Given the odds, ethics, would you go for the extra credit?

By Barry Balof
Assistant Professor of Mathematics

Barry Balof adds a touch of psychology to his teaching exercises to illustrate to students how it can affect their thought processes and mathematical decisions.

When faced with an election, are we better off choosing a candidate who will raise taxes to implement more governmental programs? Will our individual votes match what we feel is best for the community? On an international scale, what roles do calculation and psychology play in our economic and military deci- sions with regard to other nations? How should an overcommitted student, as so many of them are these days, decide to allot his or her precious time?

Quandaries like these require a measure of mathematical ability as well as psychological awareness. During my fourth year of graduate school, I left the confines of my mathematical immersion to take a course in the psychology department on decision-making. During this course, we spent the first day and a half discussing the optimal way to make a decision and the rest of the term discussing why people don’t do it that way. Mathematics plays a large role in the “optimal way,” but we often find that mathematical considerations are trumped by our psychological tendencies. I’ve incorporated some decision-making exercises into my classes both to help me gain insight into how my students think and to show them something about how psychology affects their own thought processes and mathematical decisions in areas beyond my classroom.

Below are three examples of these exercises. I’ll often give such a question at the end of a quiz. We’ll typically analyze it in the next class, discussing the optimal solution, the outcome chosen by the class and the (many) differences between the two.

What Would My Neighbor Do?

Extra credit: Would you like one point or three points of extra credit? Note: If more than 25 percent of you choose three points, no one gets anything. Students have just toiled through calculations of derivatives, integrals or vectors and are now faced with the chance to improve their grade. A response of one point increases the likeli- hood that the whole group will get some- thing, but as individuals, they might be able to afford to go for the three points, trusting their peers to be more benev- olent (or conservative) in their decisions.

So, what would you do? How do you think they did?

In general, my classes will not get any extra credit, and, in fact, they will miss it by a mile. In the last class that had this question, 50 percent of the students opted for three points. One student gave the following rationale for his decision: “I know that I should choose only one point for the benefit of the group, and that my grade could probably afford it (it could), but I would rather that no one get anything than that other students get more than I do.” Not too many in the class were surprised not to get extra credit on this one.

I have also given the slightly more humane variant:

All for One and One for One

Extra credit: You may take one extra credit point for yourself, or zero extra credit points for yourself and give one-tenth of an extra credit point to everyone else in the class (40 other students between two sections of the same class). Your extra credit will be the sum of what you take for yourself and what others give to you.

This is another example where the optimal strategy for the group differs from the optimal strategy for the individual. It seems here that each student will be guaranteed something … unless he or she is the lone student who opts to give out points to everyone!

Some calculations on the problems bear discussion. You stand to make as many as five points of extra credit on this one if everyone acts in your best interests. That is, if you are the only student who chooses to take the point for yourself and everyone else opts to give rather than to take. You might not even feel too guilty if this happens, as everyone else in the class stands to make nearly four points of extra credit. From a purely “game-theoretic” point of view, you’re always better off taking the point for yourself. If everyone follows this logic, however, the scores will plummet.

What should you do? What would you do? How did the class do?

Again, approximately 50 percent of the group opted for “greed,” earning three points, while the other half opted for benevolence, earning two points. Strikingly, there was no correlation between student choices and such factors as year in school (the group was mostly first-years, still new to college, and sophomores, who may be more attuned to the way their peers think), gender or even standing in the course. (Those who were doing well in the semester were equally likely to choose for themselves or choose for the group). Again, students commented on wanting to do well for themselves and not have to watch others do better than they did.

Half Full?

Lest you think that all Whitman students are so cynical when it comes to matters of how their peers think, I present the following example based on a short-lived game show, “Friend or Foe” (which itself was based on the classic strategy game Prisoners’ Dilemma):

Extra credit: Complete the following quiz with a partner. The score that the two of you earn is extra credit, but you must decide individually how much of the credit you’d like. You can opt for either “half” or “all” of the credit. If you both choose “half,” then you’ll split the points. If one of you chooses “half” while the other chooses “all,” then the person who chose “all” gets all the credit, while the other gets nothing. If both of you choose “all,” then neither of you gets any credit.

This example continues the theme of the difference between individual and group strategy, but the fact that the group size is now only two strengthens the psychological component. Individually, each person is better off trying for the whole amount of extra credit, but if both partners follow this individual strategy, both walk away empty-handed. For this exercise, students were paired randomly, so that the “friend” incentive to cooperate was lessened. As contestants on the game show were able to do, the students were given a brief time after the quiz to talk to their partners and convince them to “do the right thing.” It is easy to see how a carefully constructed mathematical argument in this scenario might give way to a “gut feeling” about one’s partner.

How did the class fare?

The arguments must have been convincing, because the students univer- sally opted to split the credit with their partners. Perhaps the earlier exercises had opened the group to the considerations of the whole over themselves, or perhaps, students find it harder to act selfishly when dealing with one individual rather than the nebulous “group.” What is striking is that these data differ from other groups performing similar tasks. In previous studies, participants opted nearly half of the time to act selfishly, which is also in accordance with the amount of money given out by the game show during its short run. As a result, I don’t necessarily expect the cooperation to be quite so pronounced in future groups, but I am heartened by the results thus far.

While these exercises are self-contained and may seem largely mathematical on a first reading, their relevance to other areas is quite clear. The examples on tax policy, foreign relations and time management have both mathematical and psychological considerations, and it’s important to understand how the two are interrelated.

So much of a liberal arts education revolves around cross-disciplinary thinking. Through these exercises, students see a different applicability of mathematics — and (if they trust their peers) earn a little extra credit as well.

Barry Balof joined the Whitman faculty in 2003. He received his master’s and Ph.D. degrees from Dartmouth College.