# Tag: Bayesian

## Gaming the System

Some college students used game theory to get an A by exploiting a loophole in the grading curve.

Catherine Rampell explains:

In several computer science courses at Johns Hopkins University, the grading curve was set by giving the highest score on the final an A, and then adjusting all lower scores accordingly. The students determined that if they collectively boycotted, then the highest score would be a zero, and so everyone would get an A.

Inside Higher Ed, writes:

The students refused to come into the room and take the exam, so we sat there for a while: me on the inside, they on the outside,” [Peter Fröhlich, the professor,] said. “After about 20-30 minutes I would give up…. Then we all left.” The students waited outside the rooms to make sure that others honored the boycott, and were poised to go in if someone had. No one did, though.

Andrew Kelly, a student in Fröhlich’s Introduction to Programming class who was one of the boycott’s key organizers, explained the logic of the students’ decision via e-mail: “Handing out 0’s to your classmates will not improve your performance in this course,” Kelly said.

“So if you can walk in with 100 percent confidence of answering every question correctly, then your payoff would be the same for either decision. Just consider the impact on your other exam performances if you studied for [the final] at the level required to guarantee yourself 100. Otherwise, it’s best to work with your colleagues to ensure a 100 for all and a very pleasant start to the holidays.”

Bayesian Nash equilibria

In this one-off final exam, there are at least two Bayesian Nash equilibria (a stable outcome, where no student has an incentive to change his strategy after considering the other students’ strategies). Equilibrium #1 is that no one takes the test, and equilibrium #2 is that everyone takes the test. Both equilibria depend on what all the students believe their peers will do.

If all students believe that everyone will boycott with 100 percent certainty, then everyone should boycott (#1). But if anyone suspects that even one person will break the boycott, then at least someone will break the boycott, and everyone else will update their choices and decide to take the exam (#2).

Two incomplete thoughts

First, exploiting loopholes ensures increasing rules, laws, and language (to close previous loopholes), which lead to creating more complexity. More complexity, in turn, leads to more loopholes (among other things). … you see where this is going.

Second, ‘gaming the system’ is a form of game theory. What's best for you, the individual (or in this case, a small group), may not be best for society.

Today's college kids are tomorrow's bankers and CEO's. Just because you can do something doesn't mean you should.

Update (via metafilter): In 2009, Peter Fröhlich, the instructor mentioned above, published Game Design: Tricking Students into Learning More.

## Blindness to the Benefits of Ambiguity

“Decision makers,” write Stefan Trautmann and Richard Zeckhauser in their paper Blindness to the Benefits of Ambiguity, “often prove to be blind to the learning opportunities offered by ambiguous probabilities. Such decision makers violate rational decision making and forgo significant expected payoffs.”

Trautmann and Zeckhauser argue that we often don't recognize the benefits in commonly occurring ambiguous situations. In part this is because we often treated repeated decisions involving ambiguity as one-shot decisions. In doing so, we ignore the opportunity for learning when we encounter ambiguity in decisions that offer repeat choices.

To put this in context, the authors offer the following example:

A patient is prescribed a drug for high cholesterol. It is successful, lowering her total cholesterol from 230 to 190, and her only side effect is a mild case of sweaty palms. The physician is likely to keep the patient on this drug as long as her cholesterol stays low. Yet, there are many medications for treating cholesterol. Another might lower her cholesterol even more effectively or impose no side effects. Trying an alternative would seem to make sense, since the patient is likely to be on a cholesterol medication for the rest of her life.

In situations of ambiguity with repeated choices we often gravitate towards the first decision that offers a positive payoff. Once we've found a positive payoff we're likely to stick with that decision when given the opportunity to make the same choice again rather than experiment in an attempt to optimize payoffs. We ignore the opportunity for learning and favor the status quo. Another way to think of this is uncertainty avoidance (or ambiguity aversion).

Few individuals recognize that ambiguity offers the opportunity for learning. If a choice situation is to be repeated, ambiguity brings benefits, since one can change one’s choice if one learns the ambiguous choice is superior.

“We observe,” they offer, “that people's lack of a clear understanding of learning under ambiguity leads them to adopt non-Bayesian rules.”

Another example of how this manifests itself in the real world:

In the summer of 2010, the consensus estimate is that there are five applicants for every job opening, yet major employers who expect to hire significant numbers of workers once the economy turns up are sitting by the sidelines and having current workers do overtime. The favorability of the hiring situation is unprecedented in recent years. Thus, it would seem to make sense to hire a few workers, see how they perform relative to the norm. If the finding is much better, suggesting that the ability to select in a very tough labor market and among five applicants is a big advantage, then hire many more. This situation, where the payoff to the first-round decision is highly ambiguous, but perhaps well worthwhile once learning is taken into account, is a real world exemplar of the laboratory situations investigated in this paper.

According to Tolstoi, happy families are all alike, while every unhappy family is unhappy in its own way. A similar observation seems to hold true for situations involving ambiguity: There is only one way to capitalize correctly on learning opportunities under ambiguity, but there are many ways to violate reasonable learning strategies.

From an evolutionary perspective, why would learning avoidance persist if the benefits from learning are large?

Psychological findings suggest that negative experiences are crucial to learning, while good experiences have virtually no pedagogic power. In the current setting, ambiguous options would need to be sampled repeatedly in order to obtain sufficient information on whether to switch from the status quo. Both bad and good outcomes would be experienced along the way, but only good ones could trigger switching. Bad outcomes would also weigh much more heavily, leading people to require too much positive evidence before shifting to ambiguous options. In individual decision situations, losses often weigh 2 to 3 times as much as gains.

In addition, if one does not know what returns would have come from an ambiguous alternative, one cannot feel remorse from not having chosen it. Blame from others also plays an important role. In principal-agent relationships, bad outcomes often lead to criticism, and possibly legal consequences because of responsibility and accountability. Therefore, agents, such as financial advisors or medical practitioners may experience an even higher asymmetry from bad and good payoffs. Most people, for that reason, have had many fewer positive learning experiences with ambiguity than rational sampling would provide.

It might be a good idea to try a new brand the next time you're at the store rather than just making the same choice over and over. Who knows, you might discover you like it better.