[bh3]

Only thing that comes to mind is to add some sort of blind element. For example having two people, one flips two coins but both remain ignorant of the actual outcome and assume that they have opposing interests (and so would not be likely to try to help the other in any way). Depending on the outcome of the preliminary flip, the one of the second person is either kept the same or inversed. Again still not perfect but removes some level of manipulation. Which is sort of interesting because you'd have to wonder should the preliminary tosser try to make the probability as even or skewed as possible. If he is likely to skew it it becomes a whole new scenario.

[nfnaaron]

Since the bias is in the toss, not the coin:

Put the coin in the ref's hands, which are cupped against each other in a closed sphere. The ref shakes his hands to the satisfaction of all (and we assume an honest ref). He then places the coin on his thumb, but uses the opposite hand to cover the placing operation and the final launch configuration.

After the coin is positioned, but before it's uncovered, one of the interested parties calls it. Then the coin is tossed.

Fair, random, and evenly distributed as far as I can tell.

Edit: I think this works as well with two interested parties and an impartial ref, or only two opponents where the non-tosser calls the coin before it's tossed.

[kelnos]

In the case of an honest 3rd party doing the toss, you can even have someone call it before the ref shakes the coin around in his cupped hands. That way you eliminate the possibility of someone seeing the coin's orientation as it's placed on the thumb. The ref doesn't even need to hide the starting position after that.

If one of the interested parties is also the coin tosser, this could offer an opportunity to game the system, though, so one might want to just use your proposal in all cases to avoid complexity.

Thinking about this a bit more, though, I wonder if this really has the desired effect.

Let's assume for a moment that the act of the ref shaking the coin around in his cupped hands actually generates a random starting position. Ok, so starting position is 50/50 heads/tails, and the caller of the coin doesn't know the starting position.

In that case, the final heads/tails probability then depends on the initial starting position -- to use the bad-case numbers from the article, say the final probability is 60/40, favoring the side that was initially facing up.

So what have we really gained here? All we've done is made sure the initial positioning of the coin is random. The bias in the toss itself is still there, but we're hiding it by making sure the caller doesn't know the initial state of the coin.

If that's the case, then why not:

a) Don't even toss the coin at all. If we're convinced that the act of the ref shaking the coin around in his cupped hands gives a random starting position, then why not use that in place of the coin toss?

... or...

b) Avoid the whole cupped-hands shake thing entirely. Just have the interested party call the toss before the ref places the coin on his thumb. Obviously then you have to have an impartial ref who won't then place the coin on his thumb in a way to benefit one of the interested parties. (You could just specify that the ref is required to reach into his pocket, pull out the coin, and place it on his thumb, all without looking at it at all or feeling its surface sufficiently to figure out which side is which.)

I guess it also depends on what we care about to make this "random." Personally I think it's random enough if the caller simply doesn't know the starting position when calling the coin, assuming the coin tosser doesn't use the call to game the toss.

Heh, or we can just admit that tossing a coin isn't sufficiently random, and use something else... like radioactive isotope decay... or even a PC's PRNG (though that of course opens a big computer security debate).

[nfnaaron]

"So what have we really gained here? All we've done is made sure the initial positioning of the coin is random. The bias in the toss itself is still there, but we're hiding it by making sure the caller doesn't know the initial state of the coin."

Assuming the cupped hands shake produces a random starting position, you've evenly distributed the toss bias, so the result of a series of tosses should be evenly distributed.

"If that's the case, then why not: a) Don't even toss the coin at all. If we're convinced that the act of the ref shaking the coin around in his cupped hands gives a random starting position, then why not use that in place of the coin toss?"

That's an excellent simplification, but it just doesn't feel as dramatic and traditional to decide on shaking cupped hands. You need the toss for effect.