Even-Split Strategy in Sequential Colonel Blotto Games

Abstract

Klumpp, Konrad, and Solomon (2019, GEB) showed that in a two-player sequential Colonel Blotto game with majoritarian objective, for a large class of contest success functions, there is a unique pure strategy Subgame Perfect Nash Equilibrium characterized by even-split strategy. We generalize this work to multi-player sequential Colonel Blotto games with prize functions where any contestant’s prizes only depend on this contestant’s own number of winning rounds. We show that with weakly monotonic prize functions and CSFs satisfying decreasing success rate condition, there exists a Subgame Perfect Nash Equilibrium supported by the even-split strategy profile. We also show that with strictly monotonic prize functions, even-split strategy profile is the unique Subgame Perfect Nash Equilibrium for Tullock CSFs with r <=1. In a constant-sum game, if there are only 2 contestants with CSFs satisfying decreasing success rate condition, or if there are more than 2 contestants with Tullock CSF r<=1, then weakly monotonic prize functions result in uniqueness of pure strategy Subgame Perfect Nash Equilibrium before any contestant reaches a settled prize. Our work extends Klumpp et al. (2019)’s result in three aspects: prize structure, number of contestants and uniqueness of pure strategy Subgame Perfect Nash Equilibrium. Our work provides a better understanding on the applicability of the simple even-split strategy in Colonel Blotto games where equilibrium strategies are typically complex.