William Spaniel

If you have watched an NFL game over the last six weeks, you doubtlessly know that NFL referees are in a labor dispute, and the NFL is using replacement referees for the time being. USA Today has an interesting story about the incentives these replacement refs face. Specifically, they are more vulnerable to being bought off by illicit gambling manipulation.

Among gamblers, there is obvious demand for referees willing to take bribes to alter the outcome of the game. For example, suppose the Chargers and Falcons are an even line. (All you have to do is pick the winner to win the bet.) A gambling crew could place a large sum of money on the Chargers, say $1,000,000. They could then pay $100,000 to the referee to ensure the calls go the Chargers’ way such that San Diego wins. The gamblers stand to make hundreds of thousands of dollars.

Besides the threat of criminal punishment, referees have incentive to refuse these bribes due to future benefits from continued officiating. Making terrible calls or being getting caught will get you fired, thus denying you the benefits of continued employment. All other things being equal, if you expect the NFL to continue employing you, you are less likely to take the bribe. Regular NFL officials have this type of long time horizon. They may not be completely unbribe-able, but they are darn resistant.

The replacement refs? Not so much. Their time horizon is extremely small. Once the NFL and the referees resolve their labor dispute, the replacement refs will be gone for good. Rather than years, this time horizon is probably better calculated in weeks or months. Taking a $100,000 bribe doesn’t sound so bad when you are very likely to be unemployed by Halloween, especially when you are making at most $3500 a game.

I find this argument is intuitive and compelling. Moreover, it made me rethink the reasonableness of the referees’ previous contract, which paid about $150,000 for roughly fifty days’ work last year. Such a salary seems ridiculously high given the large supply of potential referee labor. However, the NFL needs to keep the actions of the referees in line with the NFL’s wishes. We can’t just ask potential referees how much they need to be paid to not accept bribes, and then employ the cheapest labor. One way to resolve this issue is to promise continued high pay all referees. Put differently, the high salaries bridge the principal-agent problem.

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.