VotingModelRecap

(Explains Darius's VotingModel who's results are on SimulationResults)

To recap, ci is an individual's 'cost' of voting. Costs are normally distributed with mean m and standard deviation s. (Note - some agents may have negative costs - this could be interpreted as the 'warm glow' they get from doing their bit for the democratic process.)

As the 'benefit' of the vote being 'won' is 1, can think of the cost as a cost/benefit ratio. eg if m = 0.1 this means on average individuals gain ten times more from 'winning' than it costs them to vote.

N is the group size, and n^ is the threshold number required to 'win'. I've set this cell as a fixed proportion n^/N so you can change the group size and see what happens.

column a gives the group's possible 'guesstimate' c' of the 'swing' probability

column f gives the proportion of agents who will vote if they make guesstimate c' (i.e. normal cumulative distribution function for column a)

column b gives the actual swing probability p given the guesstimate c'

(i.e. binomial probability for the proportion in col f)

column c gives the actual number who will vote if they make guesstimate c' (i.e. column f times group size)

if c' = p (col a = col b) we have an equilibrium

but it will only be a 'productive' equilibrium where the vote is 'won' if n > N (col c > group size).

By playing with these parameters you can get all kinds of different shapes and combinations of equilibrium. (If you're really interested, next step might be to talk about dynamics and 'stability' of these equilibria.)

eg. mean of 0.1, group size 20, and (completely arbitrarily) setting s = 2m.

there is a productive equilibrium with c' = p (c') = 0.132 or thereabouts, and eleven group members contributing

in fact with these values of m, s, and n/N get a productive equilibrium for groups with up to 68 members

Group size is the most obvious factor effecting existence of productive equilibrium. You can increase group size and still get productive equilibrium with a more spread distribution of costs - eg s = 1 ( = 10m) then get productive equilibrium for a group of 80 with c' = p = 0.089

Otherwise the obvious way to 'improve' the result for a given size is to reduce the cost of contributing. eg for 1000 members there is a productive equilibrium around c' = p = 0.021 with m = 0.02 (i.e. possible gain fifty times greater than cost)