L.A. Math: Romance, Crime, and Mathematics in the City of Angels

by James D. Stein

  I can’t say that I was happy about my love life depending upon the second hand of a clock. “Perhaps you could explain a little further, Pete, without going into the gory details.”

  He sighed. “I’ll have to go into some details. Suppose she wants to hear from you, and you have three opportunities to call. If you call once and don’t call twice, you get ten points the time you call and two points each of the two times you don’t call, for a total of fourteen points. Got that?”

  “I can handle that 14 is 10 plus 2 times 2.”

  “Okay. Now if she doesn’t want you to call, you get zero points the one time you call, and seven points each of the two times you don’t. Again, a total of fourteen points. So, no matter how she’s feeling, in the long run calling one time out of three gives you the same reward. That comes out to 14/3 points per situation, or four and two-thirds points every time this comes up.”

  It took me a moment or so with pen and paper to digest all this. “Very ingenious, Pete. But why don’t I just call every time and hope she wants to hear from me?”

  “You could do that. But suppose that she never wants to hear from you. In this case, you never get any points. However, by adopting the strategy of calling once every three times, and using a randomizing device to select whether you call or not, you have guaranteed a long-term average of four and two-thirds points per opportunity, no matter how she feels about your calling.”

  (2 × 2 games continued on p. 214)

  I won’t say that I followed it all perfectly. On the other hand, Dolores scored sixty-some thousand bucks following Pete’s advice, and it cost her a bunch of Mexican dinners. I was getting the same advice for free. So, with some trepidation, I lifted the receiver and punched up the number.

  It was with clammy palms that I heard her pick up the receiver.

  “’Lo.”

  “Hi, Lisa, it’s Freddy.”

  “Freddy! You must have just gotten back. Pete said you were out.”

  “That’s funny. He never said a thing about you’re having called. Gee, it’s good to talk to you. What’s up?”

  “Well, I may have some really good news about a new job opportunity, among other things. But it’s still somewhat in limbo, and I don’t want to jinx it by going into specifics. But cross your fingers and wish me luck.”

  “You know I always do, Lisa.”

  We went on from there to other topics of a more personal nature. Half an hour later, I signed off, feeling a lot more than four and two-thirds points to the good.

  As soon as I hung up the phone, I had some urgent business to transact. I went downstairs, where I happened to catch Pete flipping the channel back and forth between pro and college basketball games.

  “You know, Pete, I don’t know whether to thank you or read you the riot act.”

  He paused in mid-channel-change. “Why would you want to read me the riot act?”

  “When I spoke to Lisa, she said that she had called. You never gave me the message, and you knew she wanted to talk to me.”

  He got a little huffy. “In case you don’t remember, Freddy, when you got back, I met you at the door. I was just opening my mouth to give you the message when you interrupted me. I never had a chance.”

  I thought back. He was right. Maybe I should back off my high horse. Then I realized that he had left my love life up to the digital readout of a watch.

  “Okay,” I said, “I’m willing to admit that I interrupted you, and that you would have told me Lisa wanted to hear from me. However, it has just occurred to me that I was very lucky that the digital readout on my watch was between zero and twenty seconds when you asked me to look at it. Had I read my watch some seven seconds later, you would have advised me not to call Lisa.” I started to get mad all over again. “It seems to me you were willing to risk my love life just to get an opportunity to haul out one of your pet theories.”

  “Not a chance, Freddy. You were guaranteed to make that call.”

  I was a little surprised. “Do I remember correctly? I thought you said that, if the number of seconds read between zero and twenty seconds, I should call; otherwise, I shouldn’t. And as I recall, the number of seconds read fourteen seconds.”

  “Your memory is fine. But I would have arranged things to make sure you called Lisa.”

  Silly me, thinking I understood game theory. “I thought the whole point of this game theory stuff was to randomize one’s decision so as to get the same long-term result no matter whether Lisa wanted to hear from me or not.”

  “That’s right. But I knew something you didn’t, that Lisa had already called and wanted to hear from you.”

  I was still confused. “But what would have happened if I had looked at the readout seven seconds later? It would have read twenty-one seconds.”

  “Because then, I would have said that you should call if the readout had been between twenty and forty seconds. A priori and a posteriori probabilities differ, as I may have mentioned before. And seeing as you’d gone to all the trouble of preparing that chart, and you saw why the situation differed from the problem with Dolores, I didn’t see how it would hurt you to learn a little more game theory.” He paused. “Speaking of Dolores,” he continued, “how would you feel about some Mexican food?”

  Nothing stimulates the appetite like good news on the relationship front. The way to a man’s stomach is through his heart.

  CHAPTER 13

  DIVISION OF LABOR

  I hadn’t known there was such a thing as the Bankers Club. Pete probably hadn’t known there was such a thing as the Bankers Club. But our potential client, Ellis Packard, looked every inch a banker, and at about six feet four, there were a lot of inches. His suit was appropriately gray. His Mercedes was gray. And his retainer check for $5,000 was a particularly tasteful shade of gray.

  I wasn’t going to muff a job with a $5,000 retainer because of lack of hospitality, so I offered him a choice of refreshment. He settled for a Coors and launched into his presentation.

  “I don’t know whether either of you have ever heard of the Bankers Club,” Packard began. We hadn’t. “It’s a club for members of the banking profession.” We guessed that. “It has a lot of amenities, and it also gives its members a chance to network.”

  When I become dictator of the world, one of the things I intend to do is remove “network” from the verb list and place it back where it belongs, in with the nouns. Other items high on my agenda are to remove instant coffee and nondairy creamer from the world’s grocery stores.

  But I digress, partly because Packard had paused to take a drink, it being a hot and dry day. A few sips later, he was back in stride.

  “I’ve been a member of the Bankers Club for a couple of years, and it’s a good idea for bankers on the fast track to put in some time working on club committees. So I got appointed to the Elections Committee, which didn’t seem like it was going to require an excessive amount of time or make me any enemies. Whenever you get on a committee, there’s always a chance you can make some enemies. Have either of you ever served on a committee?”

  Neither of us had had that dubious pleasure, and we said so. After finishing the remainder of his beer, Packard resumed his monologue.

  “The election for the club president is held biannually, and in the past, if no member receives a majority, the leading two candidates have a runoff. Well, nobody likes runoff elections. There’s more politicking and backbiting during this time. So I came up with what I thought was a very good idea. There were three candidates this year: Forrest Ackroyd, Helen Williams, and Artie Morris. I suggested that the ballots be printed up so that each person could indicate his or her first, second, and third choices. That way, it seemed to me we could avoid a runoff election.”

  At this stage, I thought I saw Pete wince, but evidently Packard didn’t pick up on it. He squirmed around in his chair a bit. It occurred to me that he might not think it was seemly for a banker to ask for another beer, so I asked him if he wanted one. He nodded g
ratefully and continued after his vocal chords were sufficiently lubricated.

  “Fifty-four ballots were cast, but there were only three different types of ballots that actually appeared.” He handed us each a sheet of paper. “Here is a copy of the election results, which I posted on the club bulletin board after we had tallied them.”

  I’ll present those results in a table, so you can look at them as the story progresses.

  While we were scanning the results, Packard continued his tale. “It was only now that I was beginning to get an inkling that trouble might be looming on the horizon,” he said. “There didn’t seem to be a clear-cut winner. The Elections Committee wasn’t scheduled to meet until the following week, so I figured we’d thrash it out then. I left the club, played a few games of racquetball, and returned later in the evening.”

  “I was sitting in the club bar with a Perrier, when suddenly I noticed that Forrest Ackroyd had occupied the seat next to me. Forrest is one of the more influential vice presidents of Transcontinental Trust. He patted me on the back, and to my surprise offered me an invitation to a golfing weekend at his home up near Monterey. We would play Pebble Beach, and I’ve always wanted to play Pebble Beach.”

  I could understand that. If you’ve ever seen a telecast of the golf tournament formerly known as the Crosby Clambake, you’d know why. I wasn’t much of a golfer, but I wouldn’t mind playing Pebble Beach myself. However, I seemed to recall that it’s a closed club now, so you need an invitation.

  Packard didn’t waste any time. I made a note that if I ever needed a loan, I’d call him, for I didn’t think I’d get any of this “the Loan Committee meets a week from Tuesday” stuff. “Forrest then observed that he hadn’t actually seen a notice of who the winner was,” he continued. “I told him that all we’d done was post the actual results of the balloting and that the Elections Committee was scheduled to meet next week. He then said that it seemed pretty clear that he had won the election, as he received the majority of the first-place votes, more than 40%. He told me I’d better get my short game in shape, as Pebble Beach was especially rough if you missed the green.”

  Pete was paying close attention, scanning the list of results while listening to Packard. Packard squirmed a bit more in his chair, though I wasn’t sure whether it was because the day was muggy, or the chair was uncomfortable, or Packard was born to squirm. I guess people who are abnormally tall are continually forced to occupy chairs not designed for them.

  He resumed the narrative. “The next day I was over at the club, and who should drop by during lunch but Helen Williams, who is the CEO at Consolidated Bankshares. She’s very influential in local banking circles. The next thing I knew, I had been tendered an invitation to a party that is usually attended by the Who’s Who in L.A. banking. I was astonished but gratefully accepted. Then guess what!”

  At least one of us was up to speed, for Pete said, “I think I’ve got it doped out. Williams said that she expected to be declared the winner by the Elections Committee.”

  Packard assented. “That’s what happened, all right. She said that, since Artie Morris was the first choice of less than a quarter of the voters, Artie couldn’t possibly be elected president. Fortunately, since we had allowed each voter to indicate their preferences in order, and since thirty voters preferred her to Ackroyd, whereas only twenty-four preferred Ackroyd to her, it was clear that she should be declared the winner. She then said that she hoped I would enjoy the party.”

  More squirming and fidgeting. Since the day had cooled down considerably, and it was a $700 chair he was sitting in, I was inclining to the born-to-squirm theory, although the chair was so new that the leather was still a little stiff and uncomfortable.

  Pete was still on top of things. “Let me guess what’s next. I wouldn’t be surprised if you were to hear Morris stake a claim.”

  “You’ve got a good feel for this, Mr. Lennox,” Packard allowed. “I didn’t actually hear from Morris, who works in the same bank that I do but is somewhat higher up. I heard from Celia Morris, his sister.”

  “Hold on a second,” I interjected. “Is Celia Morris about five feet two, late twenties, long auburn hair, nice smile?”

  “I take it you’ve met her,” Packard declared.

  “At a party a couple of weeks ago.”

  He looked a little embarrassed. “If you don’t mind my asking, was she at the party alone?”

  I reflected a moment. “I’m not sure. I seem to recall somebody with horn-rimmed glasses who seemed very interested in her.”

  Packard nodded grimly. “That sounds like Carroll Farnsworth. He’s a CPA. What she sees in him I don’t know.” I guess, if you’re the vice president of a bank, it’s hard to see why a CPA would offer any competition.

  “Maybe she has tax problems,” I commented.

  “Ahem!” Pete coughed. “Maybe we could get back to business.”

  “Actually, we were still on business,” Packard asserted. “I’ve been thinking of asking Celia for a date, but I was under the impression she was still seeing Farnsworth. At least, that was what I thought until I received that phone call. Needless to say, I was delighted when Celia phoned. I was about to suggest dinner and a show when she mentioned how wonderful it was that Artie was to be the club president!”

  “I tell you, I was floored. I asked her why she thought that, and she told me that Artie had told her that the club had a rather complicated ballot. However, when three points were awarded for a first-place vote, two points for a second, and one for a third, he had received 120 points. Helen got 102 points, and so did Forrest. Try as I might, Mr. Lennox, it’s hard for me to envision Celia waiting up for Artie to bring her the results of the election so that she could go through that analysis.”

  Somebody had to say it, so I did. “You think her brother put her up to it.”

  Packard nodded. “She’s in advertising and good at presenting data in the best possible light.” He broke off, and then continued.

  “You see my predicament. I want to go golfing at Pebble. It would really help my career to attend Helen Williams’s party. And I would love to get something going with Celia Morris. But it seems to me that no matter how I handle it, I am not only going to lose two out of three opportunities, but I am going to make some enemies as well. I’m at my wit’s end, Mr. Lennox. It’s worth five thousand to me to get out of this with my weekend at Pebble, the Williams’s party, and my chances with Celia Morris intact.”

  Despite the recession, bankers were still evidently doing quite nicely. However, it was time for me to butt in. Personal relations are my specialty, not Pete’s.

  I cast a longing glance at the retainer check, but you have to draw the line somewhere. “We’re here to solve problems, but we’re not a dating service.”

  Packard nodded. “Celia’s my problem. It’s just that she’s tangled up with the election thing. Anyway, do you think you can figure something out?”

  Pete stood up. “We’ll let you know how it stands in a day or so, Mr. Packard. Thank you for coming.” We shook hands, and Packard departed.

  I felt optimistic. Packard’s problem certainly dealt with numbers, and fortunately we had a house specialist in numbers.

  “All right, Pete,” I declared, “how does mathematics deal with this one? It seems to me that all three candidates can put up a pretty good argument.”

  Pete grimaced. “Yeah, they sure can. And even though mathematics has a lot to say about it, it can’t do anything to solve the problem.”

  I was more than a little surprised. I was so used to Pete’s pulling rabbits out of hats that I was expecting to see the tips of floppy little ears as one emerged.

  “How can mathematics have something to say about a problem and not be able to solve it?” I demanded.

  Pete shook his head. “There’s a very important result known as Arrow’s Impossibility Theorem. It won Kenneth Arrow the Nobel Prize for Economics in 1972.”

  It’s really a shame that quiz show
s are no longer in vogue because Pete knows these things the way kids know the lyrics of the latest hits. However, my attention was focused on the word “impossibility,” which had an ominous ring to it. I commented on this.

  “I don’t like the sound of that, Pete. What is Arrow’s Impossibility Theorem?”

  “The exact statement of it is a little formal. But the upshot of the theorem is that it is impossible to devise a scheme for assigning preferences for a society such that the chosen scheme accurately incorporates the preferences of the individuals in that society. And that’s precisely what we’ve got here.”

  (Voting methods continued on p. 219)

  “I’m not sure that I fully understand, but it sounds like sayonara to five thousand or so bucks.”

  “Let me sleep on it.” Pete’s solution to practically everything is to sleep on it. He has told me that he believes that the purpose of the left side of the brain is to gather data on which to base a decision, and the purpose of the right side of the brain is to supply the insight to reach the correct decision. He has also told me he believes that sleep unlocks the hidden power of the right side of the brain. Me, I believe that he’s just rationalizing the fact that he likes to sleep.

  Well, I wasn’t going to sleep on it. I like to think that problems are solved by working on them, not by sleeping on them. And it would be a feather in my cap if I solved one that Pete couldn’t. So, while Pete was sleeping, I was thinking.

  I had plenty of uninterrupted time for thinking, as Pete slept on this one for sixteen consecutive hours. When he had finally achieved sufficient consciousness, I could tell from the sour expression on his face that the much-vaunted right side had come up empty. He confessed as much.

  “Sorry, Freddy. I guess we’d better call up Packard and explain the situation.”

  It was time to play my card. “That might not be necessary. I’ve been thinking about this, and I might have a way out.”