In broad daylight: Fuller information and higher-order punishment opportunities can promote cooperation

•We conduct voluntary contribution experiments with opportunities to punish.•Most treatments also permit higher-order punishment.•In most cases, higher-order punishing opportunities do not harm cooperation.•When subjects know only who punished them this period, designated opportunities to counter-punish are harmful to cooperation and efficiency.•Symmetric higher-order punishing opportunities with fuller information and history display aids cooperation significantly.

The expectation that non-cooperators will be punished can help to sustain cooperation, but there are competing claims about whether opportunities to engage in higher-order punishment (punishing punishment or failure to punish) help or undermine cooperation in social dilemmas. Varying treatments of a voluntary contribution experiment, we find that availability of higher-order punishment opportunities increases cooperation and efficiency when subjects have full information on the pattern of punishing and its history, when any subject can punish any other, and when the numbers of punishment and of contribution stages are not too unequal.