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.