In Gelman and Nolan’s paper “You Can Load a Die, But You Can’t Bias a Coin” The American Statistician, November 2002, Vol. 56, No. 4 it is argued you can’t easily produce a coin that is biased when flipped (and caught). A number of variations that can be easily biased (such as spinning) are also discussed.

Obviously Gelman and Nolan are smart and careful people. And we are discussing a well-regarded peer-reviewed article. So we don’t expect there is a major error. What we say is the abstraction they are using doesn’t match the physical abstraction I would pick. I pick a different one and I get different results. This is what I would like to discuss.

In colloquial use a coin flip is when a coin is tossed into space and tumbles along one of its inertial axes parallel to the face of the coin (so it is not spinning like a frisbee). There is some uncertainty in the initial energy imparted and some uncertainty of when the motion is stopped. The coin is either then caught by hand, or allowed to come to rest on a hard or soft surface. The face up is then the outcome of the flip. We idealize and assume the coin is flipped in a vacuum and stays in motion as long as we need.

I personally don’t feel the “caught coin” model is completely specified. People do flip coins in this manner, but I don’t think we have a good description of what is done when one attempts to catch a coin that is edge down. We can assume they take the next face in spin order, but that still leaves us a problem.

The original paper uses a physics abstraction that I think implicitly disallows an obvious way of biasing a coin: moving the center of mass away from the center of geometry. We quote from the paper:

The law of conservation of angular momentum tells us that once the coin is in the air, it spins at a nearly constant rate (slowing down very slightly due to air resistance). At any rate of spin, it spends half the time with heads facing up and half the time with heads facing down, so when it lands, the two sides are equally likely (with minor corrections due to the nonzero thickness of the edge of the coin); see Figure 3. Jaynes (1996) explained why weighting the coin has no effect here (unless, of course, the coin is so light that it floats like a feather): a lopsided coin spins around an axis that passes through its center of gravity, and although the axis does not go through the geometrical center of the coin, there is no difference in the way the biased and symmetric coins spin about their axes.

We argue that assuming away “minor corrections due to the nonzero thickness of the edge of the coin” is exactly assuming away a useful mechanism for biasing the coin: moving the center of mass away from the center of symmetry so the coin experiences an unequal amount of time heads-up versus tails-up. There are differences, and let’s try to exploit them.

Consider a coin made by two layers, one much denser and heavier than the other. In edge-on cross section our coin would look like the following.

This coin is essentially the “pickle jar lid” described in the original paper. We have moved the center of mass away from the center of geometry. And I am going to argue it should show some bias even in flipping. Flipping defined here as tossing the coin in the air so it rotates along an axis perpendicular to the drawn cross-section (pretty much how coins tend to flip).

Notice as we rotate the coin around the center of mass each face is pointing clearly down a different amount of time. The tails side is down nearly 180 degrees, and the heads side is down is down an amount that is noticeably less than 180 degrees. The missing geometry is when the edge is down (which was assumed out in the original paper). So if we stop the coin mid-air at a random time chosen uniformly at random from some large interval we expect to observe it in the “tails down” configuration a bit more in the “heads down” configuration (again, the difference being “edge down”). So the only way the coin is “fair” is if we assign just the right majority of the edge-cases to “heads down.” For a “catch the coin” protocol, we need to assign what it means to observe the coin in the edge configuration. In edge-down cases even if the catch moves to the next face in spin order we still don’t get even odds (as the edge subtend the same angles and we assign one side to one face and the other to the second face).

The posited bias is proportional to coin thickness over coin diameter and is going to be very small, so it would take a very large experiment to reliably estimate it empirically. So this is not my favorite choice for a classroom demonstration. Also you can build an unfair “coin” by taking a six-sided die strongly biased towards one; we re-label “one” as “heads” the opposite side labeled as “tails” and all other sides labeled “edge, do-over.”

A coin that isn’t caught, but allowed to bounce around on a hard surface brings in additional concerns. Such a coin may be biased, but some part of its bias may come from statistical mechanical concerns. The same coin could potentially show different biases when flipped and caught or flipped and allowed to bounce on a hard surface.

Consider the following new model of a “coin flip.” Suppose we place a coin in a large hard can and shake the can vigorously. We then open the can and see which side the coin has come to rest on (assuming it is unlikely the coin stops edge-on or leaning against the wall of the can). Then by heuristic use of Boltzmann statistical mechanics style arguments the probability we expect to see the coin in a given state should proportional to `exp(-E/(k T))`

where `E`

is the energy of the state (and we treat `k T`

as a mere distributional constant). That is: since the two states (heads-up, tails-up) have different potential energies we expect the higher potential energy state to be harder to access. And the coin heads-up versus heads-down states do have differing potential energies as in each case the center of mass is either above or below the center of symmetry (see figure).

As you can see the bias estimate depends critically on the abstraction chosen. I have not specified enough of the problem to actually calculate, but I think I have made a heuristic argument for the plausibility of biased coins.

An interesting reference on coin flipping: DYNAMICAL BIAS IN THE COIN TOSS, Persi Diaconis, Susan Holmes, Richard Montgomery.

Of particular interest is the lead and balsa coin. Exactly the kind of re-weighting we would want- but not likely to survive long in the shaken can situation.