Question

Each of n people chooses whether to contribute a fixed amount toward the provision of a public good. The good is provided if and only if at least k people contribute, where 2 ≤ k < n. If it is not provided, contributions are not refunded. Each person ranks outcomes from best to worst as follows:

(i) any outcome in which the good is provided and she does not contribute,

(ii) any outcome in which the good is provided and she contributes,

(iii) any outcome in which the good is not provided and she does not contribute,

(iv) any outcome in which the good is not provided and she contributes.

Answer the following questions:

a) Formulate this situation as a strategic game

b) Is there an equilibrium in which exactly k people contribute?

c) Is there a Nash equilibrium in which more than k people contribute?

d) Is there a Nash equilibrium in which fewer than k people contribute?

Answer 1

a)

There are n players, each of which has two strategies: Contribute (C) or don’t contribute (D). The payoffs for player i are as follows, given that α other players contribute.

• If n ≥ α ≥ k, player i is better off not contributing (since the good will be provided even without her contribution).
• If α = k − 1 people contribute, player i is better off contributing (the good will only be provided with her contribution which she values higher than the good not being provided).
• If 0 ≤ α ≤ k − 2 people contribute, player i is better off not contributing (the good won’t be provided anyway).

b)

Yes: from a contributing player i’s point of view α = k −1 people play C so he should play C as well. This is true for all players who play C. For all players who play D, α = k, so playing D is optimal as well. Note that we cannot say who contributes. Each possible combination of contributing players/defecting players is a NE.

c)

No, because this would mean that from the point of view of some player i, α ≥ k others already contributed, so the best response for this player is to play D and deviate from cooperation,

d)

Yes, there is one NE where no one contributes: if nobody else contributes a = 0 and it doesn’t pay to contribute for player i either. So D for all players is a NE.

But are there any NE where a positive number of people, say β > 0 contribute but β < k? No. If β = k − 1, then each non-contributing player would have an incentive to deviate and contribute instead. If β < k − 1, then each contributing player would want to deviate and play D. Hence this is not a NE.