Prize sharing rules in collective contests: when do social norms matter?

We consider collective contests with group-specific minimum shares, which we call group-specific “norms” of competitiveness. The minimum shares bound the incentive schemes available to a group. We fully characterize how group sizes interact with such norms. We show that the smaller group is generally “aggressive,” adopting the norm sharing rule, which helps generate maximal intra-group competition. On the other hand, the larger group is often “docile,” exceeding the norm sharing rule, which softens intra-group competition. We also examine how group welfare relates to group sizes and the norm sharing rules. We identify when the larger group fares worse in the contest, a phenomenon known as Group Size Paradox in the literature.