Keep Cooperating. We Did the Math.

Should you stop cooperating with your classmates once there are only a fixed number of A’s to go around?

Absolutely not. Under most reasonable assumptions, not cooperating will harm your chances of being in the top fifth of a class. Even more importantly, working solo will make your Harvard experience both less intellectually rewarding and less fun.

In the midst of the grade cap to come, we have heard many of our students declare that the cap will stop them from working together. We suspect that this rhetoric is just a convenient way to bash the policy. But this argument is so weak, even for that purpose, that we hope students will instead turn to the more intellectually coherent arguments against the cap given last year by Scott Duke Kominers ’09, a professor at Harvard Business School.

Today, we write neither to attack nor defend the grade cap. We write in favor of cooperation, with or without the cap, and we think that a bit of simple math makes the case.

When you collaborate, you are much more likely to move up than move down: To move up, you need to be just below the A cutoff — such that the benefits from collaborating push you over. But for you to move down as a result of working together, you need to be just above the cutoff and your partner must be just below — so that you could potentially help them jump above you, pushing yourself out of the A region in the process. But this second case is extraordinarily unlikely: Being near the cutoff is always a long shot, and for both of you to be at the cutoff is orders of magnitude less likely than just you being at the cutoff.

Let’s first consider a simple case. Assume that all of the students in your class have some baseline score that will determine their grade, and that you have the option to work together in a group. Assume further that the group will raise the scores of all the members of the group by the same amount. Can joining the group ever make you worse off?

No. If your scores go up by the same amount, then your ranking within the group will stay the same, and your ranking relative to outsiders can only go up. In this setting, your chance of getting into the top fifth of the class can only go up if you join the group.

The argument is slightly more complicated if the benefits from cooperation are asymmetric — i.e., if you help your partner more than your partner helps you. Nonetheless, the case for cooperation remains strong.

Suppose there are 100 people in the class and everyone’s baseline score is a unique integer from one to 100. Neither you nor your partner know your baseline score. Suppose that by cooperating your score will increase by 1.1, moving you up one spot, and your partner’s score will increase by 2.2, moving him up by two (and potentially past you). Consequently, if you initially were ranked 25th and your partner was ranked neither 24th nor 26th , then you join the top “20 percent plus four” by cooperating. This is the case for cooperation.

What if you were already in the top 24? Cooperation cannot push you out: If your partner was already ahead of you, then you cannot fall in the order. And if they were behind you, it doesn’t matter if they leapfrog over you — so long as your score goes up by enough to push you ahead of the one person who was ahead of you.

Cooperation only hurts your grade if two things are true: Cooperation pushes your partner up a slot and doesn’t help you at all, and you have a particularly rare combination of initial placement in the pecking order. Now assume that cooperation will independently generate for each of you no bump with probability one half, or a bump of 1.1 with probability one half. As we have already noted, you cannot lose if you both gain 1.1 or you both gain nothing.

In this setting, you lose if you are ranked 24th before cooperation, get nothing from the bump, and your partner is ranked 25th and gets a positive bump to leapfrog you. You have a one-in-a-hundred chance of being 24th, and if you are 24th, then your partner has a one-in-ninety-nine chance of being 25th. The overall odds of your partner moving past you are then 1 in 39,600 — i.e., 1/100 × 1/2 × 1/99 × 1/2.

And what are the odds cooperation helps you here? You need to start as number 25 and get a positive bump. The odds of that are 1 in 200. You also need your partner not to be ranked 24th and get a positive bump, but that is a 197 out of 198 probability (1 minus 1/99 times 1/2). Then your odds of moving up are essentially 1 in 200.

So partnering is almost 200 times more likely to benefit you than hurt you. This framework is only a stylized approximation of reality, but the point holds more generally: The probability of you gaining from being on the margin is far higher than the probability of you losing from both you and your collaborator being on the margin.

We hope that you take your classes for reasons other than getting an A, like the joy of learning or the fun of having intellectual interactions with your remarkable classmates. But even if you are just going after a straight A, working together with your classmates is still the smart thing to do.

Edward L. Glaeser is the Fred and Eleanor Glimp Professor of Economics and teaches Economics 1011a: “Intermediate Microeconomics: Advanced.” Davide Viviano is an assistant professor of Economics and teaches Economics 1126: “Quantitative Methods in Economics.”