While at the laundromat today, I saw an episode of Mathematics Illuminated, about game theory. This is a series of 13 half-hour episodes on "major themes in the field of mathematics"; the game theory episode covered Nash equilibria, the prisoner's dilemma, evolutionarily stable strategies, etc. (I may be leaving out some things, because there was laundry-machine noise.) Their intended audience seems to be high school teachers and interested but perhaps mathematically unsophisticated adult learners.
It appears you can watch the whole series online. The main mathematician involved is Dan Rockmore of Dartmouth.
(And no, I don't know what channel it was on. Like I said, it wasn't my TV.)
Showing posts with label game theory. Show all posts
Showing posts with label game theory. Show all posts
17 January 2009
02 September 2008
Randomness in football
California high school's offensive scheme adds randomness to football. I know very little about football (I think we've established pretty well here that I'm a baseball fan) but it's an interesting idea. Basically, this team runs a wider variety of plays than many teams, making them less predictable.
I'm not sure if they mean "randomness" in the sense that I would hope, though. First a team would determine a mixed strategy for what plays it should run, based on how they expect the opponents would be able to defend against those plays. Then they should just let a random number generator programmed with that strategy call the plays. If it works for diplomats and terrorists, why not football?
(And why not baseball, for that matter. In fact, there are less plays possible at any given juncture of a baseball game than at any given juncture of a football game, I think, so it would probably be easier to do this in baseball.)
I'm not sure if they mean "randomness" in the sense that I would hope, though. First a team would determine a mixed strategy for what plays it should run, based on how they expect the opponents would be able to defend against those plays. Then they should just let a random number generator programmed with that strategy call the plays. If it works for diplomats and terrorists, why not football?
(And why not baseball, for that matter. In fact, there are less plays possible at any given juncture of a baseball game than at any given juncture of a football game, I think, so it would probably be easier to do this in baseball.)
01 June 2008
Game theory and toilet seats
A game-theoretic approach to the toilet seat problem -- when someone who pees standing up and someone who pees sitting down share a bathroom, what should they do?
(My solution to this problem is living alone, but this may not work for everyone.)
(My solution to this problem is living alone, but this may not work for everyone.)
11 December 2007
Why don't sports teams use randomization?
Why don't sports teams use randomization?, a guest post by Ian Ayres at Freakonomics. It's well-known that the best strategy in a lot of game-theoretic situations is to choose at random what you'll do at each encounter, where the appropriate probabilities can be calculated. (A sports example would be a pitcher choosing what sort of pitch to make, and a batter deciding whether to swing or not.) So why doesn't anybody (so far as we know) use a random number generator to call pitches?
Although this doesn't seem to be brought up in the comments there (at least not yet), I suspect that the reason for this is that sports people are for the most part resistant to change. But more importantly, they have to answer to the media. And when your closer throws a pitch down the middle of the plate and it gets hit out of the park, and you lose, do you really want to explain to the news people that a random number generator told you to do that? There is no other line of work I can think of where it would be quite so obvious which decision led to the bad outcome; more importantly, there is no other line of work with call-in radio shows devoted to dissecting everything that happens.
Although this doesn't seem to be brought up in the comments there (at least not yet), I suspect that the reason for this is that sports people are for the most part resistant to change. But more importantly, they have to answer to the media. And when your closer throws a pitch down the middle of the plate and it gets hit out of the park, and you lose, do you really want to explain to the news people that a random number generator told you to do that? There is no other line of work I can think of where it would be quite so obvious which decision led to the bad outcome; more importantly, there is no other line of work with call-in radio shows devoted to dissecting everything that happens.
22 October 2007
Diplomats play dice
State Department Struggles To Oversee Private Army, from the Washington Post (Oct. 20)
That's a good idea. I wonder if anybody just told their employees "take different routes to work" and found that their people decided which route to take based on, say, the day of the week -- having more than one route offers some protection from ambush, but there's still a pattern involved, and eventually the would-be ambusher would figure out that if they come by a certain spot on a Tuesday morning they'll succeed.
(Of course, there are still only six possible routes. If one wanted to go really crazy, one could toss a die at each juncture to decide where to go... but then who would ever get to work?)
This reminds me of airport security people who are randomizing where they are in the airport in order to thwart terrorists.
(from Marginal Revolution.)
Marc Grossman, the U.S. ambassador to Turkey in the mid-1990s, recalled telling his staff to take their own security precautions. After losing embassy employees to attacks, he advised staffers to keep a six-sided die in their glove compartments; to thwart ambushes, they should assign a different route to work to each number, he said, and toss the die as they left home each morning.
That's a good idea. I wonder if anybody just told their employees "take different routes to work" and found that their people decided which route to take based on, say, the day of the week -- having more than one route offers some protection from ambush, but there's still a pattern involved, and eventually the would-be ambusher would figure out that if they come by a certain spot on a Tuesday morning they'll succeed.
(Of course, there are still only six possible routes. If one wanted to go really crazy, one could toss a die at each juncture to decide where to go... but then who would ever get to work?)
This reminds me of airport security people who are randomizing where they are in the airport in order to thwart terrorists.
(from Marginal Revolution.)
30 September 2007
A random weapon in the war on terror
A random weapon in the war on terror, from Newsweek.
Airport security personnel at Los Angeles International Airport are now randomly being deployed; the thinking behind it is that if there's no pattern in where the security people are, it's harder for potential attackers to find a hole in the system that can be exploited.
This is the idea of a "mixed strategy" in game theory. There are various things that each of the two parties can do, and the payoffs to each player in each situation are known. It often turns out that the optimal strategy for one player might be to do one thing sixty percent of the time, say, and another thing forty percent of the time -- but that doesn't mean they should do some sort of strict alternation. Rather, they should choose which thing to do at each time at random.
The canonical example of this, I think, is pitch choice in baseball, where the pitcher chooses his pitches at random and the batter chooses what he will expect (and therefore swing at) also at random. But I'm not sure to what extent I believe it actually applies there because the strategy is different on each count. (And it would be incredibly difficult to study. You can maybe get information on what kind of pitch the pitcher pitched. But how could you know what the batter expected?)
The theory of mixed strategies, though, assumes that both parties have perfect information. So it's not exactly the same thing; the terrorists and the airport security don't have such information. Even if they think they know where there's a hole, how do they know they just haven't been watching long enough? But the idea seems reasonable; security people develop patterns, and if you can avoid creating those patterns you can avoid having them exploited.
P.S. I had to mention baseball because the Phillies won today, and are therefore in the playoffs. And the Mets lost; they have to go home now. I was at the Phillies game. See a picture of a sad Mets fan from the New York Times. At one point the odds of the Mets not making the playoffs were 500 to 1, according to Baseball Prospectus. But that's actually not the worst collapse ever; the '95 Angels beat 8800-to-1 odds. (The '64 Phillies are only tenth, according to that list.)
Airport security personnel at Los Angeles International Airport are now randomly being deployed; the thinking behind it is that if there's no pattern in where the security people are, it's harder for potential attackers to find a hole in the system that can be exploited.
This is the idea of a "mixed strategy" in game theory. There are various things that each of the two parties can do, and the payoffs to each player in each situation are known. It often turns out that the optimal strategy for one player might be to do one thing sixty percent of the time, say, and another thing forty percent of the time -- but that doesn't mean they should do some sort of strict alternation. Rather, they should choose which thing to do at each time at random.
The canonical example of this, I think, is pitch choice in baseball, where the pitcher chooses his pitches at random and the batter chooses what he will expect (and therefore swing at) also at random. But I'm not sure to what extent I believe it actually applies there because the strategy is different on each count. (And it would be incredibly difficult to study. You can maybe get information on what kind of pitch the pitcher pitched. But how could you know what the batter expected?)
The theory of mixed strategies, though, assumes that both parties have perfect information. So it's not exactly the same thing; the terrorists and the airport security don't have such information. Even if they think they know where there's a hole, how do they know they just haven't been watching long enough? But the idea seems reasonable; security people develop patterns, and if you can avoid creating those patterns you can avoid having them exploited.
P.S. I had to mention baseball because the Phillies won today, and are therefore in the playoffs. And the Mets lost; they have to go home now. I was at the Phillies game. See a picture of a sad Mets fan from the New York Times. At one point the odds of the Mets not making the playoffs were 500 to 1, according to Baseball Prospectus. But that's actually not the worst collapse ever; the '95 Angels beat 8800-to-1 odds. (The '64 Phillies are only tenth, according to that list.)
01 August 2007
turning our backs on Bourbaki
From yesterday's New York Times (July 31): In Games, an Insight Into The Rules of Evolution, a profile of Martin Nowak, a mathematical biologist. His interests lie in trying to understand cooperation, which is important in evolution; he has been coming up with mathematical models for it; one example that's given is descendants of the Prisoner's dilemma where various members of a population interact preferentially with certain other members, instead of randomly. (The members that interact with each other more frequently are the ones that are "near" each other, either geographically or in some more abstract sense.) Cooperation turns out to emerge under conditions with are rather simple to identify.
I think that it's important for mathematicians to collaborate with people outside of mathematics, and to be exposed to ideas that at first glance don't necessarily appear mathematical. We're always going on and on to our students about how mathematics can be applied to large parts of everyday life, and I believe that's true. But at the same time, the mathematical community seems to look down on those who actually try to do so, instead lionizing people like Wiles and Perelman who have solved problems that have apparently no relevance to the real world.
The linguist Steven Pinker says that “Martin has a passion for taking informal ideas that people like me find theoretically important and framing them as mathematical models... He allows our intuitions about what leads to what to be put to a test.” I believe that this is one of the most important services we can render to the world. What mathematics is good at is stripping away the parts of a problem that are irrelevant and reducing it to its essence; seeing that that essence has something in common with many other apparently different essences; and finally using that knowledge to solve the problem. (It's like the "applications" exercises in most calculus textbooks, except not stupid. In such textbooks the translation from a real-world problem to mathematics is usually so simple as to just be an annoyance.)
Now, I believe that mathematical research that apparently has no use in the "real world" should be allowed to continue, and be funded with tax dollars, because we don't know what bits of mathematics that we're coming up with now will turn out to be "useful" a couple hundred years from now. (The canonical example here, I think, is that number theory has turned out to be very important for cryptography.) But at the same time, it seems to me that the mathematical community turns its backs on those who dare to actually think about those practical applications.
I hope I'm wrong.
I think that it's important for mathematicians to collaborate with people outside of mathematics, and to be exposed to ideas that at first glance don't necessarily appear mathematical. We're always going on and on to our students about how mathematics can be applied to large parts of everyday life, and I believe that's true. But at the same time, the mathematical community seems to look down on those who actually try to do so, instead lionizing people like Wiles and Perelman who have solved problems that have apparently no relevance to the real world.
The linguist Steven Pinker says that “Martin has a passion for taking informal ideas that people like me find theoretically important and framing them as mathematical models... He allows our intuitions about what leads to what to be put to a test.” I believe that this is one of the most important services we can render to the world. What mathematics is good at is stripping away the parts of a problem that are irrelevant and reducing it to its essence; seeing that that essence has something in common with many other apparently different essences; and finally using that knowledge to solve the problem. (It's like the "applications" exercises in most calculus textbooks, except not stupid. In such textbooks the translation from a real-world problem to mathematics is usually so simple as to just be an annoyance.)
Now, I believe that mathematical research that apparently has no use in the "real world" should be allowed to continue, and be funded with tax dollars, because we don't know what bits of mathematics that we're coming up with now will turn out to be "useful" a couple hundred years from now. (The canonical example here, I think, is that number theory has turned out to be very important for cryptography.) But at the same time, it seems to me that the mathematical community turns its backs on those who dare to actually think about those practical applications.
I hope I'm wrong.
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