# Grading curves are dollar auctions

Game theory has the concept of a
dollar auction,
which goes something like this:
participants take turns placing bids for a dollar bill
until no one is willing to top the high bid.
Then *everyone* who placed a bid
pays the auctioneer whatever their last bid was,
and then the highest bidder
(and *only* the highest bidder)
wins the dollar bill.
^{1}

This has some interesting dynamics: if no one else is willing to bid, you can win the dollar bill with a bid of a single cent, making a 99 cent profit. If someone else has bid a single cent, you might be tempted to bid two cents, to make a 98 cent profit; but then the first bidder might as well top you with a three-cent bid in order to make a 97 cent profit instead of losing their one cent and getting nothing in return.

This continues on for a while. Eventually, someone tops a 98 cent bid with a 99 cent bid, at which point the 98-cent bidder might as well go ahead and bid a dollar, so that he can break even instead of taking a 98 cent loss. But then the 99-cent bidder doesn’t want to lose his 99 cents with nothing to show for it—so he bids 101 cents, preferring a one-cent loss (paying 101 cents to win a dollar) over a 99 cent loss. The first bidder will then likely bid 102 cents in order to take a two cent loss instead of losing an entire dollar, and on it goes.

Nobody truly wins such a game other then the auctioneer, so why participate in the first place? But remember, if everyone else refuses to participate, then you can make a pretty nice profit by picking up a dollar for just a single cent!

A dollar auction as described above
is the simplest form
of what is generally known as an all-pay auction.
We could easily imagine other variants
with the same fundamental dynamics.
For instance:
you could have an auction in which a
hundred participants vie for five ten-dollar bills.
These five bills will go
to the top five bidders;
but *all* bidders must pay what they bid,
be they in the top five or not.

Or imagine a version, again with one hundred participants, where there are five twenty dollar bills which go to the five highest bidders, ten ten-dollar bills that go to the next ten highest, twenty five-dollar bills for the next twenty, and thirty one-dollar bills for the next thirty, again in an all-pay format. Though this scenario is quite a bit more intricate, the core dynamics are the same. Perhaps the effect is somewhat blunted given that you might not feel as much resistance giving up on the one prize you were hoping for to get instead the next-best—but still, if you’re just a cent short of winning the top prize, you have a strong incentive to bid that one extra cent to actually win it rather than giving up and being down a net amount of almost ten dollars.

Now, imagine you are taking a university course. During the semester, everyone chooses how much to study for the course, and then at the end, a test is given; say it’s a good test, one that actually captures how much you studied. This test will be graded on a curve—perhaps the top 5 highest-scoring students receive an A, the next 10 get a B, and so forth. And needless to say, no matter what grade one gets on the exam, the time invested studying for the course is already spent and gone.

Note that this is exactly the same as the complicated version of the all-pay auction described above! Well, at least given the (cynical, but not entirely unreasonable) assumption that students wipe their minds clean of the course material the very second they step out of the exam room.

On the other hand, perhaps you don’t want to assume that students study only to pass the test, forgetting everything by the next day. Instead, perhaps you claim that your course teaches them valuable skills for future courses or for post-university life. But if so, then the reasoning behind grading on a curve is already weak.

Insofar as what’s being learned has some true objective value outside the four corners of the classroom, then it would be quite possible for one semester’s class to be filled with slackers who learn nothing and the next to be filled with smart and dedicated students who take the material deeply to heart. The first semester’s students will be completely lacking in those skills they were supposed to take away from the course, and the next semester’s students may well all have become experts, but if you’re grading on a curve then you don’t manage to capture a record of that: each semester will have the same number of students with each grade.

So either way, grading on a curve is not a good idea. Inasmuch as there exists objective merit to be measured, having the same distribution of grades each semester means that the grades can’t always reflect that external merit. But on the other hand, if there is no external yardstick and the only way to measure accomplishment is to contrast results between students, then even worse: you’ve trapped the students in the negative-sum dynamic of a dollar auction, where any student who chooses to participate at all will end up investing more time in studying than it is actually worth, much like the bidders in an all-pay auction who keep on bidding yet another one cent more in order to limit their already-guaranteed net losses.

^{1}Some variants of the dollar auction demand that only the two highest bidders pay, if there are more than two. The distinction is not particularly important for our purposes.