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

**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