Category Archives: economics

How high can high-level programming go?

Our first prototype of Predictalot was written mainly in Mathematica with a rudimentary web front end that Dan Reeves put together (with editable source code embedded right on the page via etherpad!). It proved the concept but was ugly and horribly slow.

Screenshot of pre-alpha Predictalot: Mathematica + etherpad + web

Dan and I built a second prototype in PHP. It was even uglier but about twice as fast and somewhat useable on a small scale (at least by user willing/able to formulate their own propositions in PHP). Yet it still wasn’t good enough to serve thousands of users accustomed to simplicity and speed.

Screenshot of alpha Predictalot: PHP + YAP

The final live version of Predictalot was not only pleasing to the eye — thanks to Sudar, Navneet, and Tom — but pleasingly fast, due almost entirely to the heroic efforts of Mridul M who wrote a mini PHP parser inside of java and baked in a number of datbase and caching optimizations.

Screenshot of live beta Predictalot: Java + Javascript + YAP

It seems that high-level programming languages haven’t climbed high enough. To field a fairly constrained web app that looks good and works well, we benefit greatly from having at least three specialists, for the app front end, the app back end, and the platform back end (apache, security, etc.).

Here’s a challenge to the programming language community: anything I can whip up in Mathematica I should be able to run at web scale. Math majors should be able to create Predictalot. Dan and I can mock up the basic idea of Predictalot but it still takes tremendous talent, time, and effort to turn it into a professional looking and well behaved system.

The core market math of Predictalot — a combinatorial version of Hanson’s LMSR market maker — involves summing thousands of ex terms. Here we are in the second decade of the new millenium and in order for a sum of exponentials to execute quickly and without numeric overflow, we had to work out a transformation to conduct all our summations in log space. In other words, programming still requires me to think about how my machine represents my number. That shouldn’t qualify as “high level” thinking in 2010.

I realize I may be naively asking too much. Solving the challenge fully is AI-complete. Still, while we’re making impressive strides in artificial intelligence, programming feels much the same today as it did twenty years ago. It still requires learning specialized tricks, arcane domain knowledge, and optimizations honed only over years of experience, and the most computationally intensive applications still require that extra compilation step (i.e., it’s still often necessary to use C or Java over PHP, Perl, Python, or Ruby).

Some developments hardly seem like progress. Straightforward HTML markup like border=2 has given way to unweildy CSS like style=”border:2px solid black”. In some ways the need for specialized domain knowledge has gone up, not down.

Visual programming is an oft-tried, though so far largely unsuccessful way to lower the barrier to programming. Pipes was a great effort, but YQL proved more useful and popular. Google just announced new visual developer tools for Android in an attempt to bring mobile app creation to the masses. Content management systems are getting better and broader every day, allowing more and more complex websites to be built with less time touching source code.

I look forward to the day that computational thinking can suffice to create the majority of computational objects. I suspect that day is still fifteen to twenty years away.

Why automated market makers?

Why do prediction markets need automated market makers?

Here’s an illustration why. Abe Othman recently alerted me to intrade’s market on where basketball free agent LeBron James will sign, at the time a featured market. Take a look at this screenshot taken 2010/07/07:

Wide bid-ask spread for Lebron James contract on intrade -- needs a market maker 2010-07-07

The market says there’s between a 42 and 70% chance James will sign with Cleveland, between a 5 and 40% chance he’ll sign with Chicago, etc.

In other words, it doesn’t say much. The spreads between the best bid and ask prices are wide and so its predictions are not terribly useful. We can occasionally tighten these ranges by being smarter about handling multiple outcomes, but in the end low liquidity takes the prediction out of markets.

Even if someone does have information, they may not be able trade on it so may simply go away. (Actually, the problem goes beyond apathy. Placing a limit order is a risk — whoever accepts it will have a time advantage — and reveals information. If there is little chance the order will be accepted, the costs may outweigh any potential gain.)

Enter automated market makers. An automated market maker always stands ready to buy and sell every outcome at some price, adjusting along the way to bound its risk. The market maker injects liquidity, reducing the bid-ask spread and pinpointing the market’s prediction to a single number, say 61%, or at least a tight range, say 60-63%. From an information acquisition point of view, precision is important. For traders, the ability to trade any contract at any time is satisfying and self-reinforcing.

For combinatorial prediction markets like Predictalot with trillions or more outcomes, I simply can’t imagine them working at all without a market maker.

Abe Othman, Dan Reeves, Tuomas Sandholm, and I published a paper in EC 2010 on a new automated market maker algorithm. It’s a variation on Robin Hanson‘s popular market maker called the logarithmic market scoring rule (LMSR) market maker.

Almost anyone who implements LMSR, especially for play money, wonders how to set the liquidity parameter b. I’ve been asked this at least a dozen times. The answer is I don’t know. It’s more art than science. If b is too large, prices will hardly move. If b is too small, prices will bounce around wildly. Moreover, no matter what b is set to, prices will be exactly as responsive to the first dollar as the million and first dollar, counter to intuition.

Our market maker automatically adjusts its level of liquidity depending on trading volume. Prices start off very responsive and, as volume increases, liquidity grows, obviating the need to somehow guess the “right” level before trading even starts.

A side effect is that predictions take the form of ranges, like 60-63%, rather than exact point estimates. We prove that this is a necessary trade off. Any market maker that is path independent and sensitive to liquidity must give up on providing point estimates. In a way, our market maker works more like real bookies who maintain a vig or spread for every outcome.

The market maker algorithm is theoretically elegant and seems more practical than LMSR in many ways. However I’ve learned many times than nothing can replace implementing and testing a theory with real traders. Final word awaits such a trial. Stay tuned.

It’s official: More people are playing Predictalot than Mafia Wars

It’s true.

More people are playing Predictalot today than Mafia Wars or Zynga Poker… On Yahoo!, that is.

In fact, Predictalot is the #1 game app on Yahoo! Apps by daily count. By monthly count, we are 5th and rising.

A prediction is being made about every three minutes.

Come join the fun.

predictalot most popular game app on yahoo 2010-06-12

Predictalot for World Cup: Millions of predictions, stock market action

I just left the 2010 ACM Conference on Electronic Commerce, where six (!) out of 45 papers were about prediction markets.

Yahoo! Lab’s own Predictalot market is now live and waiting for you to place almost any prediction your heart desires about the World Cup in South Africa.

Here are some terribly useful things you can learn this time around. All numbers are subject to change, and that’s kind of the point:

  • There’s a 37% chance Brazil and Spain will both make it to the final game; there’s only a 15% chance that neither of them will make it
  • There’s is a 1 in 25 chance Portugal will win the cup; 1 in 50 for Argentina
  • 42.92% chance that a country that has never won before will win
  • 19.07% chance that Australia will advance further than England
  • 65.71% chance that Denmark, Italy, Mexico and United States all will not advance to Semifinals
  • Follow Predictalot on twitter for more

If you think these odds are wrong, place your virtual wager and earn some intangible bragging rights. You can sell your prediction any time for points, even in the middle of a match, just like the stock market.

There are millions of predictions available, yet I really believe ours is the simplest prediction market interface to date. (Do you disagree, Leslie?) We have an excellent conversion rate, or percent of people who visit the site who go on to place at least one prediction — for March Madness, that rate was about 1 in 5. One of our main goals was to hide the underlying complexity and make the app fast, easy, and fun to use. I personally am thrilled with the result, but please go judge for yourself and tell us what you think.

In the first version of Predictalot, people went well beyond picking the obvious like who will win. For example, they created almost 4,000 “three-dimensional” predictions that compared one team against two others, like “Butler will advance further than Kentucky and Purdue”.

If you’re not sure what to predict, you can now check out the streaming updates of what other people are predicting in your social circle and around the world:

Predictalot recent activity screenshot 2010-06-11 18:45

Also new this time, you can join a group and challenge your friends. You can track how you stack up in each of your groups and across the globe. We now provide live match updates right within the app for your convenience.

If you have the Yahoo! Toolbar (if not, try the World Cup toolbar), you can play Predictalot directly from the toolbar without leaving the webpage you’re on, even if it’s Google. 😉

playing predictalot from the yahoo! toolbar

Bringing Predictalot to life has been a truly interdisciplinary effort. On our team we have computer scientists and economists to work out the market math, and engineers to turn those equations into something real that is fast and easy to use. Predictalot is built on the Yahoo! Application Platform, an invaluable service (open to any developer) that makes it easy to make engaging and social apps for a huge audience with built-in distribution. And we owe a great deal to promotion from well-established Yahoo! properties like Fantasy Sports and Games.

We’re excited about this second iteration of Predictalot and hope you join us as the matches continue in South Africa. We invite everyone to join, though please do keep in mind that the game is in beta, or experimental, mode. (If you prefer a more polished experience, check out the official Yahoo! Fantasy Sports World Soccer game.) We hope it’s both fun to play and helps us learn something scientifically interesting.

Read more here, here, and here.

Or watch a screencast of how to play:

World Blogging Year

First: I did it! A perfect 16 out of 31. I completed the (ok, my) World Blogging Month challenge to blog every odd day in the month of March.

Last year WoBloMo leapt out of the gates with five participants but I fell five hours short of the goal. As far as I know only Anthony and I returned for year two. He succeeded too according to official Australian Rules.

Again, I found the exercise worthwhile, clearing a number of items out of my queue, albeit mostly the easy and inane ones (c.f. the barking), and boosting readership.

In fact, I enjoyed it so much that I’ve signed up for World Blogging Year (WoBloYe). I will blog every odd day of every month at least through the end of 2010, starting today.

In fact I have formally pledged to stickk to my goal. Moreover, I am putting my money where my mouth is, PM-style. For every odd day of the month that passes blog-post-free I will donate $100 to my anticharity, the re-election fund for Don McLeroy. If I miss two deadlines in a row, my antidonation will double. Three missed deadlines in a row and it will quadruple, etc.

I’ve enlisted kibotzer’s help and you can follow my progress there. Wish me luck!

Update 2010/04/02: April Fools!

P.S. In all seriousness, read that New York Times article about Don McLeroy. It’s one of the scariest articles I’ve read in a long time. It’s about how ultra conservatives on the Texas board of education are rewriting history and science according to biblical and republican dogma, and how standards in that enormous state can dictate what gets printed in textbooks nationwide. They’ve done things like add Newt Gingrich and delete Edward Kennedy as significant Americans. They’ve banned classic children’s books by Bill Martin Jr. because they confused him with a different Bill Martin, author of “Ethical Marxism”.

It is the most crazy-making thing to sit there and watch a dentist and an insurance salesman rewrite curriculum standards in science and history. Last year, Don McLeroy believed he was smarter than the National Academy of Sciences, and he now believes he’s smarter than professors of American history.

Prediction market news: HSX, Nadex, Predictalot, mailing lists, & EC’10

Some prediction market related news:

  1. The Hollywood Stock Exchange is the latest example of life imitating art imitating life. The venerable HSX, the second oldest play-money prediction market (and oldest that anyone has actually heard of) is getting real. You’ll soon be able bet real money on box office returns and deposit your winnings in First Life. The move has been a long time coming — Cantor Fitzgerald purchased HSX in 2001 with the explicit goal of converting it into a real-money exchange — but Cantor was decimated in the September 11 WTC attacks, and the road to regulatory approval has been slow. The real HSX will of course say goodbye to the virtual specialist and the opening weekend adjust, two facets of the game that make it fun to play, but that create significant amounts of (virtual) wealth out of thin air. The Cantor Gaming group is engaged in other interesting initiatives. They are taking over a sportsbook in Las Vegas and turning it into more of a derivatives exchange with live in-game betting, a step toward my dream of a geek-friendly casino. Interestingly, another company called Veriana Networks is close to launching a competing Hollywood derivatives market called the Trend Exchange.
  2. I’ve recently seen ads in the elevator in my building for Nadex, the new incarnation of HedgeStreet that was acquired, redesigned, and rebranded as the North American Derivatives Exchange. I haven’t checked in on the market since the rebranding — in fact I can’t remember if my HedgeStreet account transfered over. I wonder how widespread their ad campaign is and how Nadex is doing?
  3. Predictalot update: 50,000 people have logged in and 11,000 have made at least one prediction.* They’ve made 91,000 total predictions of 5,500 different kinds. We’ve been written up in NYTimes, VentureBeat, L’Atelier (“C’est ce que fait Yahoo!, avec Predictalot.”), and Wired. People are saying everything from “wicked fun”, “great idea”, and “love the game” to “disaster”. Either way, people care. It’s been a ton of fun and its popularity has wildly surpassed my expectations. The final four is this weekend. Then not much time to get in as many improvements as we can before the World Cup.
  4. If you’re not aware, there is a new prediction market mailing list that strives to be open, transparent, objective, and independent under some simple ground rules. I encourage you to join it. I abandoned the old PM list owned by John Maloney for a number of reasons. The old list clearly operates in a payola style — sponsors of Maloney’s events receive prominent billing. That might be ok, except Maloney bills his list and his events as open, non-profit affairs, yet he charges quite a bit and is evasive about financial details (actually he says he doesn’t keep records). Over the years, Maloney has asked me/Yahoo! to sponsor his events several times, occasionally berating me when the answer is no. He’s made some questionable wikipedia edits to highlight his group. Still, I’ve gone to some of his events and find that very good people go there and give interesting talks, and Maloney is very reasonable and personable face to face. My final straw came when Maloney censored me on his list for criticizing him, admittedly in a snarky way.** So I joined with Emile, Bo, Forrest, Justin, Jed, Adam, and others to start afresh.
  5. The 2010 ACM Conference on Electronic Commerce to be held June 7-11 at Harvard features at least six papers on prediction markets:
    • L. Jian and R. Sami, Aggregation and Manipulation in Prediction Markets: Effects of Trading Mechanism and Information Distribution
    • K. Iyer, R. Johari and C. Moallemi, Information Aggregation in Smooth Markets
    • A. Othman, D. Pennock, D. Reeves and T. Sandholm, A Practical Liquidity-Sensitive Automated Market Maker
    • S. Goel, D. Reeves, D. Watts and D. Pennock, Prediction Without Markets
    • A. Othman and T. Sandholm, Automated Market-Making in the Large: The Gates Hillman Prediction Market
    • Y. Chen and J. Wortman Vaughan, A New Understanding of Prediction Markets Via No-Regret Learning

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* I believe one in five logged in users placing a prediction is actually a high conversion rate for a prediction market and a testament to our user interface design. Mike Speiser told me that Bix had a tough time converting their users — who loved to vote — into traders in their American Idol prediction market, one reason why they abandoned the experiment.
** The exchange went like this: Maloney’s assistant Jennifer Hewitt announced that “Crowdcast, the leading provider of prediction market solutions for collective forecasting” was joining Maloney’s latest event. Emile asked “leading… Based on what metric, exactly?”. Then I quipped “leading recent contributor to John Maloney”, which Maloney censored. In Maloney’s defense, the “leading provider” language actually came from Crowdcast’s own website (their meta description tag) and so it appears in the snippet when you search for “crowdcast” in Google.

Gates Hillman Prediction Market: The Movie

From September 2008 to August 2009, Carnegie Mellon graduate student Abe Othman ran a prediction market to forecast when CMU’s two new computer science buildings, Gates and Hillman, would open. Abe designed the market to predict not just the magic day, but the likelihood of every possible opening day (in other words, the full probability distribution), at the time making his the largest prediction market built in terms of the number of outcomes.

Now Abe created a fascinating video showing the evolution of prices over time in his market. You can see qualitatively that the thing actually worked, zeroing in closer and closer to the actual opening day as the market progressed.

Figure 3 on page 7 of Abe’s paper with Tuomas Sandholm in the 2010 ACM Conference on Electronic Commerce conveys similar information.

Evolution of prices in the Gates Hillman prediction market

Despite plenty of precedent, and despite increasing evidence that non-market methods do surprisingly well too,* I still find it astonishing to see a bunch of people play a subtle betting game for nothing but bragging rights or a small prize and end up with something reasonably intelligent.

By implementing a working market used by over a hundred CMU students, Abe learned a great deal about practical yet important details, from the difficulty of crisply defining ground truth (when exactly is a building officially “open”?) to the black art of choosing the liquidity parameter of Hanson’s market maker.** Abe independently created an intuitive interval betting interface similar, and in some ways superior, to our own Yoopick interface and Leslie Fine’s Crowdcast interface. Abe went so far as to interview his top traders in great detail to learn about their strategies, which ran the gamut from building automated statistical arbitrage agents to calling construction crew members to learn inside information. Abe observed that interval betting using Hanson’s market maker leads to very “spiky” prices. Starting from this informal observation, Abe was able to actually prove an impossibility result of sorts that any price function with otherwise reasonable properties must be spiky in a formal sense. See Abe and Tuomas’s paper for the details.

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* Our paper “Prediction without markets”, by Sharad Goel, Daniel Reeves, Duncan Watts, and me, will be published in the 2010 ACM Conference on Electronic Commerce.
** Abe has now developed a flexible market maker that automatically adjusts liquidity to match trader activity. The paper, by Abe, Tuomas, Daniel Reeves, and me, will also be published in the 2010 ACM Conference on Electronic Commerce.

Housing arbitrage, or the $1.4 million muse

Has anyone heard of the following trick, which might be called housing arbitrage?

Buy one house at the beach and a second house near a ski resort. You live in the beach house in the winter and the ski resort in the summer. You rent out the beach house in the summer and the ski resort in the winter.* Can your earnings (rental revenue minus mortgage costs) be enough to live on?

Why it could work: the cost of each house will be roughly proportional to the average annual rental income in that location. If you didn’t live in the properties at all, you should roughly break even (income = mortgage payments). But you are living in each location during the time when rent is essentially free (not contributing to the average) so you have no housing costs. If you find good enough deals (or put money down, or have some small income like freelance writing, etc.) your income may exceed your mortgage enough to live on.

What’s the minimum you could get started with on this strategy? Probably a minimum income to live comfortably as a starting point would be $70K before taxes: see justification below. Assume you can make about 5% of a home’s value in rental income: this seems feasible. Then you need $1.4 million invested in real estate (say two $700K houses) with no mortgage (completely paid). Suppose you can also borrow at 5%. Then if you put 50% down on two $1.4 million properties ($2.8 million total), your effective mortgage rate is 2.5% and your “spread” is 2.5%, so you again earn $70K, but now you have two twice as nice houses (but more risk, need to qualify for loan, etc.). Now here is some magic. Suppose you find an incredible deal (say, in a down real estate market) and you can earn 10% in rental income. You can borrow at 5% and only want to put 20% down, still a respectable portion that the bank may be willing to go for. You buy two $600K homes ($1.2 million total) needing only $240K in cash. Now your rental revenue is $120K and your mortgage payments are $48K, so your net income is, viola, $72K!

Didn’t I forget about taxes and insurance? No, I’m just assuming these can be covered by your $70K income. I did forget about health insurance, though: that could threaten the strategy, at least in the United States. You can can hope that the new health care law helps, or keep an enjoyable day job, or purchase insurance out of the $70K.

You might say $70K pretax is not enough to live the lifestyle you want. But remember, you effectively have no housing costs, and this is just meant as a starting point. This is your “muse” as Tim Ferriss calls it: a steady reliable income that is your buffer. You still should pursue freelance ideas or business ideas that you are passionate about, and one of those just might hit it big. This just gives you freedom to pursue other ideas on your own. Hopefully even at $70K you can save some money to purchase additional properties and increase your income. Note that once your mortgage is paid off, your income will go up.

One nice thing about this strategy, and real estate investments in general, is that they are naturally inflation adjusted: rental rates should go up if inflation goes up.

This really only seems practical for people without kids in school. Although I suppose if your kids went to school in the beach location it might work. You’d only spend 2.5 months in the ski resort.

Certainly there are downsides: constantly moving, living in off-season tourist towns, living in properties that are rented half the year, dealing with renters, risk of loss or default, and managing the business headaches.

If housing arbitrage could really work, why aren’t more people doing it? Maybe it requires too much capital and maybe my math is wildly optimistic. Probably it’s no more than a fun mental exercise. I’m sure it’s been thought of. I can’t find it on a cursory web search but it seems hard to articulate to a search engine. If enough people started doing it, by definition house prices would go up to eliminate the arbitrage.

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* Maybe take a week or two in the summer at the beach and the winter at the ski house.

CS ∩ Econ news

Here are some news items about the field with no name (at least not yet, see below) that lies at the intersection of computer science and economics.

  1. The Sixth Workshop on Ad Auctions is soliciting papers. The workshop will be held June 8, 2010, in Cambridge, MA, in conjunction with the ACM Conference on Electronic Commerce (EC’10). There is a terrific organizing committee this year spanning industry and academia, CS and business schools.
  2. The EC’10 list of accepted papers is out and looks great.
  3. The first-ever Behavioral and Quantitative Game Theory Conference on Future Directions will be held May 14-16 in Newport Beach, CA. The program looks fantastic.
  4. Last fall, the University of Pennsylvania announced the first-ever undergraduate degree program in Market and Social Systems Engineering. Kudos to UPenn: the move shows impressive vision and leadership.
  5. The NSF is funding research in the CS-Econ area. They support efforts to “explore the emerging interface between computer science and economics, including algorithmic game theory, automated mechanism design, computational tractability of basic economic problems, and the role of information, trust, and reputation in markets” (page 7).
  6. The NBER Market Design working group is soliciting papers for a workshop October 8-9, 2010 in Cambridge, MA.
  7. We are now reviewing some amazing submissions to Yahoo!’s 2010 Key Scientific Challenges program. Read the challenges for the area we call Algorithmic Economics.
  8. Members of Yahoo! Labs can submit proposals to fund collaborative research with academic colleagues through the Yahoo! Faculty Research and Engagement program. If you’re interested, contact a Yahoo! Labs employee.

What should be the name? CS ∩ Econ is accurate but cryptic. At Yahoo!, we call it Algorithmic Economics. At Google, they call it Market Algorithms. The ACM Special Interest Group in this area calls it Electronic Commerce, causing complaints every year. I’ve heard people suggest Economics and Computation. The name Algorithmic Game Theory has emerged as something of a standard within the CS theory community. [Update: Noam suggests Algorithmic Game Theory and Economics and even renamed his blog accordingly.] The phrase Computational Economics makes sense but is already in use by a different field. A fun suggestion is Economatics (or Autonomics), meant to invoke a mashup of economics and automation.

Prediction markets had a similar naming/identity crisis. They’ve been called information markets, idea markets, securities markets, event markets, binary options, market in uncertainty, and more. But now almost everyone has settled on prediction markets. I’ve come to like the name and I think it’s helped establish the field in it’s own right. I hope we can settle on a good name for CS ∩ Econ in part so we can create the Journal of PerfectNameForCSEcon, an outlet sorely missing from the field.

Update 2011/10/11: The journal now exists! Called the ACM Transactions on Economics and Computation, it circumvented the naming issue.

A beautiful model (of the stock market)

Here is a histogram of the daily changes of the S&P 500 from 1950 to 2009. The x-axis is the daily log difference* and the y-axis is the number of times that difference occurred.

Histogram of log differences of S&P500 from 1950 to 2009

It turns out that a Laplace distribution is a pretty good model of the stock market. The Laplace distribution is parametrized by its median u and its average absolute difference from median b. I computed these two parameters for the S&P 500 data and plugged them into a Laplace distribution in Mathematica, then used that to generate 59 years worth of random simulated S&P 500 prices. Here is the resulting histogram.

Histogram of log differences of simulated S&P500

This is not a best fit: this is simply the same u and b computed on the data and then plugged into the distribution. Here are the two histograms on top of each other.

Histogram of log differences of S&P500 from 1950 to 2009 and simulated S&P 500

At the aggregate level the stock market is well behaved: it’s randomness is remarkably predictable. It’s amazing that this social construct — created by people for people, and itself often personified — behaves so much like a physical process, more so than any other man-made entity I can think of.

The graphs also hint at the futility of attaching reasons to price movements every single day. Today, “the prospects for continued low borrowing costs buoyed investors’ hopes for the U.S. economy”. Tomorrow, there may be “profit taking”.** Why do reporters feel obligated to explain why, or more to the point why do readers demand to be told? Why must we infer reasons, see momentum where there may be none, and assign labels like bull and bear? Why can’t we accept randomness?***

Update: Anthony Towns conducted a fascinating and puzzling follow-up analysis showing that the volatility of the stock market has been going up steadily over the years, even though the mean return has not.
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* The log difference is the log of the price on day d minus the log of the price on day d-1, or equivalently ln(d/(d-1)).
** I’d love to see a controlled experiment where financial reporters are given randomized reports about the Dow and watch them manufacture explanations, I imagine occasionally invoking the same cause for both ups and downs… (a) The prospect of higher interest rates spooked investors today; or (b) the Fed’s willingness to raise rates signals a recovering economy — investors rejoiced.
*** Let me clarify. The efficient market hypothesis implies both randomness and rationality: Randomness in the sense of statistical independence (no momentum), and rationality in the sense that price changes reflect new information. So I’m not saying that prices change for no reason, just that there is rarely an easily-stated single explanation for the overall market. (BTW, “profit taking” is never a valid explanation under EMH.)