The Effect of Participation Probability on the Expected Number of Ballotters in a Four-Person Lottery.

Participation Probability Graph

Fig. 6

Horizontal axis = Probability that an individual will participate in a lottery ("participation probability").
Vertical axis = Probability that the resulting lottery has a particular total number of individuals participating.

Each curve plots the expectation that a given number of individuals will end up balloting, over the full range (0..1) of participation probabilities. Note that to maximize the probability of exactly 1 ballotter, each individual ought to participate 25% of the time as seen by the maximum in the 1 ballotter curve at 0.25. Over-participation leads to maximizing greater numbers of ballotters. The probability of 3 ballotters, for instance is maximized with a 0.75 participation probability. Conversely, under-participation leads to near-zero numbers of ballotters. A 0.05 participation probability makes it very likely (about 80%) that 0 individuals will end up balloting in a given lottery.

From the article "Lottery: Toward a Unified Rational Strategy for Cooperative Music Making" by Nick Didkovsky