Category Archives: Game Theory

Excerpt from Game Theory 101: The Complete Textbook

Posted on September 3, 2012 | 1 comment

With school starting once again, I thought it was time to do some updating to the greater Game Theory 101 enterprise. Here’s the updated version of lesson 1.1 of Game Theory 101: The Complete Textbook. Enjoy.

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Posted in Game Theory, Publishing

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Chapter 2 of The Rationality of War

Posted on August 12, 2012 | 1 comment

The Rationality of War is now out! (Buy it on Amazon or Barnes & Noble.) You can download chapter two of the book as a free PDF by clicking here. This chapter explains the fundamental puzzle of war: if fighting is costly, why can’t two states agree to a peaceful settlement? With that puzzle in mind, the rest of the book shows why states sometimes end up in war.

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Posted in Game Theory, Political Science

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Do More Accurate Tests Lead to More Frequent Drug Testing?

Posted on August 4, 2012 | 1 comment

This Olympics has been special due to bizarre cases of “cheating” and cunningly strange strategic behavior. But regardless of the year, allegations of doping are always around. So far, four athletes have been disqualified, and a fifth was booted for failing a retest from 2004. (The Olympic statute of limitations is eight years.) More will probably get caught, as half of all competitors will be sending samples to a laboratory.

Doping has some interesting strategic dimensions. The interaction is a guessing game. Dopers only want to take drugs if they aren’t going to be tested. Athletic organizations only want to test dopers; each test costs money, so every clean test is like flushing cash down a toilet. From “matching pennies,” we know that these kinds of guessing games require the players to mix. Sometimes the dopers dope, sometimes they don’t. Sometimes they are tested, sometimes they aren’t.

But tests aren’t perfect. Sometimes a doper will shoot himself up, yet the test will come back negative. Even if we ignore false positives for this post, adding this dynamic makes each actor’s optimal strategy more difficult to find. Do more accurate drug tests lead to more frequent testing or less frequent testing? There are decent arguments both ways:

Pro-Testing: More accurate drug tests will lead to increased testing, since the organization does not have to worry about paying for bad tests, i.e. tests that come back negative but should have come up positive.

Anti-Testing: More accurate drug tests will lead to decreased testing, because athletes will be more scared of them. That leads to less incentive to dope, which in turn makes the tests less necessary.

Arguments for both sides could go on forever. Fortunately, game theory can accurately sort out the actors’ incentives and counter-strategies. As it turns out, the anti-testing side is right. The proof is in the video:

Basically, the pro-testers are wrong because they fail to account for the strategic aspect of the game. The athletic organization has to adopt its strategies based off of the player’s incentives. Increasing the accuracy of the test only changes the welfare of the player when he dopes and the organization tests. So if the organization kept testing at the same rate as the quality of the tests improved, the player would never want to dope. As such, the organization cuts back on its testing as the quality of the test increases.

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Posted in Game Theory, Sports

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Romney Should Release His Returns

Posted on July 18, 2012 | 3 comments

If you have been following the 2012 campaign, you know that Mitt Romney has not released his tax returns from before 2010. This has caused speculation that Romney did something wrong during 2008 or 2009–anything from finding interesting (but legal) tax shelters for his Bain Capital income to participating in the IRS’s 2009 amnesty program. Romney has held firm, claiming that rivals are attempting to divert attention from important campaign issues to his tax life. Of course, this has backfired, inadvertently causing more speculation.

I did not fully appreciate the strategic aspects of the situation until reading this blog post from Daniel Shaviro. In it, he makes an important insight:

Romney’s reluctance to release any pre-2010 tax return might be that what it would show is worse than all the heat he is taking for non-disclosure.

I thought that was an interesting intuition. However, I have realized that it wrong, at least in the long run. No matter how bad Romney’s pre-2010 tax returns are, they cannot be worse than the speculation he will eventually receive.

To see why, imagine that one of three things is true about Romney: (1) he did nothing wrong, (2) he did something politically embarrassing (like find ridiculous tax loopholes), or (3) he did something blatantly illegal (like something that would have made him participate in the 2009 IRS program). Option 1 is a non-issue. Option 2 is not politically expedient but not altogether damning; Romney could even take some of the heat off by claiming to be the best candidate to shut down these loopholes given his first-hand knowledge of the tax system. Option 3 is game over.

The public intuitively understands that if (1) were true, Romney would have come forward already. (Let’s define the “public” as independents who haven’t already decided who to vote for. We know that ideologues from both sides are lost causes here.) Maybe he is reluctant to let everyone know he made gazillions of dollars those years, but that is a hell of a lot better than the beating he is receiving right now. So we can eliminate option 1 from the list. This is inference #1.

That leaves us with option 2 and option 3. But if both are possible, then our rational expectation of Romney’s sleaziness falls somewhere in between 2 and 3. In other words, we believe on average that Romney is worse than the guy who found tax loopholes but better than the guy who did something illegal. However, that implies that Romney should come forward if he is type 2; releasing his returns will prove that he is not as bad as the average between 2 and 3 and thus leave Romney’s reputation in better shape.

As such, the only type who does not come forward is the worst type. Put differently, by not releasing his returns, the public should rationally infer that the worst thing is true. This is inference #2.

For now, the public seems to understand inference #1 but not inference #2. However, it is only matter of time before they draw that conclusion; after all, it is logically valid. So, for now, Romney can get away without releasing the documents. But over time, things will only get worse as the public slowly reaches inference #2.

If I were Romney’s advisor, I would have him release the documents immediately. Whatever is on them cannot be as bad as where this public speculation is headed. The decision is whether to handle the blow-back now–three months before the election–or wait until October. Clearly the former is the better option.

Video explanation below:

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Posted in Game Theory, Political Science

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Game Theory Tells Us ______

Posted on March 8, 2012 | 1 comment

I hate it when I read a sentence that says “Game theory tells us _______.” For example, “game theory tells us prisoners should confess to crimes.” If you ever see something to that effect, know that the author doesn’t really understand what game theory is about.

The example is a reference to the prisoner’s dilemma. In that model, the prisoners only want to minimize jail time and defect accordingly, which leads them to an outcome that is strictly worse for both players than had they cooperated with each other.

But game theory does not generally tell us that the prisoners should rat each other out. Whenever we solve a model, we make some assumptions, formulate those assumptions into equations, do some game theory, and produce conclusions. Game theory is nothing more than overly glorified math. (Sorry to burst your bubble.) The nice part is that game theory maps assumptions to logically valid conclusions. So, really, game theory only tells us how to solve the game.

The assumptions do all of the dirty work. In the prisoner’s dilemma example, the critical assumption is that players only want to minimize jail time. Alternatively, they might want to minimize jail time and maintain their friendship. In my textbook, I formulate this as a stag hunt. The players want to keep quiet if and only if the other keeps quiet as well, while they players want to confess if and only if the other confesses as well. Consequently, in this game, the prisoners might stay silent. Changing the assumptions changes the prediction.

Bottom line: the assumptions tell us everything.

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