Sequential choice of sharing rules in collective contests

Groups competing for a prize need to determine how to distribute it among their members in case of victory. Considering competition between two groups of different size, we show that the small group's sharing rule is a strategic complement to the large group's sharing rule in the sense that if the small group chooses a more meritocratic sharing rule, the large group wishes to choose a more meritocratic rule as well. On the contrary, the large group's sharing rule is a strategic substitute to the small group's sharing rule, hence the timing of choice is crucial. For sufficiently private prizes, a switch from a simultaneous choice to the small group being the leader consists in a Pareto improvement and reduces aggregate effort. On the contrary, when the large group is the leader, aggregate effort increases. As a result, the equilibrium timing is such that the small group chooses its sharing rule first. If the prize is not private enough, the small group retires from the competition and switching from a simultaneous to a sequential timing may reverse the results in terms of aggregate effort. The sequential timing also guarantees that the small group never outperforms the large one.