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“Gifts! Especially, gift certificates!” sneered Ebenezer Scrooge. “A finer piece of humbug the world has never seen. Why, for no charge at all, they will convert my hard-earned cash to a card that can be redeemed at one store only. No doubt the store hopes the recipient will lose it. How generous!”

“But, Uncle Scrooge,” Fred explained patiently, “when I buy my wife a gift card for Havisham’s, her favorite fine clothing shop, I know that she will spend the money there, on something she will truly enjoy, when she would hesitate to indulge herself on a purchase made with money.”

“Are there no thrift shops? No doubt she could be attired comfortably enough for a few shillings at Nickleby’s without making herself a slave to fashion. And what, may I ask, does she get for you?”

“Well, for instance,” said Fred, “last year, Clara gave me a gift certificate for Copperfield’s Jewelers. I purchased this reliable timepiece there, when I might have allowed my punctuality to continue to suffer with my sad old pocket-watch if deciding on my own account.”

“A fine watch, nay, a fine waste indeed!” harrumphed Scrooge. “I do quite well with the old pocket-watch my grandfather gave me, fifty years ago, thank you – I wind it three times a day, and have never been late a minute in my life. Now, I suppose your gifts to one another were of equal value?”

“Yes, of course. Neither of us would wish to be less generous than the other.”

“Forgive an old money-lender his arithmetic,” Scrooge began craftily, “but I wish to ensure I understand this aright. You each gave the other a gift certificate for, I shudder to think it, five pounds?”

“That’s exactly it.”

“Bah, my clerk Cratchit and his large family live quite comfortably on five pounds a month, I am sure! You proceeded to use these certificates on goods which your natural sense of thrift and clear-headedness would have told you were mere frivolity?”

“Well, perhaps…”

“Young sir, my long experience in financial matters tells me exactly what you have here: a money-laundering scheme!”

“Uncle!”

“Protest if you will; there is no other word for it. The final outcome is no different from that if you had each spent your five pounds on goods for yourselves, goods which you yourself would describe as wanton indulgences. By funneling this same money into gift certificates ‘for each other,’ you believe you have cleansed the money of its sinful use, and cleansed yourselves of the sin of waste, turning it into a so-called generous impulse. You convince yourselves you are all such noble creatures while indulging your most frivolous desires. But old Scrooge knows a bit of laundering when he sees it! A fine humbug!” In making this triumphant proclamation, Scrooge experienced the one hollow sort of pleasure of which his small old heart was capable: pleasure in his own cleverness at unmasking those who pretended to be more generous than he, at revealing them to be just as self-interested as the old miser himself, as he always knew they must be.

Fred responded in measured tones, but to those who knew this most affable of gentlemen, the slight steel in his voice would have made it resound as though he used a tone of thunder. “Those of us who believe that to be truly human, one must be a social creature, are not so poor at arithmetic as you seem to think. Yes, a child can understand your argument that purchasing gift certificates is a senseless ritual. Though a ritual may have an arbitrary aspect, this does not make it senseless. When I buy my wife a gift, it causes me to consider what brings her joy, and to empathize with her pleasure at receiving it, and it causes her to think of me when she uses it. And so it is in reverse. Gift-giving may be a ritual, but it is a ritual that brings us closer together. Of course, enhancing our consideration of those closest to us is only the first step. A truer expression of the Christmas spirit is generosity towards those most in need. Those who love and feel loved, as you do not, poor sir, are more likely to be generous to the poor who share our city. Clara and I always plan our charitable work for the year during the Christmas season.”

“Charity, humbug. Making idle people merry is no virtue.”

“There are many other purposes of charity, Uncle, but at the risk of my immortal soul, I shall debate you on your own coldhearted terms. Your logic concerning gifts appears infallible, but you have made what my dear old professor of economic philosophy would call an implicit assumption, and a most unwarranted one.”

“Eh? What’s this?”

“Why Uncle, you have assumed that each of us makes the best possible spending decisions if left entirely to our own devices, in an entirely anti-social world. As you follow the markets, you know it has been a harder few winters than usual for all of us in London. We are all a bit chary of unnecessary spending.”

“Chary, yes, cautious, of course!” Scrooge interjected. “Cautious as well you should be, and not any less so because of a supposed ‘Christmas spirit.'”

“Caution is a fine thing, Uncle, but should it lead us to consume nothing beyond the bare essentials of life? Is it rational for all to live as you do, spurning all material comforts for an ever-growing pile of gold? I am sure you take no pleasure in the thought of leaving your gold to your unworthy next of kin, so perhaps you intend to be buried with it.”

“Bah, perhaps I do, what of it? De gustibus non est disputandum. By your own description, you are no better than I. You prefer ‘comforts’ – indulgences, I call them – I prefer to keep my gold. Simply spare me your pretense of a generous spirit!”

“For the moment, spiritual worth is not the issue. To the extent that decisions are a matter of taste, who is to say that the decisions Clara or I would make as individuals are superior to the decisions we make as a harmonious unit? That point aside,” Fred continued, cutting off the retort rapidly forming on his uncle’s lips, “here is one that even you will appreciate. My dear Prof. Senyek, who is literally generations ahead of his time, acquainted me with the paradox of thrift.”

“What paradox could there be concerning the virtue of thrift?”

“Well, Uncle, if a virtue is, as the noted Prussian Mr. Kant would have had it, that which makes for a good society if followed as a universal law, then thrift, particularly in these hard times, is no virtue.”

“I beg your pardon? You mean to say thrift is an especial virtue in hard times!”

“No. In hard times, if everyone saves more, our factories are under-utilized, and goods rot away on the shelves of stores. The factories and stores reduce their workforce, precious wages are missed, and the icy hand of thrift grips the city’s purse-strings still tighter.”

For the first time, Scrooge was without an immediate retort. Was it possible that his good-for-nothing nephew was teaching him, Scrooge, about practical matters? “Then,” he continued slowly, “if every household does as you do, and bypasses their natural thrift through your most ingenious scheme of gift-giving and supposed generosity, the downward spiral is halted. Stores fill their shelves in anticipation of a great December rush! Factories are employed above their usual capacity!”

With this, Scrooge sprang up with a most uncharacteristic energy, and reached for his hat and coat. “Wherever are you going, Uncle Scrooge?” asked an astonished Fred.

“My fine, dear, boy, I am a major shareholder in Harrod’s. I must tell them at once to redouble their Christmas display, and make sure their leaflets reach all neighborhoods, from the poor to the great. Most importantly, the leaflets must emphasize the noble virtues of ‘generosity’ and ‘Christmas spirit.’ An eminently valuable humbug! Thank you, dear boy! Merry Christmas!” Concluding with this most unlikely of salutations, Scrooge leapt out the door, and he could be heard repeating “Merry Christmas!” with great enthusiasm to all he encountered as he passed through the streets of the city.

Fred was left standing alone in Scrooge’s office, a bemused half-smile on his face. He reflected that in changing his uncle’s mind, he had done all that was humanly possible. Saving the old man’s hardened soul would truly require supernatural intervention.

I’ve been watching NFL football all my life, and endgame timeout-strategy (being susceptible to analysis by non-football people) has always attracted my attention, but there is one numerical quirk with big potential consequences which I only noticed this season. Consider this scenario:

Late in an NFL game, Team A has the ball and a small lead. Team B has 1 timeout left. Team A, on 1st down, runs it up the middle for no gain. The play ends (to be precise, the 40-second play clock is reset) with 2:43 remaining. What is Team B’s strategy?

The obvious thing is to call the timeout now; when it appears not to matter, I mostly see teams use their timeouts on early downs rather than later downs. But in this case, that would be an enormous mistake! Consider the very likely continuations:

Use timeout now:

2nd down: Play begins 2:43, ends 2:38. Clock ticks down to 2:00. Two-minute warning.

3rd down: Play begins 2:00, ends 1:55. Clock runs to 1:16.

4th down: Punt, snap at 1:16.

 

Save timeout:

2nd down: Play begins at 2:04, ends 1:59. Two-minute warning.

3rd down: Play begins 1:59, ends 1:54. Team B calls final timeout.

4th down: Punt, snap at 1:54.

 

Using the timeout early costs almost 40 seconds! It costs almost the full value of a timeout, because it removes all the power of the two-minute warning. I can’t actually cite a game from memory where someone screwed this up, but I don’t think I would have been alert to it until recently. Given that I’ve never heard this mentioned on TV, and given the general propensity of NFL coaches to screw up timeout strategy in more obvious ways, I would expect a high proportion of mistakes when this situation arises.

The bottom line: If the clock resets with roughly 2:41-2:49 remaining, don’t use your timeout now. Wait until after the two-minute mark, to make sure you get proper value from the two-minute warning. Note that in college football there is no two-minute warning, so this whole discussion is moot.

One tiny caveat: The recommended strategy does give Team A a free license to throw the ball on 2nd down, since an incomplete pass won’t matter. They might do so and get a first down, ending the game. But there is no way this possibility cancels out the huge time savings. Team A always has the option to throw it anyway, if they want to take the risk of leaving 40 extra seconds. That is, the waiting strategy is (almost) dominant.

 

 

This essay started as a short post responding to an article by Greg Mankiw and grew longer than I expected, so to avoid cluttering the blog I switched formats. As you might guess from the potpourri in the title, the essay is intended to be readable by a broad audience. The portions which discuss the use or misuse of economic theory in the tax debate are, it is my fond hope, of interest to economists and accessible to non-economists, as is Mankiw’s article. I began this last week, so it doesn’t refer directly to this week’s big news of the Great Tax Compromise of 2010, but the ongoing negotiations were a major motivation.

I hope this has been enough of a teaser for you to click here.

The comments on Eilon’s last post turned into a dialogue on whether it is of any use for MBAs to learn game theory. I will make some brief comments on this.

With very few exceptions, professors at leading business schools are chosen for their research prowess and hence their skill at analytic thinking and modeling, and have spent little time if any in the business world. Many will do some consulting, but this comes later in their career and is not the basis on which they are hired. How do future managers benefit from professors who have never managed?

Experience may be everything, but it is not the only thing. Or, as Ricky puts it, “Experience is inevitable. Learning is not.” Just as important a cornerstone as experience are models by which to organize experience. The research professor will teach in part via war stories (cases), just as a business veteran would. These are not his war stories, of course, so what it the advantage of his expertise? The advantage comes via in-depth understanding of models. Why is that so important? Models organize the information from the case into well-defined causes and effects. Then you can learn not only what the successful decision was in a given case, but how to adjust this decision for related cases, and, just as important, when the concepts from the case do not apply. A good professor will be able to make clear the assumptions that go into the model, and when they do or do not apply in real life. You always may come across situations where none of the models seems to help, and have to improvise. But the same would certainly be true if you learned from non-theoretical professors; their experience might not be closely enough related to yours. Worse, it might be close enough that you think you should make similar decisions, but you miss a key difference, one which a good model would have drawn your attention to.

A source of real-life war stories that can be better understood with the help of game theory is Co-optition by Brandenburger and Nalebuff. The contents of this book are a great response to anyone who is not sure why theorists might be at a business school.

 

I voted today because I enjoy voting (and because I happen to have a fairly strong preference.) The tiny chance of being pivotal surely doesn’t compensate me for the time, but I enjoy seeing the vote totals and knowing that I influenced the final digit. I have a feeling that many people are like me, or equivalently, vote out of a sense of duty. This is a bit of a problem for the game-theory models of voting where rational people weigh the probability of being pivotal and intensity of preference against the “cost” (presumably time) of voting. These models tend to predict a tiny turnout. Now, I don’t doubt that more people vote in close elections, so the likelihood of being pivotal has *some* impact on turnout. Intensity of preference probably matters more, though. I know political scientists debate the determining factors in turnout ad infinitum; I’m not trying to break new ground here, just sharing some thoughts and perhaps starting a discussion.

Sorry to disappoint: I don’t mean inappropriate in the Jeff Ely sense, but in the overmathematical sense. This post is intended for people who have had an intro econ PhD sequence, at least as far as expected utility theory.

Nabil was showing me a question he asks the MBAs about their preference between two gambles with the same mean and variance. There is no right answer; the idea is just to show them that distinct distributions can have the same mean and variance. With some normalization, the gambles boil down to: A: (1, .5;-1, .5) vs. B:(2, .125; 0, .75; -2, .125). You could think of the units as thousands or ten-thousands to make it more interesting. When Nabil showed me the problem, I said, only half-joking, “I don’t know, I’d have to think about it; I’ve never decided whether I’m kurtosis-averse.” Indeed (as my discussion will confirm), neither gamble second-order stochastically dominates the other, i.e. being risk-averse (having a concave utility function) doesn’t tell you which to choose. I decided to see what would be preferred for various CARA or CRRA utilities, and discovered the following:

 

Theorem: Let A and B be two bounded gambles which are symmetric about the same mean. If each moment of B is at least as big as each moment of A, A is weakly preferred to B by every CARA or CRRA utility function. If at least one inequality is strict, the preference is strict.

Proof: Note that symmetry means all odd moments are 0. Expanding any CARA or CRRA utility as a power series centered at the mean, we find that all even coefficients are negative. These series converge absolutely and uniformly in the range of definition, so linearity of expectation applies. Q.E.D.

 

That is, people with such utility functions are kurtosis-averse (and 6th-moment-averse, and 8th-moment-averse…) So any CARA or CRRA person prefers A to B above. Apparently most MBAs also pick A; my sense is that a small probability of a large risk tends to loom large in one’s mind. I admit to a similar psychological bias; I would force myself to overcome it if there were a good reason, but if I can support the decision with any CARA or CRRA function, that sounds all right to me.

 

So what kind of function chooses B? By my claim above there is a concave function that does, and indeed:

Claim: Let A and B be gambles with the same mean. Normalize this mean to 0, and suppose {E[|A|]>E[|B|]}. Then any concave, piecewise linear function with unique kink at 0 will prefer B to A.

Proof: Simple calculation is left to the reader.

Such piecewise linear functions are often used in a simplified version of prospect theory. So, that part of prospect theory tends to select B, but the overweighting of small probabilities tends to select A. Florian Herold has a paper about disentangling these aspects of prospect theory. I’m getting too tired to think about how it applies; perhaps he would like to comment.

Bottom line: I’ve always thought that CRRA sounded pretty reasonable. Today I learned that if I want to stick with this, I’m kurtosis-averse, and also…what should we call the 6th moment? Sextosis-averse? (nod to Jeff.)

P.S.: An interesting not-so-technical question about reference points: On the actual homework, the mean was 5 and not 0. I know when I looked at this, my psychological reference point instantly became 5, and my feelings about gains and losses went accordingly. Would this also be true of MBAs? Or would they not be so quick to recognize symmetry and “translate” their expectations?

I’ve been sitting in on our introductory Decision Science course for MBAs, which I’m planning to teach for the first time next year. One recent topic was the “flaw of averages” (a nice catchphrase due to Sam Savage, introduced into our course by Nabil Al-Najjar.) In mathematical terms, letting X be an exogenous random variable, a be a decision parameter, and f be any function not linear in X, this says

{{\rm argmax}_a E[f(a,X)] \neq {\rm argmax}_a f(a,E[X])}

In plain English, this just means that when you make a decision, do not assume that uncertainty will always take on its average, or “expected,” value. This leads me to a related point, and to my own catchphrase (see title) which I hope the students will find useful. “Expected Value” is an awful piece of terminology, as judged by its (very weak) relationship to the English word “expected.” I’m certainly not one of the first to point this out: It is possible, of course, for X to never come close to its “expected” value. The real question is why the term persists, when we have the perfectly clear term “average value” available. No better reason than our QWERTY keyboard, I suppose; once everyone is used to writing E for expectation, it’s hard to shake. Anyway, I think I’ll be showing a slide with the motto in the title next year; hopefully the students will find that memorable. It will be tempting, I suppose, to reinforce this with a brief clip from Monty Python: “No one expects the Spanish Inquisition!”

Tonight at the department party we were continuing the debate on the usefulness of game theory. Part of the argument was about whether the math has any utility, or if you could get just as much from verbal arguments about strategy as in Schelling’s work. Rakesh had some good positive examples in pricing and auctions which perhaps he’ll write up here if he has time. Here’s one that came to my mind just now. You may not consider poker “real world” but bear with me; I think it illustrates a general point.

Consider the simplified model of poker in von Neumann-Morgenstern. In one of the versions, there is a result that the unique equilibrium strategy is to bet with the best 30% and worst 10% of hands. Now, nothing nearly so simple is optimal in real poker, and furthermore all poker players, even the least mathematical, know that they should bluff sometimes. So what was gained by this exercise? Well, we have a qualitative recommendation to bluff only with your very worst hands. This is far from obvious, and I never thought of it before doing the mathematical exercise. True, I can translate it into a verbal argument, as follows: bad hands and medium hands become equal if you bet with them, since they both lose whenever (or almost whenever) you get called. But medium hands are significantly better than bad hands when you check, since they may win the pot in a showdown. So you bluff with bad hands, not of course because they are better to bluff with, but because they are worse to check with. Note that this logic only applies fully when you are last to speak in the last round of betting. In earlier rounds, “semi-bluffs” where you hope for a fold but have chances if called are a common part of good strategy, and are more common than pure bluffs.

Enough about poker; here is the broader point of the example. Solving the problem mathematically imposes a discipline on our reasoning process which forces us to discover an important qualitative insight we could easily have missed otherwise. True, I am sure some strong poker players came to this insight intuitively over the years without formal study of Bayesian games, but many players surely missed it. The examples Rakesh was discussing seem similar to me. Yes, once you hear certain insights described in words, you may decide we never needed the math. But this is much too facile, akin to thinking that every problem is easy once you’ve seen the answer. Any illuminating chain of reasoning can be missed as easily as found, and formal models can channel our reasoning in the right direction.

Browsing through Keynes’ “A Treatise on Probability,” I came across a pretty nugget which Keynes credits to Laplace. Suppose you want to make a fair decision via coin flip, but are afraid the coin is slightly biased. Flip two coins (or the same coin twice), and call it “heads” if the flips match, tails otherwise. This procedure is practically guaranteed to have very small bias. In fact, if we call {b_i = P_i(H)-P_i(T)} the bias of flip {i}, a quick calculation shows that the bias of the double flip is {b_1b_2}, so that a 1% bias would become a near-negligible .01%.

I noticed that we can extend this; consider using {n} flips, and calling the outcome “heads” if the number of tails is even, “tails” if it is odd. An easy induction shows that the bias of this procedure is {b_1b_2...b_n}, which of course goes to 0 very quickly even if each coin is quite biased. Here also is a nice direct calculation: consider expanding the product

{b_1b_2 \ldots b_n = (P_1(H) - P_1(T)) \cdots (P_n(H) - P_n(T))}

The magnitude of each of the {2^n} terms is the probability of a certain sequence of flips; the sign is positive or negative according to whether the number of tails is even or odd. Done.

I can hardly believe it of such a simple observation, but this actually feels novel to me personally (not to intellectual history, obviously.) Not surprising exactly, but novel. I suppose examples such as the following are very intuitive: The last (binary) digit of a large integer such as “number of voters for candidate X in a national election” is uniformly random, even if we know nothing about the underlying processes determining each vote, other than independence (or just independence of a large subset of the votes.)


As theorists we work with many definitions of rationality, all of them far from perfect. Here’s a contribution from Karl Popper, from The Logic of Scientific Discovery:

 

“…I equate the rational attitude and the critical attitude. The point is that, whenever we propose a solution to a problem, we ought to try as hard as we can to overthrow our solution, rather than defend it. Few of us, unfortunately, practice this precept…”

 

All of our axiomatic definitions of rationality have the flavor of presenting certain possible avenues of self-criticism to a decision-maker, and calling him rational if he is immune to these. This attitude is typified by the famous story of Savage’s reaction to the Allais paradox: he initially made the “normal” pair of choices, which contradict the substitution axiom. When this was pointed out, he considered his decisions a mistake.

Of course, any set of formal rules will leave out some possible criticisms, i.e. be insufficient for rationality, and conversely there is almost always an argument that a principle, however compelling, is not necessary for rationality. Furthermore, each principle takes a certain amount of mental energy to check, and the art of good decision-making must involve making wise decisions as to which principles to prioritize. Decision theorists can, in principle aid this process by proving equivalence results: “If you want to follow Savage’s axioms, use a subjective probability distribution.” Unfortunately, it is difficult to conceive that there will ever be a Grand Unified decision theory which aids the decision-maker in avoiding every possible criticism. Actually, such a theory would be tantamount to “strong AI,” the problem of building a machine which mimics or exceeds human capacities, which is considered at least decades away. Decision theory is not so ambitious, but merely tries to help people avoid selected mistakes in well-defined areas.

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