjust the date Ė all you had to do is turn back the computer clock Ė I have known how to do this since the days of DOS 1.0.

    The bigger issue, Don explained, is that once the police know we have this kind of evidence, they would ask how we obtained it. Don was not entirely sure the whole scanning thing was legal.

    "I thought we discussed this before we started, and you told me it was legal," I reminded him.

    "Not exactly," Don replied. "In fact, I think I said the lawyers were not in complete agreement. I have actually spoken to a number of lawyers about this, anonymously over e-mail. Although many lawyers have told me what we do is criminal, other lawyers have had trouble finding a statute that fits this situation at all."

    "What if we call the prosecutorís office, explain how we have this evidence, and ask for immunity in return for our testimony?" I asked.

    "I donít think that will work," Don replied. "For one thing, prosecutors may grant immunity when the testimony helps their case. Our testimony would hurt their case. They have no incentive to offer us immunity."

    "For another thing, we didnít just do the scanning one time and stop. It is an ongoing thing. This would mean the end of our business, and maybe some lawsuits."

    "If Davis were telling the truth, why didnít the mallís security videotapes verify him?" I asked.

    "Remember that this happened in December," Don reasoned. "Perhaps during the crowded Christmas shopping season he happened to slip in just behind someone so that the videocamera missed him."

    "Then maybe our scanner missed him too?" I asked hopefully.

    "Nope," replied Don confidently. "The technology is different. If he were there, we scanned him and he will be on the backup tape."

    "Well, we canít let them convict an innocent person for murder. Letís do the right thing here, and let the chips fall where they may," I said reluctantly.

    Don and I agreed to call the police and offer our evidence in the case. We called and learned that the prosecutor assigned to the case was Stacey Robbins, and the police investigator was John OíConnell. We made an appointment for us to talk with them in the county police station in Newark.

    Just before we left for our appointment, it occurred to me that maybe we ought to have a lawyer with us. I didnít know any criminal lawyers, so I called the only lawyer I knew - George Robinson, an old friend who had been a lawyer in the legal department of one of the insurance companies I had worked for long ago. I quickly explained to George what was going on. George agreed to meet us in Newark.

    The three of us met with Stacey and John. John began by asking us if we knew Larry or Catherine personally. We didnít know either of them, but I thought I had better admit that I might have met one of Catherineís relatives if she were part of the Purvis insurance family. But this slight connection would not affect the evidence we had.

    Don and I explained how we might happen to have proof of Larry Davis having been at the mall on the day and time of the murder. Stacey was displeased that we were potentially going to blow her case against Larry. She threatened to have us arrested for our illegal scanning operation.

    "Youíre kidding, right?" asked George. "Itís true that there are several legal arguments you could make that Don and Andrea committed a crime. You could call the crime theft, no different than if they had physically stolen a wallet. Or you could even call it battery Ė unprivileged touching. But I think those arguments are pretty weak."

    "Not that Iím looking to go to jail," I interrupted. But in the interest of full understanding I needed a question answered. "What about invasion of privacy?" I asked.

    "First of all, thatís a civil offense, not a criminal one," George replied. "Besides, people voluntarily give up their privacy rights all the time, without worrying about it. Do you have a supermarket checkout card?" George asked all of us.

    Don shook his head no. I said that I did, actually several.

    "Then even when you pay by cash, youíre voluntarily telling the supermarket who you are. They are keeping tabs on everything you buy Ė youíve seen how when you buy bread, they automatically print out a coupon for butter, for example. Plus they sell the data about you to the big food companies. The beer and pretzel companies know if you stocked up right before the Super Bowl, for example. Did you explicitly give your supermarket permission to do all this?"

    That was a rhetorical question, so George didnít wait for an answer. "I donít suppose you actuaries study market research on your exams. You have no idea what goes on at supermarkets. And then there are department stores."

    "Although the average consumer would not be happy to discover how much is known about his individual buying habits, in my opinion there is no law on the books that makes what the supermarkets do Ė or what you do Ė illegal," George concluded.

    Stacey agreed with George Ė for now. The next step was to examine the backup tape.

    Don called the service company that maintained our backup tapes. It turns out Don had ordered a six-month tape retention, and the tape for the particular day we needed still existed. The police got the tape and brought it back to us in Newark. Don wanted to load the tape on his computer where he had some sophisticated querying software, but the police wanted it on their computer, in front of their computer expert. So we loaded it on the police computer.

    Donís tape was just a big dataset of numbers and letters. Don chuckled when he noticed that the police were several versions out of date with their database software. He ran a query on "Davis". There were many, so we queried "Larry Davis". There were no Larryís. We queried "Lawrence Davis", but we could still not find any. Finally we decided to query "L. Davis". There were none.

    Maybe Larry Davis was not innocent. Or at least his alibi did not hold up.

    Don and I were quite disappointed. We believed in our scanning system, and for some reason we believed Davis.

    George made us explain for him in detail how our scanning system worked. He asked us several questions, so he was obviously trying to understand it. He was quiet for a moment. Then he asked John when the murder occurred and how the police came to suspect Davis.

    John explained that the medical examiner placed the time of death at approximately 1 p.m. on December 19. A maid found the body and called the police. The murder weapon was a kitchen knife, with Davisís fingerprints on it. The police quickly found Davis and arrested him.

    George asked John, "When you arrested Davis, was he carrying any credit cards? Can we see them?"

    A policeman went to find Davisís belongings. He returned with a couple of credit cards. They looked like ordinary credit cards, and they did have Davisís name on them.

    "So why isnít Larryís name in your database?" Stacey asked. She answered her own question. "Because he wasnít at the Short Hills Mall that day. Because he was in Glassboro, murdering Catherine Purvis."

    That seemed logical, but there was something unsettling about it.

    Don was examining the credit cards. He ran his fingers over the credit card numbers. He pulled out his own credit cards and compared them.

    "Larryís cards are hardly used, even though they are not new according to the expiration dates," he decided. "Look at how the raised numbers on my cards are worn down."

    That was interesting, I thought, but I didnít see that it was relevant.

    Unless. "Read me one of his credit card numbers," I demanded excitedly. "Actually, read the numbers backwards."

    Don supressed a giggle. My use of the word "backwards" puzzled everyone in the room, but Don assumed it was one of those strange quirks about actuaries.

    John read off a string of sixteen digits from one of Larryís cards. I started scribbling on a piece of paper. The arithmetic was pretty simple. I added the first digit, to two times the second digit, to the third digit, to two times the fourth digit, and so on. When a multiplication exceeded nine, I added the digits from the product. The result was 51.

    "Read me another card number, also backwards, please."

    John read the next string and I did the same calculation. The result was 68.

    Everyone, even Don, thought I was a little crazy.

    "These are not valid credit card numbers," I announced. "Itís called Luhnís Formula, and the sum has to be a multiple of ten to be a valid credit card number. All the major credit card companies use it."

    "Call the credit card companies, and see if anyone tried to use these cards on the day of the murder," Stacey told John.

    "I still donít get it," Don whispered to me. "Why didnít our system scan Larry?"

    While John was making the calls, I reminded Don that we only cared about males residing within two hundred miles of the mall with a listed phone number and a valid credit card. Anyone not fitting those criteria was certainly scanned, but we had nothing in the other databases to match against. Since I couldnít use the scoring program, we never recorded those people in our system.

    For the same reason, I had decided to put in this Luhnís Formula as a check against invalid credit cards. If a card was invalid there would be no credit card history, so there would be no way to score that person either. That is why Larry never made it into our database that day.

    John reported that those credit card numbers had indeed been called in by Short Hills Mall merchants on the day of the murder. Since they were not stolen cards, the store clerks had not been told to call the police to arrest the customer, merely to ask for another card or for cash.

    The credit card companies had the times of day that the cards were used, and the times were right around the time of the murder. Glassboro and Short Hills are at least two hours away, so Larry could have not been in both places. Since the cards were in Larryís possession at the time of his arrest, he was probably the one who had tried to use them, so he couldnít have been the murderer. John speculated that maybe Larry was involved not in murder but in phony credit cards.

    I suggested that maybe Larry had found some legitimate credit cards, perhaps even from the time when he had been a waiter at the Purvis house. Rather than steal the cards, which would eventually get reported, he probably just wrote down the card numbers. Then he had some new cards made with his name and where the credit card number was off by one or two from the legitimate card number.

    Of course Larry would not have known about Luhnís Formula. Only someone from a credit card company would know about it Ė or an actuary.

    "Interesting," I concluded. "We thought we had proof Larry was at the mall from our scanning software, but it turns out we didnít have that proof at all. But instead we proved Larry was committing credit card fraud, and so we indirectly proved he could not have been the murderer."

END

Jerry Tuttle, FCAS, is a senior pricing actuary at Platinum Underwriters Reinsurance Company in New York City.