An analysis of a fair division protocol for drawing legislative districts

Landau, Reid, and Yershov [A Fair Division Solution to the Problem of Redistricting, \textit{Social Choice and Welfare}, 2008] propose a protocol for drawing legislative districts based on a two player fair division process, where each player is entitled to draw the districts for a portion of the state. We call this the \textit{LRY protocol}. Landau and Su [Fair Division and Redistricting, arXiv:1402.0862, 2014] propose a measure of the fairness of a state's districts called the \textit{geometric target}. In this paper we prove that the number of districts a party can win under the LRY protocol can be at most two fewer than their geometric target, assuming no geometric constraints on the districts, and provide examples to prove this bound is tight. We also show that if the LRY protocol is applied on a state with geometric constraints, the result can be arbitrarily far from the geometric target.