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

Bargaining for Assembly

An assembly problem refers to a situation where a buyer wants to purchase a fixed number of complementary items from sellers holding an item each. We model complementarity using graphs where nodes represent items, and edges between two nodes represent a complementary relationship between these items. The buyer wants to purchase a feasible path in the graph, i.e., a path of desired length, where the sum of valuations of the sellers owning the items do not exceed buyer’s own valuation. A seller is critical if he lies on every feasible path. We examine subgame perfect equilibria of an infinite horizon alternate-offer bargaining game between the buyer and the sellers. We show that there exist equilibria where the buyer can extract full surplus within two periods if and only if (a) there are no critical sellers and (b) there exist at least two feasible paths with minimum sum of seller valuations. We also characterize the upper bounds on buyer’s surplus when she cannot extract full surplus. Thus we characterize the trade-off between complementarity and competition in terms of buyer’s equilibrium surplus share in assembly problems

Bargaining for assembly

We study a multilateral bargaining problem, where the buyer intends to purchase a subset of available items, each owned by a seller. The subset purchased must satisfy a notion of contiguity, which is modeled using graphs. The graph theoretic approach allows us to study different degrees of complementarity and substitutability between items. It also allows us to examine how degrees of complementarity and substitutability affect the share of surplus obtained by the buyer in the equilibrium of the bargaining game. We characterize necessary and sufficient conditions on the graphs for the buyer to extract full surplus in subgame perfect equilibrium. When the conditions are not met, we provide upper bounds on the equilibrium surplus share of the buyer

On the Relative Sequencing of Internal and External Rent-Seeking Contests

We consider rent-seeking contests between and within two equal-sized groups. Each group adopts one of three sequences: first internal then external contest, first external then internal contest, and simultaneous internal and external contests. Groups cannot unilaterally postpone a contest without losing. We rank the nine possible combinations according to rent-seeking expenditure and expected utilities. Rent-seeking is maximum when both internal contests either precede, or occur simultaneously with, the external contest. These forms have identical, Pareto-dominated, welfare consequences. Among contest forms which offer both groups a positive win probability, rent-seeking is minimized if the between-group contest precedes both within-group contests; this also induces Pareto-efficiency. When the groups independently choose the contest sequence, the unique Nash equilibrium involves simultaneous occurrence of all contests. Results due to Warneryd (1998) and Amegashie (1999) fall out. When a multi-member group battles a single-member one, unity against the common enemy (an efficient sequence choice) can be sustained if the larger group can resolve its internal coordination problem. With unequal groups and symmetric contest sequencing, the one-tier contest form may be Pareto-efficient, despite generating greater rent-seeking than all symmetric two-tier forms