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August 7 - October 2, 2020
an algorithm is just a finite sequence of steps used to solve a problem,
Optimal stopping tells us when to look and when to leap. The explore/exploit tradeoff tells us how to find the balance between trying new things and enjoying our favorites. Sorting theory tells us how (and whether) to arrange our offices. Caching theory tells us how to fill our closets. Scheduling theory tells us how to fill our time.
tackling real-world tasks requires being comfortable with chance, trading off time with accuracy, and using approximations.
As we have seen, this Catch-22, this angsty freshman cri de coeur, is what mathematicians call an “optimal stopping” problem, and it may actually have an answer: 37%.
When you stop too early, you leave the best applicant undiscovered. When you stop too late, you hold out for a better applicant who doesn’t exist. The optimal strategy will clearly require finding the right balance between the two, walking the tightrope between looking too much and not enough.
Look-Then-Leap Rule: You set a predetermined amount of time for “looking”—that is, exploring your options, gathering data—in which you categorically don’t choose anyone, no matter how impressive. After that point, you enter the “leap” phase, prepared to instantly commit to anyone who outshines the best applicant you saw in the look phase.
The passion between the sexes has appeared in every age to be so nearly the same that it may always be considered, in algebraic language, as a given quantity. —THOMAS MALTHUS
there’s a certain flexibility in the 37% Rule: it can be applied to either the number of applicants or the time over which one is searching. Assuming that his search would run from ages eighteen to forty, the 37% Rule gave age 26.1 years as the point at which to switch from looking to leaping.
Threshold Rule, where we immediately accept an applicant if she is above a certain percentile. We don’t need to look at an initial group of candidates to set this threshold—but we do, however, need to be keenly aware of how much looking remains available.
No matter what, never hire someone who’s below average unless you’re totally out of options.
Selling a house is similar to the full-information game.
if we have a limited amount of savings that will run out if we don’t sell by a certain time, or if we expect to get only a limited number of offers and no more interest thereafter, then we should lower our standards as such limits approach. (There’s a reason why home buyers look for “motivated” sellers.) But if neither concern leads us to believe that our backs are against the wall, then we can simply focus on a cost-benefit analysis of the waiting game.
For instance, let’s say the range of offers we’re expecting runs from $400,000 to $500,000. First, if the cost of waiting is trivial, we’re able to be almost infinitely choosy. If the cost of getting another offer is only a dollar, we’ll maximize our earnings by waiting for someone willing to offer us $499,552.79 and not a dime less. If waiting costs $2,000 an offer, we should hold out for an even $480,000. In a slow market where waiting costs $10,000 an offer, we should take anything over $455,279. Finally, if waiting costs half or more of our expected range of offers—in this case,
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in house selling and job hunting, even if it’s possible to reconsider an earlier offer, and even if that offer is guaranteed to still be on the table, you should nonetheless never do so. If it wasn’t above your threshold then, it won’t be above your threshold now. What you’ve paid to keep searching is a sunk cost. Don’t compromise, don’t second-guess. And don’t look back.
I expect to pass through this world but once. Any good therefore that I can do, or any kindness that I can show to any fellow creature, let me do it now. Let me not defer or neglect it, for I shall not pass this way again. —STEPHEN GRELLET
Spend the afternoon. You can’t take it with you. —ANNIE DILLARD
We intuitively understand that life is a balance between novelty and tradition, between the latest and the greatest, between taking risks and savoring what we know and love.
Music lovers might imagine working in music journalism to be paradise, but when you constantly have to explore the new you can never enjoy the fruits of your connoisseurship—a particular kind of hell.
But understanding the explore/exploit tradeoff isn’t just a way to improve decisions about where to eat or what to listen to. It also provides fundamental insights into how our goals should change as we age, and why the most rational course of action isn’t always trying to choose the best.
So which of those two arms should you pull? It’s a trick question. It completely depends on something we haven’t discussed yet: how long you plan to be in the casino.
When balancing favorite experiences and new ones, nothing matters as much as the interval over which we plan to enjoy them.
A sobering property of trying new things is that the value of exploration, of finding a new favorite, can only go down over time, as the remaining opportunities to savor it dwindle. Discovering an enchanting café on your last night in town doesn’t give you the opportunity to return.
explore when you will have time to use the resulting knowledge, exploit when you’re ready to cash in. The interval makes the strategy.
Robbins specifically considered the case where there are exactly two slot machines, and proposed a solution called the Win-Stay, Lose-Shift algorithm: choose an arm at random, and keep pulling it as long as it keeps paying off. If the arm doesn’t pay off after a particular pull, then switch to the other one.
the present has a higher priority: a cured patient today is taken to be more valuable than one cured a week or a year from now, and certainly the same holds true of profits. Economists refer to this idea, of valuing the present more highly than the future, as “discounting.”
The Gittins index, then, provides a formal, rigorous justification for preferring the unknown, provided we have some opportunity to exploit the results of what we learn from exploring.
Exploration in itself has value, since trying new things increases our chances of finding the best. So taking the future into account, rather than focusing just on the present, drives us toward novelty.
Regrets, I’ve had a few. But then again, too few to mention. —FRANK SINATRA
“To try and fail is at least to learn; to fail to try is to suffer the inestimable loss of what might have been.”
“regret minimization framework.”
Regret is the result of comparing what we actually did with what would have been best in hindsight.
algorithms that offer the guarantee of minimal regret. Of the ones they’ve discovered, the most popular are known as Upper Confidence Bound algorithms.
Upper Confidence Bound algorithms implement a principle that has been dubbed “optimism in the face of uncertainty.”
The success of Upper Confidence Bound algorithms offers a formal justification for the benefit of the doubt. Following the advice of these algorithms, you should be excited to meet new people and try new things—to assume the best about them, in the absence of evidence to the contrary. In the long run, optimism is the best prevention for regret.
(Google infamously tested forty-one shades of blue for one of its toolbars in 2009.)
This procedure focuses on decisively resolving the question of which treatment is better, rather than on providing the best treatment to each patient in the trial itself. In this way it operates exactly like a website’s A/B test, with a certain fraction of people receiving an experience during the experiment that will eventually be proven inferior.
More generally, our intuitions about rationality are too often informed by exploitation rather than exploration. When we talk about decision-making, we usually focus just on the immediate payoff of a single decision—and if you treat every decision as if it were your last, then indeed only exploitation makes sense. But over a lifetime, you’re going to make a lot of decisions. And it’s actually rational to emphasize exploration—the new rather than the best, the exciting rather than the safe, the random rather than the considered—for many of those choices, particularly earlier in life. What we
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I had reached a juncture in my reading life that is familiar to those who have been there: in the allotted time left to me on earth, should I read more and more new books, or should I cease with that vain consumption—vain because it is endless—and begin to reread those books that had given me the intensest pleasure in my past. —LYDIA DAVIS
the size of people’s social networks (that is, the number of social relationships they engage in) almost invariably decreases over time.
Being sensitive to how much time you have left is exactly what the computer science of the explore/exploit dilemma suggests.
“To lower costs per unit of output, people usually increase the size of their operations,” wrote J. C. Hosken in 1955, in the first scientific article published on sorting. This is the economy of scale familiar to any business student. But with sorting, size is a recipe for disaster: perversely, as a sort grows larger, “the unit cost of sorting, instead of falling, rises.” Sorting involves steep diseconomies of scale, violating our normal intuitions about the virtues of doing things in bulk.
This is the first and most fundamental insight of sorting theory. Scale hurts.
all records in sports reflect the single best performance. Computer science, however, almost never cares about the best case.
It gets worse from there. There’s “exponential time,” O(2n), where each additional guest doubles your work. Even worse is “factorial time,” O(n!), a class of problems so truly hellish that computer scientists only talk about it when they’re joking—as we were in imagining shuffling a deck until it’s sorted—or when they really, really wish they were.
“Mergesort is as important in the history of sorting as sorting in the history of computing.”
The power of Mergesort comes from the fact that it indeed ends up with a complexity between linear and quadratic time—specifically, O(n log n), known as “linearithmic” time.
Err on the side of messiness.
As the cost of searching drops, sorting becomes less valuable.
Computer science shows that the hazards of mess and the hazards of order are quantifiable and that their costs can be measured in the same currency: time. Leaving something unsorted might be thought of as an act of procrastination—passing the buck to one’s future self, who’ll have to pay off with interest what we chose not to pay up front. But the whole story is subtler than that. Sometimes mess is more than just the easy choice. It’s the optimal choice.
move from “ordinal” numbers (which only express rank) to “cardinal” ones (which directly assign a measure to something’s caliber) naturally orders a set without requiring pairwise comparisons. Accordingly, it makes possible dominance hierarchies that don’t require direct head-to-head matchups.
