This is a more general model of 'free rider' PublicGoodsProblems. It usually comes up in discussing the ParadoxOfVoting, but it has more general application than that.

There is a benefit to group members if a public good is provided. But there is also a cost to each member who plays their part in providing it – eg. a monetary contribution, or work. If the good is provided everyone in the group gets the benefit. But only those who contributed pay the cost. The good will be provided if enough group members contribute.

Each group member will weigh up the benefit she would gain if the collective good is provided (u), and the cost of contributing (c).

If you contribute and the collective good is provided, your net gain is u – c.

If you contribute but the good is not provided,you get –c.

If you don’t contribute and the good is provided anyway, you get u.

If you don’t contribute and the good is not provided, you get 0.

The values of u and c can be different for different members. But in any case – unless the individual puts no value on the good or the cost at all – the ideal situation for the individualistic group member is to get the benefit without contributing. This is called ‘free riding’.

There are N members in the group. Keeping things simple, we suppose that the good will either be fully provided if at least n^ members contribute, and not provided at all if less than n contribute.

Each individual’s decision depends on her beliefs about how what other group members are going to do. Payoffs are set out in this table:

If you are acting only for individual gain, and you can be certain about other members’ intentions, you will contribute if and only if you know that n^{-1} others will contribute.

More often, group members will be uncertain about each others’ intentions.

To model the decision problem in this case use expected utility theory. This says that a rational self-interested agent will act to get the outcome with the highest expected value. There are two choices – to contribute or not. The expected value from either choice is the sum of: the outcomes the member expects to get depending on how many other members act, multiplied by the probability she believes there is that that many other members will act.

The expected value of contributing = prob (n < n^{-1)} (-c) + prob (n > n^{-1)} (u-c) + prob (n = n^{-1)} (u-c)

The expected value of non-contributing = prob (n > n^{-1)} (u).

A member will contribute if the expected value of contributing is greater than that of not. That is if:

U prob (n > n^{-1)} + u prob (n = n^{-1)} – c > u prob (n > n^{-1)}

Or, re-arranging:

u prob (n = n^{-1)} > c

Define p = prob (n = n^{-1).} Again, that means how probable this member believes it is that (n^{-1)} other members will contribute.

The member will contribute if:

U/c > 1/p.

Pretty obviously, you are more likely to contribute the more you want the collective good and the less you are bothered about the cost. And the more chance you think there is that your contribution will make a difference.

In a group of individualists, the good will be provided if and only if there are enough members who place a high enough value u on the good, a low enough cost c, and believe there is a high enough probability p that they could be the ‘swing’ member.

Notes:

The model above is pretty much the "calculus of voting" model (Downs 1957, Riker & Ordeshook 1968). If you face a decision between voting for one of a range of political candidates, and have a preference for one of them, the choice boils down to a decision whether or not to bother going out to vote. The result is basically as above. (See eg. Aldrich 199x).

The model can be extended into a StrategicVotingModel which 'endogenises' expectations about the likely 'turnout'.