Archive for the 'Game Theory' Category

Game Theory

Recognizing that I know no more about politics than most of you, and that I have no notable record as a political prognosticator, here is my prediction, as of about a half hour before the second Republican presidential debate: Doug Burgum breaks out of the pack with strong attacks on Donald Trump.

Why? First, he needs a Hail Mary. Second, he needs it tonight, or there’s almost no chance he makes it into the third debate. Third, among the various Hail Mary’s available to him, this seems the most likely to pay off.

Arguably, they all need Hail Mary’s. But those (i.e. most of them) who have refused to substantially attack Trump in the past can’t use this particular Hail Mary without being called out for flip-flopping. (Burgum doesn’t have to worry so much about this, because almost nobody has paid any attention to anything he’s said yet.) Also, Burgum is the one who most needs to get this done tonight, with little prospect of going on otherwise.

A couple of hours from now, you can tell me why I got this wrong.

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The Coinflipper’s Dilemma

flipperThis is the story of how I came to write a little paper called The Coinflipper’s Dilemma.

When I was in high school, my English teacher must have had a free period at the time when my math class met, because every day he would march into the math class and empty his pockets on the table, whereupon my math teacher did the same. Then whoever had put down the most money scooped up everything on the table.

I am ashamed to admit that it took me until this summer to think about computing the equilibrium strategy is in that game.

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Quantum Games

The Society of Undergraduate Math Students here at Rochester asked me to give an elementary talk on quantum game theory last week. I’m posting video of the first (non-technical) half of that talk. I’ll post the second (more technical) half after I get around to editing out the embarrassing mistake I made near the very end.

Get the Flash Player to see this content.

Click the lower right corner of the video window to expand to full screen.

Possibly better viewing here.

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False Imprisonment

The frequently insightful and usually accurate Megan McArdle gets this part quite completely wrong in her latest Bloomberg column about ObamaCare:

Democratic politicians and insurers are locked in a prisoner’s dilemma. In this classic game-theory case, you and a professional associate are both arrested for theft. If neither of you talks, then you’ll probably get off. But if just one of you talks, then the person who talks will get a reduced sentence, while the other person has the book thrown at them. If you both talk, then both of you go to jail for a long time. The equilibrium is for both of you to talk, just in case the other guy does .

I sincerely hope that anyone who’s ever taken my Principles of Economics course — or for that matter, any Principles of Economics course — can explain to McArdle how wrong this is, and why.

Exercise for the reader: To what extent, if at all, does this howler undermine the larger point of McArdle’s column?

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Albert and the Dinosaurs

If you’re wondering what I’m up to, click on the picture.

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A Sip of Monstrous Moonshine

You and a stranger have been instructed to meet up sometime tomorrow, somewhere in New York City. You (and the stranger) can decide for yourselves when and where to look for each other. But there can be no advance communication. Where do you go?

Me, I’d be at the front entrance to the Empire State Building at noon, possibly missing my counterpart, who might be under the clock at Grand Central Station. But, because there are only a small number of points in New York City that stand out as “extra-special”, we’ve at least got a chance to find each other.

A Schelling point is something that stands out from the background so sharply that we can expect people to coordinate around it. Schelling points are on my mind this week, because I’ve just heard David Friedman give a fascinating talk about the evolution of property rights, and Schelling points play a big role in his story. But that story is not the topic of this post.

Instead, I’m curious about the Schelling points that say, two mathematicians, or two economists, or two philosophers, or two poets, or two street hustlers might converge on. Suppose, for example, that you asked two mathematicians each to separately pick a number between 200 and 300, with a prize if their answers coincide. I’m guessing they both go for 256, the only power of two within range.

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Best Negotiator Ever

With a hat tip to Kenneth Anderson at the Volokh Conspiracy:

Golden Balls is a British game show where players decide, in secret, whether to adopt a strategy of “Split” or “Steal”. In this episode, they face the following payoffs (in British pounds):

This is almost, but not quite a classic Prisoner’s Dilemma situation. (To make it a true Prisoner’s Dilemma, where stealing always beats splitting, you could change the lower-right hand box to “1 each” instead of “0 each”.) As in the Prisoner’s Dilemma, you can never go wrong by stealing — though you can go horribly wrong when the other guy steals, so it makes sense to reach a no-stealing agreement — and then to violate it.

In other words you’d pretty much expect homo economicus to steal every time. But this game is far more interesting than the usual textbook version of the Prisoner’s Dilemma, because it’s played by real people for real money and they negotiate in public for half a minute before they choose their strategies. In principle, the negotiation shouldn’t change anything (unless the players come to care about each other, or about the way they’re perceived by the audience). But in this episode, the negotiation took an unexpected turn.

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Cats, Dogs and Quantum Mechanics

The game of Cats and Dogs works like this: You and your teammate are placed in separate rooms and forbidden to communicate. You are each asked a randomly chosen question: Either “Do you like cats?” or “Do you like dogs?” (Each of your questions is determined by a separate fair coin flip.)

You win if your answers agree — unless you were both asked the “cats” question, in which case you win if your answers disagree.

A little reflection should convince you that if you are allowed to meet with your partner and plot strategy before the game, then the best you can do is agree to always agree — say by both always answering “yes”. That way, you win 75% of the time, and there’s no way to do better. In particular, there’s nothing to be gained by randomizing your answers.

That, at least, is true, in a world governed by the laws of classical physics and probability theory. But in a world governed by the laws of quantum mechanics — which is to say, in the world we live in — you can in principle do better. Namely: You each carry with you one of a pair of entangled “quantum coins” (actually elementary particles, but I prefer to think of them as coins, since you’re going to use them as randomizing devices).

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Absentminded Musings

Here are some thoughts on last week’s absent-minded driver problem.

First a recap of the problem, with a bit more detail than last week:

Each day, Albert leaves his office (at the bottom of the map), gets on the Main Highway and attempts to drive home to his house on Second Street. If he turns too soon (onto First Street) or if he overshoots (going all the way to the north end of the Main Highway), he is mauled by dinosaurs.

Obviously, Albert’s best strategy is to go straight at the first intersection and turn right at the second. Unfortunately, both intersections look identical. Doubly unfortunately, Albert can never remember whether he’s already passed the first intersection.

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The Absent-Minded Driver

Until last week, I had never heard of the paradox of the absent-minded driver, but I was recently told that it has some relevance to my encyclopedia article on quantum game theory. That plus the fact that I am a notoriously absent-minded driver myself made me think I should check out the original source. Here’s what I extracted:

Each day, Albert leaves his office (at the bottom of the map), gets on the Main Highway and attempts to drive home to his house on Second Street. If he turns too soon (onto First Street) or if he overshoots (going all the way to the north end of the Main Highway), he is mauled by dinosaurs.

Obviously, Albert’s best strategy is to go straight at the first intersection and turn right at the second. Unfortunately, both intersections look identical. Doubly unfortunately, Albert can never remember whether he’s already passed the first intersection.

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