A threshold (or step-level) public good is a good that would benefit members of a group but can only be provided if there are sufficient contributions to cover the cost of the good. A local community raising funds for a new community centre is one example. Flatmates trying to get together enough money to buy a new TV is another.
There are some fundamental strategic parameters in any threshold public good game: number of group members (n), the threshold amount of money needed (T), the value of the good if provided (V), and the endowment of money that group members have available to contribute (E). Early experimental studies looked at the role of n, T and V but had little to say about E. This raised the intriguing question of whether E matters. Is it 'easier' to provide a threshold public good if group members are relatively rich or poor?
To find out we needed to run some experiments. In two recently published papers, with Federica Alberti and Anna Stepanova, we report our findings. In this blog entry I will talk about a study with Federica Alberti published in Finanz Archiv. This study is distinguished by the fact we looked at games with a refund or money-back guarantee if contributions fall short of the threshold.
The benchmark game in the experimental literature has n = 5, T = 125, V = 50 and E = 55. This means that if group members split the cost of providing the public good they would each have 55 - 125/5 = 30 left. Call this the endowment remainder. We compared this benchmark to games where, everything else the same, E = 30 and E = 70. This results in an endowment remainder of 5 and 45. You can think of these as games where group members are poor and rich, respectively. We also looked at two further games where the endowment remainder was 5 and 45 but T and V varied. This allowed us to check that our results are robust.
The figure below summarizes our overall results. You can see that there is some weak evidence of a lower success rate for the intermediate level of endowment. The differences, though, are hardly large. So, does the endowment make any difference? We claim it does.
To see why, first note that successfully providing the public good requires coordination amongst group members. They want to contribute just enough to pay for the good. If group members are relatively poor then it should be simple to coordinate because the good is only provided if all of them contribute most of their endowment - so there is nothing to disagree about. If group members are relatively rich then it should also be simple to coordinate because they have lots of money to spare - it is not worth disagreeing. The intermediate case may be more tricky.
If this conjecture is correct we would expect groups to learn to coordinate when the endowment remainder is small or large. Our subjects played the game 25 times and so we can check for this. The next figure summarizes what happened in the last five plays of the game. As we would expect the success rate is now significantly higher with a low or high endowment.
Another way to check our conjecture is to look at the variance in contributions around the threshold. The better group members are at coordinating, the lower should be the variance. The next figure summarizes the variance of contributions in the last five plays of the game. Here there is a big difference.
Our results, therefore, suggest that it is 'easier' for groups to provide the public good when group members are relatively poor or rich. The important caveat is that there should be sufficient time to learn how to coordinate. For an intermediate endowment group members seemingly could not coordinate no matter how much time they had. So, the endowment does matter.
To turn this into practical guidance for fundraisers would suggest donors need to feel either critical - the poor case - or that giving will not cost them much - the rich case. It would be dangerous to enter the intermediate territory where, say, the donor is led to believe that the good could be provided without them - they are not critical - and that there donation would involve personal sacrifice - they are not rich.