Reading the literature on collective action problems I can't find a very clear picture of exactly what they amount to. Are they just PublicGoodsProblems? Is it always about FreeRiding? Can all CAPs be represented by PrisonersDilemma?

KeithDowding says 'the collective action problem encompasses demand side problems and supply side solutions'. He gives a list - KeithDowdingList - of 11 'demand-side problems' which are 'interlinked and interrelated'. This is the closest I have come across to a comprehensive treatment, although some of the 'problems' in the list I would call 'characteristics' or 'factors' rather than problems in their own right, and the links and relations are not spelt out.

I thought I'd have a stab at developing a systematic theory of CAPs that would elucidate these links and relations. (Caveats - with any luck someone better qualified has already done this somewhere I haven't come across; I am not a professional game theorist and there may well be errors in it; this is very much work in progress.)


See also: TypologiesAndPatternLanguages

(Phil - I hope you don't mind I moved this debate to a new page.)

'One-off' CAPs

There is a group of n individuals.

The easiest case to start with is a discrete, 'one-off' CAP. That is - there is a 'group action' that either will or will not take place. To make it easier still, individuals' contributions are fixed - each either participates at a given level, or not at all. If enough individuals participate, the action will take place. If they don't, it won't. We can suppose that there is a threshold number n^ who must participate for the action to succeed.

examples - signing a social contract (SocialContractProblem), turning out to vote (VotingModel), joining a crowd to storm the winter palace, agreeing to refrain from grazing the commons (see CommonsProblem), making a fixed contribution to the production of a discrete public good with a 'one-step' production function.

  • ideal condition

The group action is, in some sense, desirable - otherwise non-occurrence wouldn't be a problem. How to formalise this?

As a minimum, I think, this condition must hold - at least n^ of the group members would benefit from the group action. I will call this the 'ideal condition'.

  • individual rationality condition

Group members are assumed to be expected utility maximisers. Any individual will participate in the group action if the expected utility of participating is greater than the expected utility of not participating.

(EUp > EUn)

The group action will be successful if n^ individuals participate. I will call this the 'individual rationality' condition.

  • definition of CAP

Thus - a Collective Action Problem occurs where the 'Ideal Condition' holds, but the 'Individual Rationality Condition' does not.

  • decision problem

There are two courses of action each individual has to decide between - participate in the group action, or not.

Story - in a situation of completely unco-ordinated individual decision-making, individuals act atomistically according to their perceived best interests. (Graze on the commons, produce no public goods, live in a 'state of nature' without social contracts, etc.) Now there is a proposed schedule of group action which sets out alternative actions for group members (restrict grazing, contribute to public good, sign social contract, etc.) (Note - for some members, it may be the group action proposal is no different from the 'spontaneous' action in any case.) The individual's choice could be expressed - participate in the group action, or do the spontaneous action.

There are two possible states of the world. n^ or more members particpate, and the action succeeds. Or less than n^ particpate, and it fails.

Group members each assign a payoff or value to the possible outcomes, which can be expressed in a 2X2 payoff matrix like this:

|| || action succeeds || action fails ||

|| participate || Ups || Upf||

|| not partipate || Uns || Unf ||

Values of these payoffs can differ for different individuals. However the Ideal Condition imposes restriction(s) on these values for at least n^ members.

How to interpret the ideal condition in terms of these payoffs? I think the following is a minimum condition:

Ups > Unf for at least n^ members.

That is - there are at least the required number of members who would be better off participating in a successful group action than they would acting spontaneously without a group action (if you like, a 'state of nature').

  • uncertainty

If group members do not know the exact payoff structures of all other group members, then they make can decide by using probabilistic estimates of other members' actions. If < n-1 other members participate then the group action must fail. If n^ or more partipate then it will succeed. But if exactly n-1 others participate the 'state of the world' is in our individual's hands - he can decide whether or not the action succeeds by deciding to participate or not. We will say the member is 'pivotal' in this case.

Uncertain individuals make estimates:

pf - prob. < n-1 others participate (prob of 'failure' with or wothout my action)

ps - prob. n^ or more participate (prob of 'success' with or without me)

p - prob. n-1 others participate (prob of being pivotal)

The expected utility of particpation is:

EUp = pf Upf + ps Ups + p Ups

The expected utility of non-participation is:

EUn = pf Unf + ps Uns + p Unf

Expected utility maximisers will participate if EUp > EUn, so if:

pf (Upf - Unf) + ps (Ups - Uns) + p (Ups - Unf) > 0.

  • success or failure

There is a CAP if:

1) ideal condition: Ups > Unf for at least n^ members

& 2) IR condition: pf (Upf - Unf) + ps (Ups - Uns) + p (Ups - Unf) > 0 for at least n^ members

A first place to look to see why this might be is at the payoffs of the n^ for whom the ideal condition holds. If there is a CAP, it must be true that the IR condition fails for at least some of these individuals.

(Note - I am not ruling out the possibility that the IR condition in fact does hold for some individuals for whom the ideal condition does not. Then we could have a successful group action even though some of the 'idealists' non-contribute? Come back to this.)

A number of different payoff structures might cause the IR condition to fail for any individual. We do not need to assume that all 'idealists' have the same payoff structure - there may be different reasons why different individuals fail to particpate.

  • payoff structures

With two possible states of the world and two possible actions there are four potentially different payoffs for each individual, and 24 possible orderings of these payoffs.

For example, an individual might have the payoff ordering: ups > upf > uns > unf. In this case all the terms in brackets in the IR condition - pf (Upf - Unf) + ps (Ups - Uns) + p (Ups - Unf) > 0 - are positive, and the condition must hold.

For individuals for whom the ideal condition - Ups > Unf - holds, there are twelve possible payoff orderings. (Leaving out ones where ups < Upf there are eight. I have set these out with example payoff matrices in the table below:

matrix / ordering / bracketed terms in IR condition

ps (Ups - Uns) + pf (upf - unf) + p (ups - unf)

1) 'paradise 1'

|| p || 4|| 3||

|| np || 2|| 1 ||

ups > upf > uns > unf S + F + p +

2) 'paradise 2'

|| p || 4 || 2||

|| np || 3|| 1 ||

ups > uns > upf > unf S + F + p +

3) 'esperanto'

|| p || 4 || 1||

|| np || 3 || 2 ||

ups > uns > unf > upf S + F - p +

4) 'chicken'

|| p || 3 || 1||

|| np || 4|| 2 ||

uns > ups > upf > unf S - F + p +

5) 'prisoners dilemma'

|| p || 3|| 1||

|| np || 4 || 2 ||

uns > ups > unf > upf S - F - p +

6) 'paradise 3'

|| p || 4|| 3||

|| np || 1 || 2 ||

ups > upf > unf > uns S + F + p +

7) 'esperanto 2'

|| p || 4|| 2||

|| np || 1|| 3 ||

ups > unf > upf > uns S + F - p +

8) 'esperanto 3'

|| p || 4|| 1||

|| np || 2 || 3 ||

ups > unf > una > upf S + F - p +

  • 3 types of problems

From the above, there are three basic forms of preference orderings of 'idealist' group members that can cause them not to participate.

Further simplifying, suppose that all 'idealist' members have the same preference orderings (though not necessarily same magnitudes). Then we can start to identify three basic types of collective action problem that could arise from the preference orderings of 'idealists':

  1. free-rider problem

Members have ordering - uns > ups > unf > upf (no. 5 above). This is the familiar ordering from the 'prisoners dilemma' game.

In the IR condition - ps (Ups - Uns) + pf (upf - unf) + p (ups - unf)> 0 - the first and second terms are both negative. That is - if the group action succeeds with or without me, or if it fails with or without me, I do better by not participating.

These members will only participate if there is a high chance of being pivotal. (In a certainty version of the game, if I can individually determine the outcome.)

An example of this is the VotingModel problem, which is more generally a model of contribution to a 'one-off' public good. There: uns = u > ups = (u - c) > unf = 0 > upf = -c.

  1. chicken problem

More exactly, 'FreeRiding' means that I do better by not participating if the group action is successful. (Uns > Ups).

In this sense the 'chicken' problem (no 4) uns > ups > upf > unf is also a 'FreeRiding' problem. The difference here is that the second 'failure' term pf (upf - unf) in the IR constraint is positive. That is, I prefer to participate but only if the group action fails.

Here there is more hope of the problem not occurring than in the classic freerider problem, because of the positive 'failure term' - if I think there is a high chance of the action failing, and/or of me being pivotal, I will participate.

A real-life example of this (other than an actual game of chicken) ...

  1. set-up problem

Three preference orderings in the table above could give rise to what I am calling the 'set-up' or 'esperanto' problem.

The problem here is that: pf (upf - unf)< 0.

I will do well from participating if enough others also participate, but I will be worse off than otherwise if I participate and the action fails.

Esperanto (see SocialContractProblem) is a good idea if enough people decide to learn it, but I am worried about investing the energy into learning esperanto only to find that I have no one else to talk to.

NOTE - This is a problem that stems from uncertainty - if I know for certain that there are n^ people who will want to participate, eg. with the same payoff ordering as me, then I am not worried.

In VotingModel I modelled contributions to a non-excludable public good. If we take the same basic model but make the good excludable then we have SetUpProblem. Non-contributors are now excluded and don't get the benefit u from the good. so the payoffs become:

ups = (u - c) > uns = unf = 0 > upf = -c.

When simulated, as in StrategicVotingModel, we can see that the problem is a real one, though less extreme (i.e., for larger numbers and costs etc.), than the FreeRider problem.

To continue:

1) Is this really a general enough model to eg. encompass PublicGoodsProblems and CommonsProblem? Am I treating CommonResources as the problem of producing a public good of inaction? If so, is there anything wrong with this? See discussion in CommonResources.

2) Look at different payoff orderings in the group, including 'non-idealists.' Also raising the issue of 'obstruction' - if a sub-group has interests to work against the group action succeeding.

3) Expand to cover 'continuous' models. The bit I find tricky in this is defining the 'optimal group outcome' or 'ideal condition' for these models. What about other 'production functions' - multi-step functions, discontinuities etc.

4) Check against KeithDowdingList.