Prize Sharing Rules in Collective Contests: When Does Group Size Matter?

In this paper we deal with situations of collective contests between two groups over a private prize. A well known way to divide the prize within the winning group is the prize sharing rule introduced by Nitzan, 1991. Since its introduction it has become a standard in the collective contests literature. We generalize this rule by introducing a restriction we call norms of competitiveness of a group. We fully characterize how group sizes interact with such norms. What we show is that the smaller group is generally aggressive, but the larger group needs to have really egalitarian norms to behave aggressively in the contest. We also take up the question of how group welfare relates to group sizes under the stated norms. We provide a complete set of conditions under which the larger group fares worse in the contest, a phenomenon called Group Size Paradox (GSP) in the literature.