Maybe I'm being dense, but bear with me for a moment....
Assume: I get X utilons from winning, Y from tying, and Z from losing, where X >= Y >= Z. Everyone plying the game has exactly the same preferences.
If I (and everyone else) play 0, I get Y utilons. Straightforward.
If I play a value that gives me W chance of winning outright, and (1-W) chance of losing (with an inconsequential chance of tying because I added a small random offset), I will gain W X - (1 - W) Z utilons on average.
Assume W is fairly low, the worst and most likely case being 1/N where N is the number of participants, since we're assuming everyone is exactly like me.
Therefore, if Y > (X/N - Z + Z/N), I (and everyone) should play 0. Otherwise, we should play the thing that gives us W chance of winning. (hopefully I did the algebra right)
So, depending on the values for X, Y, and Z (and N), we could get your scenario or mine.
If Y is close to X, we get yours. If it is greatly lower than X, we will probably get mine.
All that to say I can create a scenario where the Nash equilibrium really is for everyone to play a small positive number by tweaking the players' utility functions, even given the constraint that winning, tying, and losing are valued in that order.
If this is clear to you, then we've been talking past each other. If not, then I don't understand Nash equilibrium very well (or I'm an incredibly sucky writer).
EDIT: on second thought, I think my math is probably quite bad, esp. with respect to Z. Anyway, perhaps the central idea of my post is still intelligible, so I'll leave it be.
EDIT2: Ah, I got a sign backwards (consider that if the penalty for losing is your house gets burned down, Z is a large negative number).
W X - (1 - W) Z should be W X + (1 - W) Z
Y > (X/N - Z + Z/N) should be Y > (X/N + Z - Z/N)
There are some games that don't have a Nash equilibrium. Consider a 1-player game where the available strategies are the numbers between 0 and 1, and your payoff is 1-x if you pick x>0 and 0 if you pick x=0. There is no Nash equilibrium.
If many players assign 0 utilons to tying and losing in this game, and 1 to winning, then 0 is still a Nash equilibrium, but if there is any positive chance that some gimp will submit a nonzero answer just for the hell of it, then you definately shouldn't play zero.
By the way, I guessed 100. I'm not very good with numbers - I think 100 is the best answer, right ;-0
I'd like to play a game with you. Send me, privately, a real number between 0 and 100, inclusive. (No funny business. If you say "my age", I'm going to throw it out.) The winner of this game is the person who, after a week, guesses the number closest to 2/3 of the average guess. I will reveal the average guess, and will confirm the winner's claims to have won, but I will reveal no specific guesses.
Suppose that you're a rational person. You also know that everyone else who plays this game is rational, you know that they know that, you know that they know that, and so on. Therefore, you conclude that the best guess is P. Since P is the rational guess to make, everyone will guess P, and so the best guess to make is P*2/3. This gives an equation that we can solve to get P = 0.
I propose that this game be used as a sort of test to see how well Aumann's agreement theorem applies to a group of people. The key assumption the theorem makes--which, as taw points out, is often overlooked--is that the group members are all rational and honest and also have common knowledge of this. This same assumption implies that the average guess will be 0. The farther from the truth this assumption is, the farther the average guess is going to be from 0, and the farther Aumann's agreement theorem is from applying to the group.
Update (June 20): The game is finished; sorry for the delay in getting the results. The average guess was about 13.235418197890148 (a number which probably contains as much entropy as its length), meaning that the winning guess is the one closest to 8.823612131926765. This number appears to be significantly below the number typical for groups of ordinary people, but not dramatically so. 63% of guesses were too low, indicating that people were overall slightly optimistic about the outcome (if you interpret lower as better). Anyway, I will notify the winner ahora mismo.