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Two-thirds of the way there

Evaluating your prospects against someone else who is doing the same, and making your move, is the essence of game theory

Ever played a game in which you’d do well to think a few moves ahead?

Chess is the classic case. Before moving a piece, you spend long minutes teasing out what might happen: “If I move the bishop here, she’ll push her pawn up, then I can move my knight there and threaten her rook and her queen. But suppose she doesn’t move the pawn and instead moves her rook up?" The greats can look much further ahead than mere dabblers like me can, which is naturally why they are great. Chess computers peer even further into the future, and so they are now better than the best humans.

Evaluating your prospects against someone else who is doing the same, and then making your move, is the essence of that always intriguing subject: game theory. It plays out all over the place. For example: Last Thursday was Thanksgiving, the biggest holiday of the year in the US. The day after Thanksgiving is one of the biggest shopping days of the year, probably because everyone wants to get an early start on Christmas shopping. Intent on getting a slice of the shoppers’ frenzy, every major retailer is open on that Friday. The frenzy is why it’s called Black Friday.

Naturally, retailers began looking for competitive advantages over each other. So stores opened earlier and earlier on Friday morning; in 2010, Toys “R" Us and Wal-Mart opened at midnight. Other stores felt the pressure to keep up, and opening times have crept steadily backward. Each retailer probably thinks along these lines: if I open an hour earlier, I’ll get more customers. But then the others will open earlier still. All right, I’ll pre-empt them with a midnight opening. Surely nobody will violate tradition and open on Thanksgiving itself?

But, of course, enough retailers have done just that. By now, Friday’s shopping extravaganza has encroached into the traditional Thanksgiving holiday. A professor of marketing at Wharton, Stephen Hoch, explained the trend a few years ago: “If you stay closed, you are not going to get enough credit from employees or the public (for doing so). You have to be out there or run the risk that competitors might get the sale before you."

How long before everyone stays open on Thanksgiving?

Game theory: how do you outwit your equally smart competitors who are simultaneously engaged in outwitting you? And what’s the result?

Some months ago, The New York Times asked its readers to participate in an experiment that underlined these ideas. Choose a number between 0 and 100, they said, that is “your best guess of two-thirds of the average of all numbers chosen in the contest".

Stop a minute to think this through. If everyone just picks a number between 0 and 100, the average of those will be 50, or pretty close. So, when asked to pick a number that is the average, you’ll certainly choose 50—and so will almost everyone else.

But the newspaper asks you to pick a number that’s two-thirds of the average of the picked numbers. So you’ll probably start with a number less than 50: two-thirds of 50, in fact, or 33. Remember three things, though: one, this assumes the average is 50; two, by choosing 33, you lower the average from 50; three, other readers will reason exactly the same way and also focus on 33. So, if everyone chooses 33, that will be the average, and the two-thirds figure is actually 22.

So, why not pick 22? By now you’ve got the picture. Everyone else will think the same thing. If everyone picks 22, that’s the average, and two-thirds of that is 14. So, 14 should have been your guess; but also everyone else’s… and on and on.

Where does this end? Do you keep recalibrating to ever-smaller numbers? Or do you at some point say, “This is my best estimate of the extent to which this game will play out," and pick that number? That’s what I did. Thinking most readers would reason just one round ahead, I chose 33—but to allow for some who might reason through more than one round, I picked 30.

Turned out I had slightly underestimated the scheming ways of most New York Times readers. Clearly, enough had taken the reasoning further. But not too many. I was the 58,677th reader to pick a number. With that many entries, the average was 28—slightly below my pick—and two-thirds of that is about 19.

To me, this suggests there may be an equilibrium to the race to open for business ever-earlier on Thanksgiving. Every retailer looks only a few steps ahead and stops. So, perhaps nobody will open earlier than Thanksgiving evening.

Then again, this is the market. If everyone opens at 7pm on Thanksgiving, someone will again yearn for a competitive advantage and open earlier. Nope, my forecast for Thanksgiving remains: one year soon, every establishment in America will stay open for business right through the holiday.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners.

A Matter of Numbers explores the joy of mathematics, with occasional forays into other sciences.

Comments are welcome at dilip@livemint.com. To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza

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