SequentialGroupDecisionmaking

In most theoretical models of collective action problems group-members act simultaneously, and/or don't know what the other group members are up to.

The full-on example of this being voting in a secret ballot as in VotingModel.

At the other theoretical extreme, the 'Critical Mass' model developed by Gerald Marwell and Pamela Oliver begins with a situation where group members act in sequence. The nth actor to act knows what the previous n-1 have done.

As in the VotingModel, the collective action either 'succeeds'or 'fails'. (Or - a collective good is either provided or not, etc.) This model is however more general in that agents can make different levels of contribution/activity.

Each agent chooses a contribution level ri. The gain to agent i if the action succeeds is ui.

The overall contribution level of the group is R = sum ri. The key to the analysis is the 'production function' p(R). p is the probability that the group action succeeds, and it is assumed to be a monotonically increasing function of the overall contribution R. Basically - the higher the group's contribution, the more likely it is that the action will succeed.

The model looks at how different types of production functions effect the possibility of successful collective action. Because agents know what previous agents have done, they know 'where they are' on the production function - how their contribution will increase the likelihood of the action succeeding. Which depends on the 'shape' of the production function. For example -

Decelerating production functions

Later contributions have less effect.

Eg. 'calling city hall about a pothole in a middle class area: the first person who takes the time to call makes the probability 0.4 that the hole will be fixed, the second raises it to 0.7, the third to 0.8, ...'

Accelerating production functions

Eg. 'calling about a pothole from a poor minority urban area with little political clout: it takes 20 calls before the probability reaches even .01, another 20 to reach .1, but the next 20 calls worry city hall and make the probability .9'

Model

To start with, if there is only one agent (i.e. not a collective action problem) then the agent's decision problem is:

maximise net gain = uP(r) - cr

the contribution cost can be normalised to c = 1 then the problem is: max uP(r) - r

The model also assumes that p(R) is continuous and twice differentiable. So using calculus, the first order condition for a maximum is:

p'(r) = 1/u

... tbc ...