I think the only reasonnable way to modelize this would be a probabilistic approach. Assume that people don't like to share a deal for nothing, so that they won't share when a lot of time remains, but they will try to share right before it stops to win the prize.
What would describe your system would be the density of probability of someone sharing depending on the remaining time. If you assume you know that, and you know the number of people participating you could, for example, estimate the expected frequency of prize distribution.
The problem is that the system evolves depending on human choices, which cannot really be treated mathematically (for example, you can only hypothesize exactly how the amount of the cash prize will influence the outcome of the game, i.e. human greed).
If you want a more quantitave answer, you should specify what variables you know or are willing to model, and which ones you'd like to estimate via the model.