The Dating Game

Not only is a game a good way to start leaning a subject, there is no real harm in losing. This author, a pilot most of his life, once had he opportunity to fly a DC-10 simulator at the factory. (It was a wondrous gadget, just like the cockpit of the real airplane, swiveling and bouncing realistically, with spectacular scenery passing by in the fake windshield, but still firmly attached to the floor.) He crashed ignominiously, having lost considerable face and reputation, but walked away without a scratch. If you feel an irresistible urge to crash a DC-10 that’s the way to get it off your chest. You can lose millions at Monopoly, while learning to be a real-estate wheeler and dealer, and wake up solvent. The game of go is played with devotion throughout East Asia, especially in China and Japan (it originated in China four thousand years ago), even though it is virtually unknown in the United States. It reduces to bare essentials on a simple playing board the most basic strategies of war, and can be played without any damage to people or things (except to the egos of most Westerners who play). It is no accident that chess has kings, queens, knights, bishops, and pawns. An arcade game about tank battles was adopted by the Army as a primary training tool for future tank drivers. And so forth. Never underestimate the importance of games as introduction to life.

The first step in a book on decisions is to convince the reader that there really are ways to make decisions rationally, and that it can make a difference. The second step, of course, is to say how. Most of us will insist that we make important decisions only after “thinking it over,” but who knows what that means? When pressed we usually admit that every now and then e simply “take a chance.” Sometimes there really isn’t much karma and all that, and then we just go along for the ride. A Hobson’s choice (named after an English stable keeper) is a choice that is no choice at all—like Henry Ford’s famous comment that a customer can have any color car he wants, as long as it‘s black. But sometimes the choice is important. The stakes are and straight thinking can make a difference. Then it pays do things right.

So here’s our first puzzle, about individual decision making. The problem we’ve chosen is well known to mathematicians, has an obvious analog in real life, and is interesting in its own right. We make no apology for the fact that it is framed in the context of courtship—that’s where many of us make our most important personal decisions, and where we can use all the help can get. On the other hand, games shouldn’t be taken too literally as lessons in life—they are pale shadows of the human condition. This particular puzzle has many other names.

The Dating Game

Imagine yourself a female who has decided, for reasons we cannot possibly explain, to get married (no gender bias intended, exchange the sexes throughout if you prefer - the game’s the same), and of course you want to marry the most desirable male from a pool of say, a hundred eligible and available bachelors in social circle. Second best simply won’t do for a person of outstanding quality and refinement, let alone aspirations. But finding the best of the lot out of a hundred possibilities isn’t to be easy; you need a strategy.

Obviously, you shouldn’t marry the first guy you meet. The chance that he’s really tops in the group of a hundred is, of a chance in a hundred. That’s a pretty slim chance, and a gamble in the worst sense of the word. But the same is true of the second, and the third, and so forth. Any one of those has only a Chance in a hundred of being the best of the lot. You can’t just - one at random if you want a realistic shot at the very best. It's like picking the best apple in a barrel—you’d better start comparing them with each other. Any of them can be the best, but it can also be the worst.

So you’ll have to do some dating—how else can you check them out against each other? - but the rules of the game are not like those of the apple barrel. In that case you could look at the apples side by side, but in this game you are only allowed one date with each candidate. After each date you have to decide on the spot if this one looks like the very best, even when there are some you haven’t yet dated. (They are all eager to marry you— it’s a game—so it’s your choice.) Once you select the lucky man, you stop dating—games don't have to be entirely realistic. Another rule of the game is that if, after a date, you decide against a candidate, he is lost to you forever Imagine that he marries Someone else, enters a monastery, hurls himself off a cliff. The point is that you can’t date them all in turn, put each one on a shelf in a warehouse after the date, presumably with a rating label and then later dust off the best. No stockpiling of candidates. Statisticians call this process sequential decision-making—you decide on the spot, while you ate still collecting information.

This sort of thing happens all the time in clinical trials or drug testing, in which one group of patients is given a potentially useful drug, and another group something harmless but ineffective—a placebo. The people running the tests should be ready to decide at any moment to end the trial, giving the control group the drug (if it’s turning out to be useful) or taking the treated group off the drug (if it seems harmful). They shouldn’t go on testing any longer than it takes to decide. You shouldn’t either, unless you enjoy the dates more than the prospect of marriage-that’s another subject.

The selection problem is obvious, You want the best spouse, but how can you maximize your chance of finding him under these rules? If you plunge too early in your dating career, there may be finer, undated fish in the sea, and you may go through life regretting a hasty marriage. It happens more often than we like in real life—marry in haste and repent at leisure, says the old proverb. Yet, if you wait too long, the best may have slipped through your fingers, and it is then too late. That also happens all too often in real life. Songs, poems, and novels have been written about both misfortunes.

So what is a winning strategy—one that gives you the best chance of success? You can’t be sure, you just want the best thing This is a simple game; you know what you want, everything is in the open, you alone make the fateful decision, and you have only to optimize your selection process. Is there a best way?

You bet there is. It doesn’t give you a sure thing, but it does give you the best shot at your goal. No matter how well you organize your affairs to hit the jackpot, there is always a certain risk that you’ll simply have bad luck, and end up with the bottom of the barrel. After all, someone does. So let’s go through the reasoning.

As we said, you shouldn't choose the first applicant who sea along—it would really be an amazing coincidence (a chance in a hundred) If the best of the lot showed up first. So it would make sense to use the first group of dates, say ten of them, as samplers (as in a candy shop or bakery), and then marry the next date who rates higher than any of those. That’s a way of comparing them, and is not far from real life. You could give each date a grade in your diary (say on a scale of ten), and resolve the first one who comes along with a any those first ten is the final winner. All you’re doing is using first ten dates to gain experience, and to rate the field. That’s what dating is all about.

There are two ways you can lose badly that way. If the first ten happen to be the worst of the lot—luck of the draw—and the next one just happens to be the eleventh from the bottom, you will end up with a pretty bad choice—not the worst, but pretty bad—without ever coming close to the best. You have picked the eleventh from the bottom because he is better than of the first ten—that’s your method - while the best Is still there waiting for your call. But at this early stage in your dating career you have no way of knowing that. It’s somewhat like running around with a flaky crowd: the experience distorts your impression of what real people are like. The other way you can lose is the opposite: by pure chance the best choice may have actually been in the first ten, leading you to set an impossibly standard after your early dating experience. You will then keep going through the remaining ninety candidates without ever seeing his equal, and will have to settle for the hundredth because the pool has dried up. The hundredth will be, on the average, just average. You are then doomed to go through life fantasizing about what might have been—the one who got away. So you have a chance of winning, but also a chance of losing big. It’s not hard to show (but the mathematics involved is beyond the ambitions of this book) that you have about a chance in four of winning (marrying the best of the lot) with this strategy. Better than a random choice, but not a sure thing. The rest of the time you’ll have to settle for second best, or third best, or fiftieth best, or whatever comes up.

Can you do better than that? Well, the chance of the second kind of error, letting the best slip through your fingers, is pretty small for this case—if you’ve sampled ten candidates out of a hundred. There's only a chance in ten that the best is in that lot. So you might be willing to do a bit more sampling without too much risk of an error of this kind, and thereby improve your knowledge of what’s available. You’d get more experience. What about using exactly the same strategy, but going on twenty dates before making your choice? You’ll increase the chance that the best candidate slips through your fingers from one in ten to one in five before you are ready for marriage, but will have greatly decreased the chance that you have set too low a standard. It’s a trade-off, better in one way, and worse in another. What about thirty dates, or forty? If you go too far you’ll almost certainly miss the boat, so there must be a best choice of sample size somewhere in there.

It turns out that the best strategy in your quest for the top of the heap is just this date-rate-and-wait procedure, coolly letting exactly thirty-six suitors go by before selecting the next one who is better than any of those. You still run a risk (about a chance in three that you’ve let the best get away, but you’ve done the best you can do, and actually have about a chance in three of ending up with the one in a hundred you were seeking. A chance in three isn’t bad, when you’re seeking one in a hundred. (Incidentally, we’re not being exact when we say a chance in three. There is an exact answer, but going for six decimal places of precision makes no sense at all for this kind of real-life decision.) If you were interviewing applicants for a job, the same logic would work.

But wait, let’s stop for a moment to look at some of the other factors we’ve downplayed. Are you that sure of your own motivations and ambitions? Do you really and truly require the very best out of the hundred eager swains?

There is a downside of always aiming for the top. If the best candidate was in the first group, you’ll end up having to marry last date of the hundred, the bottom of the barrel but not necessarily the worst of the lot. For this game he would be average, in the real world he may not be even that good—others are fishing, too. You are betting on a chance in three of getting the best, against about the game chance of settling for an average Joe, worse. Like always trying for an ace in tennis.

So let’s go back to the case in which you were only allowing ten dates to set the standard, and look at it a bit more closely. What has happened is that the top prospect has only a chance in ten of having been in that first group, so he’s probably still out there in the big pool of the ninety you haven’t yet dated. The reason he isn’t sure to end up with you is that there is just as good a chance that number two is also out there, and it is even my that you’ll find him before you date the finest. Either one is better than the first ten, so according to the rules, you’ll pick whichever comes along first; In fact, there is the same chance that number three is still there, and so forth. So what is happens that by having so few premarital dates you’ve reduced your chance of passing up the top prospect, but in return have increased your chance of missing him for one of the runners-up. But is that so bad? Well, it depends on whether you think you’d miserable with second best, out of a pool of a hundred. Seems pretty arrogant, doesn’t it.

Maybe it would be better to play the game a little more conservatively. Same rules but different objective. Don’t insist on maximizing your chance at the first prize, but try to avoid having rape the bottom of the barrel. That’s called risk avoidance. If were gambling (and you are, but not for the usual stakes) you might be trying to cut your losses rather than always going for long-shot high-payoff winner. You would be betting to place or to show at the races, for a bit more security. How would your strategy change?

Once you’ve decided that, after all, the second best in a crowd of a hundred suitors is really not likely to be too bad. You don’t have to go all the way from thirty-six down to ten dates to enjoy the benefits of your more relaxed approach. It turns out that then it’s better to let only thirty potential mates go by, and, as before, pick the next who is better than any of those. Though the chance of getting the best decreases a bit, you’ll have a better than 50% chance of choosing either the best of the lot or the second best. And you’ve greatly reduced the chance of going through your dating career without a decision until the final last gasp. That makes a lot of sense.

You can carry this further. If you’ll settle for any of the top five in the pool, it turns out that you should sample only twenty dates; then you have about a 70% chance of landing one of the top five. Imagine, nearly three-to-one odds in favor of marrying one of the top five candidates in a pool of a hundred suitors, just by using your head. In fact, this more conservative strategy doesn’t reduce your chance of getting the cream of the crop all that much; it goes down from about 37% to 33%, which is hardly noticeable. You’ve greatly improved your average performance, and you’ve cut the chance of running out of suitors nearly in half, just by giving up a wee bit of your chance at the grand prize. But be careful not to go too far; if you push this to the extreme you’ll be back to marrying the first guy you date.

There are many different possible strategies for this game, and the optimal one for you—the rule through which you ought to make your own personal decision—depends entirely on how well you can specify your objectives. You can gamble for the best—the first strategy—and accept the chance of losing badly. Or you can relax your criteria a little bit, to cut your losses if that doesn’t work out. You have to know, and be ready to declare in advance, what it is you are looking for. For every set of objectives, clearly recognized and stated, there is an optimal strategy of the date-rate-and-wait variety. That shouldn’t be a surprise-people do it in real life. You can’t have everything—sorry about that—so you’d better be realistic about your goals. (Schoolteachers and preachers may tell you to always aim for the top, but that’s a bad strategy for anything important. The better is the worst enemy of the plenty good enough.) You might in fact adjust your goals as you learn more, either raising or lowering your standards as you gain experience, or as the supply runs out. Most do that instinctively - it’s called a dynamic strategy. If you can say what you really want, there is a best way to get there. On the average, of course—nothing is certain. It may have been Damon Runyon who said, “The race isn’t always to the swift— but that's the way to bet.”