Help or pointers - Pool payout algorithm

I'm hoping someone can help point me in the right direction. Lets say that I wanted to set up an office pool to bet on which day a co-worker will deliver a baby. I have a table of probabilities that a baby will deliver on the nth day of pregnancy (if that's relevant, or might add an interesting twist) and the date of conception is known. I'd like to reward all participants in the pool with significantly more going to the person(s) with the most accurate prediction, and those with less accurate predictions receiving back some fraction of their initial bet. There would be no restrictions on the number of people betting on each date, nor on the number of dates that an individual couple place a bet on. The total reward would not exceed the total of the betting pool.

I'm positive that I'm not the first person to want to do something like this, however I haven't found a set of keywords that pulls up anything relevant on Google. I'm looking for an example or a walk though, and maybe some discussion of finer points I'm not considering. Could someone suggest what I should search for or perhaps post some links to relevant information?

My office does something similar, except that we do half days (am/pm) and each value can only be picked once. That solution is pretty simple. The winner gets x% of the pot, the people off by 1 period on either side get less, the people off by 2 on either side get less, and so on. You just set the payoff percentages, rank the performance of the participants, and you're good to go. The only correction that is needed is a small adjustment if/when a two sided distribution does not work. Like, if the delivery is very early and there are more people who picked after the delivery than before.

Your situation is more complicated because multiple people can pick the same time period. I think the best way to handle that exception is to set up the payouts as though there will be only one winner. So, say 1st gets 30%, 2nd gets 20%, 3rd gets 15%, 4th gets 10%, etc. Then if 4 people pick the winning day, they split 75% of the pot (30+20+15+10), and the 5th person (who is in 2nd place) gets the next-best remaining payout which is ~5% of the pool.