Category Archives: Sports

Do More Accurate Tests Lead to More Frequent Drug Testing?

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.

Olympic Rules Shenanigans: Dolphin Kick Edition

Fresh off the silliness of the badminton play-to-lose scandal comes this lovely piece on dolphin kicks. Last weekend, South African Cameron van der Burgh won gold in the 100m breaststroke.

However, Australia’s Olympic committee is putting up a fuss, as video footage of van der Burgh clearly shows him executing three dolphin kicks after diving into the water. (An Australian swimmer finished in second.) Breaststroke competitions allow only one.

And van der Burgh does not give a damn. From the link:

If you’re not doing it, you’re falling behind. It’s not obviously–shall we say–the moral thing to do, but I’m not willing to sacrifice my personal performance and four years of hard work for someone that is willing to do it and get away with it.

You see, FINA (the governing body of swimming) does not use cameras underwater to check for illegal dolphin kicks. Moreover, Australia cannot formally appeal van der Burgh’s finish, as there is no formal appeal process.

Of course, an appeal probably wouldn’t do much good, considering the Australian swimmer did the exact same thing.

As with the badminton scandal, the real moral of the story is about institutional design. If you build a bad institution, it will lead to more bad things. Here, you should not create rules that you do not plan to enforce. The players who wish to abide by those rules face a stark choice: play “fair” or let the “unfair” win. So even those wishing to play fair break the rules, and we end up in a situation as though the rule does not exist.

Strangely, the dolphin kick rule could be enforced. FINA used underwater technology at the swimming World Cup in 2010. Everyone knew that dolphin kicks were prohibited and breaking the rules would not go unnoticed, so no one broke them.

Derp! Badminton Could Learn from Political Science (Or, Winning By Losing)

Political science doesn’t have many “laws” the way physics does. But here’s one of them:

Law: People will strategize according to the institutional features put in front of them.

Here’s a corollary that I think should follow from that:

Corollary: If one creates stupid institutional rules, one loses the right to object to people taking advantage of them.

Apparently the Olympic organizing group of badminton could learn from this law and its corollary. Yesterday, you see, eight players intentionally played to lose. Full story here.

The gist of it is this: Early in the day, the #2 team in the world lost their last group game, sending them to the bottom of the teams qualified for the quarterfinals. Later on, teams that were already qualified for the quarterfinals played to lose, concerned that a win would propel them to a high seed that force them to play the #2 team sooner in the elimination bracket. Oops.

Badminton officials were shocked–shocked!–that the players would resort to such a cunningly intelligent strategy. Furthermore, the officials complained that the players had violated a rule that protects against athletes “not using one’s best efforts to win a match”–as though one could reasonably discern what qualifies as “best effort” versus “a little bit less than best effort, but still enough effort to convince everyone that we actually care even though we don’t.”

Here are a couple of solutions for the Olympic badminton committee. First, you could schedule all of the final games group play simultaneously, to make it harder for teams to know to throw matches from the start. (Soccer pulls a similar trick in the Euro and World Cup, albeit for slightly different purposes.) Or you could have a single elimination tournament from the start.

Just don’t be surprised when players try to win…by losing.

Update: The players have been disqualified. Next time, I suggest feigning an injury.

The USA Today story also reports that the Japanese women’s soccer team intentionally sought to draw yesterday, as to avoid playing the United States in the quarterfinals.

New Paper: The Hometown Discount Paradox

If you read the sports media much, you often hear about the concept of a “hometown discount”—when a player signs with his hometown team (or any team with a desirable location to the player) for less money than he could receive elsewhere. Unfortunately, misconceptions regarding the hometown discount run rampant. My latest working paper (more of a research note) dispels a couple of them.

First, we often hear players making public declarations that they will refuse to give a particular team a hometown discount. Sports reporters treat these words as gospel; if the player says it is true, then it must be true! This is absolute nonsense. Anyone who has sat through a round of poker can see right through it. From a financial standpoint, the player clearly benefits if the team offers him a contract without a hometown discount. Thus, even a player who would accept the largest hometown discount in the history of mankind has incentive to pretend like he would accept no hometown discount. There is simply no way to tell whether the player is bluffing or not.

The second misconception is that hometown discounts unequivocally make it more likely that the team will sign the player. In truth, reality is much more complicated than that. Hometown discounts can either help or hurt, depending on the nature of the discount.

Although the paper delves much deeper into this issue (and includes a lot of easy to follow pictures!), I will present an example to explain what I mean. Suppose the hometown team values a player worth $120 million and the player is being offered a $100 million contract from a rival. If no hometown discount exists, the team should be able to sign the player without problem—a contract worth $101 million, for instance, leaves both the hometown team and the player better off than if the player signed with the rival.

Now suppose the team only valued the player worth $95 million. Without a hometown discount, the situation is hopeless; the most the team would be willing to offer is $95 million, but the opposing contract is worth $5 million more. But if the player is willing to offer the hometown team a $10 million discount, then they can reach an agreement. For example, the team could offer a contract worth $93 million. The team still profits for $2 million. Likewise, the player prefers signing with the hometown team, since a $93 million contract with them is functionally worth $103 million after including the hometown benefit, which is more than the $100 million he would receive elsewhere.

This case reflects the traditional notion of the hometown discount—hometown teams can sign hometown players more easily, since the hometown player is willing to accept less money. However, this effect only holds up when the team actually knows how much of a hometown discount the player is willing to receive. This is a ridiculously strong assumption. The team should have a ballpark idea on how much of a discount the player should accept, but to know the exact amount would require actually being in the head of the player.

Allowing for uncertainty makes the situation much harder to analyze, which takes up a bulk of the paper. But to summarize the results, if the outside contract offer is extremely competitive, the team gambles with its contract offer. Players willing to accept large hometown discounts accept, while the others accept the offer from the rival.

Why is this? Well, suppose the team values the player worth $100 million, and the outside offer is also worth $100 million. In addition, suppose the team believes the player is willing to offer somewhere between $0 and $10 million of a hometown discount, but is not sure of the exact amount. The team could match the $100 million offer and induce all of the players to sign, but the team makes literally no profit on the contract; it values the player worth $100 million, but pays $100 million to the player.

In contrast, the team could offer $95 million to the player. If exactly half of the players are willing to give a hometown discount of $5 million or more, then the team profits by $5 million half of the time, for a net gain of $2.5 million. This is worth more than offering $100 million, yet it means that the player signs with the rival half of the time!

On the other hand, when opposing offers are low, the team has no reason to make this gamble. Suppose the outside offer is only worth $20 million but the hometown team still values the player worth $100 million. At this point, matching the $20 million offer is optimal for the team; its profit margin is $80 million, which is large enough to make the gamble not worth the risk.

So, in conclusion, the nature of the hometown discount determines whether the player is actually more likely to sign with the team. If everyone knows what is going on, then it can only help. But when the team is uncertain of the player’s hometown discount, things can go haywire.

New Working Paper: Risk Aversion in Sports Contract Negotiations

I got curious about sports free agency after Jered Weaver resigned with the Angels for five years and $85 million. Pitchers of his caliber on the free agent market pick up much larger contracts. Why didn’t Weaver wait it out until free agency? And, given that he was going to resign with the Angels, why did he sign when he did and not a month before or a month after?

After working out the logic, I think I have a reasonable answer. I will be presenting my findings at the SABR conference at the end of the month, but the paper is available here right now.

The basic concept involves risk aversion. Weaver alerted me to the idea when he asked “how much more money do you really need?” Although Weaver would have certainly made more money had he made it to free agency, there was inherent risk in waiting. What if he suffered a catastrophic injury in the meantime? He would go from making nine figures to virtually nothing in the blink of an eye. As such, Jered willingly accepted a smaller amount in expectation; the luxury of guaranteed money was worth paying the premium.

My conference paper expands on this idea. I show that if a player is risk averse enough, he will always resign with his team no matter how much money he would make on the free agent market. This is good news for teams, as the exclusive bargaining rights allow them to leverage the risk of injury to get their players to accept contracts for smaller amounts—which is exactly what we saw in the case with Jered Weaver.

However, risk aversion is an internal trait that is difficult to directly observe. Teams might not know exactly how risk averse a player is, which makes it hard to know how much the team should offer the player. My paper investigates such a dynamic. In this case, the team increases its offers over the course of the season. The player accepts when the team finally meets his requirements. Meanwhile, the team is willing to increase its offers because playing games credibly demonstrates the player’s tolerance for risk. This explains why we see variation in the timing of contract signings; more risk averse players sign earlier in the season, while more risk tolerant players sign later on.

The paper also proves three other neat findings. First, you may intuitively believe the team would only benefit from increasing player safety. However, the model shows that the team actually benefits in contract extension negotiations from some level of danger—otherwise, the team cannot exploit the player’s risk aversion at all. Second, it questions the wisdom of suspending contract negotiations during the season, as it prevents the team from learning more about the player as the season progresses. Finally, it advises teams to pay more attention to players’ level of risk aversion during the draft season; all other things being equal, more risk averse players sign for less money, which allows the team to concentrate its resources on other players.

You can view a version of my presentation on YouTube below. If you are going to the SABR conference, then please drop in to Ballroom 3 from 3:00 to 3:25 on Thursday to see me make the presentation in person. And, once again, you can click here to download the full paper.

Game Theory of Baseball Talk

Today, I’m giving a talk at the University of Rochester about game theory in baseball. You can see the slides by clicking here. Topics include throwing breaking balls with a runner on third base, optimal defensive positioning against a bunt, and how to catch a ball at a game. Enjoy.

New Working Paper on Bunting

If you are interested in the game theory of baseball, you should head over to my working papers page. I just uploaded a new article on optimal hitting strategy during a no hitter. As it turns out, a batter can never bunt with a no hitter in progress and still maximize his team’s probability of victory. The article has a full proof of the claim.