I Love Math, a site for math teachers, has an interesting page on probability & statistics. The Game of Greed is a precursor to Diamant — although I don't know which came first. You have five rounds. In each round, you start with some point (they use 2d6). Each student can sit (and score) or risk. Once everyone's decided, the teacher rolls a die. One number 'craps out.' Any other number adds to the current score.
They have a lesson on statistics with that, but lets ignore that.
(Interesting aside — the break even point (the score at which sitting or standing shows the same Expected Value) doesn't depend type of die, assuming only one side "craps out", and is just equal to the total of the other sides. Proof is left as an exercise for the reader.)
Assume you were playing for extra credit, but for the winner only. It's the first round, simultaneous decisions. The score is at 14. Expected Value says to risk. (We'll assume the sides are 1,2,3,4,5,Bust.
Suppose you sit. The ideal outcome has everyone else craps out. You are up 14 points with four rounds remaining. More likely, you'll be down ~3 on everyone. Presumably (given EV) most people will now sit. Some may push on and be 6-10 ahead of you. Let's assume someone gets to 10 ahead. Not good, an almost certain loss. But with 25 students, losing is expected. Being +14 gives you good odds of winning (and a 1/6th shot). So others should also sit out.
In Bridge, this is like matchpoints tournament strategy. A solid 60% game is good, but won't win. If you add risk to your game (bringing your average score down but increasing volatility), you stand a better chance of winning. [I remember an article in Bridge World about this].
The more I ponder this, the more intrigued I get. As the number of players increases, you can either risk by pushing past everyone, or by dropping out and hoping for a bust. An early drop mirrors a late push. [Of course, here the payoffs or the same, which isn't true for Diamant, and the odds change with tile draws in the real game.]
Consider the final round. Abby leads, with Betty one point back and Carla eleven points back. Everyone else is hopeless. Opening roll is a twelve. Who do you like in that situation?
You know, Carla has a shot. She passes and the roll survives. Can Abby and Betty pass now? if either one does the other just needs to survive one roll (5/6). [Technically Betty may need to survive two rolls to win outright, but lets assume that the numbers preclude a tie ... make it 2,3,4,5,6,Bust]. So both risk and survive. Do they have to keep going? Either one stopping gives it to another. At three rolls the score should be ~22, but they've only got a 50/50 shot of making it there. Great odds for Carla, because the leaders won't stop.
If both Abby and Betty keep rolling forever, they lose. Who should give ground first? Suppose they keep going for three more rolls, then Betty (being in the inferior position) stands down and takes the 1/6 shot of the next roll crapping out.
Under this scenario: Carla wins 131/256 (~50%) of the time (Crap out in the middle) [51.1%] Betty wins 1/6 * (125/256) [8.1%] Abby wins 5/6 * (125/256) [40.7%]
Not bad for Carla at all.
But Betty can improve. Suppose that Betty stands down the round after Carla. Carla's got a 1/6 chance. Betty has a 5/36 shot (makes the roll, sits down, Abby busts) and Abby gets 25/36. Carla has better odds than Betty, but not obscenely better.
Another finesse. Betty, knowing that Carla will sit, sits at the same time. Carla can't win, Betty gains 1/36th, and Abby pockets the rest. But Abby, figuring the same thing, may sit at the same time. Abby wins automatically.
Remove Carla, and the situation is simple -- Abby pushes. If Betty pushes, then Abby wins 1/6th of the time and we're back to start, so Betty just has to sit and give Abby the 5/6th shot. Carla confuses the issue.
I'm not even sure how to handle this (assuming Carla sits and we're down to two decision makers). It's not a zero sum game, it's a negative sum game (the excess value going to Carla). There's a recursive entry (both risking gives Carla a 1/6th shot, removing value from the Abby/Betty matrix).
So if Carla sits, what do you do as Betty. You sit most of the time, but you have to risk sometimes, to prevent Abby from sitting at the same time.
Putting it all together -- how many people (out of the class of 25) should take wild risks in the first round? This splits the pack into two groups, one of which is probably 15 points ahead of the other. Then how many of that group should take a high risk strategy? Meanwhile, the losing group has to go riskier, since they are behind ... should some of them play for a minor gain, let the rest of the group (probably) fail, and then try again next round. I wonder if these questions have been answered in the literature?
I'm terrible at Diamant, by the way. But I think that I may try my "Drop out slightly early and pray for disaster" idea. [Of course, Diamant lets you score extra depending on when you drop out, complicating the issue]